KILLED proof of input_9mBoYnXA0p.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 251 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 68 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 136 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 422 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 136 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 913 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 131 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (64) CdtProblem (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0) -> s(0) fact(s(x)) -> *(s(x), fact(p(s(x)))) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0') -> s(0') fact(s(x)) -> *'(s(x), fact(p(s(x)))) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y), y) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(s(x)) -> x fact(0) -> s(0) fact(s(x)) -> *(s(x), fact(p(s(x)))) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> *(s(x), fact(p(s(x)))) [1] *(0, y) -> 0 [1] *(s(x), y) -> +(*(x, y), y) [1] +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: * => times + => plus ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> times(s(x), fact(p(s(x)))) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> times(s(x), fact(p(s(x)))) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] The TRS has the following type information: p :: s:0 -> s:0 s :: s:0 -> s:0 fact :: s:0 -> s:0 0 :: s:0 times :: s:0 -> s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: times_2 fact_1 p_1 plus_2 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (12) 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: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> times(s(x), fact(p(s(x)))) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] p(v0) -> 0 [0] The TRS has the following type information: p :: s:0 -> s:0 s :: s:0 -> s:0 fact :: s:0 -> s:0 0 :: s:0 times :: s:0 -> s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> times(s(x), fact(x)) [2] fact(s(x)) -> times(s(x), fact(0)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(0, y) [2] times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] p(v0) -> 0 [0] The TRS has the following type information: p :: s:0 -> s:0 s :: s:0 -> s:0 fact :: s:0 -> s:0 0 :: s:0 times :: s:0 -> s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 Rewrite Strategy: INNERMOST ---------------------------------------- (15) 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 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> times(1 + x, fact(x)) :|: x >= 0, z = 1 + x fact(z) -{ 1 }-> times(1 + x, fact(0)) :|: x >= 0, z = 1 + x fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 2 }-> plus(plus(times(x', y), y), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> plus(0, y) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { p } { times } { fact } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fact} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fact} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fact} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {p}, {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 4*z + 2*z*z' + z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fact after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fact: runtime: ?, size: INF ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fact after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0 fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s :|: s >= 0, s <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {fact} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(1) [1], size: O(n^1) [z] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fact: runtime: INF, size: INF ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: p(v0) -> null_p [0] fact(v0) -> null_fact [0] times(v0, v1) -> null_times [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_p, null_fact, null_times, null_plus ---------------------------------------- (46) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: p(s(x)) -> x [1] fact(0) -> s(0) [1] fact(s(x)) -> times(s(x), fact(p(s(x)))) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] p(v0) -> null_p [0] fact(v0) -> null_fact [0] times(v0, v1) -> null_times [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: p :: s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus s :: s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus fact :: s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus 0 :: s:0:null_p:null_fact:null_times:null_plus times :: s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus plus :: s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus -> s:0:null_p:null_fact:null_times:null_plus null_p :: s:0:null_p:null_fact:null_times:null_plus null_fact :: s:0:null_p:null_fact:null_times:null_plus null_times :: s:0:null_p:null_fact:null_times:null_plus null_plus :: s:0:null_p:null_fact:null_times:null_plus Rewrite Strategy: INNERMOST ---------------------------------------- (47) 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 null_p => 0 null_fact => 0 null_times => 0 null_plus => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> times(1 + x, fact(p(1 + x))) :|: x >= 0, z = 1 + x fact(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 fact(z) -{ 1 }-> 1 + 0 :|: z = 0 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> plus(times(x, y), y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: P(s(z0)) -> c FACT(0) -> c1 FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0))) *'(0, z0) -> c3 *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, 0) -> c5 +'(z0, s(z1)) -> c6(+'(z0, z1)) S tuples: P(s(z0)) -> c FACT(0) -> c1 FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0))) *'(0, z0) -> c3 *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, 0) -> c5 +'(z0, s(z1)) -> c6(+'(z0, z1)) K tuples:none Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: P_1, FACT_1, *'_2, +'_2 Compound Symbols: c, c1, c2_3, c3, c4_2, c5, c6_1 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: FACT(0) -> c1 *'(0, z0) -> c3 P(s(z0)) -> c +'(z0, 0) -> c5 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0))) *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) S tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0))) *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) K tuples:none Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: FACT_1, *'_2, +'_2 Compound Symbols: c2_3, c4_2, c6_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0)))) S tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0)))) K tuples:none Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, +'_2, FACT_1 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0)))) by FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0)))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0)))) S tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0)))) K tuples:none Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, +'_2, FACT_1 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0)))) by FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) S tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) K tuples:none Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, +'_2, FACT_1 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) We considered the (Usable) Rules:none And the Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = x_2 POL(*'(x_1, x_2)) = [2] POL(+(x_1, x_2)) = [3] + [3]x_2 POL(+'(x_1, x_2)) = 0 POL(0) = [2] POL(FACT(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(fact(x_1)) = [1] + [3]x_1 POL(p(x_1)) = [1] POL(s(x_1)) = [3] + x_1 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) S tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) K tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, +'_2, FACT_1 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) We considered the (Usable) Rules:none And the Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [2]x_1 POL(*'(x_1, x_2)) = [2] + [2]x_1 POL(+(x_1, x_2)) = [1] + x_2 + x_2^2 POL(+'(x_1, x_2)) = 0 POL(0) = [2] POL(FACT(x_1)) = [2]x_1^2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(fact(x_1)) = [2] + [2]x_1^2 POL(p(x_1)) = 0 POL(s(x_1)) = [2] + x_1 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) +'(z0, s(z1)) -> c6(+'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) S tuples: +'(z0, s(z1)) -> c6(+'(z0, z1)) K tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, +'_2, FACT_1 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(z1)) -> c6(+'(z0, z1)) by +'(z0, s(s(y1))) -> c6(+'(z0, s(y1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) +'(z0, s(s(y1))) -> c6(+'(z0, s(y1))) S tuples: +'(z0, s(s(y1))) -> c6(+'(z0, s(y1))) K tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, FACT_1, +'_2 Compound Symbols: c4_2, c2_2, c6_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(s(y1))) -> c6(+'(z0, s(y1))) by +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: p(s(z0)) -> z0 fact(0) -> s(0) fact(s(z0)) -> *(s(z0), fact(p(s(z0)))) *(0, z0) -> 0 *(s(z0), z1) -> +(*(z0, z1), z1) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1)))) S tuples: +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1)))) K tuples: FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0)) *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1)) Defined Rule Symbols: p_1, fact_1, *_2, +_2 Defined Pair Symbols: *'_2, FACT_1, +'_2 Compound Symbols: c4_2, c2_2, c6_1