WORST_CASE(?,O(n^3)) proof of input_PzlPSpsDxr.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRNTS (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) IntTrsBoundProof [UPPER BOUND(ID), 156 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 303 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 1256 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 132 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) BOUNDS(1, n^3) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(p(x), y) -> p(+(x, y)) minus(0) -> 0 minus(s(x)) -> p(minus(x)) minus(p(x)) -> s(minus(x)) *(0, y) -> 0 *(s(x), y) -> +(*(x, y), y) *(p(x), y) -> +(*(x, y), minus(y)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: +(0, y) -> y [1] +(s(x), y) -> s(+(x, y)) [1] +(p(x), y) -> p(+(x, y)) [1] minus(0) -> 0 [1] minus(s(x)) -> p(minus(x)) [1] minus(p(x)) -> s(minus(x)) [1] *(0, y) -> 0 [1] *(s(x), y) -> +(*(x, y), y) [1] *(p(x), y) -> +(*(x, y), minus(y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus * => times ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] plus(p(x), y) -> p(plus(x, y)) [1] minus(0) -> 0 [1] minus(s(x)) -> p(minus(x)) [1] minus(p(x)) -> s(minus(x)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] times(p(x), y) -> plus(times(x, y), minus(y)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] plus(p(x), y) -> p(plus(x, y)) [1] minus(0) -> 0 [1] minus(s(x)) -> p(minus(x)) [1] minus(p(x)) -> s(minus(x)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] times(p(x), y) -> plus(times(x, y), minus(y)) [1] The TRS has the following type information: plus :: 0:s:p -> 0:s:p -> 0:s:p 0 :: 0:s:p s :: 0:s:p -> 0:s:p p :: 0:s:p -> 0:s:p minus :: 0:s:p -> 0:s:p times :: 0:s:p -> 0:s:p -> 0:s:p Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: times_2 minus_1 plus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (8) 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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] plus(p(x), y) -> p(plus(x, y)) [1] minus(0) -> 0 [1] minus(s(x)) -> p(minus(x)) [1] minus(p(x)) -> s(minus(x)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] times(p(x), y) -> plus(times(x, y), minus(y)) [1] The TRS has the following type information: plus :: 0:s:p -> 0:s:p -> 0:s:p 0 :: 0:s:p s :: 0:s:p -> 0:s:p p :: 0:s:p -> 0:s:p minus :: 0:s:p -> 0:s:p times :: 0:s:p -> 0:s:p -> 0:s:p Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] plus(p(x), y) -> p(plus(x, y)) [1] minus(0) -> 0 [1] minus(s(x)) -> p(minus(x)) [1] minus(p(x)) -> s(minus(x)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(0, y) [2] times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2] times(s(p(x'')), y) -> plus(plus(times(x'', y), minus(y)), y) [2] times(p(0), 0) -> plus(0, 0) [3] times(p(0), s(x3)) -> plus(0, p(minus(x3))) [3] times(p(0), p(x4)) -> plus(0, s(minus(x4))) [3] times(p(s(x1)), 0) -> plus(plus(times(x1, 0), 0), 0) [3] times(p(s(x1)), s(x5)) -> plus(plus(times(x1, s(x5)), s(x5)), p(minus(x5))) [3] times(p(s(x1)), p(x6)) -> plus(plus(times(x1, p(x6)), p(x6)), s(minus(x6))) [3] times(p(p(x2)), 0) -> plus(plus(times(x2, 0), minus(0)), 0) [3] times(p(p(x2)), s(x7)) -> plus(plus(times(x2, s(x7)), minus(s(x7))), p(minus(x7))) [3] times(p(p(x2)), p(x8)) -> plus(plus(times(x2, p(x8)), minus(p(x8))), s(minus(x8))) [3] The TRS has the following type information: plus :: 0:s:p -> 0:s:p -> 0:s:p 0 :: 0:s:p s :: 0:s:p -> 0:s:p p :: 0:s:p -> 0:s:p minus :: 0:s:p -> 0:s:p times :: 0:s:p -> 0:s:p -> 0:s:p Rewrite Strategy: INNERMOST ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(x) :|: x >= 0, z = 1 + x plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> plus(plus(times(x', y), y), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> plus(plus(times(x'', y), minus(y)), y) :|: y >= 0, x'' >= 0, z = 1 + (1 + x''), z' = y times(z, z') -{ 3 }-> plus(plus(times(x1, 0), 0), 0) :|: z = 1 + (1 + x1), x1 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x5), 1 + x5), 1 + minus(x5)) :|: z = 1 + (1 + x1), x1 >= 0, x5 >= 0, z' = 1 + x5 times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x6), 1 + x6), 1 + minus(x6)) :|: z' = 1 + x6, z = 1 + (1 + x1), x1 >= 0, x6 >= 0 times(z, z') -{ 3 }-> plus(plus(times(x2, 0), minus(0)), 0) :|: z = 1 + (1 + x2), x2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x7), minus(1 + x7)), 1 + minus(x7)) :|: z' = 1 + x7, z = 1 + (1 + x2), x7 >= 0, x2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x8), minus(1 + x8)), 1 + minus(x8)) :|: z = 1 + (1 + x2), z' = 1 + x8, x8 >= 0, x2 >= 0 times(z, z') -{ 2 }-> plus(0, y) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(x3)) :|: z' = 1 + x3, z = 1 + 0, x3 >= 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(x4)) :|: x4 >= 0, z = 1 + 0, z' = 1 + x4 times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (13) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { plus } { times } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {minus}, {plus}, {times} ---------------------------------------- (17) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {minus}, {plus}, {times} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {minus}, {plus}, {times} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0 times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0 times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + 3*z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0 times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times} Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [2*z*z' + 3*z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 12 + 31*z + z*z' + 16*z^2*z' + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: minus(z) -{ 1 }-> 0 :|: z = 0 minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0 times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0 times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0 times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0 times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: Previous analysis results are: minus: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^3) [12 + 31*z + z*z' + 16*z^2*z' + z'], size: O(n^2) [2*z*z' + 3*z'] ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^3)