/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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), 4 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), 367 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 343 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 113 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 1172 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (34) CpxRNTS (35) FinalProof [FINISHED, 0 ms] (36) BOUNDS(1, n^2) (37) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxTRS (39) SlicingProof [LOWER BOUND(ID), 0 ms] (40) CpxTRS (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (42) typed CpxTrs (43) OrderProof [LOWER BOUND(ID), 0 ms] (44) typed CpxTrs (45) RewriteLemmaProof [LOWER BOUND(ID), 1985 ms] (46) BEST (47) proven lower bound (48) LowerBoundPropagationProof [FINISHED, 0 ms] (49) BOUNDS(n^1, INF) (50) typed CpxTrs (51) RewriteLemmaProof [LOWER BOUND(ID), 276 ms] (52) typed CpxTrs (53) RewriteLemmaProof [LOWER BOUND(ID), 16 ms] (54) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: exp(x, 0) -> s(0) exp(x, s(y)) -> *(x, exp(x, y)) *(0, y) -> 0 *(s(x), y) -> +(y, *(x, y)) -(0, y) -> 0 -(x, 0) -> x -(s(x), s(y)) -> -(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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^2). The TRS R consists of the following rules: exp(x, 0) -> s(0) [1] exp(x, s(y)) -> *(x, exp(x, y)) [1] *(0, y) -> 0 [1] *(s(x), y) -> +(y, *(x, y)) [1] -(0, y) -> 0 [1] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: * => times - => minus ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> +(y, times(x, y)) [1] minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, 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: exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> +(y, times(x, y)) [1] minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] The TRS has the following type information: exp :: 0:s:+ -> 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ times :: 0:s:+ -> 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ minus :: 0:s:+ -> 0:s:+ -> 0:s:+ 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: minus_2 (c) The following functions are completely defined: exp_2 times_2 Due to the following rules being added: exp(v0, v1) -> 0 [0] times(v0, v1) -> 0 [0] 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: exp(x, 0) -> s(0) [1] exp(x, s(y)) -> times(x, exp(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> +(y, times(x, y)) [1] minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] exp(v0, v1) -> 0 [0] times(v0, v1) -> 0 [0] The TRS has the following type information: exp :: 0:s:+ -> 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ times :: 0:s:+ -> 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ minus :: 0:s:+ -> 0:s:+ -> 0:s:+ 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: exp(x, 0) -> s(0) [1] exp(x, s(0)) -> times(x, s(0)) [2] exp(x, s(s(y'))) -> times(x, times(x, exp(x, y'))) [2] exp(x, s(y)) -> times(x, 0) [1] times(0, y) -> 0 [1] times(s(x), y) -> +(y, times(x, y)) [1] minus(0, y) -> 0 [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] exp(v0, v1) -> 0 [0] times(v0, v1) -> 0 [0] The TRS has the following type information: exp :: 0:s:+ -> 0:s:+ -> 0:s:+ 0 :: 0:s:+ s :: 0:s:+ -> 0:s:+ times :: 0:s:+ -> 0:s:+ -> 0:s:+ + :: 0:s:+ -> 0:s:+ -> 0:s:+ minus :: 0:s:+ -> 0:s:+ -> 0:s:+ 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: exp(z, z') -{ 2 }-> times(x, times(x, exp(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x exp(z, z') -{ 1 }-> times(x, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x exp(z, z') -{ 2 }-> times(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = x exp(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 exp(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 times(z, z') -{ 1 }-> 1 + y + times(x, y) :|: x >= 0, y >= 0, z = 1 + x, 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: exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 1 }-> times(z, 0) :|: z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, 1 + 0) :|: z >= 0, z' = 1 + 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 }-> 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { times } { minus } { exp } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 1 }-> times(z, 0) :|: z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, 1 + 0) :|: z >= 0, z' = 1 + 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 }-> 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {times}, {minus}, {exp} ---------------------------------------- (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: exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 1 }-> times(z, 0) :|: z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, 1 + 0) :|: z >= 0, z' = 1 + 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 }-> 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {times}, {minus}, {exp} ---------------------------------------- (19) 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: z + z*z' ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 1 }-> times(z, 0) :|: z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, 1 + 0) :|: z >= 0, z' = 1 + 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 }-> 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {times}, {minus}, {exp} Previous analysis results are: times: runtime: ?, size: O(n^2) [z + z*z'] ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: times 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: exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 1 }-> times(z, 0) :|: z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, 1 + 0) :|: z >= 0, z' = 1 + 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 }-> 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*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: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] minus: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: exp after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: {exp} Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] minus: runtime: O(n^1) [2 + z'], size: O(n^1) [z] exp: runtime: ?, size: INF ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: exp after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 6 + 2*z + 2*z*z' + 4*z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0 exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0 exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z'] minus: runtime: O(n^1) [2 + z'], size: O(n^1) [z] exp: runtime: O(n^2) [6 + 2*z + 2*z*z' + 4*z'], size: INF ---------------------------------------- (35) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (36) BOUNDS(1, n^2) ---------------------------------------- (37) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (38) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(y, *'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (39) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: +'/0 ---------------------------------------- (40) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (41) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (42) Obligation: Innermost TRS: Rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) Types: exp :: 0':s:+' -> 0':s:+' -> 0':s:+' 0' :: 0':s:+' s :: 0':s:+' -> 0':s:+' *' :: 0':s:+' -> 0':s:+' -> 0':s:+' +' :: 0':s:+' -> 0':s:+' - :: 0':s:+' -> 0':s:+' -> 0':s:+' hole_0':s:+'1_0 :: 0':s:+' gen_0':s:+'2_0 :: Nat -> 0':s:+' ---------------------------------------- (43) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: exp, *', - They will be analysed ascendingly in the following order: *' < exp ---------------------------------------- (44) Obligation: Innermost TRS: Rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) Types: exp :: 0':s:+' -> 0':s:+' -> 0':s:+' 0' :: 0':s:+' s :: 0':s:+' -> 0':s:+' *' :: 0':s:+' -> 0':s:+' -> 0':s:+' +' :: 0':s:+' -> 0':s:+' - :: 0':s:+' -> 0':s:+' -> 0':s:+' hole_0':s:+'1_0 :: 0':s:+' gen_0':s:+'2_0 :: Nat -> 0':s:+' Generator Equations: gen_0':s:+'2_0(0) <=> 0' gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x)) The following defined symbols remain to be analysed: *', exp, - They will be analysed ascendingly in the following order: *' < exp ---------------------------------------- (45) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: *'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) -> *3_0, rt in Omega(n4_0) Induction Base: *'(gen_0':s:+'2_0(+(1, 0)), gen_0':s:+'2_0(b)) Induction Step: *'(gen_0':s:+'2_0(+(1, +(n4_0, 1))), gen_0':s:+'2_0(b)) ->_R^Omega(1) +'(*'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b))) ->_IH +'(*3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (46) Complex Obligation (BEST) ---------------------------------------- (47) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) Types: exp :: 0':s:+' -> 0':s:+' -> 0':s:+' 0' :: 0':s:+' s :: 0':s:+' -> 0':s:+' *' :: 0':s:+' -> 0':s:+' -> 0':s:+' +' :: 0':s:+' -> 0':s:+' - :: 0':s:+' -> 0':s:+' -> 0':s:+' hole_0':s:+'1_0 :: 0':s:+' gen_0':s:+'2_0 :: Nat -> 0':s:+' Generator Equations: gen_0':s:+'2_0(0) <=> 0' gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x)) The following defined symbols remain to be analysed: *', exp, - They will be analysed ascendingly in the following order: *' < exp ---------------------------------------- (48) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (49) BOUNDS(n^1, INF) ---------------------------------------- (50) Obligation: Innermost TRS: Rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) Types: exp :: 0':s:+' -> 0':s:+' -> 0':s:+' 0' :: 0':s:+' s :: 0':s:+' -> 0':s:+' *' :: 0':s:+' -> 0':s:+' -> 0':s:+' +' :: 0':s:+' -> 0':s:+' - :: 0':s:+' -> 0':s:+' -> 0':s:+' hole_0':s:+'1_0 :: 0':s:+' gen_0':s:+'2_0 :: Nat -> 0':s:+' Lemmas: *'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_0':s:+'2_0(0) <=> 0' gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x)) The following defined symbols remain to be analysed: exp, - ---------------------------------------- (51) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1117_0)) -> *3_0, rt in Omega(n1117_0) Induction Base: exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(0)) Induction Step: exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(+(n1117_0, 1))) ->_R^Omega(1) *'(gen_0':s:+'2_0(0), exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1117_0))) ->_IH *'(gen_0':s:+'2_0(0), *3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (52) Obligation: Innermost TRS: Rules: exp(x, 0') -> s(0') exp(x, s(y)) -> *'(x, exp(x, y)) *'(0', y) -> 0' *'(s(x), y) -> +'(*'(x, y)) -(0', y) -> 0' -(x, 0') -> x -(s(x), s(y)) -> -(x, y) Types: exp :: 0':s:+' -> 0':s:+' -> 0':s:+' 0' :: 0':s:+' s :: 0':s:+' -> 0':s:+' *' :: 0':s:+' -> 0':s:+' -> 0':s:+' +' :: 0':s:+' -> 0':s:+' - :: 0':s:+' -> 0':s:+' -> 0':s:+' hole_0':s:+'1_0 :: 0':s:+' gen_0':s:+'2_0 :: Nat -> 0':s:+' Lemmas: *'(gen_0':s:+'2_0(+(1, n4_0)), gen_0':s:+'2_0(b)) -> *3_0, rt in Omega(n4_0) exp(gen_0':s:+'2_0(0), gen_0':s:+'2_0(n1117_0)) -> *3_0, rt in Omega(n1117_0) Generator Equations: gen_0':s:+'2_0(0) <=> 0' gen_0':s:+'2_0(+(x, 1)) <=> s(gen_0':s:+'2_0(x)) The following defined symbols remain to be analysed: - ---------------------------------------- (53) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_0':s:+'2_0(n6315_0), gen_0':s:+'2_0(n6315_0)) -> gen_0':s:+'2_0(0), rt in Omega(1 + n6315_0) Induction Base: -(gen_0':s:+'2_0(0), gen_0':s:+'2_0(0)) ->_R^Omega(1) 0' Induction Step: -(gen_0':s:+'2_0(+(n6315_0, 1)), gen_0':s:+'2_0(+(n6315_0, 1))) ->_R^Omega(1) -(gen_0':s:+'2_0(n6315_0), gen_0':s:+'2_0(n6315_0)) ->_IH gen_0':s:+'2_0(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (54) BOUNDS(1, INF)