/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).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), 206 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), 9 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 1028 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 1076 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 198 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 208 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 12 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 168 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 370 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 90 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 1342 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 424 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 1326 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 444 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 391 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 257 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 83 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 258 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 82 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 125 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 299 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 259 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (96) CpxRNTS (97) FinalProof [FINISHED, 0 ms] (98) BOUNDS(1, n^2) (99) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CpxRelTRS (101) SlicingProof [LOWER BOUND(ID), 0 ms] (102) CpxRelTRS (103) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (104) typed CpxTrs (105) OrderProof [LOWER BOUND(ID), 5 ms] (106) typed CpxTrs (107) RewriteLemmaProof [LOWER BOUND(ID), 1992 ms] (108) typed CpxTrs (109) RewriteLemmaProof [LOWER BOUND(ID), 394 ms] (110) BEST (111) proven lower bound (112) LowerBoundPropagationProof [FINISHED, 0 ms] (113) BOUNDS(n^1, INF) (114) typed CpxTrs (115) RewriteLemmaProof [LOWER BOUND(ID), 431 ms] (116) typed CpxTrs (117) RewriteLemmaProof [LOWER BOUND(ID), 464 ms] (118) typed CpxTrs (119) RewriteLemmaProof [LOWER BOUND(ID), 441 ms] (120) typed CpxTrs (121) RewriteLemmaProof [LOWER BOUND(ID), 547 ms] (122) typed CpxTrs (123) RewriteLemmaProof [LOWER BOUND(ID), 562 ms] (124) BOUNDS(1, INF) ---------------------------------------- (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: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort(@x) -> insertionsort(testList(#unit)) testInsertionsortD(@x) -> insertionsortD(testList(#unit)) testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) The (relative) TRS S consists of the following rules: #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) 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: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort(@x) -> insertionsort(testList(#unit)) testInsertionsortD(@x) -> insertionsortD(testList(#unit)) testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) The (relative) TRS S consists of the following rules: #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) 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: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertD(@x, @l) -> insertD#1(@l, @x) [1] insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1] insertD#1(nil, @x) -> ::(@x, nil) [1] insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] insertionsortD(@l) -> insertionsortD#1(@l) [1] insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1] insertionsortD#1(nil) -> nil [1] testInsertionsort(@x) -> insertionsort(testList(#unit)) [1] testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1] testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1] #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] 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: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertD(@x, @l) -> insertD#1(@l, @x) [1] insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1] insertD#1(nil, @x) -> ::(@x, nil) [1] insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] insertionsortD(@l) -> insertionsortD#1(@l) [1] insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1] insertionsortD#1(nil) -> nil [1] testInsertionsort(@x) -> insertionsort(testList(#unit)) [1] testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1] testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1] #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] The TRS has the following type information: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: a -> :::nil testList :: #unit -> :::nil #unit :: #unit testInsertionsortD :: b -> :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT 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: testInsertionsort_1 testInsertionsortD_1 (c) The following functions are completely defined: insertionsortD_1 insertionsort_1 #less_2 testList_1 #abs_1 insertionsortD#1_1 insertionsort#1_1 insertD_2 insert_2 insertD#1_2 insertD#2_4 insert#1_2 insert#2_4 #cklt_1 #compare_2 Due to the following rules being added: #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insertD#2(v0, v1, v2, v3) -> nil [0] insert#2(v0, v1, v2, v3) -> nil [0] And the following fresh constants: null_#cklt, null_#compare, const, const1 ---------------------------------------- (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: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [1] #less(@x, @y) -> #cklt(#compare(@x, @y)) [1] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertD(@x, @l) -> insertD#1(@l, @x) [1] insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1] insertD#1(nil, @x) -> ::(@x, nil) [1] insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1] insertionsort#1(nil) -> nil [1] insertionsortD(@l) -> insertionsortD#1(@l) [1] insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1] insertionsortD#1(nil) -> nil [1] testInsertionsort(@x) -> insertionsort(testList(#unit)) [1] testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1] testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1] #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] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insertD#2(v0, v1, v2, v3) -> nil [0] insert#2(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: a -> :::nil testList :: #unit -> :::nil #unit :: #unit testInsertionsortD :: b -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare const :: a const1 :: b 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: #abs(#0) -> #0 [1] #abs(#neg(@x)) -> #pos(@x) [1] #abs(#pos(@x)) -> #pos(@x) [1] #abs(#s(@x)) -> #pos(#s(@x)) [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] insert(@x, @l) -> insert#1(@l, @x) [1] insert#1(::(@y, @ys), @x) -> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) [2] insert#1(nil, @x) -> ::(@x, nil) [1] insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1] insertD(@x, @l) -> insertD#1(@l, @x) [1] insertD#1(::(@y, @ys), @x) -> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) [2] insertD#1(nil, @x) -> ::(@x, nil) [1] insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1] insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1] insertionsort(@l) -> insertionsort#1(@l) [1] insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort#1(@xs)) [2] insertionsort#1(nil) -> nil [1] insertionsortD(@l) -> insertionsortD#1(@l) [1] insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD#1(@xs)) [2] insertionsortD#1(nil) -> nil [1] testInsertionsort(@x) -> insertionsort(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2] testInsertionsortD(@x) -> insertionsortD(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2] testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1] #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] #cklt(v0) -> null_#cklt [0] #compare(v0, v1) -> null_#compare [0] insertD#2(v0, v1, v2, v3) -> nil [0] insert#2(v0, v1, v2, v3) -> nil [0] The TRS has the following type information: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt #cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true:null_#cklt #true :: #false:#true:null_#cklt insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: a -> :::nil testList :: #unit -> :::nil #unit :: #unit testInsertionsortD :: b -> :::nil #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare null_#cklt :: #false:#true:null_#cklt null_#compare :: #EQ:#GT:#LT:null_#compare const :: a const1 :: b 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: #0 => 0 nil => 0 #false => 1 #true => 2 #unit => 0 #EQ => 1 #GT => 2 #LT => 3 null_#cklt => 0 null_#compare => 0 const => 0 const1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x #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 #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 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, 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) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertD(z, z') -{ 1 }-> insertD#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insertD(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(@l) :|: z = @l, @l >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: @_ >= 0, z = @_ ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: #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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x #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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, 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) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertD(z, z') -{ 1 }-> insertD#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insertD(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(@l) :|: z = @l, @l >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: @_ >= 0, z = @_ ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #compare } { #cklt } { #abs } { #less } { insertD#2, insertD#1, insertD } { insert#2, insert, insert#1 } { testList } { insertionsortD#1 } { insertionsort#1 } { insertionsortD } { insertionsort } { testInsertionsortD } { testInsertionsort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} 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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} 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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: ?, size: O(1) [2] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: 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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] ---------------------------------------- (39) 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 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: ?, size: O(1) [2] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: 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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1 Computed SIZE bound using CoFloCo for: insertD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' Computed SIZE bound using CoFloCo for: insertD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: ?, size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: ?, size: O(n^1) [1 + z + z'] insertD: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insertD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 4*z1 Computed RUNTIME bound using CoFloCo for: insertD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 4*z Computed RUNTIME bound using CoFloCo for: insertD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 4*z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0 insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] ---------------------------------------- (51) 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: 2 + 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: 1 + 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: 1 + z + z' ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: ?, size: O(n^1) [2 + z' + z'' + z1] insert: runtime: ?, size: O(n^1) [1 + z + z'] insert#1: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (53) 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: 3 + 4*z1 Computed RUNTIME bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 4*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: 1 + 4*z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0 insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: testList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 74 ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: ?, size: O(1) [74] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: testList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 11 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsortD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insertionsortD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z^2 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0 insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: ?, size: O(n^1) [z] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insertionsort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z^2 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0 insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsortD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: ?, size: O(n^1) [z] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: insertionsortD after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 4*z^2 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insertionsort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: ?, size: O(n^1) [z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: insertionsort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 4*z^2 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: testInsertionsortD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 74 ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] testInsertionsortD: runtime: ?, size: O(1) [74] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: testInsertionsortD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 21918 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] testInsertionsortD: runtime: O(1) [21918], size: O(1) [74] ---------------------------------------- (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: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] testInsertionsortD: runtime: O(1) [21918], size: O(1) [74] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: testInsertionsort after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 74 ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: {testInsertionsort} Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] testInsertionsortD: runtime: O(1) [21918], size: O(1) [74] testInsertionsort: runtime: ?, size: O(1) [74] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: testInsertionsort after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 21918 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: #abs(z) -{ 1 }-> 0 :|: z = 0 #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0 #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 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 #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 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 = 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 insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0 insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 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' >= 0, z = 1, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0 insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0 insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0 insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0 insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0 insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsort#1(z) -{ 1 }-> 0 :|: z = 0 insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0 insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0 testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0 Function symbols to be analyzed: Previous analysis results are: #compare: runtime: O(1) [0], size: O(1) [3] #cklt: runtime: O(1) [0], size: O(1) [2] #abs: runtime: O(1) [1], size: O(n^1) [1 + z] #less: runtime: O(1) [1], size: O(1) [2] insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1] insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z'] insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [11], size: O(1) [74] insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z] insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z] testInsertionsortD: runtime: O(1) [21918], size: O(1) [74] testInsertionsort: runtime: O(1) [21918], size: O(1) [74] ---------------------------------------- (97) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (98) BOUNDS(1, n^2) ---------------------------------------- (99) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (100) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort(@x) -> insertionsort(testList(#unit)) testInsertionsortD(@x) -> insertionsortD(testList(#unit)) testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) The (relative) TRS S consists of the following rules: #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) Rewrite Strategy: INNERMOST ---------------------------------------- (101) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: testInsertionsort/0 testList/0 testInsertionsortD/0 ---------------------------------------- (102) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) The (relative) TRS S consists of the following rules: #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) Rewrite Strategy: INNERMOST ---------------------------------------- (103) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (104) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil ---------------------------------------- (105) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: #compare, insert, insert#1, insertD, insertD#1, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertD = insertD#1 insertD < insertionsortD#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (106) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: #compare, insert, insert#1, insertD, insertD#1, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertD = insertD#1 insertD < insertionsortD#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (107) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) Induction Base: #compare(gen_#0:#neg:#pos:#s5_3(0), gen_#0:#neg:#pos:#s5_3(0)) ->_R^Omega(0) #EQ Induction Step: #compare(gen_#0:#neg:#pos:#s5_3(+(n8_3, 1)), gen_#0:#neg:#pos:#s5_3(+(n8_3, 1))) ->_R^Omega(0) #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) ->_IH #EQ We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (108) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertD#1, insert, insert#1, insertD, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertD = insertD#1 insertD < insertionsortD#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (109) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3) Induction Base: insertionsortD#1(gen_:::nil6_3(0)) Induction Step: insertionsortD#1(gen_:::nil6_3(+(n319123_3, 1))) ->_R^Omega(1) insertD(#0, insertionsortD(gen_:::nil6_3(n319123_3))) ->_R^Omega(1) insertD(#0, insertionsortD#1(gen_:::nil6_3(n319123_3))) ->_IH insertD(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (110) Complex Obligation (BEST) ---------------------------------------- (111) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertionsortD#1, insert, insert#1, insertionsort, insertionsort#1, insertionsortD They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (112) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (113) BOUNDS(n^1, INF) ---------------------------------------- (114) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertionsortD, insert, insert#1, insertionsort, insertionsort#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (115) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3) Induction Base: insertionsortD(gen_:::nil6_3(0)) Induction Step: insertionsortD(gen_:::nil6_3(+(n322764_3, 1))) ->_R^Omega(1) insertionsortD#1(gen_:::nil6_3(+(n322764_3, 1))) ->_R^Omega(1) insertD(#0, insertionsortD(gen_:::nil6_3(n322764_3))) ->_IH insertD(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (116) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3) insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertionsortD#1, insert, insert#1, insertionsort, insertionsort#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertionsort = insertionsort#1 insertionsortD = insertionsortD#1 ---------------------------------------- (117) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3) Induction Base: insertionsortD#1(gen_:::nil6_3(0)) Induction Step: insertionsortD#1(gen_:::nil6_3(+(n331845_3, 1))) ->_R^Omega(1) insertD(#0, insertionsortD(gen_:::nil6_3(n331845_3))) ->_R^Omega(1) insertD(#0, insertionsortD#1(gen_:::nil6_3(n331845_3))) ->_IH insertD(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (118) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3) insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insert#1, insert, insertionsort, insertionsort#1 They will be analysed ascendingly in the following order: insert = insert#1 insert < insertionsort#1 insertionsort = insertionsort#1 ---------------------------------------- (119) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3) Induction Base: insertionsort#1(gen_:::nil6_3(0)) Induction Step: insertionsort#1(gen_:::nil6_3(+(n346798_3, 1))) ->_R^Omega(1) insert(#0, insertionsort(gen_:::nil6_3(n346798_3))) ->_R^Omega(1) insert(#0, insertionsort#1(gen_:::nil6_3(n346798_3))) ->_IH insert(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (120) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3) insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3) insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertionsort They will be analysed ascendingly in the following order: insertionsort = insertionsort#1 ---------------------------------------- (121) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort(gen_:::nil6_3(n361251_3)) -> *7_3, rt in Omega(n361251_3) Induction Base: insertionsort(gen_:::nil6_3(0)) Induction Step: insertionsort(gen_:::nil6_3(+(n361251_3, 1))) ->_R^Omega(1) insertionsort#1(gen_:::nil6_3(+(n361251_3, 1))) ->_R^Omega(1) insert(#0, insertionsort(gen_:::nil6_3(n361251_3))) ->_IH insert(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (122) Obligation: Innermost TRS: Rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #less(@x, @y) -> #cklt(#compare(@x, @y)) insert(@x, @l) -> insert#1(@l, @x) insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) insert#1(nil, @x) -> ::(@x, nil) insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) insertD(@x, @l) -> insertD#1(@l, @x) insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) insertD#1(nil, @x) -> ::(@x, nil) insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) insertionsort(@l) -> insertionsort#1(@l) insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) insertionsort#1(nil) -> nil insertionsortD(@l) -> insertionsortD#1(@l) insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) insertionsortD#1(nil) -> nil testInsertionsort -> insertionsort(testList) testInsertionsortD -> insertionsortD(testList) testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #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) Types: #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #0 :: #0:#neg:#pos:#s #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s #less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #cklt :: #EQ:#GT:#LT -> #false:#true #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT insert :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil :: :: #0:#neg:#pos:#s -> :::nil -> :::nil insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil nil :: :::nil #false :: #false:#true #true :: #false:#true insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil insertionsort :: :::nil -> :::nil insertionsort#1 :: :::nil -> :::nil insertionsortD :: :::nil -> :::nil insertionsortD#1 :: :::nil -> :::nil testInsertionsort :: :::nil testList :: :::nil testInsertionsortD :: :::nil #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s hole_#false:#true2_3 :: #false:#true hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT hole_:::nil4_3 :: :::nil gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s gen_:::nil6_3 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0) insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3) insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3) insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3) insertionsort(gen_:::nil6_3(n361251_3)) -> *7_3, rt in Omega(n361251_3) Generator Equations: gen_#0:#neg:#pos:#s5_3(0) <=> #0 gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x)) gen_:::nil6_3(0) <=> nil gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x)) The following defined symbols remain to be analysed: insertionsort#1 They will be analysed ascendingly in the following order: insertionsort = insertionsort#1 ---------------------------------------- (123) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: insertionsort#1(gen_:::nil6_3(n380522_3)) -> *7_3, rt in Omega(n380522_3) Induction Base: insertionsort#1(gen_:::nil6_3(0)) Induction Step: insertionsort#1(gen_:::nil6_3(+(n380522_3, 1))) ->_R^Omega(1) insert(#0, insertionsort(gen_:::nil6_3(n380522_3))) ->_R^Omega(1) insert(#0, insertionsort#1(gen_:::nil6_3(n380522_3))) ->_IH insert(#0, *7_3) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (124) BOUNDS(1, INF)