KILLED proof of input_Texo5DXy9Q.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 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), 268 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 20 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 2030 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 383 ms] (30) CpxRNTS (31) CompletionProof [UPPER BOUND(ID), 0 ms] (32) CpxTypedWeightedCompleteTrs (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 77 ms] (40) CdtProblem (41) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) 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: int(0, 0) -> .(0, nil) int(0, s(y)) -> .(0, int(s(0), s(y))) int(s(x), 0) -> nil int(s(x), s(y)) -> int_list(int(x, y)) int_list(nil) -> nil int_list(.(x, y)) -> .(s(x), int_list(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: int(0', 0') -> .(0', nil) int(0', s(y)) -> .(0', int(s(0'), s(y))) int(s(x), 0') -> nil int(s(x), s(y)) -> int_list(int(x, y)) int_list(nil) -> nil int_list(.(x, y)) -> .(s(x), int_list(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: int(0, 0) -> .(0, nil) int(0, s(y)) -> .(0, int(s(0), s(y))) int(s(x), 0) -> nil int(s(x), s(y)) -> int_list(int(x, y)) int_list(nil) -> nil int_list(.(x, y)) -> .(s(x), int_list(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: int(0, 0) -> .(0, nil) [1] int(0, s(y)) -> .(0, int(s(0), s(y))) [1] int(s(x), 0) -> nil [1] int(s(x), s(y)) -> int_list(int(x, y)) [1] int_list(nil) -> nil [1] int_list(.(x, y)) -> .(s(x), int_list(y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: int(0, 0) -> .(0, nil) [1] int(0, s(y)) -> .(0, int(s(0), s(y))) [1] int(s(x), 0) -> nil [1] int(s(x), s(y)) -> int_list(int(x, y)) [1] int_list(nil) -> nil [1] int_list(.(x, y)) -> .(s(x), int_list(y)) [1] The TRS has the following type information: int :: 0:s -> 0:s -> nil:. 0 :: 0:s . :: 0:s -> nil:. -> nil:. nil :: nil:. s :: 0:s -> 0:s int_list :: nil:. -> nil:. Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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: int_2 int_list_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (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: int(0, 0) -> .(0, nil) [1] int(0, s(y)) -> .(0, int(s(0), s(y))) [1] int(s(x), 0) -> nil [1] int(s(x), s(y)) -> int_list(int(x, y)) [1] int_list(nil) -> nil [1] int_list(.(x, y)) -> .(s(x), int_list(y)) [1] The TRS has the following type information: int :: 0:s -> 0:s -> nil:. 0 :: 0:s . :: 0:s -> nil:. -> nil:. nil :: nil:. s :: 0:s -> 0:s int_list :: nil:. -> nil:. Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: int(0, 0) -> .(0, nil) [1] int(0, s(y)) -> .(0, int(s(0), s(y))) [1] int(s(x), 0) -> nil [1] int(s(0), s(0)) -> int_list(.(0, nil)) [2] int(s(0), s(s(y'))) -> int_list(.(0, int(s(0), s(y')))) [2] int(s(s(x')), s(0)) -> int_list(nil) [2] int(s(s(x'')), s(s(y''))) -> int_list(int_list(int(x'', y''))) [2] int_list(nil) -> nil [1] int_list(.(x, y)) -> .(s(x), int_list(y)) [1] The TRS has the following type information: int :: 0:s -> 0:s -> nil:. 0 :: 0:s . :: 0:s -> nil:. -> nil:. nil :: nil:. s :: 0:s -> 0:s int_list :: nil:. -> nil:. Rewrite Strategy: INNERMOST ---------------------------------------- (13) 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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 2 }-> int_list(int_list(int(x'', y''))) :|: z' = 1 + (1 + y''), y'' >= 0, x'' >= 0, z = 1 + (1 + x'') int(z, z') -{ 2 }-> int_list(0) :|: x' >= 0, z = 1 + (1 + x'), z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + y')) :|: z' = 1 + (1 + y'), z = 1 + 0, y' >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + y) :|: z' = 1 + y, y >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (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: int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { int_list } { int } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int_list}, {int} ---------------------------------------- (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: int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int_list}, {int} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: int_list after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int_list}, {int} Previous analysis results are: int_list: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: int_list after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(0) :|: z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + 0) :|: z = 1 + 0, z' = 1 + 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int} Previous analysis results are: int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (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: int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0 int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int} Previous analysis results are: int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: int after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0 int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int} Previous analysis results are: int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] int: runtime: ?, size: EXP ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: int after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 4 }-> s :|: s >= 0, s <= 2 * (1 + 0 + 0), z = 1 + 0, z' = 1 + 0 int(z, z') -{ 3 }-> s' :|: s' >= 0, s' <= 2 * 0, z - 2 >= 0, z' = 1 + 0 int(z, z') -{ 2 }-> int_list(int_list(int(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0 int(z, z') -{ 2 }-> int_list(1 + 0 + int(1 + 0, 1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0 int(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 2 + y }-> 1 + (1 + x) + s'' :|: s'' >= 0, s'' <= 2 * y, z = 1 + x + y, x >= 0, y >= 0 Function symbols to be analyzed: {int} Previous analysis results are: int_list: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] int: runtime: INF, size: EXP ---------------------------------------- (31) 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: none And the following fresh constants: none ---------------------------------------- (32) 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: int(0, 0) -> .(0, nil) [1] int(0, s(y)) -> .(0, int(s(0), s(y))) [1] int(s(x), 0) -> nil [1] int(s(x), s(y)) -> int_list(int(x, y)) [1] int_list(nil) -> nil [1] int_list(.(x, y)) -> .(s(x), int_list(y)) [1] The TRS has the following type information: int :: 0:s -> 0:s -> nil:. 0 :: 0:s . :: 0:s -> nil:. -> nil:. nil :: nil:. s :: 0:s -> 0:s int_list :: nil:. -> nil:. Rewrite Strategy: INNERMOST ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: int(z, z') -{ 1 }-> int_list(int(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x int(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 int(z, z') -{ 1 }-> 1 + 0 + int(1 + 0, 1 + y) :|: z' = 1 + y, y >= 0, z = 0 int(z, z') -{ 1 }-> 1 + 0 + 0 :|: z = 0, z' = 0 int_list(z) -{ 1 }-> 0 :|: z = 0 int_list(z) -{ 1 }-> 1 + (1 + x) + int_list(y) :|: z = 1 + x + y, x >= 0, y >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (35) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, 0) -> c INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), 0) -> c2 INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(nil) -> c4 INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) S tuples: INT(0, 0) -> c INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), 0) -> c2 INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(nil) -> c4 INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples:none Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c, c1_1, c2, c3_2, c4, c5_1 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: INT_LIST(nil) -> c4 INT(s(z0), 0) -> c2 INT(0, 0) -> c ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) S tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples:none Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c3_2, c5_1 ---------------------------------------- (39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) We considered the (Usable) Rules:none And the Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [1] + x_1 POL(0) = [1] POL(INT(x_1, x_2)) = x_2 POL(INT_LIST(x_1)) = 0 POL(c1(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(int(x_1, x_2)) = [1] POL(int_list(x_1)) = [1] POL(nil) = [1] POL(s(x_1)) = [1] + x_1 ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) S tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c3_2, c5_1 ---------------------------------------- (41) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c3_2, c5_1 ---------------------------------------- (43) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) by INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0)) INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0)) INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0)) INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0)) INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(s(z0), s(z1)) -> c3(INT_LIST(int(z0, z1)), INT(z0, z1)) INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_2 ---------------------------------------- (45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: INT(s(s(z0)), s(0)) -> c3(INT_LIST(nil), INT(s(z0), 0)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil)), INT(0, 0)) INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_2 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_2, c3_1 ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(s(0), s(s(z0))) -> c3(INT_LIST(.(0, int(s(0), s(z0)))), INT(0, s(z0))) by INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_2, c3_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(s(s(z0)), s(s(z1))) -> c3(INT_LIST(int_list(int(z0, z1))), INT(s(z0), s(z1))) by INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0)))) INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)), INT(s(s(z0)), s(0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0)))) INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil)), INT(s(s(z0)), s(0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_1, c3_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(0)), s(s(0))) -> c3(INT_LIST(int_list(.(0, nil))), INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c3(INT_LIST(int_list(.(0, int(s(0), s(z0))))), INT(s(0), s(s(z0)))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_1, c3_2 ---------------------------------------- (55) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_1, c3_2, c_1 ---------------------------------------- (57) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(int_list(nil))) by INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil)) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_1, c3_2, c_1 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: INT(s(s(s(z0))), s(s(0))) -> c3(INT_LIST(nil)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(z0)) -> c1(INT(s(0), s(z0))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c1_1, c5_1, c3_1, c3_2, c_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INT(0, s(z0)) -> c1(INT(s(0), s(z0))) by INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) S tuples: INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) K tuples: INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_LIST_1, INT_2 Compound Symbols: c5_1, c3_1, c3_2, c_1, c1_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INT_LIST(.(z0, z1)) -> c5(INT_LIST(z1)) by INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_1, c3_2, c_1, c1_1, c5_1 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: INT(0, s(0)) -> c1(INT(s(0), s(0))) INT(s(0), s(0)) -> c3(INT_LIST(.(0, nil))) INT(s(s(0)), s(s(0))) -> c(INT(s(0), s(0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c3_1, c_1, c1_1, c5_1 ---------------------------------------- (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace INT(s(s(0)), s(s(0))) -> c(INT_LIST(int_list(.(0, nil)))) by INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c3_1, c_1, c1_1, c5_1 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(int_list(.(0, int(s(0), s(z0)))))) by INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c3_1, c_1, c1_1, c5_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INT(s(0), s(s(x0))) -> c3(INT(0, s(x0))) by INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c_1, c1_1, c5_1, c3_1 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), int_list(nil)))) by INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c_1, c1_1, c5_1, c3_1 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: INT(s(s(0)), s(s(0))) -> c(INT_LIST(.(s(0), nil))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) S tuples: INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c_1, c1_1, c5_1, c3_1 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INT_LIST(.(z0, .(y0, y1))) -> c5(INT_LIST(.(y0, y1))) by INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2)))) S tuples: INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2)))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c_1, c1_1, c3_1, c5_1 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int(s(0), s(z0)))))) by INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int_list(int(0, z0)))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: int(0, 0) -> .(0, nil) int(0, s(z0)) -> .(0, int(s(0), s(z0))) int(s(z0), 0) -> nil int(s(z0), s(z1)) -> int_list(int(z0, z1)) int_list(nil) -> nil int_list(.(z0, z1)) -> .(s(z0), int_list(z1)) Tuples: INT(s(0), s(s(z1))) -> c3(INT_LIST(.(0, int_list(int(0, z1)))), INT(0, s(z1))) INT(s(s(s(z0))), s(s(s(z1)))) -> c3(INT_LIST(int_list(int_list(int(z0, z1)))), INT(s(s(z0)), s(s(z1)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT(s(0), s(s(z0)))) INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) INT(s(0), s(s(s(y0)))) -> c3(INT(0, s(s(y0)))) INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2)))) INT(s(s(0)), s(s(s(z0)))) -> c(INT_LIST(.(s(0), int_list(int_list(int(0, z0)))))) S tuples: INT_LIST(.(z0, .(z1, .(y1, y2)))) -> c5(INT_LIST(.(z1, .(y1, y2)))) K tuples: INT(0, s(s(y0))) -> c1(INT(s(0), s(s(y0)))) Defined Rule Symbols: int_2, int_list_1 Defined Pair Symbols: INT_2, INT_LIST_1 Compound Symbols: c3_2, c_1, c1_1, c3_1, c5_1