MAYBE proof of input_lRp6Pz7aEP.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 59 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 912 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 165 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms] (42) CdtProblem (43) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 9 ms] (44) CdtProblem (45) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (62) 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: fst(0, Z) -> nil fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) from(X) -> cons(X, from(s(X))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) len(nil) -> 0 len(cons(X, Z)) -> s(len(Z)) 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: fst(0', Z) -> nil fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) from(X) -> cons(X, from(s(X))) add(0', X) -> X add(s(X), Y) -> s(add(X, Y)) len(nil) -> 0' len(cons(X, Z)) -> s(len(Z)) 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: fst(0, Z) -> nil fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) from(X) -> cons(X, from(s(X))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) len(nil) -> 0 len(cons(X, Z)) -> s(len(Z)) 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: fst(0, Z) -> nil [1] fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] len(nil) -> 0 [1] len(cons(X, Z)) -> s(len(Z)) [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: fst(0, Z) -> nil [1] fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] len(nil) -> 0 [1] len(cons(X, Z)) -> s(len(Z)) [1] The TRS has the following type information: fst :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons add :: 0:s -> 0:s -> 0:s len :: nil:cons -> 0:s 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: fst_2 from_1 add_2 len_1 (c) The following functions are completely defined: none 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: fst(0, Z) -> nil [1] fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] len(nil) -> 0 [1] len(cons(X, Z)) -> s(len(Z)) [1] The TRS has the following type information: fst :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons add :: 0:s -> 0:s -> 0:s len :: nil:cons -> 0:s 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: fst(0, Z) -> nil [1] fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] len(nil) -> 0 [1] len(cons(X, Z)) -> s(len(Z)) [1] The TRS has the following type information: fst :: 0:s -> nil:cons -> nil:cons 0 :: 0:s nil :: nil:cons s :: 0:s -> 0:s cons :: 0:s -> nil:cons -> nil:cons from :: 0:s -> nil:cons add :: 0:s -> 0:s -> 0:s len :: nil:cons -> 0:s 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: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X fst(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + 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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { len } { from } { add } { fst } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {len}, {from}, {add}, {fst} ---------------------------------------- (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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {len}, {from}, {add}, {fst} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: len after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {len}, {from}, {add}, {fst} Previous analysis results are: len: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: len 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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {from}, {add}, {fst} Previous analysis results are: len: runtime: O(n^1) [1 + z], size: O(n^1) [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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 2 + Z }-> 1 + s :|: s >= 0, s <= Z, Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {from}, {add}, {fst} Previous analysis results are: len: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 2 + Z }-> 1 + s :|: s >= 0, s <= Z, Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {from}, {add}, {fst} Previous analysis results are: len: runtime: O(n^1) [1 + z], size: O(n^1) [z] from: runtime: ?, size: O(1) [0] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: from after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 fst(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 fst(z, z') -{ 1 }-> 1 + Y + fst(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 2 + Z }-> 1 + s :|: s >= 0, s <= Z, Z >= 0, X >= 0, z = 1 + X + Z Function symbols to be analyzed: {from}, {add}, {fst} Previous analysis results are: len: runtime: O(n^1) [1 + z], size: O(n^1) [z] from: runtime: INF, size: O(1) [0] ---------------------------------------- (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: fst(v0, v1) -> null_fst [0] len(v0) -> null_len [0] add(v0, v1) -> null_add [0] And the following fresh constants: null_fst, null_len, null_add ---------------------------------------- (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: fst(0, Z) -> nil [1] fst(s(X), cons(Y, Z)) -> cons(Y, fst(X, Z)) [1] from(X) -> cons(X, from(s(X))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] len(nil) -> 0 [1] len(cons(X, Z)) -> s(len(Z)) [1] fst(v0, v1) -> null_fst [0] len(v0) -> null_len [0] add(v0, v1) -> null_add [0] The TRS has the following type information: fst :: 0:s:null_len:null_add -> nil:cons:null_fst -> nil:cons:null_fst 0 :: 0:s:null_len:null_add nil :: nil:cons:null_fst s :: 0:s:null_len:null_add -> 0:s:null_len:null_add cons :: 0:s:null_len:null_add -> nil:cons:null_fst -> nil:cons:null_fst from :: 0:s:null_len:null_add -> nil:cons:null_fst add :: 0:s:null_len:null_add -> 0:s:null_len:null_add -> 0:s:null_len:null_add len :: nil:cons:null_fst -> 0:s:null_len:null_add null_fst :: nil:cons:null_fst null_len :: 0:s:null_len:null_add null_add :: 0:s:null_len:null_add 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 null_fst => 0 null_len => 0 null_add => 0 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X fst(z, z') -{ 1 }-> 0 :|: Z >= 0, z' = Z, z = 0 fst(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 fst(z, z') -{ 1 }-> 1 + Y + fst(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z len(z) -{ 1 }-> 0 :|: z = 0 len(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 len(z) -{ 1 }-> 1 + len(Z) :|: Z >= 0, X >= 0, z = 1 + X + Z 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: fst(0, z0) -> nil fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2)) from(z0) -> cons(z0, from(s(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) len(nil) -> 0 len(cons(z0, z1)) -> s(len(z1)) Tuples: FST(0, z0) -> c FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(nil) -> c5 LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FST(0, z0) -> c FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(0, z0) -> c3 ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(nil) -> c5 LEN(cons(z0, z1)) -> c6(LEN(z1)) K tuples:none Defined Rule Symbols: fst_2, from_1, add_2, len_1 Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c, c1_1, c2_1, c3, c4_1, c5, c6_1 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: ADD(0, z0) -> c3 FST(0, z0) -> c LEN(nil) -> c5 ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: fst(0, z0) -> nil fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2)) from(z0) -> cons(z0, from(s(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) len(nil) -> 0 len(cons(z0, z1)) -> s(len(z1)) Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) K tuples:none Defined Rule Symbols: fst_2, from_1, add_2, len_1 Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: fst(0, z0) -> nil fst(s(z0), cons(z1, z2)) -> cons(z1, fst(z0, z2)) from(z0) -> cons(z0, from(s(z0))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) len(nil) -> 0 len(cons(z0, z1)) -> s(len(z1)) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) We considered the (Usable) Rules:none And the Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD(x_1, x_2)) = 0 POL(FROM(x_1)) = 0 POL(FST(x_1, x_2)) = x_1 + x_2 POL(LEN(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) K tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) Defined Rule Symbols:none Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LEN(cons(z0, z1)) -> c6(LEN(z1)) We considered the (Usable) Rules:none And the Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD(x_1, x_2)) = x_1 POL(FROM(x_1)) = x_1 POL(FST(x_1, x_2)) = x_1 + x_2 POL(LEN(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = x_1 ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) K tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) LEN(cons(z0, z1)) -> c6(LEN(z1)) Defined Rule Symbols:none Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. ADD(s(z0), z1) -> c4(ADD(z0, z1)) We considered the (Usable) Rules:none And the Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(ADD(x_1, x_2)) = x_1 POL(FROM(x_1)) = 0 POL(FST(x_1, x_2)) = x_2 POL(LEN(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = x_2 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) S tuples: FROM(z0) -> c2(FROM(s(z0))) K tuples: FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) LEN(cons(z0, z1)) -> c6(LEN(z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) Defined Rule Symbols:none Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FST(s(z0), cons(z1, z2)) -> c1(FST(z0, z2)) by FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(z0) -> c2(FROM(s(z0))) ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) S tuples: FROM(z0) -> c2(FROM(s(z0))) K tuples: LEN(cons(z0, z1)) -> c6(LEN(z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, ADD_2, LEN_1, FST_2 Compound Symbols: c2_1, c4_1, c6_1, c1_1 ---------------------------------------- (49) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by FROM(s(x0)) -> c2(FROM(s(s(x0)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) FROM(s(x0)) -> c2(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c2(FROM(s(s(x0)))) K tuples: LEN(cons(z0, z1)) -> c6(LEN(z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: ADD_2, LEN_1, FST_2, FROM_1 Compound Symbols: c4_1, c6_1, c1_1, c2_1 ---------------------------------------- (51) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: ADD(s(z0), z1) -> c4(ADD(z0, z1)) LEN(cons(z0, z1)) -> c6(LEN(z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: LEN(cons(z0, z1)) -> c6(LEN(z1)) ADD(s(z0), z1) -> c4(ADD(z0, z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: ADD_2, LEN_1, FST_2, FROM_1 Compound Symbols: c4_1, c6_1, c1_1, c2_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(z0), z1) -> c4(ADD(z0, z1)) by ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: LEN(cons(z0, z1)) -> c6(LEN(z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: LEN(cons(z0, z1)) -> c6(LEN(z1)) FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) Defined Rule Symbols:none Defined Pair Symbols: LEN_1, FST_2, FROM_1, ADD_2 Compound Symbols: c6_1, c1_1, c2_1, c4_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LEN(cons(z0, z1)) -> c6(LEN(z1)) by LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) Defined Rule Symbols:none Defined Pair Symbols: FST_2, FROM_1, ADD_2, LEN_1 Compound Symbols: c1_1, c2_1, c4_1, c6_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FST(s(s(y0)), cons(z1, cons(y1, y2))) -> c1(FST(s(y0), cons(y1, y2))) by FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, ADD_2, LEN_1, FST_2 Compound Symbols: c2_1, c4_1, c6_1, c1_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) by ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, LEN_1, FST_2, ADD_2 Compound Symbols: c2_1, c6_1, c1_1, c4_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LEN(cons(z0, cons(y0, y1))) -> c6(LEN(cons(y0, y1))) by LEN(cons(z0, cons(z1, cons(y1, y2)))) -> c6(LEN(cons(z1, cons(y1, y2)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) LEN(cons(z0, cons(z1, cons(y1, y2)))) -> c6(LEN(cons(z1, cons(y1, y2)))) S tuples: FROM(s(s(x0))) -> c2(FROM(s(s(s(x0))))) K tuples: FST(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c1(FST(s(s(y0)), cons(z2, cons(y2, y3)))) ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1)) LEN(cons(z0, cons(z1, cons(y1, y2)))) -> c6(LEN(cons(z1, cons(y1, y2)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, FST_2, ADD_2, LEN_1 Compound Symbols: c2_1, c1_1, c4_1, c6_1