76.80/20.68 WORST_CASE(Omega(n^1), O(n^1)) 76.80/20.70 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 76.80/20.70 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 76.80/20.70 76.80/20.70 76.80/20.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 76.80/20.70 76.80/20.70 (0) CpxTRS 76.80/20.70 (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (2) CpxWeightedTrs 76.80/20.70 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (4) CpxTypedWeightedTrs 76.80/20.70 (5) CompletionProof [UPPER BOUND(ID), 0 ms] 76.80/20.70 (6) CpxTypedWeightedCompleteTrs 76.80/20.70 (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (8) CpxTypedWeightedCompleteTrs 76.80/20.70 (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms] 76.80/20.70 (10) CpxRNTS 76.80/20.70 (11) InliningProof [UPPER BOUND(ID), 421 ms] 76.80/20.70 (12) CpxRNTS 76.80/20.70 (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (14) CpxRNTS 76.80/20.70 (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (16) CpxRNTS 76.80/20.70 (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 76.80/20.70 (18) CpxRNTS 76.80/20.70 (19) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] 76.80/20.70 (20) CpxRNTS 76.80/20.70 (21) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] 76.80/20.70 (22) CpxRNTS 76.80/20.70 (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 76.80/20.70 (24) CpxRNTS 76.80/20.70 (25) IntTrsBoundProof [UPPER BOUND(ID), 202 ms] 76.80/20.70 (26) CpxRNTS 76.80/20.70 (27) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] 76.80/20.70 (28) CpxRNTS 76.80/20.70 (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 76.80/20.70 (30) CpxRNTS 76.80/20.70 (31) IntTrsBoundProof [UPPER BOUND(ID), 291 ms] 76.80/20.70 (32) CpxRNTS 76.80/20.70 (33) IntTrsBoundProof [UPPER BOUND(ID), 130 ms] 76.80/20.70 (34) CpxRNTS 76.80/20.70 (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 76.80/20.70 (36) CpxRNTS 76.80/20.70 (37) IntTrsBoundProof [UPPER BOUND(ID), 3559 ms] 76.80/20.70 (38) CpxRNTS 76.80/20.70 (39) IntTrsBoundProof [UPPER BOUND(ID), 1515 ms] 76.80/20.70 (40) CpxRNTS 76.80/20.70 (41) FinalProof [FINISHED, 0 ms] 76.80/20.70 (42) BOUNDS(1, n^1) 76.80/20.70 (43) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (44) CpxTRS 76.80/20.70 (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 76.80/20.70 (46) typed CpxTrs 76.80/20.70 (47) OrderProof [LOWER BOUND(ID), 0 ms] 76.80/20.70 (48) typed CpxTrs 76.80/20.70 (49) RewriteLemmaProof [LOWER BOUND(ID), 309 ms] 76.80/20.70 (50) BEST 76.80/20.70 (51) proven lower bound 76.80/20.70 (52) LowerBoundPropagationProof [FINISHED, 0 ms] 76.80/20.70 (53) BOUNDS(n^1, INF) 76.80/20.70 (54) typed CpxTrs 76.80/20.70 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (0) 76.80/20.70 Obligation: 76.80/20.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 76.80/20.70 76.80/20.70 76.80/20.70 The TRS R consists of the following rules: 76.80/20.70 76.80/20.70 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.70 cond2(true, x, y) -> cond3(gr(x, 0), x, y) 76.80/20.70 cond2(false, x, y) -> cond4(gr(y, 0), x, y) 76.80/20.70 cond3(true, x, y) -> cond3(gr(x, 0), p(x), y) 76.80/20.70 cond3(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) 76.80/20.70 cond4(true, x, y) -> cond4(gr(y, 0), x, p(y)) 76.80/20.70 cond4(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) 76.80/20.70 gr(0, x) -> false 76.80/20.70 gr(s(x), 0) -> true 76.80/20.70 gr(s(x), s(y)) -> gr(x, y) 76.80/20.70 and(true, true) -> true 76.80/20.70 and(false, x) -> false 76.80/20.70 and(x, false) -> false 76.80/20.70 p(0) -> 0 76.80/20.70 p(s(x)) -> x 76.80/20.70 76.80/20.70 S is empty. 76.80/20.70 Rewrite Strategy: INNERMOST 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 76.80/20.70 Transformed relative TRS to weighted TRS 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (2) 76.80/20.70 Obligation: 76.80/20.70 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 76.80/20.70 76.80/20.70 76.80/20.70 The TRS R consists of the following rules: 76.80/20.70 76.80/20.70 cond1(true, x, y) -> cond2(gr(x, y), x, y) [1] 76.80/20.70 cond2(true, x, y) -> cond3(gr(x, 0), x, y) [1] 76.80/20.70 cond2(false, x, y) -> cond4(gr(y, 0), x, y) [1] 76.80/20.70 cond3(true, x, y) -> cond3(gr(x, 0), p(x), y) [1] 76.80/20.70 cond3(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 cond4(true, x, y) -> cond4(gr(y, 0), x, p(y)) [1] 76.80/20.70 cond4(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 gr(0, x) -> false [1] 76.80/20.70 gr(s(x), 0) -> true [1] 76.80/20.70 gr(s(x), s(y)) -> gr(x, y) [1] 76.80/20.70 and(true, true) -> true [1] 76.80/20.70 and(false, x) -> false [1] 76.80/20.70 and(x, false) -> false [1] 76.80/20.70 p(0) -> 0 [1] 76.80/20.70 p(s(x)) -> x [1] 76.80/20.70 76.80/20.70 Rewrite Strategy: INNERMOST 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 76.80/20.70 Infered types. 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (4) 76.80/20.70 Obligation: 76.80/20.70 Runtime Complexity Weighted TRS with Types. 76.80/20.70 The TRS R consists of the following rules: 76.80/20.70 76.80/20.70 cond1(true, x, y) -> cond2(gr(x, y), x, y) [1] 76.80/20.70 cond2(true, x, y) -> cond3(gr(x, 0), x, y) [1] 76.80/20.70 cond2(false, x, y) -> cond4(gr(y, 0), x, y) [1] 76.80/20.70 cond3(true, x, y) -> cond3(gr(x, 0), p(x), y) [1] 76.80/20.70 cond3(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 cond4(true, x, y) -> cond4(gr(y, 0), x, p(y)) [1] 76.80/20.70 cond4(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 gr(0, x) -> false [1] 76.80/20.70 gr(s(x), 0) -> true [1] 76.80/20.70 gr(s(x), s(y)) -> gr(x, y) [1] 76.80/20.70 and(true, true) -> true [1] 76.80/20.70 and(false, x) -> false [1] 76.80/20.70 and(x, false) -> false [1] 76.80/20.70 p(0) -> 0 [1] 76.80/20.70 p(s(x)) -> x [1] 76.80/20.70 76.80/20.70 The TRS has the following type information: 76.80/20.70 cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 true :: true:false 76.80/20.70 cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 gr :: 0:s -> 0:s -> true:false 76.80/20.70 cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 0 :: 0:s 76.80/20.70 false :: true:false 76.80/20.70 cond4 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 p :: 0:s -> 0:s 76.80/20.70 and :: true:false -> true:false -> true:false 76.80/20.70 s :: 0:s -> 0:s 76.80/20.70 76.80/20.70 Rewrite Strategy: INNERMOST 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (5) CompletionProof (UPPER BOUND(ID)) 76.80/20.70 The transformation into a RNTS is sound, since: 76.80/20.70 76.80/20.70 (a) The obligation is a constructor system where every type has a constant constructor, 76.80/20.70 76.80/20.70 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 76.80/20.70 76.80/20.70 cond1_3 76.80/20.70 cond2_3 76.80/20.70 cond3_3 76.80/20.70 cond4_3 76.80/20.70 76.80/20.70 (c) The following functions are completely defined: 76.80/20.70 76.80/20.70 and_2 76.80/20.70 gr_2 76.80/20.70 p_1 76.80/20.70 76.80/20.70 Due to the following rules being added: 76.80/20.70 none 76.80/20.70 76.80/20.70 And the following fresh constants: const 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (6) 76.80/20.70 Obligation: 76.80/20.70 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 76.80/20.70 76.80/20.70 Runtime Complexity Weighted TRS with Types. 76.80/20.70 The TRS R consists of the following rules: 76.80/20.70 76.80/20.70 cond1(true, x, y) -> cond2(gr(x, y), x, y) [1] 76.80/20.70 cond2(true, x, y) -> cond3(gr(x, 0), x, y) [1] 76.80/20.70 cond2(false, x, y) -> cond4(gr(y, 0), x, y) [1] 76.80/20.70 cond3(true, x, y) -> cond3(gr(x, 0), p(x), y) [1] 76.80/20.70 cond3(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 cond4(true, x, y) -> cond4(gr(y, 0), x, p(y)) [1] 76.80/20.70 cond4(false, x, y) -> cond1(and(gr(x, 0), gr(y, 0)), x, y) [1] 76.80/20.70 gr(0, x) -> false [1] 76.80/20.70 gr(s(x), 0) -> true [1] 76.80/20.70 gr(s(x), s(y)) -> gr(x, y) [1] 76.80/20.70 and(true, true) -> true [1] 76.80/20.70 and(false, x) -> false [1] 76.80/20.70 and(x, false) -> false [1] 76.80/20.70 p(0) -> 0 [1] 76.80/20.70 p(s(x)) -> x [1] 76.80/20.70 76.80/20.70 The TRS has the following type information: 76.80/20.70 cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 true :: true:false 76.80/20.70 cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 gr :: 0:s -> 0:s -> true:false 76.80/20.70 cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 0 :: 0:s 76.80/20.70 false :: true:false 76.80/20.70 cond4 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 p :: 0:s -> 0:s 76.80/20.70 and :: true:false -> true:false -> true:false 76.80/20.70 s :: 0:s -> 0:s 76.80/20.70 const :: cond1:cond2:cond3:cond4 76.80/20.70 76.80/20.70 Rewrite Strategy: INNERMOST 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (7) NarrowingProof (BOTH BOUNDS(ID, ID)) 76.80/20.70 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (8) 76.80/20.70 Obligation: 76.80/20.70 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 76.80/20.70 76.80/20.70 Runtime Complexity Weighted TRS with Types. 76.80/20.70 The TRS R consists of the following rules: 76.80/20.70 76.80/20.70 cond1(true, 0, y) -> cond2(false, 0, y) [2] 76.80/20.70 cond1(true, s(x'), 0) -> cond2(true, s(x'), 0) [2] 76.80/20.70 cond1(true, s(x''), s(y')) -> cond2(gr(x'', y'), s(x''), s(y')) [2] 76.80/20.70 cond2(true, 0, y) -> cond3(false, 0, y) [2] 76.80/20.70 cond2(true, s(x1), y) -> cond3(true, s(x1), y) [2] 76.80/20.70 cond2(false, x, 0) -> cond4(false, x, 0) [2] 76.80/20.70 cond2(false, x, s(x2)) -> cond4(true, x, s(x2)) [2] 76.80/20.70 cond3(true, 0, y) -> cond3(false, 0, y) [3] 76.80/20.70 cond3(true, s(x3), y) -> cond3(true, x3, y) [3] 76.80/20.70 cond3(false, 0, 0) -> cond1(and(false, false), 0, 0) [3] 76.80/20.70 cond3(false, 0, s(x5)) -> cond1(and(false, true), 0, s(x5)) [3] 76.80/20.70 cond3(false, s(x4), 0) -> cond1(and(true, false), s(x4), 0) [3] 76.80/20.70 cond3(false, s(x4), s(x6)) -> cond1(and(true, true), s(x4), s(x6)) [3] 76.80/20.70 cond4(true, x, 0) -> cond4(false, x, 0) [3] 76.80/20.70 cond4(true, x, s(x7)) -> cond4(true, x, x7) [3] 76.80/20.70 cond4(false, 0, 0) -> cond1(and(false, false), 0, 0) [3] 76.80/20.70 cond4(false, 0, s(x9)) -> cond1(and(false, true), 0, s(x9)) [3] 76.80/20.70 cond4(false, s(x8), 0) -> cond1(and(true, false), s(x8), 0) [3] 76.80/20.70 cond4(false, s(x8), s(x10)) -> cond1(and(true, true), s(x8), s(x10)) [3] 76.80/20.70 gr(0, x) -> false [1] 76.80/20.70 gr(s(x), 0) -> true [1] 76.80/20.70 gr(s(x), s(y)) -> gr(x, y) [1] 76.80/20.70 and(true, true) -> true [1] 76.80/20.70 and(false, x) -> false [1] 76.80/20.70 and(x, false) -> false [1] 76.80/20.70 p(0) -> 0 [1] 76.80/20.70 p(s(x)) -> x [1] 76.80/20.70 76.80/20.70 The TRS has the following type information: 76.80/20.70 cond1 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 true :: true:false 76.80/20.70 cond2 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 gr :: 0:s -> 0:s -> true:false 76.80/20.70 cond3 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 0 :: 0:s 76.80/20.70 false :: true:false 76.80/20.70 cond4 :: true:false -> 0:s -> 0:s -> cond1:cond2:cond3:cond4 76.80/20.70 p :: 0:s -> 0:s 76.80/20.70 and :: true:false -> true:false -> true:false 76.80/20.70 s :: 0:s -> 0:s 76.80/20.70 const :: cond1:cond2:cond3:cond4 76.80/20.70 76.80/20.70 Rewrite Strategy: INNERMOST 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 76.80/20.70 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 76.80/20.70 The constant constructors are abstracted as follows: 76.80/20.70 76.80/20.70 true => 1 76.80/20.70 0 => 0 76.80/20.70 false => 0 76.80/20.70 const => 0 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (10) 76.80/20.70 Obligation: 76.80/20.70 Complexity RNTS consisting of the following rules: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(gr(x'', y'), 1 + x'', 1 + y') :|: z' = 1 + x'', z = 1, y' >= 0, x'' >= 0, z'' = 1 + y' 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(1, x, 1 + x2) :|: z' = x, x >= 0, z'' = 1 + x2, z = 0, x2 >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + x1, y) :|: x1 >= 0, z'' = y, z = 1, y >= 0, z' = 1 + x1 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(1, x3, y) :|: z'' = y, z = 1, z' = 1 + x3, y >= 0, x3 >= 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond1(and(1, 1), 1 + x4, 1 + x6) :|: x4 >= 0, z' = 1 + x4, x6 >= 0, z'' = 1 + x6, z = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond1(and(1, 0), 1 + x4, 0) :|: z'' = 0, x4 >= 0, z' = 1 + x4, z = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond1(and(0, 1), 0, 1 + x5) :|: x5 >= 0, z'' = 1 + x5, z = 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond1(and(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(1, x, x7) :|: z' = x, z = 1, x7 >= 0, x >= 0, z'' = 1 + x7 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond1(and(1, 1), 1 + x8, 1 + x10) :|: z' = 1 + x8, x8 >= 0, z'' = 1 + x10, x10 >= 0, z = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond1(and(1, 0), 1 + x8, 0) :|: z'' = 0, z' = 1 + x8, x8 >= 0, z = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond1(and(0, 1), 0, 1 + x9) :|: z = 0, x9 >= 0, z' = 0, z'' = 1 + x9 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond1(and(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0 76.80/20.70 gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 76.80/20.70 gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 76.80/20.70 gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 76.80/20.70 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 76.80/20.70 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.70 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (11) InliningProof (UPPER BOUND(ID)) 76.80/20.70 Inlined the following terminating rules on right-hand sides where appropriate: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (12) 76.80/20.70 Obligation: 76.80/20.70 Complexity RNTS consisting of the following rules: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(gr(x'', y'), 1 + x'', 1 + y') :|: z' = 1 + x'', z = 1, y' >= 0, x'' >= 0, z'' = 1 + y' 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(1, x, 1 + x2) :|: z' = x, x >= 0, z'' = 1 + x2, z = 0, x2 >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + x1, y) :|: x1 >= 0, z'' = y, z = 1, y >= 0, z' = 1 + x1 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(1, x3, y) :|: z'' = y, z = 1, z' = 1 + x3, y >= 0, x3 >= 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + x4, 1 + x6) :|: x4 >= 0, z' = 1 + x4, x6 >= 0, z'' = 1 + x6, z = 0, 1 = 1 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + x5) :|: x5 >= 0, z'' = 1 + x5, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + x4, 0) :|: z'' = 0, x4 >= 0, z' = 1 + x4, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(1, x, x7) :|: z' = x, z = 1, x7 >= 0, x >= 0, z'' = 1 + x7 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + x8, 1 + x10) :|: z' = 1 + x8, x8 >= 0, z'' = 1 + x10, x10 >= 0, z = 0, 1 = 1 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + x9) :|: z = 0, x9 >= 0, z' = 0, z'' = 1 + x9, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + x8, 0) :|: z'' = 0, z' = 1 + x8, x8 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x 76.80/20.70 gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 76.80/20.70 gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 76.80/20.70 p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x 76.80/20.70 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.70 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (13) SimplificationProof (BOTH BOUNDS(ID, ID)) 76.80/20.70 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (14) 76.80/20.70 Obligation: 76.80/20.70 Complexity RNTS consisting of the following rules: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.70 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.70 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.70 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.70 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.70 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 76.80/20.70 Found the following analysis order by SCC decomposition: 76.80/20.70 76.80/20.70 { and } 76.80/20.70 { p } 76.80/20.70 { gr } 76.80/20.70 { cond2, cond1, cond3, cond4 } 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (16) 76.80/20.70 Obligation: 76.80/20.70 Complexity RNTS consisting of the following rules: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.70 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.70 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.70 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.70 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.70 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.70 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.70 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.70 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.70 76.80/20.70 Function symbols to be analyzed: {and}, {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.70 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (17) ResultPropagationProof (UPPER BOUND(ID)) 76.80/20.70 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 76.80/20.70 ---------------------------------------- 76.80/20.70 76.80/20.70 (18) 76.80/20.70 Obligation: 76.80/20.70 Complexity RNTS consisting of the following rules: 76.80/20.70 76.80/20.70 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.70 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.70 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.70 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {and}, {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (19) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: and 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 1 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (20) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {and}, {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: ?, size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (21) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed RUNTIME bound using CoFloCo for: and 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 1 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (22) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (23) ResultPropagationProof (UPPER BOUND(ID)) 76.80/20.71 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (24) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (25) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed SIZE bound using KoAT for: p 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(n^1) with polynomial bound: z 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (26) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {p}, {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: ?, size: O(n^1) [z] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (27) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed RUNTIME bound using CoFloCo for: p 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 1 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (28) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (29) ResultPropagationProof (UPPER BOUND(ID)) 76.80/20.71 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (30) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (31) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: gr 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 1 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (32) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {gr}, {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.71 gr: runtime: ?, size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (33) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed RUNTIME bound using KoAT for: gr 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(n^1) with polynomial bound: 2 + z' 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (34) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(gr(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.71 gr: runtime: O(n^1) [2 + z'], size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (35) ResultPropagationProof (UPPER BOUND(ID)) 76.80/20.71 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (36) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 3 + z'' }-> cond2(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.71 gr: runtime: O(n^1) [2 + z'], size: O(1) [1] 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (37) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: cond2 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 0 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: cond1 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 0 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: cond3 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 0 76.80/20.71 76.80/20.71 Computed SIZE bound using CoFloCo for: cond4 76.80/20.71 after applying outer abstraction to obtain an ITS, 76.80/20.71 resulting in: O(1) with polynomial bound: 0 76.80/20.71 76.80/20.71 ---------------------------------------- 76.80/20.71 76.80/20.71 (38) 76.80/20.71 Obligation: 76.80/20.71 Complexity RNTS consisting of the following rules: 76.80/20.71 76.80/20.71 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.71 cond1(z, z', z'') -{ 3 + z'' }-> cond2(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.71 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.71 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.71 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.71 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.71 gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 76.80/20.71 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.71 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.71 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.71 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.71 76.80/20.71 Function symbols to be analyzed: {cond2,cond1,cond3,cond4} 76.80/20.71 Previous analysis results are: 76.80/20.71 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.71 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.72 gr: runtime: O(n^1) [2 + z'], size: O(1) [1] 76.80/20.72 cond2: runtime: ?, size: O(1) [0] 76.80/20.72 cond1: runtime: ?, size: O(1) [0] 76.80/20.72 cond3: runtime: ?, size: O(1) [0] 76.80/20.72 cond4: runtime: ?, size: O(1) [0] 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (39) IntTrsBoundProof (UPPER BOUND(ID)) 76.80/20.72 76.80/20.72 Computed RUNTIME bound using CoFloCo for: cond2 76.80/20.72 after applying outer abstraction to obtain an ITS, 76.80/20.72 resulting in: O(n^1) with polynomial bound: 12 + 3*z' + 3*z'' 76.80/20.72 76.80/20.72 Computed RUNTIME bound using CoFloCo for: cond1 76.80/20.72 after applying outer abstraction to obtain an ITS, 76.80/20.72 resulting in: O(n^1) with polynomial bound: 15 + 3*z' + 4*z'' 76.80/20.72 76.80/20.72 Computed RUNTIME bound using CoFloCo for: cond3 76.80/20.72 after applying outer abstraction to obtain an ITS, 76.80/20.72 resulting in: O(n^1) with polynomial bound: 22 + 3*z' + 4*z'' 76.80/20.72 76.80/20.72 Computed RUNTIME bound using CoFloCo for: cond4 76.80/20.72 after applying outer abstraction to obtain an ITS, 76.80/20.72 resulting in: O(n^1) with polynomial bound: 22 + 3*z' + 4*z'' 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (40) 76.80/20.72 Obligation: 76.80/20.72 Complexity RNTS consisting of the following rules: 76.80/20.72 76.80/20.72 and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1 76.80/20.72 and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.72 and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 76.80/20.72 cond1(z, z', z'') -{ 3 + z'' }-> cond2(s, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0 76.80/20.72 cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), 0) :|: z'' = 0, z = 1, z' - 1 >= 0 76.80/20.72 cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.72 cond2(z, z', z'') -{ 2 }-> cond4(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0 76.80/20.72 cond2(z, z', z'') -{ 2 }-> cond4(0, z', 0) :|: z'' = 0, z' >= 0, z = 0 76.80/20.72 cond2(z, z', z'') -{ 2 }-> cond3(1, 1 + (z' - 1), z'') :|: z' - 1 >= 0, z = 1, z'' >= 0 76.80/20.72 cond2(z, z', z'') -{ 2 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.72 cond3(z, z', z'') -{ 3 }-> cond3(1, z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0 76.80/20.72 cond3(z, z', z'') -{ 3 }-> cond3(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0 76.80/20.72 cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.72 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.72 cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.72 cond3(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.72 cond4(z, z', z'') -{ 3 }-> cond4(1, z', z'' - 1) :|: z = 1, z'' - 1 >= 0, z' >= 0 76.80/20.72 cond4(z, z', z'') -{ 3 }-> cond4(0, z', 0) :|: z'' = 0, z = 1, z' >= 0 76.80/20.72 cond4(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0, 1 = 1 76.80/20.72 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = x, x >= 0, 0 = 0 76.80/20.72 cond4(z, z', z'') -{ 4 }-> cond1(0, 0, 1 + (z'' - 1)) :|: z = 0, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0 76.80/20.72 cond4(z, z', z'') -{ 4 }-> cond1(0, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, x >= 0, 1 = x, 0 = 0 76.80/20.72 gr(z, z') -{ 2 + z' }-> s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0 76.80/20.72 gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0 76.80/20.72 gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 76.80/20.72 p(z) -{ 1 }-> 0 :|: z = 0 76.80/20.72 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 76.80/20.72 76.80/20.72 Function symbols to be analyzed: 76.80/20.72 Previous analysis results are: 76.80/20.72 and: runtime: O(1) [1], size: O(1) [1] 76.80/20.72 p: runtime: O(1) [1], size: O(n^1) [z] 76.80/20.72 gr: runtime: O(n^1) [2 + z'], size: O(1) [1] 76.80/20.72 cond2: runtime: O(n^1) [12 + 3*z' + 3*z''], size: O(1) [0] 76.80/20.72 cond1: runtime: O(n^1) [15 + 3*z' + 4*z''], size: O(1) [0] 76.80/20.72 cond3: runtime: O(n^1) [22 + 3*z' + 4*z''], size: O(1) [0] 76.80/20.72 cond4: runtime: O(n^1) [22 + 3*z' + 4*z''], size: O(1) [0] 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (41) FinalProof (FINISHED) 76.80/20.72 Computed overall runtime complexity 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (42) 76.80/20.72 BOUNDS(1, n^1) 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (43) RenamingProof (BOTH BOUNDS(ID, ID)) 76.80/20.72 Renamed function symbols to avoid clashes with predefined symbol. 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (44) 76.80/20.72 Obligation: 76.80/20.72 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 76.80/20.72 76.80/20.72 76.80/20.72 The TRS R consists of the following rules: 76.80/20.72 76.80/20.72 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.72 cond2(true, x, y) -> cond3(gr(x, 0'), x, y) 76.80/20.72 cond2(false, x, y) -> cond4(gr(y, 0'), x, y) 76.80/20.72 cond3(true, x, y) -> cond3(gr(x, 0'), p(x), y) 76.80/20.72 cond3(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 cond4(true, x, y) -> cond4(gr(y, 0'), x, p(y)) 76.80/20.72 cond4(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 gr(0', x) -> false 76.80/20.72 gr(s(x), 0') -> true 76.80/20.72 gr(s(x), s(y)) -> gr(x, y) 76.80/20.72 and(true, true) -> true 76.80/20.72 and(false, x) -> false 76.80/20.72 and(x, false) -> false 76.80/20.72 p(0') -> 0' 76.80/20.72 p(s(x)) -> x 76.80/20.72 76.80/20.72 S is empty. 76.80/20.72 Rewrite Strategy: INNERMOST 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (45) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 76.80/20.72 Infered types. 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (46) 76.80/20.72 Obligation: 76.80/20.72 Innermost TRS: 76.80/20.72 Rules: 76.80/20.72 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.72 cond2(true, x, y) -> cond3(gr(x, 0'), x, y) 76.80/20.72 cond2(false, x, y) -> cond4(gr(y, 0'), x, y) 76.80/20.72 cond3(true, x, y) -> cond3(gr(x, 0'), p(x), y) 76.80/20.72 cond3(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 cond4(true, x, y) -> cond4(gr(y, 0'), x, p(y)) 76.80/20.72 cond4(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 gr(0', x) -> false 76.80/20.72 gr(s(x), 0') -> true 76.80/20.72 gr(s(x), s(y)) -> gr(x, y) 76.80/20.72 and(true, true) -> true 76.80/20.72 and(false, x) -> false 76.80/20.72 and(x, false) -> false 76.80/20.72 p(0') -> 0' 76.80/20.72 p(s(x)) -> x 76.80/20.72 76.80/20.72 Types: 76.80/20.72 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 true :: true:false 76.80/20.72 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 gr :: 0':s -> 0':s -> true:false 76.80/20.72 cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 0' :: 0':s 76.80/20.72 false :: true:false 76.80/20.72 cond4 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 p :: 0':s -> 0':s 76.80/20.72 and :: true:false -> true:false -> true:false 76.80/20.72 s :: 0':s -> 0':s 76.80/20.72 hole_cond1:cond2:cond3:cond41_0 :: cond1:cond2:cond3:cond4 76.80/20.72 hole_true:false2_0 :: true:false 76.80/20.72 hole_0':s3_0 :: 0':s 76.80/20.72 gen_0':s4_0 :: Nat -> 0':s 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (47) OrderProof (LOWER BOUND(ID)) 76.80/20.72 Heuristically decided to analyse the following defined symbols: 76.80/20.72 cond1, cond2, gr, cond3, cond4 76.80/20.72 76.80/20.72 They will be analysed ascendingly in the following order: 76.80/20.72 cond1 = cond2 76.80/20.72 gr < cond1 76.80/20.72 cond1 = cond3 76.80/20.72 cond1 = cond4 76.80/20.72 gr < cond2 76.80/20.72 cond2 = cond3 76.80/20.72 cond2 = cond4 76.80/20.72 gr < cond3 76.80/20.72 gr < cond4 76.80/20.72 cond3 = cond4 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (48) 76.80/20.72 Obligation: 76.80/20.72 Innermost TRS: 76.80/20.72 Rules: 76.80/20.72 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.72 cond2(true, x, y) -> cond3(gr(x, 0'), x, y) 76.80/20.72 cond2(false, x, y) -> cond4(gr(y, 0'), x, y) 76.80/20.72 cond3(true, x, y) -> cond3(gr(x, 0'), p(x), y) 76.80/20.72 cond3(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 cond4(true, x, y) -> cond4(gr(y, 0'), x, p(y)) 76.80/20.72 cond4(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 gr(0', x) -> false 76.80/20.72 gr(s(x), 0') -> true 76.80/20.72 gr(s(x), s(y)) -> gr(x, y) 76.80/20.72 and(true, true) -> true 76.80/20.72 and(false, x) -> false 76.80/20.72 and(x, false) -> false 76.80/20.72 p(0') -> 0' 76.80/20.72 p(s(x)) -> x 76.80/20.72 76.80/20.72 Types: 76.80/20.72 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 true :: true:false 76.80/20.72 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 gr :: 0':s -> 0':s -> true:false 76.80/20.72 cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 0' :: 0':s 76.80/20.72 false :: true:false 76.80/20.72 cond4 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 p :: 0':s -> 0':s 76.80/20.72 and :: true:false -> true:false -> true:false 76.80/20.72 s :: 0':s -> 0':s 76.80/20.72 hole_cond1:cond2:cond3:cond41_0 :: cond1:cond2:cond3:cond4 76.80/20.72 hole_true:false2_0 :: true:false 76.80/20.72 hole_0':s3_0 :: 0':s 76.80/20.72 gen_0':s4_0 :: Nat -> 0':s 76.80/20.72 76.80/20.72 76.80/20.72 Generator Equations: 76.80/20.72 gen_0':s4_0(0) <=> 0' 76.80/20.72 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 76.80/20.72 76.80/20.72 76.80/20.72 The following defined symbols remain to be analysed: 76.80/20.72 gr, cond1, cond2, cond3, cond4 76.80/20.72 76.80/20.72 They will be analysed ascendingly in the following order: 76.80/20.72 cond1 = cond2 76.80/20.72 gr < cond1 76.80/20.72 cond1 = cond3 76.80/20.72 cond1 = cond4 76.80/20.72 gr < cond2 76.80/20.72 cond2 = cond3 76.80/20.72 cond2 = cond4 76.80/20.72 gr < cond3 76.80/20.72 gr < cond4 76.80/20.72 cond3 = cond4 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (49) RewriteLemmaProof (LOWER BOUND(ID)) 76.80/20.72 Proved the following rewrite lemma: 76.80/20.72 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 76.80/20.72 76.80/20.72 Induction Base: 76.80/20.72 gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1) 76.80/20.72 false 76.80/20.72 76.80/20.72 Induction Step: 76.80/20.72 gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1) 76.80/20.72 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH 76.80/20.72 false 76.80/20.72 76.80/20.72 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (50) 76.80/20.72 Complex Obligation (BEST) 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (51) 76.80/20.72 Obligation: 76.80/20.72 Proved the lower bound n^1 for the following obligation: 76.80/20.72 76.80/20.72 Innermost TRS: 76.80/20.72 Rules: 76.80/20.72 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.72 cond2(true, x, y) -> cond3(gr(x, 0'), x, y) 76.80/20.72 cond2(false, x, y) -> cond4(gr(y, 0'), x, y) 76.80/20.72 cond3(true, x, y) -> cond3(gr(x, 0'), p(x), y) 76.80/20.72 cond3(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 cond4(true, x, y) -> cond4(gr(y, 0'), x, p(y)) 76.80/20.72 cond4(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 gr(0', x) -> false 76.80/20.72 gr(s(x), 0') -> true 76.80/20.72 gr(s(x), s(y)) -> gr(x, y) 76.80/20.72 and(true, true) -> true 76.80/20.72 and(false, x) -> false 76.80/20.72 and(x, false) -> false 76.80/20.72 p(0') -> 0' 76.80/20.72 p(s(x)) -> x 76.80/20.72 76.80/20.72 Types: 76.80/20.72 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 true :: true:false 76.80/20.72 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 gr :: 0':s -> 0':s -> true:false 76.80/20.72 cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 0' :: 0':s 76.80/20.72 false :: true:false 76.80/20.72 cond4 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 p :: 0':s -> 0':s 76.80/20.72 and :: true:false -> true:false -> true:false 76.80/20.72 s :: 0':s -> 0':s 76.80/20.72 hole_cond1:cond2:cond3:cond41_0 :: cond1:cond2:cond3:cond4 76.80/20.72 hole_true:false2_0 :: true:false 76.80/20.72 hole_0':s3_0 :: 0':s 76.80/20.72 gen_0':s4_0 :: Nat -> 0':s 76.80/20.72 76.80/20.72 76.80/20.72 Generator Equations: 76.80/20.72 gen_0':s4_0(0) <=> 0' 76.80/20.72 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 76.80/20.72 76.80/20.72 76.80/20.72 The following defined symbols remain to be analysed: 76.80/20.72 gr, cond1, cond2, cond3, cond4 76.80/20.72 76.80/20.72 They will be analysed ascendingly in the following order: 76.80/20.72 cond1 = cond2 76.80/20.72 gr < cond1 76.80/20.72 cond1 = cond3 76.80/20.72 cond1 = cond4 76.80/20.72 gr < cond2 76.80/20.72 cond2 = cond3 76.80/20.72 cond2 = cond4 76.80/20.72 gr < cond3 76.80/20.72 gr < cond4 76.80/20.72 cond3 = cond4 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (52) LowerBoundPropagationProof (FINISHED) 76.80/20.72 Propagated lower bound. 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (53) 76.80/20.72 BOUNDS(n^1, INF) 76.80/20.72 76.80/20.72 ---------------------------------------- 76.80/20.72 76.80/20.72 (54) 76.80/20.72 Obligation: 76.80/20.72 Innermost TRS: 76.80/20.72 Rules: 76.80/20.72 cond1(true, x, y) -> cond2(gr(x, y), x, y) 76.80/20.72 cond2(true, x, y) -> cond3(gr(x, 0'), x, y) 76.80/20.72 cond2(false, x, y) -> cond4(gr(y, 0'), x, y) 76.80/20.72 cond3(true, x, y) -> cond3(gr(x, 0'), p(x), y) 76.80/20.72 cond3(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 cond4(true, x, y) -> cond4(gr(y, 0'), x, p(y)) 76.80/20.72 cond4(false, x, y) -> cond1(and(gr(x, 0'), gr(y, 0')), x, y) 76.80/20.72 gr(0', x) -> false 76.80/20.72 gr(s(x), 0') -> true 76.80/20.72 gr(s(x), s(y)) -> gr(x, y) 76.80/20.72 and(true, true) -> true 76.80/20.72 and(false, x) -> false 76.80/20.72 and(x, false) -> false 76.80/20.72 p(0') -> 0' 76.80/20.72 p(s(x)) -> x 76.80/20.72 76.80/20.72 Types: 76.80/20.72 cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 true :: true:false 76.80/20.72 cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 gr :: 0':s -> 0':s -> true:false 76.80/20.72 cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 0' :: 0':s 76.80/20.72 false :: true:false 76.80/20.72 cond4 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3:cond4 76.80/20.72 p :: 0':s -> 0':s 76.80/20.72 and :: true:false -> true:false -> true:false 76.80/20.72 s :: 0':s -> 0':s 76.80/20.72 hole_cond1:cond2:cond3:cond41_0 :: cond1:cond2:cond3:cond4 76.80/20.72 hole_true:false2_0 :: true:false 76.80/20.72 hole_0':s3_0 :: 0':s 76.80/20.72 gen_0':s4_0 :: Nat -> 0':s 76.80/20.72 76.80/20.72 76.80/20.72 Lemmas: 76.80/20.72 gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0) 76.80/20.72 76.80/20.72 76.80/20.72 Generator Equations: 76.80/20.72 gen_0':s4_0(0) <=> 0' 76.80/20.72 gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x)) 76.80/20.72 76.80/20.72 76.80/20.72 The following defined symbols remain to be analysed: 76.80/20.72 cond2, cond1, cond3, cond4 76.80/20.72 76.80/20.72 They will be analysed ascendingly in the following order: 76.80/20.72 cond1 = cond2 76.80/20.72 cond1 = cond3 76.80/20.72 cond1 = cond4 76.80/20.72 cond2 = cond3 76.80/20.72 cond2 = cond4 76.80/20.72 cond3 = cond4 77.06/20.76 EOF