/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 209 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 114 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 1399 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 1024 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 230 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 274 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 261 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 75 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 262 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 251 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 312 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 582 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 120 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 753 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 296 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 633 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 295 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 1809 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 667 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 390 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (102) CpxRNTS (103) FinalProof [FINISHED, 0 ms] (104) BOUNDS(1, n^2) (105) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (106) TRS for Loop Detection (107) DecreasingLoopProof [LOWER BOUND(ID), 16 ms] (108) BEST (109) proven lower bound (110) LowerBoundPropagationProof [FINISHED, 0 ms] (111) BOUNDS(n^1, INF) (112) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #less(@x, @y) -> #cklt(#compare(@x, @y)) and(@x, @y) -> #and(@x, @y) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) isortlist(@l) -> isortlist#1(@l) isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) isortlist#1(nil) -> nil leq(@l1, @l2) -> leq#1(@l1, @l2) leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) leq#1(nil, @l2) -> #true leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) leq#2(nil, @x, @xs) -> #false or(@x, @y) -> #or(@x, @y) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #or(#false, #false) -> #false #or(#false, #true) -> #true #or(#true, #false) -> #true #or(#true, #true) -> #true Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #less(@x, @y) -> #cklt(#compare(@x, @y)) and(@x, @y) -> #and(@x, @y) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) isortlist(@l) -> isortlist#1(@l) isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) isortlist#1(nil) -> nil leq(@l1, @l2) -> leq#1(@l1, @l2) leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) leq#1(nil, @l2) -> #true leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) leq#2(nil, @x, @xs) -> #false or(@x, @y) -> #or(@x, @y) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #or(#false, #false) -> #false #or(#false, #true) -> #true #or(#true, #false) -> #true #or(#true, #true) -> #true Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] and(@x, @y) -> #and(@x, @y) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] isortlist(@l) -> isortlist#1(@l) [1] isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) [1] isortlist#1(nil) -> nil [1] leq(@l1, @l2) -> leq#1(@l1, @l2) [1] leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) [1] leq#1(nil, @l2) -> #true [1] leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1] leq#2(nil, @x, @xs) -> #false [1] or(@x, @y) -> #or(@x, @y) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #or(#false, #false) -> #false [0] #or(#false, #true) -> #true [0] #or(#true, #false) -> #true [0] #or(#true, #true) -> #true [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] and(@x, @y) -> #and(@x, @y) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] isortlist(@l) -> isortlist#1(@l) [1] isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) [1] isortlist#1(nil) -> nil [1] leq(@l1, @l2) -> leq#1(@l1, @l2) [1] leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) [1] leq#1(nil, @l2) -> #true [1] leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1] leq#2(nil, @x, @xs) -> #false [1] or(@x, @y) -> #or(@x, @y) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #or(#false, #false) -> #false [0] #or(#false, #true) -> #true [0] #or(#true, #false) -> #true [0] #or(#true, #true) -> #true [0] The TRS has the following type information: #equal :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true #eq :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true #less :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #EQ:#GT:#LT and :: #false:#true -> #false:#true -> #false:#true #and :: #false:#true -> #false:#true -> #false:#true insert :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s insert#1 :: :::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 insert#2 :: #false:#true -> :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s leq :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true nil :: :::nil:#0:#neg:#pos:#s #false :: #false:#true #true :: #false:#true isortlist :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s isortlist#1 :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s leq#1 :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true leq#2 :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s -> #false:#true or :: #false:#true -> #false:#true -> #false:#true #or :: #false:#true -> #false:#true -> #false:#true #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: :::nil:#0:#neg:#pos:#s #neg :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s #pos :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s #s :: :::nil:#0:#neg:#pos:#s -> :::nil:#0:#neg:#pos:#s Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: leq_2 isortlist_1 #less_2 and_2 #equal_2 leq#1_2 isortlist#1_1 leq#2_3 insert_2 or_2 insert#1_2 insert#2_4 #and_2 #cklt_1 #compare_2 #eq_2 #or_2 Due to the following rules being added: #and(v0, v1) -> null_#and [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #or(v0, v1) -> null_#or [0] leq#1(v0, v1) -> null_leq#1 [0] isortlist#1(v0) -> null_isortlist#1 [0] leq#2(v0, v1, v2) -> null_leq#2 [0] insert#1(v0, v1) -> null_insert#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] And the following fresh constants: null_#and, null_#cklt, null_#compare, null_#eq, null_#or, null_leq#1, null_isortlist#1, null_leq#2, null_insert#1, null_insert#2 ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] and(@x, @y) -> #and(@x, @y) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] isortlist(@l) -> isortlist#1(@l) [1] isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) [1] isortlist#1(nil) -> nil [1] leq(@l1, @l2) -> leq#1(@l1, @l2) [1] leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) [1] leq#1(nil, @l2) -> #true [1] leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1] leq#2(nil, @x, @xs) -> #false [1] or(@x, @y) -> #or(@x, @y) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #or(#false, #false) -> #false [0] #or(#false, #true) -> #true [0] #or(#true, #false) -> #true [0] #or(#true, #true) -> #true [0] #and(v0, v1) -> null_#and [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #or(v0, v1) -> null_#or [0] leq#1(v0, v1) -> null_leq#1 [0] isortlist#1(v0) -> null_isortlist#1 [0] leq#2(v0, v1, v2) -> null_leq#2 [0] insert#1(v0, v1) -> null_insert#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] The TRS has the following type information: #equal :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #eq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #less :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #compare :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #EQ:#GT:#LT:null_#compare and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 insert :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 :: :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 insert#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 leq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 nil :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #false :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #true :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 isortlist :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 leq#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 leq#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #neg :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #pos :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #s :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#cklt :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_leq#1 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_leq#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_insert#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #less(#0, #0) -> #cklt(#EQ) [1] #less(#0, #neg(@y')) -> #cklt(#GT) [1] #less(#0, #pos(@y'')) -> #cklt(#LT) [1] #less(#0, #s(@y1)) -> #cklt(#LT) [1] #less(#neg(@x'), #0) -> #cklt(#LT) [1] #less(#neg(@x''), #neg(@y2)) -> #cklt(#compare(@y2, @x'')) [1] #less(#neg(@x1), #pos(@y3)) -> #cklt(#LT) [1] #less(#pos(@x2), #0) -> #cklt(#GT) [1] #less(#pos(@x3), #neg(@y4)) -> #cklt(#GT) [1] #less(#pos(@x4), #pos(@y5)) -> #cklt(#compare(@x4, @y5)) [1] #less(#s(@x5), #0) -> #cklt(#GT) [1] #less(#s(@x6), #s(@y6)) -> #cklt(#compare(@x6, @y6)) [1] #less(@x, @y) -> #cklt(null_#compare) [1] and(@x, @y) -> #and(@x, @y) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(leq#1(@x, @y), @x, @y, @ys) [2] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] isortlist(@l) -> isortlist#1(@l) [1] isortlist#1(::(@x, @xs)) -> insert(@x, isortlist#1(@xs)) [2] isortlist#1(nil) -> nil [1] leq(@l1, @l2) -> leq#1(@l1, @l2) [1] leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) [1] leq#1(nil, @l2) -> #true [1] leq#2(::(@y, @ys), @x, @xs) -> or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) [4] leq#2(nil, @x, @xs) -> #false [1] or(@x, @y) -> #or(@x, @y) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #cklt(#EQ) -> #false [0] #cklt(#GT) -> #false [0] #cklt(#LT) -> #true [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #or(#false, #false) -> #false [0] #or(#false, #true) -> #true [0] #or(#true, #false) -> #true [0] #or(#true, #true) -> #true [0] #and(v0, v1) -> null_#and [0] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] #or(v0, v1) -> null_#or [0] leq#1(v0, v1) -> null_leq#1 [0] isortlist#1(v0) -> null_isortlist#1 [0] leq#2(v0, v1, v2) -> null_leq#2 [0] insert#1(v0, v1) -> null_insert#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] The TRS has the following type information: #equal :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #eq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #less :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #compare :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #EQ:#GT:#LT:null_#compare and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 insert :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 :: :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 insert#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 leq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 nil :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #false :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #true :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 isortlist :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 leq#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 leq#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 -> #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #neg :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #pos :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 #s :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 -> :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#cklt :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_leq#1 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_leq#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 null_insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 null_insert#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 1 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 #0 => 0 null_#and => 0 null_#cklt => 0 null_#compare => 0 null_#eq => 0 null_#or => 0 null_leq#1 => 0 null_isortlist#1 => 0 null_leq#2 => 0 null_insert#1 => 0 null_insert#2 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #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 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 #less(z, z') -{ 1 }-> #cklt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 #less(z, z') -{ 1 }-> #cklt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 #less(z, z') -{ 1 }-> #cklt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(1) :|: z = 0, z' = 0 #less(z, z') -{ 1 }-> #cklt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 1 }-> #and(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(leq#1(@x, @y), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 insert#1(z, z') -{ 1 }-> 1 + @x + 1 :|: @x >= 0, z = 1, z' = @x insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y isortlist(z) -{ 1 }-> isortlist#1(@l) :|: z = @l, @l >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 leq(z, z') -{ 1 }-> leq#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 leq#1(z, z') -{ 1 }-> leq#2(@l2, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z' = @l2, z = 1, @l2 >= 0 leq#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) :|: z = 1 + @y + @ys, @x >= 0, @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: @x >= 0, z = 1, @xs >= 0, z' = @x, z'' = @xs leq#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 or(z, z') -{ 1 }-> #or(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 2 :|: z = 3 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 and(z, z') -{ 1 }-> #and(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 or(z, z') -{ 1 }-> #or(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #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 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 and(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 2 and(z, z') -{ 1 }-> 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @y = 2, @x = 1 and(z, z') -{ 1 }-> 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 1 and(z, z') -{ 1 }-> 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 1, @y = 1 and(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, v1 >= 0, @x = v0, @y = v1 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(leq#1(@x, @y), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 insert#1(z, z') -{ 1 }-> 1 + @x + 1 :|: @x >= 0, z = 1, z' = @x insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y isortlist(z) -{ 1 }-> isortlist#1(@l) :|: z = @l, @l >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 leq(z, z') -{ 1 }-> leq#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1 leq#1(z, z') -{ 1 }-> leq#2(@l2, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z' = @l2, z = 1, @l2 >= 0 leq#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) :|: z = 1 + @y + @ys, @x >= 0, @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: @x >= 0, z = 1, @xs >= 0, z' = @x, z'' = @xs leq#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 or(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @y = 2, @x = 1 or(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 1 or(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 2 or(z, z') -{ 1 }-> 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 1, @y = 1 or(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, v1 >= 0, @x = v0, @y = v1 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #compare } { #or } { #and } { and } { #cklt } { or } { #less } { #eq } { #equal } { leq#1, leq#2 } { leq } { insert#2, insert, insert#1 } { isortlist#1 } { isortlist } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#compare}, {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#compare}, {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#compare}, {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: ?, size: O(1) [2] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: ?, size: O(1) [2] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: ?, size: O(1) [2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: ?, size: O(1) [2] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #cklt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: ?, size: O(1) [2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: or after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: ?, size: O(1) [2] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #less after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: ?, size: O(1) [2] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #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 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: ?, size: O(1) [2] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: leq#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 Computed SIZE bound using CoFloCo for: leq#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: ?, size: O(1) [2] leq#2: runtime: ?, size: O(1) [2] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: leq#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 9 + 7*z' Computed RUNTIME bound using CoFloCo for: leq#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 8 + 7*z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 1 }-> leq#1(z, z') :|: z >= 0, z' >= 0 leq#1(z, z') -{ 1 }-> leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 4 }-> or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: leq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: ?, size: O(1) [2] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: leq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 10 + 7*z' ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + z' + z'' + z1 Computed SIZE bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' Computed SIZE bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: ?, size: O(n^1) [3 + z' + z'' + z1] insert: runtime: ?, size: O(n^1) [2 + z + z'] insert#1: runtime: ?, size: O(n^1) [2 + z + z'] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 16 + 4*z'' + 13*z1 Computed RUNTIME bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 15 + 13*z' Computed RUNTIME bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 14 + 13*z ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 11 + 7*@y }-> insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: isortlist#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist#1}, {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] isortlist#1: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: isortlist#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 17*z + 26*z^2 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 1 }-> isortlist#1(z) :|: z >= 0 isortlist#1(z) -{ 2 }-> insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] isortlist#1: runtime: O(n^2) [1 + 17*z + 26*z^2], size: O(n^1) [2*z] ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 2 + 17*z + 26*z^2 }-> s21 :|: s21 >= 0, s21 <= 2 * z, z >= 0 isortlist#1(z) -{ 18 + 17*@xs + 26*@xs^2 + 13*s22 }-> s23 :|: s22 >= 0, s22 <= 2 * @xs, s23 >= 0, s23 <= @x + s22 + 2, @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] isortlist#1: runtime: O(n^2) [1 + 17*z + 26*z^2], size: O(n^1) [2*z] ---------------------------------------- (99) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: isortlist after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 2 + 17*z + 26*z^2 }-> s21 :|: s21 >= 0, s21 <= 2 * z, z >= 0 isortlist#1(z) -{ 18 + 17*@xs + 26*@xs^2 + 13*s22 }-> s23 :|: s22 >= 0, s22 <= 2 * @xs, s23 >= 0, s23 <= @x + s22 + 2, @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {isortlist} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] isortlist#1: runtime: O(n^2) [1 + 17*z + 26*z^2], size: O(n^1) [2*z] isortlist: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: isortlist after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 17*z + 26*z^2 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #cklt(z) -{ 0 }-> 2 :|: z = 3 #cklt(z) -{ 0 }-> 1 :|: z = 1 #cklt(z) -{ 0 }-> 1 :|: z = 2 #cklt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 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 #eq(z, z') -{ 0 }-> s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0 #less(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3 #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 1 #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #or(z, z') -{ 0 }-> 2 :|: z' = 2, z = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 1 #or(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #or(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #or(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 and(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z' = 2, z = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 2, z' = 1 and(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 insert(z, z') -{ 15 + 13*z' }-> s18 :|: s18 >= 0, s18 <= z' + z + 2, z' >= 0, z >= 0 insert#1(z, z') -{ 27 + 11*@y + 13*@ys }-> s19 :|: s19 >= 0, s19 <= z' + @y + @ys + 3, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 1 :|: z' >= 0, z = 1 insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 16 + 13*z1 }-> 1 + z'' + s20 :|: s20 >= 0, s20 <= z' + z1 + 2, z' >= 0, z = 1, z'' >= 0, z1 >= 0 isortlist(z) -{ 2 + 17*z + 26*z^2 }-> s21 :|: s21 >= 0, s21 <= 2 * z, z >= 0 isortlist#1(z) -{ 18 + 17*@xs + 26*@xs^2 + 13*s22 }-> s23 :|: s22 >= 0, s22 <= 2 * @xs, s23 >= 0, s23 <= @x + s22 + 2, @x >= 0, z = 1 + @x + @xs, @xs >= 0 isortlist#1(z) -{ 1 }-> 1 :|: z = 1 isortlist#1(z) -{ 0 }-> 0 :|: z >= 0 leq(z, z') -{ 10 + 7*z' }-> s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0 leq#1(z, z') -{ 9 + 7*z' }-> s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0 leq#1(z, z') -{ 1 }-> 2 :|: z = 1, z' >= 0 leq#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 leq#2(z, z', z'') -{ 15 + 7*@ys }-> s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0 leq#2(z, z', z'') -{ 1 }-> 1 :|: z' >= 0, z = 1, z'' >= 0 leq#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z' = 2, z = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 1 or(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, z = 2, z' = 2 or(z, z') -{ 1 }-> 1 :|: z >= 0, z' >= 0, z = 1, z' = 1 or(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #or: runtime: O(1) [0], size: O(1) [2] #and: runtime: O(1) [0], size: O(1) [2] and: runtime: O(1) [1], size: O(1) [2] #cklt: runtime: O(1) [0], size: O(1) [2] or: runtime: O(1) [1], size: O(1) [2] #less: runtime: O(1) [1], size: O(1) [2] #eq: runtime: O(1) [0], size: O(1) [2] #equal: runtime: O(1) [1], size: O(1) [2] leq#1: runtime: O(n^1) [9 + 7*z'], size: O(1) [2] leq#2: runtime: O(n^1) [8 + 7*z], size: O(1) [2] leq: runtime: O(n^1) [10 + 7*z'], size: O(1) [2] insert#2: runtime: O(n^1) [16 + 4*z'' + 13*z1], size: O(n^1) [3 + z' + z'' + z1] insert: runtime: O(n^1) [15 + 13*z'], size: O(n^1) [2 + z + z'] insert#1: runtime: O(n^1) [14 + 13*z], size: O(n^1) [2 + z + z'] isortlist#1: runtime: O(n^2) [1 + 17*z + 26*z^2], size: O(n^1) [2*z] isortlist: runtime: O(n^2) [2 + 17*z + 26*z^2], size: O(n^1) [2*z] ---------------------------------------- (103) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (104) BOUNDS(1, n^2) ---------------------------------------- (105) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (106) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #less(@x, @y) -> #cklt(#compare(@x, @y)) and(@x, @y) -> #and(@x, @y) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) isortlist(@l) -> isortlist#1(@l) isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) isortlist#1(nil) -> nil leq(@l1, @l2) -> leq#1(@l1, @l2) leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) leq#1(nil, @l2) -> #true leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) leq#2(nil, @x, @xs) -> #false or(@x, @y) -> #or(@x, @y) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #or(#false, #false) -> #false #or(#false, #true) -> #true #or(#true, #false) -> #true #or(#true, #true) -> #true Rewrite Strategy: INNERMOST ---------------------------------------- (107) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence isortlist(::(@x1_0, @xs2_0)) ->^+ insert(@x1_0, isortlist(@xs2_0)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [@xs2_0 / ::(@x1_0, @xs2_0)]. The result substitution is [ ]. ---------------------------------------- (108) Complex Obligation (BEST) ---------------------------------------- (109) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #less(@x, @y) -> #cklt(#compare(@x, @y)) and(@x, @y) -> #and(@x, @y) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) isortlist(@l) -> isortlist#1(@l) isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) isortlist#1(nil) -> nil leq(@l1, @l2) -> leq#1(@l1, @l2) leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) leq#1(nil, @l2) -> #true leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) leq#2(nil, @x, @xs) -> #false or(@x, @y) -> #or(@x, @y) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #or(#false, #false) -> #false #or(#false, #true) -> #true #or(#true, #false) -> #true #or(#true, #true) -> #true Rewrite Strategy: INNERMOST ---------------------------------------- (110) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (111) BOUNDS(n^1, INF) ---------------------------------------- (112) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #less(@x, @y) -> #cklt(#compare(@x, @y)) and(@x, @y) -> #and(@x, @y) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) isortlist(@l) -> isortlist#1(@l) isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) isortlist#1(nil) -> nil leq(@l1, @l2) -> leq#1(@l1, @l2) leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) leq#1(nil, @l2) -> #true leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) leq#2(nil, @x, @xs) -> #false or(@x, @y) -> #or(@x, @y) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #cklt(#EQ) -> #false #cklt(#GT) -> #false #cklt(#LT) -> #true #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #or(#false, #false) -> #false #or(#false, #true) -> #true #or(#true, #false) -> #true #or(#true, #true) -> #true Rewrite Strategy: INNERMOST