MAYBE proof of input_3NnPTIQEPR.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 184 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 161 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 294 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 131 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 43 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 416 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 287 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 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 ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> X if(false, X, Y) -> Y diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(0') -> 0' p(s(X)) -> X leq(0', Y) -> true leq(s(X), 0') -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> X if(false, X, Y) -> Y diff(X, Y) -> if(leq(X, Y), 0', s(diff(p(X), Y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(0) -> 0 p(s(X)) -> X leq(0, Y) -> true leq(s(X), 0) -> false leq(s(X), s(Y)) -> leq(X, Y) if(true, X, Y) -> X if(false, X, Y) -> Y diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: p(0) -> 0 [1] p(s(X)) -> X [1] leq(0, Y) -> true [1] leq(s(X), 0) -> false [1] leq(s(X), s(Y)) -> leq(X, Y) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), 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: p(0) -> 0 [1] p(s(X)) -> X [1] leq(0, Y) -> true [1] leq(s(X), 0) -> false [1] leq(s(X), s(Y)) -> leq(X, Y) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1] The TRS has the following type information: p :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s leq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if :: true:false -> 0:s -> 0:s -> 0:s diff :: 0:s -> 0:s -> 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: none (c) The following functions are completely defined: leq_2 diff_2 p_1 if_3 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: p(0) -> 0 [1] p(s(X)) -> X [1] leq(0, Y) -> true [1] leq(s(X), 0) -> false [1] leq(s(X), s(Y)) -> leq(X, Y) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1] The TRS has the following type information: p :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s leq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if :: true:false -> 0:s -> 0:s -> 0:s diff :: 0:s -> 0:s -> 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: p(0) -> 0 [1] p(s(X)) -> X [1] leq(0, Y) -> true [1] leq(s(X), 0) -> false [1] leq(s(X), s(Y)) -> leq(X, Y) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] diff(0, Y) -> if(true, 0, s(diff(0, Y))) [3] diff(s(X'), 0) -> if(false, 0, s(diff(X', 0))) [3] diff(s(X''), s(Y')) -> if(leq(X'', Y'), 0, s(diff(X'', s(Y')))) [3] The TRS has the following type information: p :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s leq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if :: true:false -> 0:s -> 0:s -> 0:s diff :: 0:s -> 0:s -> 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 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(X'', Y'), 0, 1 + diff(X'', 1 + Y')) :|: z = 1 + X'', Y' >= 0, z' = 1 + Y', X'' >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, Y)) :|: z' = Y, Y >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(X', 0)) :|: X' >= 0, z = 1 + X', z' = 0 if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(X'', Y'), 0, 1 + diff(X'', 1 + Y')) :|: z = 1 + X'', Y' >= 0, z' = 1 + Y', X'' >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, Y)) :|: z' = Y, Y >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(X', 0)) :|: X' >= 0, z = 1 + X', z' = 0 if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 1 }-> 0 :|: z = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { leq } { if } { p } { diff } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {leq}, {if}, {p}, {diff} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {leq}, {if}, {p}, {diff} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: leq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {leq}, {if}, {p}, {diff} Previous analysis results are: leq: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: leq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 3 }-> if(leq(z - 1, z' - 1), 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 1 }-> leq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: diff after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] diff: runtime: ?, size: O(1) [0] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: diff after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 4 + z' }-> if(s', 0, 1 + diff(z - 1, 1 + (z' - 1))) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0 diff(z, z') -{ 3 }-> if(1, 0, 1 + diff(0, z')) :|: z' >= 0, z = 0 diff(z, z') -{ 3 }-> if(0, 0, 1 + diff(z - 1, 0)) :|: z - 1 >= 0, z' = 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 leq(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0 leq(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 Function symbols to be analyzed: {diff} Previous analysis results are: leq: runtime: O(n^1) [2 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n^1) [z' + z''] p: runtime: O(1) [1], size: O(n^1) [z] diff: runtime: INF, size: O(1) [0] ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (46) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: p(0) -> 0 [1] p(s(X)) -> X [1] leq(0, Y) -> true [1] leq(s(X), 0) -> false [1] leq(s(X), s(Y)) -> leq(X, Y) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] diff(X, Y) -> if(leq(X, Y), 0, s(diff(p(X), Y))) [1] The TRS has the following type information: p :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s leq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if :: true:false -> 0:s -> 0:s -> 0:s diff :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: diff(z, z') -{ 1 }-> if(leq(X, Y), 0, 1 + diff(p(X), Y)) :|: z' = Y, Y >= 0, X >= 0, z = X if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 leq(z, z') -{ 1 }-> leq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 leq(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0 leq(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 1 }-> 0 :|: z = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 leq(0, z0) -> true leq(s(z0), 0) -> false leq(s(z0), s(z1)) -> leq(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1))) Tuples: P(0) -> c P(s(z0)) -> c1 LEQ(0, z0) -> c2 LEQ(s(z0), 0) -> c3 LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) IF(true, z0, z1) -> c5 IF(false, z0, z1) -> c6 DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1)) DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0)) S tuples: P(0) -> c P(s(z0)) -> c1 LEQ(0, z0) -> c2 LEQ(s(z0), 0) -> c3 LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) IF(true, z0, z1) -> c5 IF(false, z0, z1) -> c6 DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1)) DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0)) K tuples:none Defined Rule Symbols: p_1, leq_2, if_3, diff_2 Defined Pair Symbols: P_1, LEQ_2, IF_3, DIFF_2 Compound Symbols: c, c1, c2, c3, c4_1, c5, c6, c7_2, c8_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: IF(false, z0, z1) -> c6 LEQ(0, z0) -> c2 P(s(z0)) -> c1 LEQ(s(z0), 0) -> c3 P(0) -> c IF(true, z0, z1) -> c5 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 leq(0, z0) -> true leq(s(z0), 0) -> false leq(s(z0), s(z1)) -> leq(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1))) Tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1)) DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0)) S tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), LEQ(z0, z1)) DIFF(z0, z1) -> c8(IF(leq(z0, z1), 0, s(diff(p(z0), z1))), DIFF(p(z0), z1), P(z0)) K tuples:none Defined Rule Symbols: p_1, leq_2, if_3, diff_2 Defined Pair Symbols: LEQ_2, DIFF_2 Compound Symbols: c4_1, c7_2, c8_3 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 leq(0, z0) -> true leq(s(z0), 0) -> false leq(s(z0), s(z1)) -> leq(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1))) Tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) S tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples:none Defined Rule Symbols: p_1, leq_2, if_3, diff_2 Defined Pair Symbols: LEQ_2, DIFF_2 Compound Symbols: c4_1, c7_1, c8_1 ---------------------------------------- (55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: leq(0, z0) -> true leq(s(z0), 0) -> false leq(s(z0), s(z1)) -> leq(z0, z1) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 diff(z0, z1) -> if(leq(z0, z1), 0, s(diff(p(z0), z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) S tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples:none Defined Rule Symbols: p_1 Defined Pair Symbols: LEQ_2, DIFF_2 Compound Symbols: c4_1, c7_1, c8_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) We considered the (Usable) Rules: p(0) -> 0 p(s(z0)) -> z0 And the Tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DIFF(x_1, x_2)) = [1] + x_1 + x_2 POL(LEQ(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) S tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples: LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) DIFF(z0, z1) -> c7(LEQ(z0, z1)) Defined Rule Symbols: p_1 Defined Pair Symbols: LEQ_2, DIFF_2 Compound Symbols: c4_1, c7_1, c8_1 ---------------------------------------- (59) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LEQ(s(z0), s(z1)) -> c4(LEQ(z0, z1)) by LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: DIFF(z0, z1) -> c7(LEQ(z0, z1)) DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) S tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples: DIFF(z0, z1) -> c7(LEQ(z0, z1)) LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) Defined Rule Symbols: p_1 Defined Pair Symbols: DIFF_2, LEQ_2 Compound Symbols: c7_1, c8_1, c4_1 ---------------------------------------- (61) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DIFF(z0, z1) -> c7(LEQ(z0, z1)) by DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) S tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples: LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: p_1 Defined Pair Symbols: DIFF_2, LEQ_2 Compound Symbols: c8_1, c4_1, c7_1 ---------------------------------------- (63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LEQ(s(s(y0)), s(s(y1))) -> c4(LEQ(s(y0), s(y1))) by LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1)))) S tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples: DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: p_1 Defined Pair Symbols: DIFF_2, LEQ_2 Compound Symbols: c8_1, c7_1, c4_1 ---------------------------------------- (65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DIFF(s(s(y0)), s(s(y1))) -> c7(LEQ(s(s(y0)), s(s(y1)))) by DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: p(0) -> 0 p(s(z0)) -> z0 Tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1)))) DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1))))) S tuples: DIFF(z0, z1) -> c8(DIFF(p(z0), z1)) K tuples: LEQ(s(s(s(y0))), s(s(s(y1)))) -> c4(LEQ(s(s(y0)), s(s(y1)))) DIFF(s(s(s(y0))), s(s(s(y1)))) -> c7(LEQ(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: p_1 Defined Pair Symbols: DIFF_2, LEQ_2 Compound Symbols: c8_1, c4_1, c7_1