MAYBE proof of input_THdKtk6nEZ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 805 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 117 ms] (24) CpxRNTS (25) CompletionProof [UPPER BOUND(ID), 0 ms] (26) CpxTypedWeightedCompleteTrs (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms] (36) CdtProblem (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtForwardInstantiationProof [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), 1 ms] (58) CdtProblem (59) CdtForwardInstantiationProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (68) 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: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) 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: dbl(0') -> 0' dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0', cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) 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: dbl(0) -> 0 dbl(s(X)) -> s(s(dbl(X))) dbls(nil) -> nil dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) sel(0, cons(X, Y)) -> X sel(s(X), cons(Y, Z)) -> sel(X, Z) indx(nil, X) -> nil indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) from(X) -> cons(X, from(s(X))) 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: dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] dbls(nil) -> nil [1] dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] indx(nil, X) -> nil [1] indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1] from(X) -> cons(X, from(s(X))) [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: dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] dbls(nil) -> nil [1] dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] indx(nil, X) -> nil [1] indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: dbl :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s dbls :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons sel :: 0:s -> nil:cons -> 0:s indx :: nil:cons -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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: dbl_1 dbls_1 sel_2 indx_2 from_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: dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] dbls(nil) -> nil [1] dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] indx(nil, X) -> nil [1] indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: dbl :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s dbls :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons sel :: 0:s -> nil:cons -> 0:s indx :: nil:cons -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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: dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] dbls(nil) -> nil [1] dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] indx(nil, X) -> nil [1] indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1] from(X) -> cons(X, from(s(X))) [1] The TRS has the following type information: dbl :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s dbls :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons sel :: 0:s -> nil:cons -> 0:s indx :: nil:cons -> nil:cons -> nil:cons from :: 0:s -> nil:cons 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: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X indx(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, Z) + indx(Y, Z) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + 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: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { from } { sel } { dbl } { indx } { dbls } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls} ---------------------------------------- (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: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls} ---------------------------------------- (21) 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 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls} Previous analysis results are: from: runtime: ?, size: O(1) [0] ---------------------------------------- (23) 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: ? ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 1 }-> 1 + (1 + dbl(z - 1)) :|: z - 1 >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0 indx(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, z') + indx(Y, z') :|: z' >= 0, Y >= 0, z = 1 + X + Y, X >= 0 sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z Function symbols to be analyzed: {from}, {sel}, {dbl}, {indx}, {dbls} Previous analysis results are: from: runtime: INF, size: O(1) [0] ---------------------------------------- (25) 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: sel(v0, v1) -> null_sel [0] dbl(v0) -> null_dbl [0] And the following fresh constants: null_sel, null_dbl ---------------------------------------- (26) 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: dbl(0) -> 0 [1] dbl(s(X)) -> s(s(dbl(X))) [1] dbls(nil) -> nil [1] dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) [1] sel(0, cons(X, Y)) -> X [1] sel(s(X), cons(Y, Z)) -> sel(X, Z) [1] indx(nil, X) -> nil [1] indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) [1] from(X) -> cons(X, from(s(X))) [1] sel(v0, v1) -> null_sel [0] dbl(v0) -> null_dbl [0] The TRS has the following type information: dbl :: 0:s:null_sel:null_dbl -> 0:s:null_sel:null_dbl 0 :: 0:s:null_sel:null_dbl s :: 0:s:null_sel:null_dbl -> 0:s:null_sel:null_dbl dbls :: nil:cons -> nil:cons nil :: nil:cons cons :: 0:s:null_sel:null_dbl -> nil:cons -> nil:cons sel :: 0:s:null_sel:null_dbl -> nil:cons -> 0:s:null_sel:null_dbl indx :: nil:cons -> nil:cons -> nil:cons from :: 0:s:null_sel:null_dbl -> nil:cons null_sel :: 0:s:null_sel:null_dbl null_dbl :: 0:s:null_sel:null_dbl Rewrite Strategy: INNERMOST ---------------------------------------- (27) 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_sel => 0 null_dbl => 0 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: dbl(z) -{ 1 }-> 0 :|: z = 0 dbl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 dbl(z) -{ 1 }-> 1 + (1 + dbl(X)) :|: z = 1 + X, X >= 0 dbls(z) -{ 1 }-> 0 :|: z = 0 dbls(z) -{ 1 }-> 1 + dbl(X) + dbls(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X indx(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 indx(z, z') -{ 1 }-> 1 + sel(X, Z) + indx(Y, Z) :|: Z >= 0, Y >= 0, z = 1 + X + Y, X >= 0, z' = Z sel(z, z') -{ 1 }-> X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0 sel(z, z') -{ 1 }-> sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (29) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) dbls(nil) -> nil dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) indx(nil, z0) -> nil indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2)) from(z0) -> cons(z0, from(s(z0))) Tuples: DBL(0) -> c DBL(s(z0)) -> c1(DBL(z0)) DBLS(nil) -> c2 DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(nil, z0) -> c7 INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) S tuples: DBL(0) -> c DBL(s(z0)) -> c1(DBL(z0)) DBLS(nil) -> c2 DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(0, cons(z0, z1)) -> c5 SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(nil, z0) -> c7 INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) K tuples:none Defined Rule Symbols: dbl_1, dbls_1, sel_2, indx_2, from_1 Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c, c1_1, c2, c3_1, c4_1, c5, c6_1, c7, c8_1, c9_1, c10_1 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: INDX(nil, z0) -> c7 DBLS(nil) -> c2 DBL(0) -> c SEL(0, cons(z0, z1)) -> c5 ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) dbls(nil) -> nil dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) indx(nil, z0) -> nil indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2)) from(z0) -> cons(z0, from(s(z0))) Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) S tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) K tuples:none Defined Rule Symbols: dbl_1, dbls_1, sel_2, indx_2, from_1 Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1 ---------------------------------------- (33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: dbl(0) -> 0 dbl(s(z0)) -> s(s(dbl(z0))) dbls(nil) -> nil dbls(cons(z0, z1)) -> cons(dbl(z0), dbls(z1)) sel(0, cons(z0, z1)) -> z0 sel(s(z0), cons(z1, z2)) -> sel(z0, z2) indx(nil, z0) -> nil indx(cons(z0, z1), z2) -> cons(sel(z0, z2), indx(z1, z2)) from(z0) -> cons(z0, from(s(z0))) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) S tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) K tuples:none Defined Rule Symbols:none Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1 ---------------------------------------- (35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) We considered the (Usable) Rules:none And the Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DBL(x_1)) = [1] + x_1 POL(DBLS(x_1)) = [1] + x_1 POL(FROM(x_1)) = 0 POL(INDX(x_1, x_2)) = x_1 + x_2 POL(SEL(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(z0) -> c10(FROM(s(z0))) S tuples: FROM(z0) -> c10(FROM(s(z0))) K tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) Defined Rule Symbols:none Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1 ---------------------------------------- (37) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(z0) -> c10(FROM(s(z0))) by FROM(s(x0)) -> c10(FROM(s(s(x0)))) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(x0)) -> c10(FROM(s(s(x0)))) S tuples: FROM(s(x0)) -> c10(FROM(s(s(x0)))) K tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) Defined Rule Symbols:none Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1 ---------------------------------------- (39) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace FROM(s(x0)) -> c10(FROM(s(s(x0)))) by FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBL(s(z0)) -> c1(DBL(z0)) DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) Defined Rule Symbols:none Defined Pair Symbols: DBL_1, DBLS_1, SEL_2, INDX_2, FROM_1 Compound Symbols: c1_1, c3_1, c4_1, c6_1, c8_1, c9_1, c10_1 ---------------------------------------- (41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBL(s(z0)) -> c1(DBL(z0)) by DBL(s(s(y0))) -> c1(DBL(s(y0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBLS(cons(z0, z1)) -> c3(DBL(z0)) DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) DBL(s(s(y0))) -> c1(DBL(s(y0))) Defined Rule Symbols:none Defined Pair Symbols: DBLS_1, SEL_2, INDX_2, FROM_1, DBL_1 Compound Symbols: c3_1, c4_1, c6_1, c8_1, c9_1, c10_1, c1_1 ---------------------------------------- (43) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBLS(cons(z0, z1)) -> c3(DBL(z0)) by DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBLS(cons(z0, z1)) -> c4(DBLS(z1)) SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) Defined Rule Symbols:none Defined Pair Symbols: DBLS_1, SEL_2, INDX_2, FROM_1, DBL_1 Compound Symbols: c4_1, c6_1, c8_1, c9_1, c10_1, c1_1, c3_1 ---------------------------------------- (45) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBLS(cons(z0, z1)) -> c4(DBLS(z1)) by DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) Defined Rule Symbols:none Defined Pair Symbols: SEL_2, INDX_2, FROM_1, DBL_1, DBLS_1 Compound Symbols: c6_1, c8_1, c9_1, c10_1, c1_1, c3_1, c4_1 ---------------------------------------- (47) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, z2)) by SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) Defined Rule Symbols:none Defined Pair Symbols: INDX_2, FROM_1, DBL_1, DBLS_1, SEL_2 Compound Symbols: c8_1, c9_1, c10_1, c1_1, c3_1, c4_1, c6_1 ---------------------------------------- (49) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INDX(cons(z0, z1), z2) -> c8(SEL(z0, z2)) by INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: INDX_2, FROM_1, DBL_1, DBLS_1, SEL_2 Compound Symbols: c9_1, c10_1, c1_1, c3_1, c4_1, c6_1, c8_1 ---------------------------------------- (51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INDX(cons(z0, z1), z2) -> c9(INDX(z1, z2)) by INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBL(s(s(y0))) -> c1(DBL(s(y0))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, DBL_1, DBLS_1, SEL_2, INDX_2 Compound Symbols: c10_1, c1_1, c3_1, c4_1, c6_1, c8_1, c9_1 ---------------------------------------- (53) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBL(s(s(y0))) -> c1(DBL(s(y0))) by DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1 Compound Symbols: c10_1, c3_1, c4_1, c6_1, c8_1, c9_1, c1_1 ---------------------------------------- (55) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBLS(cons(s(s(y0)), z1)) -> c3(DBL(s(s(y0)))) by DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1 Compound Symbols: c10_1, c4_1, c6_1, c8_1, c9_1, c1_1, c3_1 ---------------------------------------- (57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBLS(cons(z0, cons(y0, y1))) -> c4(DBLS(cons(y0, y1))) by DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, DBLS_1, SEL_2, INDX_2, DBL_1 Compound Symbols: c10_1, c4_1, c6_1, c8_1, c9_1, c1_1, c3_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DBLS(cons(z0, cons(s(s(y0)), y1))) -> c4(DBLS(cons(s(s(y0)), y1))) by DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, SEL_2, INDX_2, DBL_1, DBLS_1 Compound Symbols: c10_1, c6_1, c8_1, c9_1, c1_1, c3_1, c4_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) by SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2 Compound Symbols: c10_1, c8_1, c9_1, c1_1, c3_1, c4_1, c6_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INDX(cons(s(s(y0)), z1), cons(y1, cons(y2, y3))) -> c8(SEL(s(s(y0)), cons(y1, cons(y2, y3)))) by INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2 Compound Symbols: c10_1, c9_1, c1_1, c3_1, c4_1, c6_1, c8_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INDX(cons(z0, cons(y0, y1)), z2) -> c9(INDX(cons(y0, y1), z2)) by INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3)) INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3)) INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5))))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3)) INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5))))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, INDX_2, DBL_1, DBLS_1, SEL_2 Compound Symbols: c10_1, c9_1, c1_1, c3_1, c4_1, c6_1, c8_1 ---------------------------------------- (67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace INDX(cons(z0, cons(s(s(y0)), y1)), cons(y2, cons(y3, y4))) -> c9(INDX(cons(s(s(y0)), y1), cons(y2, cons(y3, y4)))) by INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(z3, cons(z4, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(z3, cons(z4, cons(y4, y5))))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules:none Tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3)) INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5))))) INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6))))) S tuples: FROM(s(s(x0))) -> c10(FROM(s(s(s(x0))))) K tuples: DBL(s(s(s(y0)))) -> c1(DBL(s(s(y0)))) DBLS(cons(s(s(s(y0))), z1)) -> c3(DBL(s(s(s(y0))))) DBLS(cons(z0, cons(z1, cons(y1, y2)))) -> c4(DBLS(cons(z1, cons(y1, y2)))) DBLS(cons(z0, cons(z1, cons(s(s(y1)), y2)))) -> c4(DBLS(cons(z1, cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(s(y0))), z2))) -> c4(DBLS(cons(s(s(s(y0))), z2))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(y1)), y2)))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(y2, y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) -> c4(DBLS(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))))) DBLS(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2)))) -> c4(DBLS(cons(s(s(z1)), cons(s(s(s(y1))), y2)))) SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3)))) INDX(cons(s(s(s(y0))), z1), cons(z2, cons(z3, cons(y3, y4)))) -> c8(SEL(s(s(s(y0))), cons(z2, cons(z3, cons(y3, y4))))) INDX(cons(z0, cons(z1, cons(y1, y2))), z3) -> c9(INDX(cons(z1, cons(y1, y2)), z3)) INDX(cons(z0, cons(z1, cons(s(s(y1)), y2))), cons(y3, cons(y4, y5))) -> c9(INDX(cons(z1, cons(s(s(y1)), y2)), cons(y3, cons(y4, y5)))) INDX(cons(z0, cons(s(s(s(y0))), z2)), cons(y2, cons(y3, cons(y4, y5)))) -> c9(INDX(cons(s(s(s(y0))), z2), cons(y2, cons(y3, cons(y4, y5))))) INDX(cons(z0, cons(s(s(z1)), cons(s(s(y1)), y2))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(s(s(y1)), y2)), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(y2, y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(y2, y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3)))), cons(z3, cons(z4, z5))) -> c9(INDX(cons(s(s(z1)), cons(y1, cons(s(s(y2)), y3))), cons(z3, cons(z4, z5)))) INDX(cons(z0, cons(s(s(z1)), cons(s(s(s(y1))), y2))), cons(z3, cons(z4, cons(y5, y6)))) -> c9(INDX(cons(s(s(z1)), cons(s(s(s(y1))), y2)), cons(z3, cons(z4, cons(y5, y6))))) Defined Rule Symbols:none Defined Pair Symbols: FROM_1, DBL_1, DBLS_1, SEL_2, INDX_2 Compound Symbols: c10_1, c1_1, c3_1, c4_1, c6_1, c8_1, c9_1