WORST_CASE(?,O(n^1)) proof of input_EHGoiqnEkG.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) InliningProof [UPPER BOUND(ID), 2 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), 163 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 130 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 253 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 312 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 114 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 412 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 220 ms] (40) CpxRNTS (41) FinalProof [FINISHED, 0 ms] (42) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: minus(0, Y) -> 0 minus(s(X), s(Y)) -> minus(X, Y) geq(X, 0) -> true geq(0, s(Y)) -> false geq(s(X), s(Y)) -> geq(X, Y) div(0, s(Y)) -> 0 div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) if(true, X, Y) -> X if(false, X, Y) -> 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^1). The TRS R consists of the following rules: minus(0, Y) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] geq(X, 0) -> true [1] geq(0, s(Y)) -> false [1] geq(s(X), s(Y)) -> geq(X, Y) [1] div(0, s(Y)) -> 0 [1] div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: minus(0, Y) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] geq(X, 0) -> true [1] geq(0, s(Y)) -> false [1] geq(s(X), s(Y)) -> geq(X, Y) [1] div(0, s(Y)) -> 0 [1] div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s geq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false div :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (5) 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: geq_2 div_2 minus_2 if_3 Due to the following rules being added: div(v0, v1) -> 0 [0] minus(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (6) 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: minus(0, Y) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] geq(X, 0) -> true [1] geq(0, s(Y)) -> false [1] geq(s(X), s(Y)) -> geq(X, Y) [1] div(0, s(Y)) -> 0 [1] div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] div(v0, v1) -> 0 [0] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s geq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false div :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (7) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (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: minus(0, Y) -> 0 [1] minus(s(X), s(Y)) -> minus(X, Y) [1] geq(X, 0) -> true [1] geq(0, s(Y)) -> false [1] geq(s(X), s(Y)) -> geq(X, Y) [1] div(0, s(Y)) -> 0 [1] div(s(0), s(0)) -> if(true, s(div(0, s(0))), 0) [3] div(s(X), s(0)) -> if(true, s(div(0, s(0))), 0) [2] div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [3] div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [2] div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3] div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] div(v0, v1) -> 0 [0] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s geq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false div :: 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) 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 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X') div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X') div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0 geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, 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 minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (11) 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 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X') div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X') div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0 geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, 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 minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (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: div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ---------------------------------------- (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { if } { geq } { div } ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {if}, {geq}, {div} ---------------------------------------- (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: div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {if}, {geq}, {div} ---------------------------------------- (19) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {minus}, {if}, {geq}, {div} Previous analysis results are: minus: runtime: ?, size: O(1) [0] ---------------------------------------- (21) 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: 1 + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] ---------------------------------------- (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: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] ---------------------------------------- (25) 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'' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {if}, {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], 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: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: geq 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: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {geq}, {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] geq: runtime: ?, size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: geq after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] geq: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] geq: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: div 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: div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: {div} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] geq: runtime: O(n^1) [2 + z'], size: O(1) [1] div: runtime: ?, size: O(1) [1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: div after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + 2*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0 div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0 div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0 div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0 div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0 geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 geq(z, 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 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 Function symbols to be analyzed: Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n^1) [z' + z''] geq: runtime: O(n^1) [2 + z'], size: O(1) [1] div: runtime: O(n^1) [5 + 2*z'], size: O(1) [1] ---------------------------------------- (41) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (42) BOUNDS(1, n^1)