1120.21/292.00 WORST_CASE(Omega(n^1), O(n^2)) 1121.48/292.31 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1121.48/292.31 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1121.48/292.31 1121.48/292.31 1121.48/292.31 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1121.48/292.31 1121.48/292.31 (0) CpxRelTRS 1121.48/292.31 (1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 815 ms] 1121.48/292.31 (2) CpxRelTRS 1121.48/292.31 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 1121.48/292.31 (4) CpxWeightedTrs 1121.48/292.31 (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 1121.48/292.31 (6) CpxWeightedTrs 1121.48/292.31 (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 1 ms] 1121.48/292.31 (8) CpxTypedWeightedTrs 1121.48/292.31 (9) CompletionProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (10) CpxTypedWeightedCompleteTrs 1121.48/292.31 (11) NarrowingProof [BOTH BOUNDS(ID, ID), 80 ms] 1121.48/292.31 (12) CpxTypedWeightedCompleteTrs 1121.48/292.31 (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 10 ms] 1121.48/292.31 (14) CpxRNTS 1121.48/292.31 (15) InliningProof [UPPER BOUND(ID), 2162 ms] 1121.48/292.31 (16) CpxRNTS 1121.48/292.31 (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 1121.48/292.31 (18) CpxRNTS 1121.48/292.31 (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 1121.48/292.31 (20) CpxRNTS 1121.48/292.31 (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (22) CpxRNTS 1121.48/292.31 (23) IntTrsBoundProof [UPPER BOUND(ID), 357 ms] 1121.48/292.31 (24) CpxRNTS 1121.48/292.31 (25) IntTrsBoundProof [UPPER BOUND(ID), 7 ms] 1121.48/292.31 (26) CpxRNTS 1121.48/292.31 (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (28) CpxRNTS 1121.48/292.31 (29) IntTrsBoundProof [UPPER BOUND(ID), 161 ms] 1121.48/292.31 (30) CpxRNTS 1121.48/292.31 (31) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] 1121.48/292.31 (32) CpxRNTS 1121.48/292.31 (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (34) CpxRNTS 1121.48/292.31 (35) IntTrsBoundProof [UPPER BOUND(ID), 371 ms] 1121.48/292.31 (36) CpxRNTS 1121.48/292.31 (37) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] 1121.48/292.31 (38) CpxRNTS 1121.48/292.31 (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (40) CpxRNTS 1121.48/292.31 (41) IntTrsBoundProof [UPPER BOUND(ID), 1054 ms] 1121.48/292.31 (42) CpxRNTS 1121.48/292.31 (43) IntTrsBoundProof [UPPER BOUND(ID), 245 ms] 1121.48/292.31 (44) CpxRNTS 1121.48/292.31 (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (46) CpxRNTS 1121.48/292.31 (47) IntTrsBoundProof [UPPER BOUND(ID), 167 ms] 1121.48/292.31 (48) CpxRNTS 1121.48/292.31 (49) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] 1121.48/292.31 (50) CpxRNTS 1121.48/292.31 (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (52) CpxRNTS 1121.48/292.31 (53) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] 1121.48/292.31 (54) CpxRNTS 1121.48/292.31 (55) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] 1121.48/292.31 (56) CpxRNTS 1121.48/292.31 (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (58) CpxRNTS 1121.48/292.31 (59) IntTrsBoundProof [UPPER BOUND(ID), 422 ms] 1121.48/292.31 (60) CpxRNTS 1121.48/292.31 (61) IntTrsBoundProof [UPPER BOUND(ID), 150 ms] 1121.48/292.31 (62) CpxRNTS 1121.48/292.31 (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (64) CpxRNTS 1121.48/292.31 (65) IntTrsBoundProof [UPPER BOUND(ID), 310 ms] 1121.48/292.31 (66) CpxRNTS 1121.48/292.31 (67) IntTrsBoundProof [UPPER BOUND(ID), 14 ms] 1121.48/292.31 (68) CpxRNTS 1121.48/292.31 (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (70) CpxRNTS 1121.48/292.31 (71) IntTrsBoundProof [UPPER BOUND(ID), 189 ms] 1121.48/292.31 (72) CpxRNTS 1121.48/292.31 (73) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] 1121.48/292.31 (74) CpxRNTS 1121.48/292.31 (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (76) CpxRNTS 1121.48/292.31 (77) IntTrsBoundProof [UPPER BOUND(ID), 218 ms] 1121.48/292.31 (78) CpxRNTS 1121.48/292.31 (79) IntTrsBoundProof [UPPER BOUND(ID), 62 ms] 1121.48/292.31 (80) CpxRNTS 1121.48/292.31 (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (82) CpxRNTS 1121.48/292.31 (83) IntTrsBoundProof [UPPER BOUND(ID), 289 ms] 1121.48/292.31 (84) CpxRNTS 1121.48/292.31 (85) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (86) CpxRNTS 1121.48/292.31 (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (88) CpxRNTS 1121.48/292.31 (89) IntTrsBoundProof [UPPER BOUND(ID), 804 ms] 1121.48/292.31 (90) CpxRNTS 1121.48/292.31 (91) IntTrsBoundProof [UPPER BOUND(ID), 165 ms] 1121.48/292.31 (92) CpxRNTS 1121.48/292.31 (93) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (94) CpxRNTS 1121.48/292.31 (95) IntTrsBoundProof [UPPER BOUND(ID), 178 ms] 1121.48/292.31 (96) CpxRNTS 1121.48/292.31 (97) IntTrsBoundProof [UPPER BOUND(ID), 23 ms] 1121.48/292.31 (98) CpxRNTS 1121.48/292.31 (99) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (100) CpxRNTS 1121.48/292.31 (101) IntTrsBoundProof [UPPER BOUND(ID), 616 ms] 1121.48/292.31 (102) CpxRNTS 1121.48/292.31 (103) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] 1121.48/292.31 (104) CpxRNTS 1121.48/292.31 (105) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (106) CpxRNTS 1121.48/292.31 (107) IntTrsBoundProof [UPPER BOUND(ID), 2300 ms] 1121.48/292.31 (108) CpxRNTS 1121.48/292.31 (109) IntTrsBoundProof [UPPER BOUND(ID), 162 ms] 1121.48/292.31 (110) CpxRNTS 1121.48/292.31 (111) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (112) CpxRNTS 1121.48/292.31 (113) IntTrsBoundProof [UPPER BOUND(ID), 268 ms] 1121.48/292.31 (114) CpxRNTS 1121.48/292.31 (115) IntTrsBoundProof [UPPER BOUND(ID), 54 ms] 1121.48/292.31 (116) CpxRNTS 1121.48/292.31 (117) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (118) CpxRNTS 1121.48/292.31 (119) IntTrsBoundProof [UPPER BOUND(ID), 189 ms] 1121.48/292.31 (120) CpxRNTS 1121.48/292.31 (121) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] 1121.48/292.31 (122) CpxRNTS 1121.48/292.31 (123) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (124) CpxRNTS 1121.48/292.31 (125) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] 1121.48/292.31 (126) CpxRNTS 1121.48/292.31 (127) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] 1121.48/292.31 (128) CpxRNTS 1121.48/292.31 (129) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (130) CpxRNTS 1121.48/292.31 (131) IntTrsBoundProof [UPPER BOUND(ID), 156 ms] 1121.48/292.31 (132) CpxRNTS 1121.48/292.31 (133) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] 1121.48/292.31 (134) CpxRNTS 1121.48/292.31 (135) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (136) CpxRNTS 1121.48/292.31 (137) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] 1121.48/292.31 (138) CpxRNTS 1121.48/292.31 (139) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 1121.48/292.31 (140) CpxRNTS 1121.48/292.31 (141) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (142) CpxRNTS 1121.48/292.31 (143) IntTrsBoundProof [UPPER BOUND(ID), 389 ms] 1121.48/292.31 (144) CpxRNTS 1121.48/292.31 (145) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] 1121.48/292.31 (146) CpxRNTS 1121.48/292.31 (147) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (148) CpxRNTS 1121.48/292.31 (149) IntTrsBoundProof [UPPER BOUND(ID), 399 ms] 1121.48/292.31 (150) CpxRNTS 1121.48/292.31 (151) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] 1121.48/292.31 (152) CpxRNTS 1121.48/292.31 (153) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (154) CpxRNTS 1121.48/292.31 (155) IntTrsBoundProof [UPPER BOUND(ID), 730 ms] 1121.48/292.31 (156) CpxRNTS 1121.48/292.31 (157) IntTrsBoundProof [UPPER BOUND(ID), 123 ms] 1121.48/292.31 (158) CpxRNTS 1121.48/292.31 (159) ResultPropagationProof [UPPER BOUND(ID), 2 ms] 1121.48/292.31 (160) CpxRNTS 1121.48/292.31 (161) IntTrsBoundProof [UPPER BOUND(ID), 832 ms] 1121.48/292.31 (162) CpxRNTS 1121.48/292.31 (163) IntTrsBoundProof [UPPER BOUND(ID), 26 ms] 1121.48/292.31 (164) CpxRNTS 1121.48/292.31 (165) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (166) CpxRNTS 1121.48/292.31 (167) IntTrsBoundProof [UPPER BOUND(ID), 4081 ms] 1121.48/292.31 (168) CpxRNTS 1121.48/292.31 (169) IntTrsBoundProof [UPPER BOUND(ID), 411 ms] 1121.48/292.31 (170) CpxRNTS 1121.48/292.31 (171) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (172) CpxRNTS 1121.48/292.31 (173) IntTrsBoundProof [UPPER BOUND(ID), 697 ms] 1121.48/292.31 (174) CpxRNTS 1121.48/292.31 (175) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] 1121.48/292.31 (176) CpxRNTS 1121.48/292.31 (177) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (178) CpxRNTS 1121.48/292.31 (179) IntTrsBoundProof [UPPER BOUND(ID), 1369 ms] 1121.48/292.31 (180) CpxRNTS 1121.48/292.31 (181) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] 1121.48/292.31 (182) CpxRNTS 1121.48/292.31 (183) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (184) CpxRNTS 1121.48/292.31 (185) IntTrsBoundProof [UPPER BOUND(ID), 1045 ms] 1121.48/292.31 (186) CpxRNTS 1121.48/292.31 (187) IntTrsBoundProof [UPPER BOUND(ID), 73 ms] 1121.48/292.31 (188) CpxRNTS 1121.48/292.31 (189) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (190) CpxRNTS 1121.48/292.31 (191) IntTrsBoundProof [UPPER BOUND(ID), 967 ms] 1121.48/292.31 (192) CpxRNTS 1121.48/292.31 (193) IntTrsBoundProof [UPPER BOUND(ID), 145 ms] 1121.48/292.31 (194) CpxRNTS 1121.48/292.31 (195) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (196) CpxRNTS 1121.48/292.31 (197) IntTrsBoundProof [UPPER BOUND(ID), 907 ms] 1121.48/292.31 (198) CpxRNTS 1121.48/292.31 (199) IntTrsBoundProof [UPPER BOUND(ID), 1 ms] 1121.48/292.31 (200) CpxRNTS 1121.48/292.31 (201) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 1121.48/292.31 (202) CpxRNTS 1121.48/292.31 (203) IntTrsBoundProof [UPPER BOUND(ID), 856 ms] 1121.48/292.31 (204) CpxRNTS 1121.48/292.31 (205) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] 1121.48/292.31 (206) CpxRNTS 1121.48/292.31 (207) FinalProof [FINISHED, 0 ms] 1121.48/292.31 (208) BOUNDS(1, n^2) 1121.48/292.31 (209) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1121.48/292.31 (210) TRS for Loop Detection 1121.48/292.31 (211) DecreasingLoopProof [LOWER BOUND(ID), 88 ms] 1121.48/292.31 (212) BEST 1121.48/292.31 (213) proven lower bound 1121.48/292.31 (214) LowerBoundPropagationProof [FINISHED, 0 ms] 1121.48/292.31 (215) BOUNDS(n^1, INF) 1121.48/292.31 (216) TRS for Loop Detection 1121.48/292.31 1121.48/292.31 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (0) 1121.48/292.31 Obligation: 1121.48/292.31 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1121.48/292.31 1121.48/292.31 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 1121.48/292.31 +(@x, @y) -> #add(@x, @y) 1121.48/292.31 firstline(@l) -> firstline#1(@l) 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) 1121.48/292.31 firstline#1(nil) -> nil 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) 1121.48/292.31 lcs#2(nil) -> #abs(#0) 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len 1121.48/292.31 lcs#3(nil) -> #abs(#0) 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) 1121.48/292.31 max#1(#false, @a, @b) -> @b 1121.48/292.31 max#1(#true, @a, @b) -> @a 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) 1121.48/292.31 right(@l) -> right#1(@l) 1121.48/292.31 right#1(::(@x, @xs)) -> @x 1121.48/292.31 right#1(nil) -> #abs(#0) 1121.48/292.31 1121.48/292.31 The (relative) TRS S consists of the following rules: 1121.48/292.31 1121.48/292.31 #add(#0, @y) -> @y 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) 1121.48/292.31 #and(#false, #false) -> #false 1121.48/292.31 #and(#false, #true) -> #false 1121.48/292.31 #and(#true, #false) -> #false 1121.48/292.31 #and(#true, #true) -> #true 1121.48/292.31 #ckgt(#EQ) -> #false 1121.48/292.31 #ckgt(#GT) -> #true 1121.48/292.31 #ckgt(#LT) -> #false 1121.48/292.31 #compare(#0, #0) -> #EQ 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT 1121.48/292.31 #compare(#0, #s(@y)) -> #LT 1121.48/292.31 #compare(#neg(@x), #0) -> #LT 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT 1121.48/292.31 #compare(#pos(@x), #0) -> #GT 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 1121.48/292.31 #compare(#s(@x), #0) -> #GT 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 1121.48/292.31 #eq(#0, #0) -> #true 1121.48/292.31 #eq(#0, #neg(@y)) -> #false 1121.48/292.31 #eq(#0, #pos(@y)) -> #false 1121.48/292.31 #eq(#0, #s(@y)) -> #false 1121.48/292.31 #eq(#neg(@x), #0) -> #false 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false 1121.48/292.31 #eq(#pos(@x), #0) -> #false 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(#s(@x), #0) -> #false 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false 1121.48/292.31 #eq(nil, nil) -> #true 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) 1121.48/292.31 1121.48/292.31 Rewrite Strategy: INNERMOST 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) 1121.48/292.31 proved termination of relative rules 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (2) 1121.48/292.31 Obligation: 1121.48/292.31 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1121.48/292.31 1121.48/292.31 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 1121.48/292.31 +(@x, @y) -> #add(@x, @y) 1121.48/292.31 firstline(@l) -> firstline#1(@l) 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) 1121.48/292.31 firstline#1(nil) -> nil 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) 1121.48/292.31 lcs#2(nil) -> #abs(#0) 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len 1121.48/292.31 lcs#3(nil) -> #abs(#0) 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) 1121.48/292.31 max#1(#false, @a, @b) -> @b 1121.48/292.31 max#1(#true, @a, @b) -> @a 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) 1121.48/292.31 right(@l) -> right#1(@l) 1121.48/292.31 right#1(::(@x, @xs)) -> @x 1121.48/292.31 right#1(nil) -> #abs(#0) 1121.48/292.31 1121.48/292.31 The (relative) TRS S consists of the following rules: 1121.48/292.31 1121.48/292.31 #add(#0, @y) -> @y 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) 1121.48/292.31 #and(#false, #false) -> #false 1121.48/292.31 #and(#false, #true) -> #false 1121.48/292.31 #and(#true, #false) -> #false 1121.48/292.31 #and(#true, #true) -> #true 1121.48/292.31 #ckgt(#EQ) -> #false 1121.48/292.31 #ckgt(#GT) -> #true 1121.48/292.31 #ckgt(#LT) -> #false 1121.48/292.31 #compare(#0, #0) -> #EQ 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT 1121.48/292.31 #compare(#0, #s(@y)) -> #LT 1121.48/292.31 #compare(#neg(@x), #0) -> #LT 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT 1121.48/292.31 #compare(#pos(@x), #0) -> #GT 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 1121.48/292.31 #compare(#s(@x), #0) -> #GT 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 1121.48/292.31 #eq(#0, #0) -> #true 1121.48/292.31 #eq(#0, #neg(@y)) -> #false 1121.48/292.31 #eq(#0, #pos(@y)) -> #false 1121.48/292.31 #eq(#0, #s(@y)) -> #false 1121.48/292.31 #eq(#neg(@x), #0) -> #false 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false 1121.48/292.31 #eq(#pos(@x), #0) -> #false 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(#s(@x), #0) -> #false 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false 1121.48/292.31 #eq(nil, nil) -> #true 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) 1121.48/292.31 1121.48/292.31 Rewrite Strategy: INNERMOST 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 1121.48/292.31 Transformed relative TRS to weighted TRS 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (4) 1121.48/292.31 Obligation: 1121.48/292.31 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 1121.48/292.31 1121.48/292.31 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 [1] 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) [1] 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) [1] 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] 1121.48/292.31 +(@x, @y) -> #add(@x, @y) [1] 1121.48/292.31 firstline(@l) -> firstline#1(@l) [1] 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] 1121.48/292.31 firstline#1(nil) -> nil [1] 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) [1] 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] 1121.48/292.31 lcs#2(nil) -> #abs(#0) [1] 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len [1] 1121.48/292.31 lcs#3(nil) -> #abs(#0) [1] 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil [1] 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] 1121.48/292.31 max#1(#false, @a, @b) -> @b [1] 1121.48/292.31 max#1(#true, @a, @b) -> @a [1] 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil [1] 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil [1] 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) [1] 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) [1] 1121.48/292.31 right(@l) -> right#1(@l) [1] 1121.48/292.31 right#1(::(@x, @xs)) -> @x [1] 1121.48/292.31 right#1(nil) -> #abs(#0) [1] 1121.48/292.31 #add(#0, @y) -> @y [0] 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) [0] 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) [0] 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #and(#false, #false) -> #false [0] 1121.48/292.31 #and(#false, #true) -> #false [0] 1121.48/292.31 #and(#true, #false) -> #false [0] 1121.48/292.31 #and(#true, #true) -> #true [0] 1121.48/292.31 #ckgt(#EQ) -> #false [0] 1121.48/292.31 #ckgt(#GT) -> #true [0] 1121.48/292.31 #ckgt(#LT) -> #false [0] 1121.48/292.31 #compare(#0, #0) -> #EQ [0] 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#0, #s(@y)) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #0) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#pos(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #compare(#s(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #eq(#0, #0) -> #true [0] 1121.48/292.31 #eq(#0, #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #s(@y)) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #0) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #0) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#s(@x), #0) -> #false [0] 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false [0] 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false [0] 1121.48/292.31 #eq(nil, nil) -> #true [0] 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) [0] 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 [0] 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) [0] 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 [0] 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] 1121.48/292.31 1121.48/292.31 Rewrite Strategy: INNERMOST 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) 1121.48/292.31 Renamed defined symbols to avoid conflicts with arithmetic symbols: 1121.48/292.31 1121.48/292.31 + => plus 1121.48/292.31 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (6) 1121.48/292.31 Obligation: 1121.48/292.31 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). 1121.48/292.31 1121.48/292.31 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 [1] 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) [1] 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) [1] 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] 1121.48/292.31 plus(@x, @y) -> #add(@x, @y) [1] 1121.48/292.31 firstline(@l) -> firstline#1(@l) [1] 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] 1121.48/292.31 firstline#1(nil) -> nil [1] 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) [1] 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] 1121.48/292.31 lcs#2(nil) -> #abs(#0) [1] 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len [1] 1121.48/292.31 lcs#3(nil) -> #abs(#0) [1] 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil [1] 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] 1121.48/292.31 max#1(#false, @a, @b) -> @b [1] 1121.48/292.31 max#1(#true, @a, @b) -> @a [1] 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil [1] 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil [1] 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) [1] 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] 1121.48/292.31 right(@l) -> right#1(@l) [1] 1121.48/292.31 right#1(::(@x, @xs)) -> @x [1] 1121.48/292.31 right#1(nil) -> #abs(#0) [1] 1121.48/292.31 #add(#0, @y) -> @y [0] 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) [0] 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) [0] 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #and(#false, #false) -> #false [0] 1121.48/292.31 #and(#false, #true) -> #false [0] 1121.48/292.31 #and(#true, #false) -> #false [0] 1121.48/292.31 #and(#true, #true) -> #true [0] 1121.48/292.31 #ckgt(#EQ) -> #false [0] 1121.48/292.31 #ckgt(#GT) -> #true [0] 1121.48/292.31 #ckgt(#LT) -> #false [0] 1121.48/292.31 #compare(#0, #0) -> #EQ [0] 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#0, #s(@y)) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #0) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#pos(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #compare(#s(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #eq(#0, #0) -> #true [0] 1121.48/292.31 #eq(#0, #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #s(@y)) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #0) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #0) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#s(@x), #0) -> #false [0] 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false [0] 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false [0] 1121.48/292.31 #eq(nil, nil) -> #true [0] 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) [0] 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 [0] 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) [0] 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 [0] 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] 1121.48/292.31 1121.48/292.31 Rewrite Strategy: INNERMOST 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 1121.48/292.31 Infered types. 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (8) 1121.48/292.31 Obligation: 1121.48/292.31 Runtime Complexity Weighted TRS with Types. 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 [1] 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) [1] 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) [1] 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] 1121.48/292.31 plus(@x, @y) -> #add(@x, @y) [1] 1121.48/292.31 firstline(@l) -> firstline#1(@l) [1] 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] 1121.48/292.31 firstline#1(nil) -> nil [1] 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) [1] 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] 1121.48/292.31 lcs#2(nil) -> #abs(#0) [1] 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len [1] 1121.48/292.31 lcs#3(nil) -> #abs(#0) [1] 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil [1] 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] 1121.48/292.31 max#1(#false, @a, @b) -> @b [1] 1121.48/292.31 max#1(#true, @a, @b) -> @a [1] 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil [1] 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil [1] 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) [1] 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] 1121.48/292.31 right(@l) -> right#1(@l) [1] 1121.48/292.31 right#1(::(@x, @xs)) -> @x [1] 1121.48/292.31 right#1(nil) -> #abs(#0) [1] 1121.48/292.31 #add(#0, @y) -> @y [0] 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) [0] 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) [0] 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #and(#false, #false) -> #false [0] 1121.48/292.31 #and(#false, #true) -> #false [0] 1121.48/292.31 #and(#true, #false) -> #false [0] 1121.48/292.31 #and(#true, #true) -> #true [0] 1121.48/292.31 #ckgt(#EQ) -> #false [0] 1121.48/292.31 #ckgt(#GT) -> #true [0] 1121.48/292.31 #ckgt(#LT) -> #false [0] 1121.48/292.31 #compare(#0, #0) -> #EQ [0] 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#0, #s(@y)) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #0) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#pos(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #compare(#s(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #eq(#0, #0) -> #true [0] 1121.48/292.31 #eq(#0, #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #s(@y)) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #0) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #0) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#s(@x), #0) -> #false [0] 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false [0] 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false [0] 1121.48/292.31 #eq(nil, nil) -> #true [0] 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) [0] 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 [0] 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) [0] 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 [0] 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] 1121.48/292.31 1121.48/292.31 The TRS has the following type information: 1121.48/292.31 #abs :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #0 :: #0:#neg:#pos:#s::::nil 1121.48/292.31 #neg :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #pos :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #s :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #equal :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true 1121.48/292.31 #eq :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true 1121.48/292.31 #greater :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #false:#true 1121.48/292.31 #ckgt :: #EQ:#GT:#LT -> #false:#true 1121.48/292.31 #compare :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #EQ:#GT:#LT 1121.48/292.31 plus :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #add :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 firstline :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 firstline#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 :: :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 nil :: #0:#neg:#pos:#s::::nil 1121.48/292.31 lcs :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcs#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcstable :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcs#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcs#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcstable#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcstable#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 lcstable#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 max :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 max#1 :: #false:#true -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #false :: #false:#true 1121.48/292.31 #true :: #false:#true 1121.48/292.31 newline#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#2 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#3 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#4 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 right :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#5 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#6 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 newline#7 :: #false:#true -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 right#1 :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #pred :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #succ :: #0:#neg:#pos:#s::::nil -> #0:#neg:#pos:#s::::nil 1121.48/292.31 #and :: #false:#true -> #false:#true -> #false:#true 1121.48/292.31 #EQ :: #EQ:#GT:#LT 1121.48/292.31 #GT :: #EQ:#GT:#LT 1121.48/292.31 #LT :: #EQ:#GT:#LT 1121.48/292.31 1121.48/292.31 Rewrite Strategy: INNERMOST 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (9) CompletionProof (UPPER BOUND(ID)) 1121.48/292.31 The transformation into a RNTS is sound, since: 1121.48/292.31 1121.48/292.31 (a) The obligation is a constructor system where every type has a constant constructor, 1121.48/292.31 1121.48/292.31 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 1121.48/292.31 1121.48/292.31 lcs_2 1121.48/292.31 lcs#1_1 1121.48/292.31 lcs#2_1 1121.48/292.31 lcs#3_1 1121.48/292.31 1121.48/292.31 (c) The following functions are completely defined: 1121.48/292.31 1121.48/292.31 lcstable_2 1121.48/292.31 #greater_2 1121.48/292.31 newline#7_4 1121.48/292.31 #equal_2 1121.48/292.31 right_1 1121.48/292.31 newline_3 1121.48/292.31 newline#1_3 1121.48/292.31 plus_2 1121.48/292.31 right#1_1 1121.48/292.31 lcstable#1_2 1121.48/292.31 firstline_1 1121.48/292.31 #abs_1 1121.48/292.31 firstline#1_1 1121.48/292.31 newline#2_4 1121.48/292.31 max_2 1121.48/292.31 lcstable#2_3 1121.48/292.31 max#1_3 1121.48/292.31 newline#3_5 1121.48/292.31 lcstable#3_3 1121.48/292.31 newline#4_6 1121.48/292.31 newline#5_6 1121.48/292.31 newline#6_2 1121.48/292.31 #add_2 1121.48/292.31 #and_2 1121.48/292.31 #ckgt_1 1121.48/292.31 #compare_2 1121.48/292.31 #eq_2 1121.48/292.31 #pred_1 1121.48/292.31 #succ_1 1121.48/292.31 1121.48/292.31 Due to the following rules being added: 1121.48/292.31 1121.48/292.31 #add(v0, v1) -> null_#add [0] 1121.48/292.31 #and(v0, v1) -> null_#and [0] 1121.48/292.31 #ckgt(v0) -> null_#ckgt [0] 1121.48/292.31 #compare(v0, v1) -> null_#compare [0] 1121.48/292.31 #eq(v0, v1) -> null_#eq [0] 1121.48/292.31 #pred(v0) -> null_#pred [0] 1121.48/292.31 #succ(v0) -> null_#succ [0] 1121.48/292.31 newline#7(v0, v1, v2, v3) -> null_newline#7 [0] 1121.48/292.31 newline#1(v0, v1, v2) -> null_newline#1 [0] 1121.48/292.31 right#1(v0) -> null_right#1 [0] 1121.48/292.31 lcstable#1(v0, v1) -> null_lcstable#1 [0] 1121.48/292.31 #abs(v0) -> null_#abs [0] 1121.48/292.31 firstline#1(v0) -> null_firstline#1 [0] 1121.48/292.31 newline#2(v0, v1, v2, v3) -> null_newline#2 [0] 1121.48/292.31 max#1(v0, v1, v2) -> null_max#1 [0] 1121.48/292.31 lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] 1121.48/292.31 1121.48/292.31 And the following fresh constants: null_#add, null_#and, null_#ckgt, null_#compare, null_#eq, null_#pred, null_#succ, null_newline#7, null_newline#1, null_right#1, null_lcstable#1, null_#abs, null_firstline#1, null_newline#2, null_max#1, null_lcstable#3 1121.48/292.31 1121.48/292.31 ---------------------------------------- 1121.48/292.31 1121.48/292.31 (10) 1121.48/292.31 Obligation: 1121.48/292.31 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1121.48/292.31 1121.48/292.31 Runtime Complexity Weighted TRS with Types. 1121.48/292.31 The TRS R consists of the following rules: 1121.48/292.31 1121.48/292.31 #abs(#0) -> #0 [1] 1121.48/292.31 #abs(#neg(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#pos(@x)) -> #pos(@x) [1] 1121.48/292.31 #abs(#s(@x)) -> #pos(#s(@x)) [1] 1121.48/292.31 #equal(@x, @y) -> #eq(@x, @y) [1] 1121.48/292.31 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] 1121.48/292.31 plus(@x, @y) -> #add(@x, @y) [1] 1121.48/292.31 firstline(@l) -> firstline#1(@l) [1] 1121.48/292.31 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] 1121.48/292.31 firstline#1(nil) -> nil [1] 1121.48/292.31 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) [1] 1121.48/292.31 lcs#1(@m) -> lcs#2(@m) [1] 1121.48/292.31 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] 1121.48/292.31 lcs#2(nil) -> #abs(#0) [1] 1121.48/292.31 lcs#3(::(@len, @_@1)) -> @len [1] 1121.48/292.31 lcs#3(nil) -> #abs(#0) [1] 1121.48/292.31 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] 1121.48/292.31 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) [1] 1121.48/292.31 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] 1121.48/292.31 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] 1121.48/292.31 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] 1121.48/292.31 lcstable#3(nil, @l2, @x) -> nil [1] 1121.48/292.31 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) [1] 1121.48/292.31 max#1(#false, @a, @b) -> @b [1] 1121.48/292.31 max#1(#true, @a, @b) -> @a [1] 1121.48/292.31 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] 1121.48/292.31 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] 1121.48/292.31 newline#1(nil, @lastline, @y) -> nil [1] 1121.48/292.31 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) [1] 1121.48/292.31 newline#2(nil, @x, @xs, @y) -> nil [1] 1121.48/292.31 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) [1] 1121.48/292.31 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [1] 1121.48/292.31 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [1] 1121.48/292.31 newline#6(@elem, @nl) -> ::(@elem, @nl) [1] 1121.48/292.31 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] 1121.48/292.31 newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] 1121.48/292.31 right(@l) -> right#1(@l) [1] 1121.48/292.31 right#1(::(@x, @xs)) -> @x [1] 1121.48/292.31 right#1(nil) -> #abs(#0) [1] 1121.48/292.31 #add(#0, @y) -> @y [0] 1121.48/292.31 #add(#neg(#s(#0)), @y) -> #pred(@y) [0] 1121.48/292.31 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #add(#pos(#s(#0)), @y) -> #succ(@y) [0] 1121.48/292.31 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) [0] 1121.48/292.31 #and(#false, #false) -> #false [0] 1121.48/292.31 #and(#false, #true) -> #false [0] 1121.48/292.31 #and(#true, #false) -> #false [0] 1121.48/292.31 #and(#true, #true) -> #true [0] 1121.48/292.31 #ckgt(#EQ) -> #false [0] 1121.48/292.31 #ckgt(#GT) -> #true [0] 1121.48/292.31 #ckgt(#LT) -> #false [0] 1121.48/292.31 #compare(#0, #0) -> #EQ [0] 1121.48/292.31 #compare(#0, #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#0, #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#0, #s(@y)) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #0) -> #LT [0] 1121.48/292.31 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] 1121.48/292.31 #compare(#neg(@x), #pos(@y)) -> #LT [0] 1121.48/292.31 #compare(#pos(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #neg(@y)) -> #GT [0] 1121.48/292.31 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #compare(#s(@x), #0) -> #GT [0] 1121.48/292.31 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] 1121.48/292.31 #eq(#0, #0) -> #true [0] 1121.48/292.31 #eq(#0, #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#0, #s(@y)) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #0) -> #false [0] 1121.48/292.31 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#neg(@x), #pos(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #0) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #neg(@y)) -> #false [0] 1121.48/292.31 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(#s(@x), #0) -> #false [0] 1121.48/292.31 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] 1121.48/292.31 #eq(::(@x_1, @x_2), nil) -> #false [0] 1121.48/292.31 #eq(nil, ::(@y_1, @y_2)) -> #false [0] 1121.48/292.31 #eq(nil, nil) -> #true [0] 1121.48/292.31 #pred(#0) -> #neg(#s(#0)) [0] 1121.48/292.31 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] 1121.48/292.31 #pred(#pos(#s(#0))) -> #0 [0] 1121.48/292.31 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] 1121.48/292.31 #succ(#0) -> #pos(#s(#0)) [0] 1121.48/292.31 #succ(#neg(#s(#0))) -> #0 [0] 1121.48/292.31 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] 1121.48/292.31 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] 1121.48/292.31 #add(v0, v1) -> null_#add [0] 1121.48/292.31 #and(v0, v1) -> null_#and [0] 1121.48/292.31 #ckgt(v0) -> null_#ckgt [0] 1121.48/292.31 #compare(v0, v1) -> null_#compare [0] 1121.48/292.31 #eq(v0, v1) -> null_#eq [0] 1121.48/292.31 #pred(v0) -> null_#pred [0] 1121.48/292.31 #succ(v0) -> null_#succ [0] 1121.48/292.31 newline#7(v0, v1, v2, v3) -> null_newline#7 [0] 1121.48/292.31 newline#1(v0, v1, v2) -> null_newline#1 [0] 1121.48/292.31 right#1(v0) -> null_right#1 [0] 1121.48/292.31 lcstable#1(v0, v1) -> null_lcstable#1 [0] 1121.48/292.31 #abs(v0) -> null_#abs [0] 1121.48/292.31 firstline#1(v0) -> null_firstline#1 [0] 1121.48/292.31 newline#2(v0, v1, v2, v3) -> null_newline#2 [0] 1121.48/292.31 max#1(v0, v1, v2) -> null_max#1 [0] 1121.48/292.31 lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] 1121.48/292.31 1121.48/292.31 The TRS has the following type information: 1121.48/292.31 #abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #0 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #neg :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #pos :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #s :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #equal :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 #eq :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 #greater :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 #compare :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #EQ:#GT:#LT:null_#compare 1121.48/292.31 plus :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 #add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 firstline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 :: :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 nil :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 lcs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 lcs#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 lcstable :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 lcs#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 lcs#3 :: 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#0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.31 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 null_#compare :: #EQ:#GT:#LT:null_#compare 1121.48/292.31 null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.31 null_#pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_#succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_newline#7 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_#abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_newline#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_max#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 null_lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 1121.48/292.32 Rewrite Strategy: INNERMOST 1121.48/292.32 ---------------------------------------- 1121.48/292.32 1121.48/292.32 (11) NarrowingProof (BOTH BOUNDS(ID, ID)) 1121.48/292.32 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 1121.48/292.32 ---------------------------------------- 1121.48/292.32 1121.48/292.32 (12) 1121.48/292.32 Obligation: 1121.48/292.32 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 1121.48/292.32 1121.48/292.32 Runtime Complexity Weighted TRS with Types. 1121.48/292.32 The TRS R consists of the following rules: 1121.48/292.32 1121.48/292.32 #abs(#0) -> #0 [1] 1121.48/292.32 #abs(#neg(@x)) -> #pos(@x) [1] 1121.48/292.32 #abs(#pos(@x)) -> #pos(@x) [1] 1121.48/292.32 #abs(#s(@x)) -> #pos(#s(@x)) [1] 1121.48/292.32 #equal(@x, @y) -> #eq(@x, @y) [1] 1121.48/292.32 #greater(#0, #0) -> #ckgt(#EQ) [1] 1121.48/292.32 #greater(#0, #neg(@y')) -> #ckgt(#GT) [1] 1121.48/292.32 #greater(#0, #pos(@y'')) -> #ckgt(#LT) [1] 1121.48/292.32 #greater(#0, #s(@y1)) -> #ckgt(#LT) [1] 1121.48/292.32 #greater(#neg(@x'), #0) -> #ckgt(#LT) [1] 1121.48/292.32 #greater(#neg(@x''), #neg(@y2)) -> #ckgt(#compare(@y2, @x'')) [1] 1121.48/292.32 #greater(#neg(@x1), #pos(@y3)) -> #ckgt(#LT) [1] 1121.48/292.32 #greater(#pos(@x2), #0) -> #ckgt(#GT) [1] 1121.48/292.32 #greater(#pos(@x3), #neg(@y4)) -> #ckgt(#GT) [1] 1121.48/292.32 #greater(#pos(@x4), #pos(@y5)) -> #ckgt(#compare(@x4, @y5)) [1] 1121.48/292.32 #greater(#s(@x5), #0) -> #ckgt(#GT) [1] 1121.48/292.32 #greater(#s(@x6), #s(@y6)) -> #ckgt(#compare(@x6, @y6)) [1] 1121.48/292.32 #greater(@x, @y) -> #ckgt(null_#compare) [1] 1121.48/292.32 plus(@x, @y) -> #add(@x, @y) [1] 1121.48/292.32 firstline(@l) -> firstline#1(@l) [1] 1121.48/292.32 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) [1] 1121.48/292.32 firstline#1(nil) -> nil [1] 1121.48/292.32 lcs(@l1, @l2) -> lcs#1(lcstable#1(@l1, @l2)) [2] 1121.48/292.32 lcs#1(@m) -> lcs#2(@m) [1] 1121.48/292.32 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) [1] 1121.48/292.32 lcs#2(nil) -> #abs(#0) [1] 1121.48/292.32 lcs#3(::(@len, @_@1)) -> @len [1] 1121.48/292.32 lcs#3(nil) -> #abs(#0) [1] 1121.48/292.32 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) [1] 1121.48/292.32 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) [2] 1121.48/292.32 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) [1] 1121.48/292.32 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) [1] 1121.48/292.32 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) [1] 1121.48/292.32 lcstable#3(nil, @l2, @x) -> nil [1] 1121.48/292.32 max(@a, @b) -> max#1(#ckgt(#compare(@a, @b)), @a, @b) [2] 1121.48/292.32 max#1(#false, @a, @b) -> @b [1] 1121.48/292.32 max#1(#true, @a, @b) -> @a [1] 1121.48/292.32 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) [1] 1121.48/292.32 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) [1] 1121.48/292.32 newline#1(nil, @lastline, @y) -> nil [1] 1121.48/292.32 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) [2] 1121.48/292.32 newline#2(nil, @x, @xs, @y) -> nil [1] 1121.48/292.32 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right#1(@nl), @belowVal, @lastline', @nl, @x, @y) [2] 1121.48/292.32 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right#1(@lastline'), @belowVal, @nl, @rightVal, @x, @y) [2] 1121.48/292.32 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) [2] 1121.48/292.32 newline#6(@elem, @nl) -> ::(@elem, @nl) [1] 1121.48/292.32 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) [1] 1121.48/292.32 newline#7(#true, @belowVal, @diagVal, @rightVal) -> plus(@diagVal, #pos(#s(#0))) [1] 1121.48/292.32 right(@l) -> right#1(@l) [1] 1121.48/292.32 right#1(::(@x, @xs)) -> @x [1] 1121.48/292.32 right#1(nil) -> #abs(#0) [1] 1121.48/292.32 #add(#0, @y) -> @y [0] 1121.48/292.32 #add(#neg(#s(#0)), @y) -> #pred(@y) [0] 1121.48/292.32 #add(#neg(#s(#s(#0))), @y) -> #pred(#succ(@y)) [0] 1121.48/292.32 #add(#neg(#s(#s(#s(@x7)))), @y) -> #pred(#succ(#add(#pos(#s(@x7)), @y))) [0] 1121.48/292.32 #add(#neg(#s(#s(@x))), @y) -> #pred(null_#add) [0] 1121.48/292.32 #add(#pos(#s(#0)), @y) -> #succ(@y) [0] 1121.48/292.32 #add(#pos(#s(#s(#0))), @y) -> #succ(#succ(@y)) [0] 1121.48/292.32 #add(#pos(#s(#s(#s(@x8)))), @y) -> #succ(#succ(#add(#pos(#s(@x8)), @y))) [0] 1121.48/292.32 #add(#pos(#s(#s(@x))), @y) -> #succ(null_#add) [0] 1121.48/292.32 #and(#false, #false) -> #false [0] 1121.48/292.32 #and(#false, #true) -> #false [0] 1121.48/292.32 #and(#true, #false) -> #false [0] 1121.48/292.32 #and(#true, #true) -> #true [0] 1121.48/292.32 #ckgt(#EQ) -> #false [0] 1121.48/292.32 #ckgt(#GT) -> #true [0] 1121.48/292.32 #ckgt(#LT) -> #false [0] 1121.48/292.32 #compare(#0, #0) -> #EQ [0] 1121.48/292.32 #compare(#0, #neg(@y)) -> #GT [0] 1121.48/292.32 #compare(#0, #pos(@y)) -> #LT [0] 1121.48/292.32 #compare(#0, #s(@y)) -> #LT [0] 1121.48/292.32 #compare(#neg(@x), #0) -> #LT [0] 1121.48/292.32 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] 1121.48/292.32 #compare(#neg(@x), #pos(@y)) -> #LT [0] 1121.48/292.32 #compare(#pos(@x), #0) -> #GT [0] 1121.48/292.32 #compare(#pos(@x), #neg(@y)) -> #GT [0] 1121.48/292.32 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] 1121.48/292.32 #compare(#s(@x), #0) -> #GT [0] 1121.48/292.32 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] 1121.48/292.32 #eq(#0, #0) -> #true [0] 1121.48/292.32 #eq(#0, #neg(@y)) -> #false [0] 1121.48/292.32 #eq(#0, #pos(@y)) -> #false [0] 1121.48/292.32 #eq(#0, #s(@y)) -> #false [0] 1121.48/292.32 #eq(#neg(@x), #0) -> #false [0] 1121.48/292.32 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] 1121.48/292.32 #eq(#neg(@x), #pos(@y)) -> #false [0] 1121.48/292.32 #eq(#pos(@x), #0) -> #false [0] 1121.48/292.32 #eq(#pos(@x), #neg(@y)) -> #false [0] 1121.48/292.32 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] 1121.48/292.32 #eq(#s(@x), #0) -> #false [0] 1121.48/292.32 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] 1121.48/292.32 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] 1121.48/292.32 #eq(::(@x_1, @x_2), nil) -> #false [0] 1121.48/292.32 #eq(nil, ::(@y_1, @y_2)) -> #false [0] 1121.48/292.32 #eq(nil, nil) -> #true [0] 1121.48/292.32 #pred(#0) -> #neg(#s(#0)) [0] 1121.48/292.32 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) [0] 1121.48/292.32 #pred(#pos(#s(#0))) -> #0 [0] 1121.48/292.32 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) [0] 1121.48/292.32 #succ(#0) -> #pos(#s(#0)) [0] 1121.48/292.32 #succ(#neg(#s(#0))) -> #0 [0] 1121.48/292.32 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) [0] 1121.48/292.32 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) [0] 1121.48/292.32 #add(v0, v1) -> null_#add [0] 1121.48/292.32 #and(v0, v1) -> null_#and [0] 1121.48/292.32 #ckgt(v0) -> null_#ckgt [0] 1121.48/292.32 #compare(v0, v1) -> null_#compare [0] 1121.48/292.32 #eq(v0, v1) -> null_#eq [0] 1121.48/292.32 #pred(v0) -> null_#pred [0] 1121.48/292.32 #succ(v0) -> null_#succ [0] 1121.48/292.32 newline#7(v0, v1, v2, v3) -> null_newline#7 [0] 1121.48/292.32 newline#1(v0, v1, v2) -> null_newline#1 [0] 1121.48/292.32 right#1(v0) -> null_right#1 [0] 1121.48/292.32 lcstable#1(v0, v1) -> null_lcstable#1 [0] 1121.48/292.32 #abs(v0) -> null_#abs [0] 1121.48/292.32 firstline#1(v0) -> null_firstline#1 [0] 1121.48/292.32 newline#2(v0, v1, v2, v3) -> null_newline#2 [0] 1121.48/292.32 max#1(v0, v1, v2) -> null_max#1 [0] 1121.48/292.32 lcstable#3(v0, v1, v2) -> null_lcstable#3 [0] 1121.48/292.32 1121.48/292.32 The TRS has the following type information: 1121.48/292.32 #abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #0 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #neg :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #pos :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #s :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #equal :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.32 #eq :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.32 #greater :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.32 #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.32 #compare :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #EQ:#GT:#LT:null_#compare 1121.48/292.32 plus :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.32 #add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 firstline :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 :: :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 nil :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 lcs :: 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#0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 #pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 #succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 -> #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 #and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.33 #EQ :: #EQ:#GT:#LT:null_#compare 1121.48/292.33 #GT :: #EQ:#GT:#LT:null_#compare 1121.48/292.33 #LT :: #EQ:#GT:#LT:null_#compare 1121.48/292.33 null_#add :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.33 null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.33 null_#compare :: #EQ:#GT:#LT:null_#compare 1121.48/292.33 null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq 1121.48/292.33 null_#pred :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_#succ :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_newline#7 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_newline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_right#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_lcstable#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_#abs :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_firstline#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_newline#2 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_max#1 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 null_lcstable#3 :: #0:#neg:#pos:#s::::nil:null_#add:null_#pred:null_#succ:null_newline#7:null_newline#1:null_right#1:null_lcstable#1:null_#abs:null_firstline#1:null_newline#2:null_max#1:null_lcstable#3 1121.48/292.33 1121.48/292.33 Rewrite Strategy: INNERMOST 1121.48/292.33 ---------------------------------------- 1121.48/292.33 1121.48/292.33 (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 1121.48/292.33 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 1121.48/292.33 The constant constructors are abstracted as follows: 1121.48/292.33 1121.48/292.33 #0 => 0 1121.48/292.33 nil => 1 1121.48/292.33 #false => 1 1121.48/292.33 #true => 2 1121.48/292.33 #EQ => 1 1121.48/292.33 #GT => 2 1121.48/292.33 #LT => 3 1121.48/292.33 null_#add => 0 1121.48/292.33 null_#and => 0 1121.48/292.33 null_#ckgt => 0 1121.48/292.33 null_#compare => 0 1121.48/292.33 null_#eq => 0 1121.48/292.33 null_#pred => 0 1121.48/292.33 null_#succ => 0 1121.48/292.33 null_newline#7 => 0 1121.48/292.33 null_newline#1 => 0 1121.48/292.33 null_right#1 => 0 1121.48/292.33 null_lcstable#1 => 0 1121.48/292.33 null_#abs => 0 1121.48/292.33 null_firstline#1 => 0 1121.48/292.33 null_newline#2 => 0 1121.48/292.33 null_max#1 => 0 1121.48/292.33 null_lcstable#3 => 0 1121.48/292.33 1121.48/292.33 ---------------------------------------- 1121.48/292.33 1121.48/292.33 (14) 1121.48/292.33 Obligation: 1121.48/292.33 Complexity RNTS consisting of the following rules: 1121.48/292.33 1121.48/292.33 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.48/292.33 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x 1121.48/292.33 #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x 1121.48/292.33 #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #add(z, z') -{ 0 }-> #succ(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #succ(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #succ(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x8), @y))) :|: z = 1 + (1 + (1 + (1 + @x8))), @x8 >= 0, z' = @y, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(@y) :|: z = 1 + (1 + 0), z' = @y, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(#succ(@y)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x7), @y))) :|: z' = @y, z = 1 + (1 + (1 + (1 + @x7))), @x7 >= 0, @y >= 0 1121.48/292.33 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.48/292.33 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.33 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.33 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(1) :|: z = 0, z' = 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 firstline(z) -{ 1 }-> firstline#1(@l) :|: z = @l, @l >= 0 1121.48/292.33 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.48/292.33 firstline#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 firstline#1(z) -{ 1 }-> 1 + #abs(0) + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.33 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(@l1, @l2)) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 1121.48/292.33 lcs#1(z) -{ 1 }-> lcs#2(@m) :|: @m >= 0, z = @m 1121.48/292.33 lcs#2(z) -{ 1 }-> lcs#3(@l1) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0 1121.48/292.33 lcs#2(z) -{ 1 }-> #abs(0) :|: z = 1 1121.48/292.33 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.48/292.33 lcs#3(z) -{ 1 }-> #abs(0) :|: z = 1 1121.48/292.33 lcstable(z, z') -{ 1 }-> lcstable#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 1121.48/292.33 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 1121.48/292.33 lcstable#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 lcstable#1(z, z') -{ 1 }-> 1 + firstline(@l2) + 1 :|: z' = @l2, z = 1, @l2 >= 0 1121.48/292.33 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(@m, @l2, @x) :|: @m >= 0, z' = @l2, @x >= 0, @l2 >= 0, z = @m, z'' = @x 1121.48/292.33 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, @x >= 0, z = 1, @l2 >= 0, z'' = @x 1121.48/292.33 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(@x, @l, @l2) + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' = @l2, @x >= 0, @l2 >= 0, z'' = @x 1121.48/292.33 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(@a, @b)), @a, @b) :|: @a >= 0, z = @a, z' = @b, @b >= 0 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 newline(z, z', z'') -{ 1 }-> newline#1(@l, @lastline, @y) :|: @l >= 0, z'' = @l, @lastline >= 0, z' = @lastline, z = @y, @y >= 0 1121.48/292.33 newline#1(z, z', z'') -{ 1 }-> newline#2(@lastline, @x, @xs, @y) :|: @x >= 0, z = 1 + @x + @xs, @lastline >= 0, z' = @lastline, @xs >= 0, @y >= 0, z'' = @y 1121.48/292.33 newline#1(z, z', z'') -{ 1 }-> 1 :|: @lastline >= 0, z = 1, z' = @lastline, @y >= 0, z'' = @y 1121.48/292.33 newline#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) :|: @lastline' >= 0, @x >= 0, z1 = @y, z = 1 + @belowVal + @lastline', @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @belowVal >= 0 1121.48/292.33 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: @x >= 0, z = 1, z1 = @y, @xs >= 0, @y >= 0, z' = @x, z'' = @xs 1121.48/292.33 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 1121.48/292.33 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(right#1(@nl), @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0 1121.48/292.33 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(right#1(@lastline'), @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0 1121.48/292.33 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) :|: z = @diagVal, @x >= 0, @rightVal >= 0, z2 = @x, z' = @belowVal, z3 = @y, z'' = @nl, z1 = @rightVal, @diagVal >= 0, @y >= 0, @belowVal >= 0, @nl >= 0 1121.48/292.33 newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem 1121.48/292.33 newline#7(z, z', z'', z1) -{ 1 }-> plus(@diagVal, 1 + (1 + 0)) :|: z = 2, @rightVal >= 0, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 1121.48/292.33 newline#7(z, z', z'', z1) -{ 1 }-> max(@belowVal, @rightVal) :|: @rightVal >= 0, z = 1, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 1121.48/292.33 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 1121.48/292.33 plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 1121.48/292.33 right(z) -{ 1 }-> right#1(@l) :|: z = @l, @l >= 0 1121.48/292.33 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.33 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 right#1(z) -{ 1 }-> #abs(0) :|: z = 1 1121.48/292.33 1121.48/292.33 1121.48/292.33 ---------------------------------------- 1121.48/292.33 1121.48/292.33 (15) InliningProof (UPPER BOUND(ID)) 1121.48/292.33 Inlined the following terminating rules on right-hand sides where appropriate: 1121.48/292.33 1121.48/292.33 #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x 1121.48/292.33 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x 1121.48/292.33 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.48/292.33 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.48/292.33 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.48/292.33 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.48/292.33 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 lcs#3(z) -{ 1 }-> #abs(0) :|: z = 1 1121.48/292.33 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.48/292.33 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.33 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 right#1(z) -{ 1 }-> #abs(0) :|: z = 1 1121.48/292.33 1121.48/292.33 ---------------------------------------- 1121.48/292.33 1121.48/292.33 (16) 1121.48/292.33 Obligation: 1121.48/292.33 Complexity RNTS consisting of the following rules: 1121.48/292.33 1121.48/292.33 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.48/292.33 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x 1121.48/292.33 #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x 1121.48/292.33 #add(z, z') -{ 0 }-> @y :|: z' = @y, z = 0, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 1 + (1 + 0) 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' = @y, @y >= 0, v0 >= 0, @y = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, v0 >= 0, 0 = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, v0' >= 0, 0 = v0' 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), v0 >= 0, 1 + (1 + @x) = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), 1 + (1 + @x) = 1 + (1 + 0) 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), v0 >= 0, 1 + (1 + (1 + @x)) = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.48/292.33 #add(z, z') -{ 0 }-> 0 :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + @x) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + (1 + @x')) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + (1 + @x')) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @y = 0 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: @x >= 0, z' = @y, z = 1 + (1 + (1 + @x)), @y >= 0, 0 = 0 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0, 0 = 0 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0), 0 = 0 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + 0), z' = @y, @y >= 0, @x >= 0, @y = 1 + (1 + @x) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)), @x' >= 0, 1 + (1 + @x) = 1 + (1 + @x') 1121.48/292.33 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x), @x' >= 0, 1 + (1 + (1 + @x)) = 1 + (1 + @x') 1121.48/292.33 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + @x8), @y))) :|: z = 1 + (1 + (1 + (1 + @x8))), @x8 >= 0, z' = @y, @y >= 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, v0 >= 0, @y = v0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(0) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 1 + (1 + 0) 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(1 + (1 + @x)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + (1 + @x)) 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @y = 0 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + @x))) :|: z' = @y, z = 1 + (1 + (1 + 0)), @y >= 0, @x >= 0, @y = 1 + (1 + @x) 1121.48/292.33 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + @x7), @y))) :|: z' = @y, z = 1 + (1 + (1 + (1 + @x7))), @x7 >= 0, @y >= 0 1121.48/292.33 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.48/292.33 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.48/292.33 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.48/292.33 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.48/292.33 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.48/292.33 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.33 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 1121.48/292.33 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.33 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> 2 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 1121.48/292.33 #greater(z, z') -{ 1 }-> 2 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 1121.48/292.33 #greater(z, z') -{ 1 }-> 2 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 1121.48/292.33 #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 1121.48/292.33 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.48/292.33 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 1121.48/292.33 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 1121.48/292.33 #greater(z, z') -{ 1 }-> 1 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 1121.48/292.33 #greater(z, z') -{ 1 }-> 1 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 1121.48/292.33 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #pred(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 #pred(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.33 #succ(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + (1 + (1 + @x)) 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.33 #succ(z) -{ 0 }-> 1 + (1 + (1 + @x)) :|: @x >= 0, z = 1 + (1 + @x) 1121.48/292.33 firstline(z) -{ 1 }-> firstline#1(@l) :|: z = @l, @l >= 0 1121.48/292.33 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.48/292.33 firstline#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.33 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.48/292.33 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.48/292.33 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(@l1, @l2)) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 1121.48/292.33 lcs#1(z) -{ 1 }-> lcs#2(@m) :|: @m >= 0, z = @m 1121.48/292.33 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.48/292.33 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.33 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.33 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.48/292.33 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.48/292.33 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.33 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.33 lcstable(z, z') -{ 1 }-> lcstable#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 1121.48/292.33 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, @l2), @l2, @x) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 1121.48/292.33 lcstable#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 1121.48/292.33 lcstable#1(z, z') -{ 1 }-> 1 + firstline(@l2) + 1 :|: z' = @l2, z = 1, @l2 >= 0 1121.48/292.33 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(@m, @l2, @x) :|: @m >= 0, z' = @l2, @x >= 0, @l2 >= 0, z = @m, z'' = @x 1121.48/292.33 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z' = @l2, @x >= 0, z = 1, @l2 >= 0, z'' = @x 1121.48/292.33 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(@x, @l, @l2) + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z' = @l2, @x >= 0, @l2 >= 0, z'' = @x 1121.48/292.33 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(@a, @b)), @a, @b) :|: @a >= 0, z = @a, z' = @b, @b >= 0 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @a :|: z = 2, @a >= 0, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 max#1(z, z', z'') -{ 1 }-> @b :|: @a >= 0, z = 1, z' = @a, @b >= 0, z'' = @b 1121.48/292.33 max#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.33 newline(z, z', z'') -{ 1 }-> newline#1(@l, @lastline, @y) :|: @l >= 0, z'' = @l, @lastline >= 0, z' = @lastline, z = @y, @y >= 0 1121.48/292.34 newline#1(z, z', z'') -{ 1 }-> newline#2(@lastline, @x, @xs, @y) :|: @x >= 0, z = 1 + @x + @xs, @lastline >= 0, z' = @lastline, @xs >= 0, @y >= 0, z'' = @y 1121.48/292.34 newline#1(z, z', z'') -{ 1 }-> 1 :|: @lastline >= 0, z = 1, z' = @lastline, @y >= 0, z'' = @y 1121.48/292.34 newline#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 1121.48/292.34 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(@xs, @lastline', @y), @belowVal, @lastline', @x, @y) :|: @lastline' >= 0, @x >= 0, z1 = @y, z = 1 + @belowVal + @lastline', @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @belowVal >= 0 1121.48/292.34 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: @x >= 0, z = 1, z1 = @y, @xs >= 0, @y >= 0, z' = @x, z'' = @xs 1121.48/292.34 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, @x' >= 0, @nl = 1 + @x' + @xs, @xs >= 0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, v0 >= 0, @nl = v0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), @belowVal, @lastline', @nl, @x, @y) :|: z'' = @lastline', z1 = @x, @lastline' >= 0, @x >= 0, z = @nl, z2 = @y, z' = @belowVal, @y >= 0, @nl >= 0, @belowVal >= 0, @nl = 1 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, @x' >= 0, @lastline' = 1 + @x' + @xs, @xs >= 0 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, v0 >= 0, @lastline' = v0 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), @belowVal, @nl, @rightVal, @x, @y) :|: z'' = @lastline', @lastline' >= 0, @x >= 0, @rightVal >= 0, z1 = @nl, z2 = @x, z = @rightVal, z' = @belowVal, z3 = @y, @y >= 0, @belowVal >= 0, @nl >= 0, @lastline' = 1 1121.48/292.34 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(@x, @y), @belowVal, @diagVal, @rightVal), @nl) :|: z = @diagVal, @x >= 0, @rightVal >= 0, z2 = @x, z' = @belowVal, z3 = @y, z'' = @nl, z1 = @rightVal, @diagVal >= 0, @y >= 0, @belowVal >= 0, @nl >= 0 1121.48/292.34 newline#6(z, z') -{ 1 }-> 1 + @elem + @nl :|: @elem >= 0, z' = @nl, @nl >= 0, z = @elem 1121.48/292.34 newline#7(z, z', z'', z1) -{ 1 }-> plus(@diagVal, 1 + (1 + 0)) :|: z = 2, @rightVal >= 0, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 1121.48/292.34 newline#7(z, z', z'', z1) -{ 1 }-> max(@belowVal, @rightVal) :|: @rightVal >= 0, z = 1, z' = @belowVal, z1 = @rightVal, @diagVal >= 0, z'' = @diagVal, @belowVal >= 0 1121.48/292.34 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 1121.48/292.34 plus(z, z') -{ 1 }-> #add(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 1121.48/292.34 right(z) -{ 2 }-> @x :|: z = @l, @l >= 0, @x >= 0, @l = 1 + @x + @xs, @xs >= 0 1121.48/292.34 right(z) -{ 1 }-> 0 :|: z = @l, @l >= 0, v0 >= 0, @l = v0 1121.48/292.34 right(z) -{ 2 }-> #abs(0) :|: z = @l, @l >= 0, @l = 1 1121.48/292.34 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.34 right#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 1121.48/292.34 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.34 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.34 1121.48/292.34 1121.48/292.34 ---------------------------------------- 1121.48/292.34 1121.48/292.34 (17) SimplificationProof (BOTH BOUNDS(ID, ID)) 1121.48/292.34 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 1121.48/292.34 ---------------------------------------- 1121.48/292.34 1121.48/292.34 (18) 1121.48/292.34 Obligation: 1121.48/292.34 Complexity RNTS consisting of the following rules: 1121.48/292.34 1121.48/292.34 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.48/292.34 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.48/292.34 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.48/292.34 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.48/292.34 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.48/292.34 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.48/292.34 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.48/292.34 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.48/292.34 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.48/292.34 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.48/292.34 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.48/292.34 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.48/292.34 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.48/292.34 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.48/292.34 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.48/292.34 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.48/292.34 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.48/292.34 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.34 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.48/292.34 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.48/292.34 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.48/292.34 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.48/292.34 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.48/292.34 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.48/292.34 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.48/292.34 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.48/292.34 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.48/292.34 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.48/292.34 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.48/292.34 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.48/292.34 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.34 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.34 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.48/292.34 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.48/292.34 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.48/292.34 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.48/292.34 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.48/292.34 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.48/292.34 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.48/292.34 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.48/292.34 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.48/292.34 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.48/292.34 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.48/292.34 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.48/292.34 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.48/292.34 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.34 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.34 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.48/292.34 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.48/292.34 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.34 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.34 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.48/292.34 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.48/292.34 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.48/292.34 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.48/292.34 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.48/292.34 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.48/292.34 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.48/292.34 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.48/292.34 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.48/292.34 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.48/292.34 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.48/292.34 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.48/292.34 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.48/292.34 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.48/292.34 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.48/292.34 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.48/292.34 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.48/292.34 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.48/292.34 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.48/292.34 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.48/292.34 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.48/292.34 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.48/292.34 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.48/292.34 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.48/292.34 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.48/292.34 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.48/292.34 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.34 right(z) -{ 1 }-> 0 :|: z >= 0 1121.48/292.34 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.48/292.34 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.48/292.34 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.48/292.34 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.48/292.34 1121.48/292.34 1121.48/292.34 ---------------------------------------- 1121.48/292.34 1121.48/292.34 (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 1121.48/292.34 Found the following analysis order by SCC decomposition: 1121.48/292.34 1121.48/292.34 { #and } 1121.48/292.34 { right#1 } 1121.48/292.34 { max#1 } 1121.48/292.34 { #compare } 1121.48/292.34 { #ckgt } 1121.48/292.34 { newline#6 } 1121.48/292.34 { firstline#1, firstline } 1121.48/292.34 { #pred } 1121.48/292.34 { lcs#3 } 1121.48/292.34 { #abs } 1121.48/292.34 { #succ } 1121.48/292.34 { #eq } 1121.48/292.34 { max } 1121.48/292.34 { #greater } 1121.48/292.34 { #add } 1121.48/292.34 { right } 1121.48/292.34 { lcs#2 } 1121.48/292.34 { #equal } 1121.48/292.34 { plus } 1121.48/292.34 { lcs#1 } 1121.48/292.34 { newline#7 } 1121.48/292.34 { newline#5 } 1121.48/292.34 { newline#4 } 1121.48/292.34 { newline#3 } 1121.48/292.34 { newline#1, newline#2 } 1121.48/292.34 { newline } 1121.48/292.34 { lcstable#3 } 1121.48/292.34 { lcstable#2 } 1121.48/292.34 { lcstable#1 } 1121.48/292.34 { lcstable } 1121.48/292.34 { lcs } 1121.48/292.34 1121.48/292.34 ---------------------------------------- 1121.48/292.34 1121.48/292.34 (20) 1121.48/292.34 Obligation: 1121.48/292.34 Complexity RNTS consisting of the following rules: 1121.48/292.34 1121.48/292.34 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.48/292.34 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.48/292.34 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.48/292.34 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.48/292.34 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.36 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.36 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.36 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.36 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.36 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.36 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.36 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.36 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.36 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.36 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.36 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.36 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.36 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.36 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.36 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.36 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.36 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.36 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.36 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.36 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.36 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.36 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.36 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.36 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.36 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.36 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.36 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.36 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.36 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.36 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.36 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.36 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.36 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.36 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.36 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.36 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.36 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.36 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.36 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.36 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.36 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.36 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.36 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.36 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.36 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.36 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.36 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.36 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.36 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.36 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.36 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.36 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.36 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.36 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.36 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.36 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.36 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.36 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.36 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.36 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.36 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.36 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.36 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.36 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.36 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.36 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.36 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.36 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.36 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.36 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.36 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.36 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.36 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.36 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.36 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.36 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.36 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.36 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.38 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.38 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.38 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.38 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.38 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.38 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.38 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.38 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.38 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (21) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.40 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (22) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (23) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed SIZE bound using CoFloCo for: #and 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(1) with polynomial bound: 2 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (24) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {#and}, {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: ?, size: O(1) [2] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (25) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed RUNTIME bound using CoFloCo for: #and 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(1) with polynomial bound: 0 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (26) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (27) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.40 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (28) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (29) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed SIZE bound using KoAT for: right#1 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(n^1) with polynomial bound: z 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (30) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {right#1}, {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 right#1: runtime: ?, size: O(n^1) [z] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (31) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed RUNTIME bound using CoFloCo for: right#1 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(1) with polynomial bound: 2 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (32) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (33) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.40 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (34) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (35) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed SIZE bound using CoFloCo for: max#1 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(n^1) with polynomial bound: z' + z'' 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (36) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {max#1}, {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.40 max#1: runtime: ?, size: O(n^1) [z' + z''] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (37) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.40 1121.59/292.40 Computed RUNTIME bound using CoFloCo for: max#1 1121.59/292.40 after applying outer abstraction to obtain an ITS, 1121.59/292.40 resulting in: O(1) with polynomial bound: 1 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (38) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.40 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.40 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.40 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.40 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.40 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.40 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.40 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.40 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.40 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.40 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.40 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.40 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 1121.59/292.40 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.40 Previous analysis results are: 1121.59/292.40 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.40 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.40 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.40 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (39) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.40 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.40 ---------------------------------------- 1121.59/292.40 1121.59/292.40 (40) 1121.59/292.40 Obligation: 1121.59/292.40 Complexity RNTS consisting of the following rules: 1121.59/292.40 1121.59/292.40 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.40 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.40 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.40 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.40 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.40 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.40 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.40 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.40 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.40 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.40 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.40 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.40 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.40 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.40 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.40 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.40 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.40 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.40 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.40 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.40 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.40 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.40 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.40 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.40 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.40 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.40 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.40 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.40 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.40 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (41) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed SIZE bound using CoFloCo for: #compare 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(1) with polynomial bound: 3 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (42) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {#compare}, {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: ?, size: O(1) [3] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (43) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed RUNTIME bound using CoFloCo for: #compare 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(1) with polynomial bound: 0 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (44) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(#compare(z, z')), z, z') :|: z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (45) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.41 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (46) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (47) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed SIZE bound using CoFloCo for: #ckgt 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(1) with polynomial bound: 2 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (48) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {#ckgt}, {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 #ckgt: runtime: ?, size: O(1) [2] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (49) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed RUNTIME bound using CoFloCo for: #ckgt 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(1) with polynomial bound: 0 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (50) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 2 }-> max#1(#ckgt(s2), z, z') :|: s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (51) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.41 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (52) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (53) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed SIZE bound using CoFloCo for: newline#6 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(n^1) with polynomial bound: 1 + z + z' 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (54) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.41 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.41 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.41 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.41 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.41 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.41 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.41 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.41 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.41 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.41 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.41 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.41 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.41 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.41 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.41 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.41 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.41 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.41 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.41 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.41 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.41 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.41 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.41 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.41 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.41 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.41 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.41 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.41 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.41 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.41 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.41 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.41 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.41 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.41 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.41 1121.59/292.41 Function symbols to be analyzed: {newline#6}, {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.41 Previous analysis results are: 1121.59/292.41 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.41 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.41 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.41 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.41 newline#6: runtime: ?, size: O(n^1) [1 + z + z'] 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (55) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.41 1121.59/292.41 Computed RUNTIME bound using CoFloCo for: newline#6 1121.59/292.41 after applying outer abstraction to obtain an ITS, 1121.59/292.41 resulting in: O(1) with polynomial bound: 1 1121.59/292.41 1121.59/292.41 ---------------------------------------- 1121.59/292.41 1121.59/292.41 (56) 1121.59/292.41 Obligation: 1121.59/292.41 Complexity RNTS consisting of the following rules: 1121.59/292.41 1121.59/292.41 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.41 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.41 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.41 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.41 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.42 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.42 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.42 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.42 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.42 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.42 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.42 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.42 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.42 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.42 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.42 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.42 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.42 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.42 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.42 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.42 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.42 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.42 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.42 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.42 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.42 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.42 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.42 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.42 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.42 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.42 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.42 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.42 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.42 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.42 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.42 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.42 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.42 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.42 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.42 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.42 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.42 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.42 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.42 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.42 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.42 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.42 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.42 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.42 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.42 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.42 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.42 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.42 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.42 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.42 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.42 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.42 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.42 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.42 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.42 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.42 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.42 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.42 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.42 1121.59/292.42 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.42 Previous analysis results are: 1121.59/292.42 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.42 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.42 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.42 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.42 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.42 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.59/292.42 1121.59/292.42 ---------------------------------------- 1121.59/292.42 1121.59/292.42 (57) ResultPropagationProof (UPPER BOUND(ID)) 1121.59/292.42 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.59/292.42 ---------------------------------------- 1121.59/292.42 1121.59/292.42 (58) 1121.59/292.42 Obligation: 1121.59/292.42 Complexity RNTS consisting of the following rules: 1121.59/292.42 1121.59/292.42 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.42 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.42 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.42 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.42 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.42 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.42 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.42 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.42 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.42 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.42 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.42 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.42 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.42 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.42 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.42 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.42 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.42 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.42 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.42 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.42 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.42 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.42 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.42 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.42 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.42 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.42 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.42 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.42 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.42 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.42 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.42 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.42 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.42 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.42 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.42 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.43 1121.59/292.43 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.43 Previous analysis results are: 1121.59/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.59/292.43 1121.59/292.43 ---------------------------------------- 1121.59/292.43 1121.59/292.43 (59) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.43 1121.59/292.43 Computed SIZE bound using KoAT for: firstline#1 1121.59/292.43 after applying outer abstraction to obtain an ITS, 1121.59/292.43 resulting in: O(n^1) with polynomial bound: z 1121.59/292.43 1121.59/292.43 Computed SIZE bound using CoFloCo for: firstline 1121.59/292.43 after applying outer abstraction to obtain an ITS, 1121.59/292.43 resulting in: O(n^1) with polynomial bound: z 1121.59/292.43 1121.59/292.43 ---------------------------------------- 1121.59/292.43 1121.59/292.43 (60) 1121.59/292.43 Obligation: 1121.59/292.43 Complexity RNTS consisting of the following rules: 1121.59/292.43 1121.59/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.59/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.59/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.59/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.59/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.59/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.59/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.59/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.59/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.59/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.59/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.59/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.59/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.59/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.59/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.59/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.59/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.59/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.59/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.59/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.59/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.59/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.59/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.59/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.59/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.59/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.59/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.59/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.59/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.59/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.59/292.43 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.59/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.59/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.59/292.43 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.59/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.59/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.59/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.59/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.59/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.59/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.59/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.59/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.59/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.59/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.59/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.59/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.59/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.59/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.59/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.59/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.59/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.59/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.59/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.59/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.59/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.59/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.59/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.59/292.43 1121.59/292.43 Function symbols to be analyzed: {firstline#1,firstline}, {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.59/292.43 Previous analysis results are: 1121.59/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.59/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.59/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.59/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.59/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.59/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.59/292.43 firstline#1: runtime: ?, size: O(n^1) [z] 1121.59/292.43 firstline: runtime: ?, size: O(n^1) [z] 1121.59/292.43 1121.59/292.43 ---------------------------------------- 1121.59/292.43 1121.59/292.43 (61) IntTrsBoundProof (UPPER BOUND(ID)) 1121.59/292.43 1121.59/292.43 Computed RUNTIME bound using CoFloCo for: firstline#1 1121.59/292.43 after applying outer abstraction to obtain an ITS, 1121.59/292.43 resulting in: O(n^1) with polynomial bound: 5 + 3*z 1121.59/292.43 1121.59/292.43 Computed RUNTIME bound using CoFloCo for: firstline 1121.59/292.43 after applying outer abstraction to obtain an ITS, 1121.59/292.43 resulting in: O(n^1) with polynomial bound: 6 + 3*z 1121.59/292.43 1121.59/292.43 ---------------------------------------- 1121.59/292.43 1121.59/292.43 (62) 1121.59/292.43 Obligation: 1121.59/292.43 Complexity RNTS consisting of the following rules: 1121.59/292.43 1121.59/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.59/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.59/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.59/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.59/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 1 }-> firstline#1(z) :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 2 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 + 0 + firstline(@xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 1 }-> 1 + firstline(z') + 1 :|: z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (63) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.43 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (64) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (65) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed SIZE bound using CoFloCo for: #pred 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(n^1) with polynomial bound: 2 + z 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (66) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#pred}, {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: ?, size: O(n^1) [2 + z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (67) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed RUNTIME bound using CoFloCo for: #pred 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(1) with polynomial bound: 0 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (68) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + 0)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (z' - 3))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(1 + (1 + (1 + (z' - 2)))) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (69) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.43 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (70) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (71) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed SIZE bound using KoAT for: lcs#3 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(n^1) with polynomial bound: z 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (72) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {lcs#3}, {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 lcs#3: runtime: ?, size: O(n^1) [z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (73) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed RUNTIME bound using CoFloCo for: lcs#3 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(1) with polynomial bound: 2 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (74) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (75) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.43 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (76) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (77) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed SIZE bound using CoFloCo for: #abs 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(n^1) with polynomial bound: 1 + z 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (78) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.43 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.43 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.43 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.43 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.43 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 1121.89/292.43 Function symbols to be analyzed: {#abs}, {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.43 Previous analysis results are: 1121.89/292.43 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.43 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.43 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.43 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.43 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.43 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.43 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.43 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.43 #abs: runtime: ?, size: O(n^1) [1 + z] 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (79) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.43 1121.89/292.43 Computed RUNTIME bound using CoFloCo for: #abs 1121.89/292.43 after applying outer abstraction to obtain an ITS, 1121.89/292.43 resulting in: O(1) with polynomial bound: 1 1121.89/292.43 1121.89/292.43 ---------------------------------------- 1121.89/292.43 1121.89/292.43 (80) 1121.89/292.43 Obligation: 1121.89/292.43 Complexity RNTS consisting of the following rules: 1121.89/292.43 1121.89/292.43 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.43 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.43 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.43 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.43 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.43 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.43 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.43 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.43 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.43 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.43 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.43 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.43 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.43 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.43 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.43 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.43 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.43 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.43 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.43 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.43 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.43 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.43 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcs#2(z) -{ 2 }-> #abs(0) :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.43 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.43 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.43 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.43 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.43 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.43 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.43 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.43 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.43 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.43 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.43 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(#abs(0), z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.43 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(#abs(0), z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.44 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.44 right(z) -{ 2 }-> #abs(0) :|: z >= 0, z = 1 1121.89/292.44 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 1121.89/292.44 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.44 Previous analysis results are: 1121.89/292.44 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.44 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.44 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.44 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.44 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.44 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.44 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (81) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.44 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (82) 1121.89/292.44 Obligation: 1121.89/292.44 Complexity RNTS consisting of the following rules: 1121.89/292.44 1121.89/292.44 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.44 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.44 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.44 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.44 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.44 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.44 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.44 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.44 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.44 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.44 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.44 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.44 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.44 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.44 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 1121.89/292.44 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.44 Previous analysis results are: 1121.89/292.44 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.44 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.44 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.44 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.44 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.44 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.44 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (83) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.44 1121.89/292.44 Computed SIZE bound using CoFloCo for: #succ 1121.89/292.44 after applying outer abstraction to obtain an ITS, 1121.89/292.44 resulting in: O(n^1) with polynomial bound: 2 + z 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (84) 1121.89/292.44 Obligation: 1121.89/292.44 Complexity RNTS consisting of the following rules: 1121.89/292.44 1121.89/292.44 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.44 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.44 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.44 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.44 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.44 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.44 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.44 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.44 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.44 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.44 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.44 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.44 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.44 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.44 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 1121.89/292.44 Function symbols to be analyzed: {#succ}, {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.44 Previous analysis results are: 1121.89/292.44 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.44 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.44 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.44 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.44 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.44 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.44 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.44 #succ: runtime: ?, size: O(n^1) [2 + z] 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (85) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.44 1121.89/292.44 Computed RUNTIME bound using CoFloCo for: #succ 1121.89/292.44 after applying outer abstraction to obtain an ITS, 1121.89/292.44 resulting in: O(1) with polynomial bound: 0 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (86) 1121.89/292.44 Obligation: 1121.89/292.44 Complexity RNTS consisting of the following rules: 1121.89/292.44 1121.89/292.44 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.44 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.44 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.44 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.44 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.44 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.44 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.44 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.44 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.44 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.44 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.44 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.44 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.44 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.44 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.44 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.44 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.44 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 1121.89/292.44 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.44 Previous analysis results are: 1121.89/292.44 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.44 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.44 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.44 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.44 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.44 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.44 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.44 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.44 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.44 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.44 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (87) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.44 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.44 ---------------------------------------- 1121.89/292.44 1121.89/292.44 (88) 1121.89/292.44 Obligation: 1121.89/292.44 Complexity RNTS consisting of the following rules: 1121.89/292.44 1121.89/292.44 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.44 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.44 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.44 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.44 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.44 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.44 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.44 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.44 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.44 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.44 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.44 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.44 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.44 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.44 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.44 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.44 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.44 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.44 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.44 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.44 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.44 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.44 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.44 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.44 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.44 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.44 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.44 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.44 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.44 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.44 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.44 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.44 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.44 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.44 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (89) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.45 1121.89/292.45 Computed SIZE bound using CoFloCo for: #eq 1121.89/292.45 after applying outer abstraction to obtain an ITS, 1121.89/292.45 resulting in: O(1) with polynomial bound: 2 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (90) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.45 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.45 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.45 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.45 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.45 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.45 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.45 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.45 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.45 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.45 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.45 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {#eq}, {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 #eq: runtime: ?, size: O(1) [2] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (91) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.45 1121.89/292.45 Computed RUNTIME bound using CoFloCo for: #eq 1121.89/292.45 after applying outer abstraction to obtain an ITS, 1121.89/292.45 resulting in: O(1) with polynomial bound: 0 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (92) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.45 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.45 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.45 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.45 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.45 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.45 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.45 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.45 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.45 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.45 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.45 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(#eq(z2, z3), z', z, z1), z'') :|: z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (93) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.45 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (94) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.45 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.45 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.45 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.45 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.45 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.45 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.45 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.45 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.45 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.45 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.45 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (95) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.45 1121.89/292.45 Computed SIZE bound using CoFloCo for: max 1121.89/292.45 after applying outer abstraction to obtain an ITS, 1121.89/292.45 resulting in: O(n^1) with polynomial bound: z + z' 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (96) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.45 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.45 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.45 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.45 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.45 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.45 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.45 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.45 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.45 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.45 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.45 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {max}, {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 max: runtime: ?, size: O(n^1) [z + z'] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (97) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.45 1121.89/292.45 Computed RUNTIME bound using CoFloCo for: max 1121.89/292.45 after applying outer abstraction to obtain an ITS, 1121.89/292.45 resulting in: O(1) with polynomial bound: 3 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (98) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.45 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.45 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.45 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.45 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.45 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.45 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.45 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.45 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.45 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.45 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.45 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.45 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.45 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.45 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.45 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.45 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.45 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.45 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.45 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.45 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.45 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.45 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.45 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.45 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.45 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.45 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.45 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.45 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 1 }-> max(z', z1) :|: z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.45 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.45 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.45 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.45 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.45 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.45 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.45 1121.89/292.45 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.45 Previous analysis results are: 1121.89/292.45 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.45 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.45 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.45 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.45 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.45 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.45 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.45 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.45 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.45 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.45 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (99) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.45 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.45 ---------------------------------------- 1121.89/292.45 1121.89/292.45 (100) 1121.89/292.45 Obligation: 1121.89/292.45 Complexity RNTS consisting of the following rules: 1121.89/292.45 1121.89/292.45 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.45 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.45 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.45 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.45 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.46 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.46 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.46 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.46 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.46 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.46 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.46 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.46 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.46 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.46 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.46 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.46 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.46 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.46 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.46 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.46 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.46 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.46 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.46 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.46 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.46 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.46 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.46 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.46 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.46 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.46 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.46 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.46 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.46 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (101) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed SIZE bound using CoFloCo for: #greater 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(1) with polynomial bound: 2 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (102) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {#greater}, {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: ?, size: O(1) [2] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (103) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed RUNTIME bound using CoFloCo for: #greater 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(1) with polynomial bound: 1 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (104) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (105) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.47 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (106) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (107) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed SIZE bound using KoAT for: #add 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(n^1) with polynomial bound: 2*z + z' 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (108) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {#add}, {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: ?, size: O(n^1) [2*z + z'] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (109) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed RUNTIME bound using CoFloCo for: #add 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(1) with polynomial bound: 0 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (110) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #succ(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> #pred(#succ(#add(1 + (1 + (z - 4)), z'))) :|: z - 4 >= 0, z' >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> #add(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (111) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.47 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (112) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (113) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed SIZE bound using KoAT for: right 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(n^1) with polynomial bound: z 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (114) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {right}, {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 right: runtime: ?, size: O(n^1) [z] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (115) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed RUNTIME bound using CoFloCo for: right 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(1) with polynomial bound: 3 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (116) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (117) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.47 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (118) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (119) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed SIZE bound using KoAT for: lcs#2 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(n^1) with polynomial bound: z 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (120) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.47 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.47 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.47 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.47 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.47 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.47 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.47 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.47 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.47 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.47 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.47 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.47 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.47 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.47 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.47 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.47 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.47 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.47 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.47 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.47 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.47 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.47 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.47 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.47 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.47 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.47 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.47 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.47 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.47 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.47 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.47 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.47 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.47 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.47 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.47 1121.89/292.47 Function symbols to be analyzed: {lcs#2}, {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.47 Previous analysis results are: 1121.89/292.47 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.47 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.47 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.47 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.47 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.47 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.47 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.47 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.47 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.47 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.47 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.47 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.47 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.47 lcs#2: runtime: ?, size: O(n^1) [z] 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (121) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.47 1121.89/292.47 Computed RUNTIME bound using CoFloCo for: lcs#2 1121.89/292.47 after applying outer abstraction to obtain an ITS, 1121.89/292.47 resulting in: O(1) with polynomial bound: 3 1121.89/292.47 1121.89/292.47 ---------------------------------------- 1121.89/292.47 1121.89/292.47 (122) 1121.89/292.47 Obligation: 1121.89/292.47 Complexity RNTS consisting of the following rules: 1121.89/292.47 1121.89/292.47 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.47 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.47 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.47 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.47 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.48 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.48 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.48 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.48 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.48 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.48 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.48 lcs#1(z) -{ 1 }-> lcs#2(z) :|: z >= 0 1121.89/292.48 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.48 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.48 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.48 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.48 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.48 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.48 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.48 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 1121.89/292.48 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.48 Previous analysis results are: 1121.89/292.48 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.48 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.48 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.48 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.48 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.48 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.48 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.48 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.48 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (123) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.48 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (124) 1121.89/292.48 Obligation: 1121.89/292.48 Complexity RNTS consisting of the following rules: 1121.89/292.48 1121.89/292.48 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.48 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.48 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.48 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.48 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.48 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.48 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.48 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.48 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1121.89/292.48 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.48 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.48 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.48 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.48 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.48 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.48 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.48 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 1121.89/292.48 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.48 Previous analysis results are: 1121.89/292.48 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.48 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.48 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.48 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.48 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.48 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.48 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.48 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.48 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (125) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.48 1121.89/292.48 Computed SIZE bound using CoFloCo for: #equal 1121.89/292.48 after applying outer abstraction to obtain an ITS, 1121.89/292.48 resulting in: O(1) with polynomial bound: 2 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (126) 1121.89/292.48 Obligation: 1121.89/292.48 Complexity RNTS consisting of the following rules: 1121.89/292.48 1121.89/292.48 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.48 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.48 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.48 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.48 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.48 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.48 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.48 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.48 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1121.89/292.48 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.48 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.48 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.48 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.48 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.48 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.48 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.48 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 1121.89/292.48 Function symbols to be analyzed: {#equal}, {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.48 Previous analysis results are: 1121.89/292.48 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.48 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.48 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.48 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.48 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.48 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.48 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.48 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.48 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 #equal: runtime: ?, size: O(1) [2] 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (127) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.48 1121.89/292.48 Computed RUNTIME bound using CoFloCo for: #equal 1121.89/292.48 after applying outer abstraction to obtain an ITS, 1121.89/292.48 resulting in: O(1) with polynomial bound: 1 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (128) 1121.89/292.48 Obligation: 1121.89/292.48 Complexity RNTS consisting of the following rules: 1121.89/292.48 1121.89/292.48 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.48 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.48 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.48 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.48 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.48 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.48 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.48 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.48 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1121.89/292.48 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.48 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.48 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.48 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.48 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.48 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.48 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.48 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 1121.89/292.48 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.48 Previous analysis results are: 1121.89/292.48 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.48 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.48 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.48 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.48 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.48 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.48 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.48 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.48 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 #equal: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (129) ResultPropagationProof (UPPER BOUND(ID)) 1121.89/292.48 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (130) 1121.89/292.48 Obligation: 1121.89/292.48 Complexity RNTS consisting of the following rules: 1121.89/292.48 1121.89/292.48 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.48 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1121.89/292.48 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1121.89/292.48 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1121.89/292.48 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1121.89/292.48 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1121.89/292.48 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1121.89/292.48 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1121.89/292.48 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1121.89/292.48 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1121.89/292.48 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1121.89/292.48 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #pred(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1121.89/292.48 #succ(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1121.89/292.48 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1121.89/292.48 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1121.89/292.48 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1121.89/292.48 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1121.89/292.48 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1121.89/292.48 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1121.89/292.48 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1121.89/292.48 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1121.89/292.48 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1121.89/292.48 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1121.89/292.48 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1121.89/292.48 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1121.89/292.48 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1121.89/292.48 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1121.89/292.48 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1121.89/292.48 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1121.89/292.48 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1121.89/292.48 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1121.89/292.48 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1121.89/292.48 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1121.89/292.48 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1121.89/292.48 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1121.89/292.48 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1121.89/292.48 right(z) -{ 1 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1121.89/292.48 right#1(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1121.89/292.48 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1121.89/292.48 1121.89/292.48 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1121.89/292.48 Previous analysis results are: 1121.89/292.48 #and: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 right#1: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1121.89/292.48 #compare: runtime: O(1) [0], size: O(1) [3] 1121.89/292.48 #ckgt: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1121.89/292.48 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1121.89/292.48 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1121.89/292.48 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1121.89/292.48 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1121.89/292.48 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1121.89/292.48 #eq: runtime: O(1) [0], size: O(1) [2] 1121.89/292.48 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1121.89/292.48 #greater: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1121.89/292.48 right: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1121.89/292.48 #equal: runtime: O(1) [1], size: O(1) [2] 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (131) IntTrsBoundProof (UPPER BOUND(ID)) 1121.89/292.48 1121.89/292.48 Computed SIZE bound using CoFloCo for: plus 1121.89/292.48 after applying outer abstraction to obtain an ITS, 1121.89/292.48 resulting in: O(n^1) with polynomial bound: 2*z + z' 1121.89/292.48 1121.89/292.48 ---------------------------------------- 1121.89/292.48 1121.89/292.48 (132) 1121.89/292.48 Obligation: 1121.89/292.48 Complexity RNTS consisting of the following rules: 1121.89/292.48 1121.89/292.48 #abs(z) -{ 1 }-> 0 :|: z = 0 1121.89/292.48 #abs(z) -{ 0 }-> 0 :|: z >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1121.89/292.48 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1121.89/292.48 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1121.89/292.48 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1121.89/292.48 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.49 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.49 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.49 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.49 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.49 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.49 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.49 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.49 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.49 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.49 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.49 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.49 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.49 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.49 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.49 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 1122.10/292.49 Function symbols to be analyzed: {plus}, {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.49 Previous analysis results are: 1122.10/292.49 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.49 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.49 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.49 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.49 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.49 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.49 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.49 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.49 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 plus: runtime: ?, size: O(n^1) [2*z + z'] 1122.10/292.49 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (133) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.49 1122.10/292.49 Computed RUNTIME bound using CoFloCo for: plus 1122.10/292.49 after applying outer abstraction to obtain an ITS, 1122.10/292.49 resulting in: O(1) with polynomial bound: 1 1122.10/292.49 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (134) 1122.10/292.49 Obligation: 1122.10/292.49 Complexity RNTS consisting of the following rules: 1122.10/292.49 1122.10/292.49 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.49 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.49 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.49 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.49 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.49 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.49 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.49 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.49 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.49 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.49 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.49 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.49 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.49 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.49 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 1 }-> plus(z'', 1 + (1 + 0)) :|: z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.49 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.49 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 1122.10/292.49 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.49 Previous analysis results are: 1122.10/292.49 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.49 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.49 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.49 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.49 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.49 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.49 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.49 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.49 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.49 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (135) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.49 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (136) 1122.10/292.49 Obligation: 1122.10/292.49 Complexity RNTS consisting of the following rules: 1122.10/292.49 1122.10/292.49 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.49 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.49 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.49 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.49 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.49 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.49 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.49 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.49 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.49 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.49 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.49 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.49 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.49 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.49 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.49 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.49 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.49 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.49 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.49 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.49 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.49 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.49 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.49 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.49 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 1122.10/292.49 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.49 Previous analysis results are: 1122.10/292.49 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.49 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.49 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.49 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.49 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.49 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.49 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.49 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.49 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.49 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.49 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.49 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.49 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.49 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.49 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (137) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.49 1122.10/292.49 Computed SIZE bound using CoFloCo for: lcs#1 1122.10/292.49 after applying outer abstraction to obtain an ITS, 1122.10/292.49 resulting in: O(n^1) with polynomial bound: z 1122.10/292.49 1122.10/292.49 ---------------------------------------- 1122.10/292.49 1122.10/292.49 (138) 1122.10/292.49 Obligation: 1122.10/292.49 Complexity RNTS consisting of the following rules: 1122.10/292.49 1122.10/292.49 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.49 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.49 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.49 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.49 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.49 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.49 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.49 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.49 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.49 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.49 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.49 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.49 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.49 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.49 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.49 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.49 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.49 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.49 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.49 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.49 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.49 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.49 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.49 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.49 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.49 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.49 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.49 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.49 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.49 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.49 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.49 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.49 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {lcs#1}, {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: ?, size: O(n^1) [z] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (139) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed RUNTIME bound using CoFloCo for: lcs#1 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(1) with polynomial bound: 4 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (140) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (141) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (142) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (143) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed SIZE bound using CoFloCo for: newline#7 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(n^1) with polynomial bound: 2 + z' + 2*z'' + z1 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (144) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#7}, {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: ?, size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (145) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed RUNTIME bound using CoFloCo for: newline#7 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(1) with polynomial bound: 4 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (146) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 2 }-> newline#6(newline#7(s22, z', z, z1), z'') :|: s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (147) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (148) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (149) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed SIZE bound using CoFloCo for: newline#5 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(n^1) with polynomial bound: 3 + 2*z + z' + z'' + z1 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (150) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#5}, {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 newline#5: runtime: ?, size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (151) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed RUNTIME bound using CoFloCo for: newline#5 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(1) with polynomial bound: 7 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (152) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 3 }-> newline#5(@x', z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 4 }-> newline#5(s18, z', z1, z, z2, z3) :|: s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 2 }-> newline#5(0, z', z1, z, z2, z3) :|: z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (153) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (154) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (155) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed SIZE bound using KoAT for: newline#4 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(n^1) with polynomial bound: 3 + z + z' + 2*z'' + z1 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (156) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#4}, {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.50 newline#4: runtime: ?, size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (157) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.50 1122.10/292.50 Computed RUNTIME bound using CoFloCo for: newline#4 1122.10/292.50 after applying outer abstraction to obtain an ITS, 1122.10/292.50 resulting in: O(1) with polynomial bound: 11 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (158) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.50 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.50 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.50 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.50 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.50 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.50 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.50 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.50 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.50 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.50 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.50 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.50 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.50 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.50 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.50 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.50 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.50 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.50 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.50 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.50 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.50 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.50 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.50 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.50 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.50 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 3 }-> newline#4(@x', z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 4 }-> newline#4(s17, z', z'', z, z1, z2) :|: s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.50 newline#3(z, z', z'', z1, z2) -{ 2 }-> newline#4(0, z', z'', z, z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.50 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.50 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.50 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.50 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.50 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.50 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.50 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.50 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.50 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.50 1122.10/292.50 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.50 Previous analysis results are: 1122.10/292.50 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.50 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.50 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.50 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.50 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.50 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.50 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.50 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.50 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.50 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.50 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.50 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.50 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.50 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.50 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.50 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.50 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.50 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.50 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (159) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.50 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.50 ---------------------------------------- 1122.10/292.50 1122.10/292.50 (160) 1122.10/292.50 Obligation: 1122.10/292.50 Complexity RNTS consisting of the following rules: 1122.10/292.50 1122.10/292.50 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.50 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.50 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.50 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.50 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (161) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed SIZE bound using KoAT for: newline#3 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: O(n^1) with polynomial bound: 3 + 2*z + z' + 2*z'' 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (162) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline#3}, {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: ?, size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (163) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed RUNTIME bound using CoFloCo for: newline#3 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: O(1) with polynomial bound: 15 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (164) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (165) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (166) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (167) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed SIZE bound using KoAT for: newline#1 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: EXP with polynomial bound: ? 1122.10/292.51 1122.10/292.51 Computed SIZE bound using CoFloCo for: newline#2 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: INF with polynomial bound: ? 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (168) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline#1,newline#2}, {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: ?, size: EXP 1122.10/292.51 newline#2: runtime: ?, size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (169) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed RUNTIME bound using CoFloCo for: newline#1 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: O(n^1) with polynomial bound: 20 + 18*z 1122.10/292.51 1122.10/292.51 Computed RUNTIME bound using CoFloCo for: newline#2 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: O(n^1) with polynomial bound: 37 + 18*z'' 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (170) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 1 }-> newline#1(z'', z', z) :|: z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> newline#2(z', @x, @xs, z'') :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 2 }-> newline#3(newline#1(z'', @lastline', z1), @belowVal, @lastline', z', z1) :|: @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.51 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (171) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (172) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.51 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (173) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed SIZE bound using CoFloCo for: newline 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: INF with polynomial bound: ? 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (174) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {newline}, {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.51 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.51 newline: runtime: ?, size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (175) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed RUNTIME bound using CoFloCo for: newline 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: O(n^1) with polynomial bound: 21 + 18*z'' 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (176) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 + newline(z'', @l, z') + (1 + @l + @ls) :|: z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.51 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.51 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (177) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.51 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (178) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.51 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.51 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.51 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.51 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.51 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.51 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.51 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.51 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.51 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.51 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.51 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.51 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.51 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.51 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.51 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.51 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.51 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.51 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.51 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.51 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.51 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.51 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.51 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.51 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.51 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.51 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.51 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.51 1122.10/292.51 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.51 Previous analysis results are: 1122.10/292.51 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.51 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.51 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.51 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.51 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.51 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.51 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.51 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.51 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.51 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.51 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.51 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.51 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.51 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.51 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.51 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.51 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.51 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.51 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.51 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.51 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.51 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (179) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.51 1122.10/292.51 Computed SIZE bound using CoFloCo for: lcstable#3 1122.10/292.51 after applying outer abstraction to obtain an ITS, 1122.10/292.51 resulting in: INF with polynomial bound: ? 1122.10/292.51 1122.10/292.51 ---------------------------------------- 1122.10/292.51 1122.10/292.51 (180) 1122.10/292.51 Obligation: 1122.10/292.51 Complexity RNTS consisting of the following rules: 1122.10/292.51 1122.10/292.51 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.51 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.51 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.51 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.51 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.51 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.51 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.51 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.51 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.51 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.51 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.51 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.51 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.51 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#3}, {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: ?, size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (181) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed RUNTIME bound using CoFloCo for: lcstable#3 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: O(n^1) with polynomial bound: 22 + 18*z' 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (182) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 1 }-> lcstable#3(z, z', z'') :|: z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (183) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.52 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (184) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (185) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed SIZE bound using CoFloCo for: lcstable#2 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: INF with polynomial bound: ? 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (186) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#2}, {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: ?, size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (187) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed RUNTIME bound using CoFloCo for: lcstable#2 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: O(n^1) with polynomial bound: 23 + 18*z' 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (188) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (189) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.52 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (190) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (191) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed SIZE bound using CoFloCo for: lcstable#1 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: INF with polynomial bound: ? 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (192) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable#1}, {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 lcstable#1: runtime: ?, size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (193) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed RUNTIME bound using KoAT for: lcstable#1 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: O(n^2) with polynomial bound: 7 + 25*z + 18*z*z' + 3*z' 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (194) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 2 }-> lcs#1(lcstable#1(z, z')) :|: z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 1 }-> lcstable#1(z, z') :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 2 }-> lcstable#2(lcstable#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (195) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.52 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (196) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (197) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed SIZE bound using CoFloCo for: lcstable 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: INF with polynomial bound: ? 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (198) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.52 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.52 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.52 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.52 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.52 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.52 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.52 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.52 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.52 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.52 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.52 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.52 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.52 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.52 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.52 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.52 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.52 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.52 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.52 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.52 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.52 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.52 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.52 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.52 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.52 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.52 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.52 1122.10/292.52 Function symbols to be analyzed: {lcstable}, {lcs} 1122.10/292.52 Previous analysis results are: 1122.10/292.52 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.52 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.52 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.52 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.52 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.52 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.52 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.52 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.52 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.52 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.52 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.52 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.52 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.52 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.52 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.52 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.52 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.52 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.52 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.52 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.52 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.52 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.52 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.52 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.52 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.52 lcstable: runtime: ?, size: INF 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (199) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.52 1122.10/292.52 Computed RUNTIME bound using KoAT for: lcstable 1122.10/292.52 after applying outer abstraction to obtain an ITS, 1122.10/292.52 resulting in: O(n^2) with polynomial bound: 8 + 25*z + 18*z*z' + 3*z' 1122.10/292.52 1122.10/292.52 ---------------------------------------- 1122.10/292.52 1122.10/292.52 (200) 1122.10/292.52 Obligation: 1122.10/292.52 Complexity RNTS consisting of the following rules: 1122.10/292.52 1122.10/292.52 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.52 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.52 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.52 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.52 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.52 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.52 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.52 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.52 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.52 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.52 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.52 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.52 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.52 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.53 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.53 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.53 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.53 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.53 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.53 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.53 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.53 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.53 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.53 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.53 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.53 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.53 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 1122.10/292.53 Function symbols to be analyzed: {lcs} 1122.10/292.53 Previous analysis results are: 1122.10/292.53 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.53 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.53 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.53 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.53 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.53 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.53 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.53 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.53 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.53 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.53 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.53 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.53 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.53 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.53 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.53 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.53 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.53 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.53 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.53 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (201) ResultPropagationProof (UPPER BOUND(ID)) 1122.10/292.53 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (202) 1122.10/292.53 Obligation: 1122.10/292.53 Complexity RNTS consisting of the following rules: 1122.10/292.53 1122.10/292.53 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.53 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.53 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.53 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.53 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.53 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.53 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.53 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.53 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.53 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.53 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.53 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.53 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.53 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.53 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.53 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.53 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 1122.10/292.53 Function symbols to be analyzed: {lcs} 1122.10/292.53 Previous analysis results are: 1122.10/292.53 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.53 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.53 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.53 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.53 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.53 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.53 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.53 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.53 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.53 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.53 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.53 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.53 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.53 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.53 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.53 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.53 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.53 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.53 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.53 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (203) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.53 1122.10/292.53 Computed SIZE bound using CoFloCo for: lcs 1122.10/292.53 after applying outer abstraction to obtain an ITS, 1122.10/292.53 resulting in: INF with polynomial bound: ? 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (204) 1122.10/292.53 Obligation: 1122.10/292.53 Complexity RNTS consisting of the following rules: 1122.10/292.53 1122.10/292.53 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.53 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.53 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.53 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.53 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.53 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.53 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.53 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.53 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.53 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.53 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.53 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.53 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.53 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.53 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.53 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.53 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 1122.10/292.53 Function symbols to be analyzed: {lcs} 1122.10/292.53 Previous analysis results are: 1122.10/292.53 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.53 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.53 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.53 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.53 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.53 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.53 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.53 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.53 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.53 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.53 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.53 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.53 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.53 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.53 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.53 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.53 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.53 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.53 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.53 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcs: runtime: ?, size: INF 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (205) IntTrsBoundProof (UPPER BOUND(ID)) 1122.10/292.53 1122.10/292.53 Computed RUNTIME bound using KoAT for: lcs 1122.10/292.53 after applying outer abstraction to obtain an ITS, 1122.10/292.53 resulting in: O(n^2) with polynomial bound: 13 + 25*z + 18*z*z' + 3*z' 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (206) 1122.10/292.53 Obligation: 1122.10/292.53 Complexity RNTS consisting of the following rules: 1122.10/292.53 1122.10/292.53 #abs(z) -{ 1 }-> 0 :|: z = 0 1122.10/292.53 #abs(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 1122.10/292.53 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s12 :|: s12 >= 0, s12 <= 1 + (1 + (z' - 3)) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s13 :|: s13 >= 0, s13 <= 1 + (1 + (1 + (z' - 2))) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s14 :|: s14 >= 0, s14 <= 0 + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> s15 :|: s15 >= 0, s15 <= 1 + (1 + 0) + 2, z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> s30 :|: s28 >= 0, s28 <= 2 * (1 + (1 + (z - 4))) + z', s29 >= 0, s29 <= s28 + 2, s30 >= 0, s30 <= s29 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> s33 :|: s31 >= 0, s31 <= 2 * (1 + (1 + (z - 4))) + z', s32 >= 0, s32 <= s31 + 2, s33 >= 0, s33 <= s32 + 2, z - 4 >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> z' :|: z = 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0, z' = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + 0), z' >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z - 3 >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, v0' >= 0, 0 = v0' 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, v0 >= 0, 1 + (1 + (z' - 3)) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, v0 >= 0, 1 + (1 + (1 + (z' - 2))) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), v0 >= 0, 0 = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, v0 >= 0, 1 + (1 + 0) = v0 1122.10/292.53 #add(z, z') -{ 0 }-> 0 :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, 1 + (1 + 0) = 1 + (1 + 0) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + @x') :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + (1 + @x')) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + 0), z' >= 0, z' = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z - 3 >= 0, z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + 0) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 1 + (1 + 0), 0 = 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (z' - 3)) :|: z = 1 + (1 + 0), z' >= 0, z' - 3 >= 0 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x)) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' = 0, @x >= 0, 1 + (1 + 0) = 1 + (1 + @x) 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 3 >= 0, @x' >= 0, 1 + (1 + (z' - 3)) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + @x')) :|: z = 1 + (1 + (1 + 0)), z' >= 0, z' - 2 >= 0, @x' >= 0, 1 + (1 + (1 + (z' - 2))) = 1 + (1 + @x') 1122.10/292.53 #add(z, z') -{ 0 }-> 1 + (1 + (1 + (z' - 2))) :|: z = 1 + (1 + 0), z' >= 0, z' - 2 >= 0 1122.10/292.53 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 1122.10/292.53 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #ckgt(z) -{ 0 }-> 2 :|: z = 2 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 1 1122.10/292.53 #ckgt(z) -{ 0 }-> 1 :|: z = 3 1122.10/292.53 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 1122.10/292.53 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s21 :|: s21 >= 0, s21 <= 2, z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> s25 :|: s23 >= 0, s23 <= 2, s24 >= 0, s24 <= 2, s25 >= 0, s25 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 1122.10/292.53 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 1122.10/292.53 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 #equal(z, z') -{ 1 }-> s20 :|: s20 >= 0, s20 <= 2, z >= 0, z' >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 1122.10/292.53 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #pred(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #pred(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z = 1 + (1 + 0) 1122.10/292.53 #succ(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + 0) :|: z = 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (z - 3)) :|: z - 3 >= 0 1122.10/292.53 #succ(z) -{ 0 }-> 1 + (1 + (1 + (z - 2))) :|: z - 2 >= 0 1122.10/292.53 firstline(z) -{ 6 + 3*z }-> s7 :|: s7 >= 0, s7 <= z, z >= 0 1122.10/292.53 firstline#1(z) -{ 1 }-> 1 :|: z = 1 1122.10/292.53 firstline#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 firstline#1(z) -{ 7 + 3*@xs }-> 1 + 0 + s10 :|: s10 >= 0, s10 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, v0 >= 0, 0 = v0 1122.10/292.53 firstline#1(z) -{ 8 + 3*@xs }-> 1 + 0 + s9 :|: s9 >= 0, s9 <= @xs, @x >= 0, z = 1 + @x + @xs, @xs >= 0, 0 = 0 1122.10/292.53 lcs(z, z') -{ 13 + 25*z + 18*z*z' + 3*z' }-> s51 :|: s50 >= 0, s50 <= inf3, s51 >= 0, s51 <= s50, z >= 0, z' >= 0 1122.10/292.53 lcs#1(z) -{ 4 }-> s34 :|: s34 >= 0, s34 <= z, z >= 0 1122.10/292.53 lcs#2(z) -{ 2 }-> @len :|: z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @_@1 >= 0, @l1 = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#2(z) -{ 3 }-> s16 :|: s16 >= 0, s16 <= 0 + 1, z = 1 + @l1 + @_@2, @l1 >= 0, @_@2 >= 0, @l1 = 1 1122.10/292.53 lcs#2(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#2(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcs#3(z) -{ 1 }-> @len :|: @_@1 >= 0, z = 1 + @len + @_@1, @len >= 0 1122.10/292.53 lcs#3(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 lcs#3(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 lcstable(z, z') -{ 8 + 25*z + 18*z*z' + 3*z' }-> s52 :|: s52 >= 0, s52 <= inf4, z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 32 + 25*@xs + 18*@xs*z' + 21*z' }-> s54 :|: s53 >= 0, s53 <= inf5, s54 >= 0, s54 <= inf6, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 1122.10/292.53 lcstable#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 1122.10/292.53 lcstable#1(z, z') -{ 7 + 3*z' }-> 1 + s8 + 1 :|: s8 >= 0, s8 <= z', z = 1, z' >= 0 1122.10/292.53 lcstable#2(z, z', z'') -{ 23 + 18*z' }-> s49 :|: s49 >= 0, s49 <= inf2, z >= 0, z'' >= 0, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 lcstable#3(z, z', z'') -{ 22 + 18*z' }-> 1 + s48 + (1 + @l + @ls) :|: s48 >= 0, s48 <= inf1, z = 1 + @l + @ls, @ls >= 0, @l >= 0, z'' >= 0, z' >= 0 1122.10/292.53 max(z, z') -{ 3 }-> s6 :|: s5 >= 0, s5 <= 2, s6 >= 0, s6 <= z' + z, s2 >= 0, s2 <= 3, z >= 0, z' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z' :|: z = 2, z' >= 0, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 max#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline(z, z', z'') -{ 21 + 18*z'' }-> s44 :|: s44 >= 0, s44 <= inf, z'' >= 0, z' >= 0, z >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 38 + 18*@xs }-> s45 :|: s45 >= 0, s45 <= inf', @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 1122.10/292.53 newline#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 37 + 18*z'' }-> s47 :|: s46 >= 0, s46 <= inf'', s47 >= 0, s47 <= 2 * s46 + @belowVal + 2 * @lastline' + 3, @lastline' >= 0, z' >= 0, z = 1 + @belowVal + @lastline', z'' >= 0, z1 >= 0, @belowVal >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 1122.10/292.53 newline#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 14 }-> s41 :|: s41 >= 0, s41 <= @x' + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, @x' >= 0, z = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 13 }-> s42 :|: s42 >= 0, s42 <= 0 + z' + 2 * z'' + z + 3, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0 1122.10/292.53 newline#3(z, z', z'', z1, z2) -{ 15 }-> s43 :|: s43 >= 0, s43 <= s17 + z' + 2 * z'' + z + 3, s17 >= 0, s17 <= 0 + 1, z'' >= 0, z1 >= 0, z2 >= 0, z >= 0, z' >= 0, z = 1 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 10 }-> s38 :|: s38 >= 0, s38 <= 2 * @x' + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, @x' >= 0, z'' = 1 + @x' + @xs, @xs >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 9 }-> s39 :|: s39 >= 0, s39 <= 2 * 0 + z' + z1 + z + 3, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0 1122.10/292.53 newline#4(z, z', z'', z1, z2, z3) -{ 11 }-> s40 :|: s40 >= 0, s40 <= 2 * s18 + z' + z1 + z + 3, s18 >= 0, s18 <= 0 + 1, z'' >= 0, z2 >= 0, z >= 0, z3 >= 0, z' >= 0, z1 >= 0, z'' = 1 1122.10/292.53 newline#5(z, z', z'', z1, z2, z3) -{ 7 }-> s37 :|: s36 >= 0, s36 <= 2 * z + 2 + z' + z1, s37 >= 0, s37 <= s36 + z'' + 1, s22 >= 0, s22 <= 2, z2 >= 0, z1 >= 0, z >= 0, z3 >= 0, z' >= 0, z'' >= 0 1122.10/292.53 newline#6(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 4 }-> s26 :|: s26 >= 0, s26 <= z' + z1, z1 >= 0, z = 1, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 2 }-> s35 :|: s35 >= 0, s35 <= 2 * z'' + (1 + (1 + 0)), z = 2, z1 >= 0, z'' >= 0, z' >= 0 1122.10/292.53 newline#7(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 1122.10/292.53 plus(z, z') -{ 1 }-> s27 :|: s27 >= 0, s27 <= 2 * z + z', z >= 0, z' >= 0 1122.10/292.53 right(z) -{ 2 }-> @x :|: z >= 0, @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right(z) -{ 3 }-> s19 :|: s19 >= 0, s19 <= 0 + 1, z >= 0, z = 1 1122.10/292.53 right(z) -{ 1 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 1 }-> @x :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 1122.10/292.53 right#1(z) -{ 0 }-> 0 :|: z >= 0 1122.10/292.53 right#1(z) -{ 2 }-> 0 :|: z = 1, 0 = 0 1122.10/292.53 right#1(z) -{ 1 }-> 0 :|: z = 1, v0 >= 0, 0 = v0 1122.10/292.53 1122.10/292.53 Function symbols to be analyzed: 1122.10/292.53 Previous analysis results are: 1122.10/292.53 #and: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 right#1: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 max#1: runtime: O(1) [1], size: O(n^1) [z' + z''] 1122.10/292.53 #compare: runtime: O(1) [0], size: O(1) [3] 1122.10/292.53 #ckgt: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 newline#6: runtime: O(1) [1], size: O(n^1) [1 + z + z'] 1122.10/292.53 firstline#1: runtime: O(n^1) [5 + 3*z], size: O(n^1) [z] 1122.10/292.53 firstline: runtime: O(n^1) [6 + 3*z], size: O(n^1) [z] 1122.10/292.53 #pred: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 lcs#3: runtime: O(1) [2], size: O(n^1) [z] 1122.10/292.53 #abs: runtime: O(1) [1], size: O(n^1) [1 + z] 1122.10/292.53 #succ: runtime: O(1) [0], size: O(n^1) [2 + z] 1122.10/292.53 #eq: runtime: O(1) [0], size: O(1) [2] 1122.10/292.53 max: runtime: O(1) [3], size: O(n^1) [z + z'] 1122.10/292.53 #greater: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 #add: runtime: O(1) [0], size: O(n^1) [2*z + z'] 1122.10/292.53 right: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 lcs#2: runtime: O(1) [3], size: O(n^1) [z] 1122.10/292.53 #equal: runtime: O(1) [1], size: O(1) [2] 1122.10/292.53 plus: runtime: O(1) [1], size: O(n^1) [2*z + z'] 1122.10/292.53 lcs#1: runtime: O(1) [4], size: O(n^1) [z] 1122.10/292.53 newline#7: runtime: O(1) [4], size: O(n^1) [2 + z' + 2*z'' + z1] 1122.10/292.53 newline#5: runtime: O(1) [7], size: O(n^1) [3 + 2*z + z' + z'' + z1] 1122.10/292.53 newline#4: runtime: O(1) [11], size: O(n^1) [3 + z + z' + 2*z'' + z1] 1122.10/292.53 newline#3: runtime: O(1) [15], size: O(n^1) [3 + 2*z + z' + 2*z''] 1122.10/292.53 newline#1: runtime: O(n^1) [20 + 18*z], size: EXP 1122.10/292.53 newline#2: runtime: O(n^1) [37 + 18*z''], size: INF 1122.10/292.53 newline: runtime: O(n^1) [21 + 18*z''], size: INF 1122.10/292.53 lcstable#3: runtime: O(n^1) [22 + 18*z'], size: INF 1122.10/292.53 lcstable#2: runtime: O(n^1) [23 + 18*z'], size: INF 1122.10/292.53 lcstable#1: runtime: O(n^2) [7 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcstable: runtime: O(n^2) [8 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 lcs: runtime: O(n^2) [13 + 25*z + 18*z*z' + 3*z'], size: INF 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (207) FinalProof (FINISHED) 1122.10/292.53 Computed overall runtime complexity 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (208) 1122.10/292.53 BOUNDS(1, n^2) 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (209) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1122.10/292.53 Transformed a relative TRS into a decreasing-loop problem. 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (210) 1122.10/292.53 Obligation: 1122.10/292.53 Analyzing the following TRS for decreasing loops: 1122.10/292.53 1122.10/292.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1122.10/292.53 1122.10/292.53 1122.10/292.53 The TRS R consists of the following rules: 1122.10/292.53 1122.10/292.53 #abs(#0) -> #0 1122.10/292.53 #abs(#neg(@x)) -> #pos(@x) 1122.10/292.53 #abs(#pos(@x)) -> #pos(@x) 1122.10/292.53 #abs(#s(@x)) -> #pos(#s(@x)) 1122.10/292.53 #equal(@x, @y) -> #eq(@x, @y) 1122.10/292.53 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 1122.10/292.53 +(@x, @y) -> #add(@x, @y) 1122.10/292.53 firstline(@l) -> firstline#1(@l) 1122.10/292.53 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) 1122.10/292.53 firstline#1(nil) -> nil 1122.10/292.53 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) 1122.10/292.53 lcs#1(@m) -> lcs#2(@m) 1122.10/292.53 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) 1122.10/292.53 lcs#2(nil) -> #abs(#0) 1122.10/292.53 lcs#3(::(@len, @_@1)) -> @len 1122.10/292.53 lcs#3(nil) -> #abs(#0) 1122.10/292.53 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) 1122.10/292.53 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) 1122.10/292.53 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) 1122.10/292.53 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) 1122.10/292.53 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) 1122.10/292.53 lcstable#3(nil, @l2, @x) -> nil 1122.10/292.53 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) 1122.10/292.53 max#1(#false, @a, @b) -> @b 1122.10/292.53 max#1(#true, @a, @b) -> @a 1122.10/292.53 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) 1122.10/292.53 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) 1122.10/292.53 newline#1(nil, @lastline, @y) -> nil 1122.10/292.53 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) 1122.10/292.53 newline#2(nil, @x, @xs, @y) -> nil 1122.10/292.53 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) 1122.10/292.53 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) 1122.10/292.53 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) 1122.10/292.53 newline#6(@elem, @nl) -> ::(@elem, @nl) 1122.10/292.53 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) 1122.10/292.53 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) 1122.10/292.53 right(@l) -> right#1(@l) 1122.10/292.53 right#1(::(@x, @xs)) -> @x 1122.10/292.53 right#1(nil) -> #abs(#0) 1122.10/292.53 1122.10/292.53 The (relative) TRS S consists of the following rules: 1122.10/292.53 1122.10/292.53 #add(#0, @y) -> @y 1122.10/292.53 #add(#neg(#s(#0)), @y) -> #pred(@y) 1122.10/292.53 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #add(#pos(#s(#0)), @y) -> #succ(@y) 1122.10/292.53 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #and(#false, #false) -> #false 1122.10/292.53 #and(#false, #true) -> #false 1122.10/292.53 #and(#true, #false) -> #false 1122.10/292.53 #and(#true, #true) -> #true 1122.10/292.53 #ckgt(#EQ) -> #false 1122.10/292.53 #ckgt(#GT) -> #true 1122.10/292.53 #ckgt(#LT) -> #false 1122.10/292.53 #compare(#0, #0) -> #EQ 1122.10/292.53 #compare(#0, #neg(@y)) -> #GT 1122.10/292.53 #compare(#0, #pos(@y)) -> #LT 1122.10/292.53 #compare(#0, #s(@y)) -> #LT 1122.10/292.53 #compare(#neg(@x), #0) -> #LT 1122.10/292.53 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 1122.10/292.53 #compare(#neg(@x), #pos(@y)) -> #LT 1122.10/292.53 #compare(#pos(@x), #0) -> #GT 1122.10/292.53 #compare(#pos(@x), #neg(@y)) -> #GT 1122.10/292.53 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 1122.10/292.53 #compare(#s(@x), #0) -> #GT 1122.10/292.53 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 1122.10/292.53 #eq(#0, #0) -> #true 1122.10/292.53 #eq(#0, #neg(@y)) -> #false 1122.10/292.53 #eq(#0, #pos(@y)) -> #false 1122.10/292.53 #eq(#0, #s(@y)) -> #false 1122.10/292.53 #eq(#neg(@x), #0) -> #false 1122.10/292.53 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#neg(@x), #pos(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #0) -> #false 1122.10/292.53 #eq(#pos(@x), #neg(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#s(@x), #0) -> #false 1122.10/292.53 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 1122.10/292.53 #eq(::(@x_1, @x_2), nil) -> #false 1122.10/292.53 #eq(nil, ::(@y_1, @y_2)) -> #false 1122.10/292.53 #eq(nil, nil) -> #true 1122.10/292.53 #pred(#0) -> #neg(#s(#0)) 1122.10/292.53 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) 1122.10/292.53 #pred(#pos(#s(#0))) -> #0 1122.10/292.53 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) 1122.10/292.53 #succ(#0) -> #pos(#s(#0)) 1122.10/292.53 #succ(#neg(#s(#0))) -> #0 1122.10/292.53 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) 1122.10/292.53 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) 1122.10/292.53 1122.10/292.53 Rewrite Strategy: INNERMOST 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (211) DecreasingLoopProof (LOWER BOUND(ID)) 1122.10/292.53 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1122.10/292.53 1122.10/292.53 The rewrite sequence 1122.10/292.53 1122.10/292.53 firstline#1(::(@x, @xs)) ->^+ ::(#abs(#0), firstline#1(@xs)) 1122.10/292.53 1122.10/292.53 gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. 1122.10/292.53 1122.10/292.53 The pumping substitution is [@xs / ::(@x, @xs)]. 1122.10/292.53 1122.10/292.53 The result substitution is [ ]. 1122.10/292.53 1122.10/292.53 1122.10/292.53 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (212) 1122.10/292.53 Complex Obligation (BEST) 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (213) 1122.10/292.53 Obligation: 1122.10/292.53 Proved the lower bound n^1 for the following obligation: 1122.10/292.53 1122.10/292.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1122.10/292.53 1122.10/292.53 1122.10/292.53 The TRS R consists of the following rules: 1122.10/292.53 1122.10/292.53 #abs(#0) -> #0 1122.10/292.53 #abs(#neg(@x)) -> #pos(@x) 1122.10/292.53 #abs(#pos(@x)) -> #pos(@x) 1122.10/292.53 #abs(#s(@x)) -> #pos(#s(@x)) 1122.10/292.53 #equal(@x, @y) -> #eq(@x, @y) 1122.10/292.53 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 1122.10/292.53 +(@x, @y) -> #add(@x, @y) 1122.10/292.53 firstline(@l) -> firstline#1(@l) 1122.10/292.53 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) 1122.10/292.53 firstline#1(nil) -> nil 1122.10/292.53 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) 1122.10/292.53 lcs#1(@m) -> lcs#2(@m) 1122.10/292.53 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) 1122.10/292.53 lcs#2(nil) -> #abs(#0) 1122.10/292.53 lcs#3(::(@len, @_@1)) -> @len 1122.10/292.53 lcs#3(nil) -> #abs(#0) 1122.10/292.53 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) 1122.10/292.53 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) 1122.10/292.53 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) 1122.10/292.53 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) 1122.10/292.53 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) 1122.10/292.53 lcstable#3(nil, @l2, @x) -> nil 1122.10/292.53 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) 1122.10/292.53 max#1(#false, @a, @b) -> @b 1122.10/292.53 max#1(#true, @a, @b) -> @a 1122.10/292.53 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) 1122.10/292.53 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) 1122.10/292.53 newline#1(nil, @lastline, @y) -> nil 1122.10/292.53 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) 1122.10/292.53 newline#2(nil, @x, @xs, @y) -> nil 1122.10/292.53 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) 1122.10/292.53 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) 1122.10/292.53 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) 1122.10/292.53 newline#6(@elem, @nl) -> ::(@elem, @nl) 1122.10/292.53 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) 1122.10/292.53 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) 1122.10/292.53 right(@l) -> right#1(@l) 1122.10/292.53 right#1(::(@x, @xs)) -> @x 1122.10/292.53 right#1(nil) -> #abs(#0) 1122.10/292.53 1122.10/292.53 The (relative) TRS S consists of the following rules: 1122.10/292.53 1122.10/292.53 #add(#0, @y) -> @y 1122.10/292.53 #add(#neg(#s(#0)), @y) -> #pred(@y) 1122.10/292.53 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #add(#pos(#s(#0)), @y) -> #succ(@y) 1122.10/292.53 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #and(#false, #false) -> #false 1122.10/292.53 #and(#false, #true) -> #false 1122.10/292.53 #and(#true, #false) -> #false 1122.10/292.53 #and(#true, #true) -> #true 1122.10/292.53 #ckgt(#EQ) -> #false 1122.10/292.53 #ckgt(#GT) -> #true 1122.10/292.53 #ckgt(#LT) -> #false 1122.10/292.53 #compare(#0, #0) -> #EQ 1122.10/292.53 #compare(#0, #neg(@y)) -> #GT 1122.10/292.53 #compare(#0, #pos(@y)) -> #LT 1122.10/292.53 #compare(#0, #s(@y)) -> #LT 1122.10/292.53 #compare(#neg(@x), #0) -> #LT 1122.10/292.53 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 1122.10/292.53 #compare(#neg(@x), #pos(@y)) -> #LT 1122.10/292.53 #compare(#pos(@x), #0) -> #GT 1122.10/292.53 #compare(#pos(@x), #neg(@y)) -> #GT 1122.10/292.53 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 1122.10/292.53 #compare(#s(@x), #0) -> #GT 1122.10/292.53 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 1122.10/292.53 #eq(#0, #0) -> #true 1122.10/292.53 #eq(#0, #neg(@y)) -> #false 1122.10/292.53 #eq(#0, #pos(@y)) -> #false 1122.10/292.53 #eq(#0, #s(@y)) -> #false 1122.10/292.53 #eq(#neg(@x), #0) -> #false 1122.10/292.53 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#neg(@x), #pos(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #0) -> #false 1122.10/292.53 #eq(#pos(@x), #neg(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#s(@x), #0) -> #false 1122.10/292.53 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 1122.10/292.53 #eq(::(@x_1, @x_2), nil) -> #false 1122.10/292.53 #eq(nil, ::(@y_1, @y_2)) -> #false 1122.10/292.53 #eq(nil, nil) -> #true 1122.10/292.53 #pred(#0) -> #neg(#s(#0)) 1122.10/292.53 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) 1122.10/292.53 #pred(#pos(#s(#0))) -> #0 1122.10/292.53 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) 1122.10/292.53 #succ(#0) -> #pos(#s(#0)) 1122.10/292.53 #succ(#neg(#s(#0))) -> #0 1122.10/292.53 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) 1122.10/292.53 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) 1122.10/292.53 1122.10/292.53 Rewrite Strategy: INNERMOST 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (214) LowerBoundPropagationProof (FINISHED) 1122.10/292.53 Propagated lower bound. 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (215) 1122.10/292.53 BOUNDS(n^1, INF) 1122.10/292.53 1122.10/292.53 ---------------------------------------- 1122.10/292.53 1122.10/292.53 (216) 1122.10/292.53 Obligation: 1122.10/292.53 Analyzing the following TRS for decreasing loops: 1122.10/292.53 1122.10/292.53 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). 1122.10/292.53 1122.10/292.53 1122.10/292.53 The TRS R consists of the following rules: 1122.10/292.53 1122.10/292.53 #abs(#0) -> #0 1122.10/292.53 #abs(#neg(@x)) -> #pos(@x) 1122.10/292.53 #abs(#pos(@x)) -> #pos(@x) 1122.10/292.53 #abs(#s(@x)) -> #pos(#s(@x)) 1122.10/292.53 #equal(@x, @y) -> #eq(@x, @y) 1122.10/292.53 #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 1122.10/292.53 +(@x, @y) -> #add(@x, @y) 1122.10/292.53 firstline(@l) -> firstline#1(@l) 1122.10/292.53 firstline#1(::(@x, @xs)) -> ::(#abs(#0), firstline(@xs)) 1122.10/292.53 firstline#1(nil) -> nil 1122.10/292.53 lcs(@l1, @l2) -> lcs#1(lcstable(@l1, @l2)) 1122.10/292.53 lcs#1(@m) -> lcs#2(@m) 1122.10/292.53 lcs#2(::(@l1, @_@2)) -> lcs#3(@l1) 1122.10/292.53 lcs#2(nil) -> #abs(#0) 1122.10/292.53 lcs#3(::(@len, @_@1)) -> @len 1122.10/292.53 lcs#3(nil) -> #abs(#0) 1122.10/292.53 lcstable(@l1, @l2) -> lcstable#1(@l1, @l2) 1122.10/292.53 lcstable#1(::(@x, @xs), @l2) -> lcstable#2(lcstable(@xs, @l2), @l2, @x) 1122.10/292.53 lcstable#1(nil, @l2) -> ::(firstline(@l2), nil) 1122.10/292.53 lcstable#2(@m, @l2, @x) -> lcstable#3(@m, @l2, @x) 1122.10/292.53 lcstable#3(::(@l, @ls), @l2, @x) -> ::(newline(@x, @l, @l2), ::(@l, @ls)) 1122.10/292.53 lcstable#3(nil, @l2, @x) -> nil 1122.10/292.53 max(@a, @b) -> max#1(#greater(@a, @b), @a, @b) 1122.10/292.53 max#1(#false, @a, @b) -> @b 1122.10/292.53 max#1(#true, @a, @b) -> @a 1122.10/292.53 newline(@y, @lastline, @l) -> newline#1(@l, @lastline, @y) 1122.10/292.53 newline#1(::(@x, @xs), @lastline, @y) -> newline#2(@lastline, @x, @xs, @y) 1122.10/292.53 newline#1(nil, @lastline, @y) -> nil 1122.10/292.53 newline#2(::(@belowVal, @lastline'), @x, @xs, @y) -> newline#3(newline(@y, @lastline', @xs), @belowVal, @lastline', @x, @y) 1122.10/292.53 newline#2(nil, @x, @xs, @y) -> nil 1122.10/292.53 newline#3(@nl, @belowVal, @lastline', @x, @y) -> newline#4(right(@nl), @belowVal, @lastline', @nl, @x, @y) 1122.10/292.53 newline#4(@rightVal, @belowVal, @lastline', @nl, @x, @y) -> newline#5(right(@lastline'), @belowVal, @nl, @rightVal, @x, @y) 1122.10/292.53 newline#5(@diagVal, @belowVal, @nl, @rightVal, @x, @y) -> newline#6(newline#7(#equal(@x, @y), @belowVal, @diagVal, @rightVal), @nl) 1122.10/292.53 newline#6(@elem, @nl) -> ::(@elem, @nl) 1122.10/292.53 newline#7(#false, @belowVal, @diagVal, @rightVal) -> max(@belowVal, @rightVal) 1122.10/292.53 newline#7(#true, @belowVal, @diagVal, @rightVal) -> +(@diagVal, #pos(#s(#0))) 1122.10/292.53 right(@l) -> right#1(@l) 1122.10/292.53 right#1(::(@x, @xs)) -> @x 1122.10/292.53 right#1(nil) -> #abs(#0) 1122.10/292.53 1122.10/292.53 The (relative) TRS S consists of the following rules: 1122.10/292.53 1122.10/292.53 #add(#0, @y) -> @y 1122.10/292.53 #add(#neg(#s(#0)), @y) -> #pred(@y) 1122.10/292.53 #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #add(#pos(#s(#0)), @y) -> #succ(@y) 1122.10/292.53 #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) 1122.10/292.53 #and(#false, #false) -> #false 1122.10/292.53 #and(#false, #true) -> #false 1122.10/292.53 #and(#true, #false) -> #false 1122.10/292.53 #and(#true, #true) -> #true 1122.10/292.53 #ckgt(#EQ) -> #false 1122.10/292.53 #ckgt(#GT) -> #true 1122.10/292.53 #ckgt(#LT) -> #false 1122.10/292.53 #compare(#0, #0) -> #EQ 1122.10/292.53 #compare(#0, #neg(@y)) -> #GT 1122.10/292.53 #compare(#0, #pos(@y)) -> #LT 1122.10/292.53 #compare(#0, #s(@y)) -> #LT 1122.10/292.53 #compare(#neg(@x), #0) -> #LT 1122.10/292.53 #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 1122.10/292.53 #compare(#neg(@x), #pos(@y)) -> #LT 1122.10/292.53 #compare(#pos(@x), #0) -> #GT 1122.10/292.53 #compare(#pos(@x), #neg(@y)) -> #GT 1122.10/292.53 #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 1122.10/292.53 #compare(#s(@x), #0) -> #GT 1122.10/292.53 #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 1122.10/292.53 #eq(#0, #0) -> #true 1122.10/292.53 #eq(#0, #neg(@y)) -> #false 1122.10/292.53 #eq(#0, #pos(@y)) -> #false 1122.10/292.53 #eq(#0, #s(@y)) -> #false 1122.10/292.53 #eq(#neg(@x), #0) -> #false 1122.10/292.53 #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#neg(@x), #pos(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #0) -> #false 1122.10/292.53 #eq(#pos(@x), #neg(@y)) -> #false 1122.10/292.53 #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(#s(@x), #0) -> #false 1122.10/292.53 #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 1122.10/292.53 #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 1122.10/292.53 #eq(::(@x_1, @x_2), nil) -> #false 1122.10/292.53 #eq(nil, ::(@y_1, @y_2)) -> #false 1122.10/292.53 #eq(nil, nil) -> #true 1122.10/292.53 #pred(#0) -> #neg(#s(#0)) 1122.10/292.53 #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) 1122.10/292.53 #pred(#pos(#s(#0))) -> #0 1122.10/292.53 #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) 1122.10/292.53 #succ(#0) -> #pos(#s(#0)) 1122.10/292.53 #succ(#neg(#s(#0))) -> #0 1122.10/292.53 #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) 1122.10/292.53 #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) 1122.10/292.53 1122.10/292.53 Rewrite Strategy: INNERMOST 1122.49/292.66 EOF