622.00/291.57 KILLED 622.13/291.59 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 622.13/291.59 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 622.13/291.59 622.13/291.59 622.13/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 (0) CpxTRS 622.13/291.59 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (2) CpxTRS 622.13/291.59 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (4) typed CpxTrs 622.13/291.59 (5) OrderProof [LOWER BOUND(ID), 0 ms] 622.13/291.59 (6) typed CpxTrs 622.13/291.59 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 622.13/291.59 (8) TRS for Loop Detection 622.13/291.59 (9) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (10) CpxTRS 622.13/291.59 (11) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (12) CpxRelTRS 622.13/291.59 (13) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (14) CpxRelTRS 622.13/291.59 (15) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (16) CpxWeightedTrs 622.13/291.59 (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (18) CpxTypedWeightedTrs 622.13/291.59 (19) CompletionProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (20) CpxTypedWeightedCompleteTrs 622.13/291.59 (21) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (22) CpxTypedWeightedCompleteTrs 622.13/291.59 (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (24) CpxRNTS 622.13/291.59 (25) InliningProof [UPPER BOUND(ID), 210 ms] 622.13/291.59 (26) CpxRNTS 622.13/291.59 (27) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (28) CpxRNTS 622.13/291.59 (29) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (30) CpxRNTS 622.13/291.59 (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (32) CpxRNTS 622.13/291.59 (33) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] 622.13/291.59 (34) CpxRNTS 622.13/291.59 (35) IntTrsBoundProof [UPPER BOUND(ID), 25 ms] 622.13/291.59 (36) CpxRNTS 622.13/291.59 (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (38) CpxRNTS 622.13/291.59 (39) IntTrsBoundProof [UPPER BOUND(ID), 4554 ms] 622.13/291.59 (40) CpxRNTS 622.13/291.59 (41) IntTrsBoundProof [UPPER BOUND(ID), 1564 ms] 622.13/291.59 (42) CpxRNTS 622.13/291.59 (43) CompletionProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (44) CpxTypedWeightedCompleteTrs 622.13/291.59 (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (46) CpxRNTS 622.13/291.59 (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] 622.13/291.59 (48) CdtProblem 622.13/291.59 (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (50) CdtProblem 622.13/291.59 (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (52) CdtProblem 622.13/291.59 (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (54) CdtProblem 622.13/291.59 (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (56) CdtProblem 622.13/291.59 (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (58) CdtProblem 622.13/291.59 (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (60) CdtProblem 622.13/291.59 (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (62) CdtProblem 622.13/291.59 (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (64) CdtProblem 622.13/291.59 (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (66) CdtProblem 622.13/291.59 (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (68) CdtProblem 622.13/291.59 (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (70) CdtProblem 622.13/291.59 (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (72) CdtProblem 622.13/291.59 (73) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (74) CdtProblem 622.13/291.59 (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (76) CdtProblem 622.13/291.59 (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (78) CdtProblem 622.13/291.59 (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] 622.13/291.59 (80) CdtProblem 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (0) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 S is empty. 622.13/291.59 Rewrite Strategy: FULL 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Renamed function symbols to avoid clashes with predefined symbol. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (2) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 S is empty. 622.13/291.59 Rewrite Strategy: FULL 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Infered types. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (4) 622.13/291.59 Obligation: 622.13/291.59 TRS: 622.13/291.59 Rules: 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 Types: 622.13/291.59 f :: i:a -> i:a -> i:a 622.13/291.59 i :: i:a -> i:a 622.13/291.59 g :: i:a -> i:a 622.13/291.59 a :: i:a 622.13/291.59 hole_i:a1_0 :: i:a 622.13/291.59 gen_i:a2_0 :: Nat -> i:a 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (5) OrderProof (LOWER BOUND(ID)) 622.13/291.59 Heuristically decided to analyse the following defined symbols: 622.13/291.59 f 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (6) 622.13/291.59 Obligation: 622.13/291.59 TRS: 622.13/291.59 Rules: 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 Types: 622.13/291.59 f :: i:a -> i:a -> i:a 622.13/291.59 i :: i:a -> i:a 622.13/291.59 g :: i:a -> i:a 622.13/291.59 a :: i:a 622.13/291.59 hole_i:a1_0 :: i:a 622.13/291.59 gen_i:a2_0 :: Nat -> i:a 622.13/291.59 622.13/291.59 622.13/291.59 Generator Equations: 622.13/291.59 gen_i:a2_0(0) <=> a 622.13/291.59 gen_i:a2_0(+(x, 1)) <=> i(gen_i:a2_0(x)) 622.13/291.59 622.13/291.59 622.13/291.59 The following defined symbols remain to be analysed: 622.13/291.59 f 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 622.13/291.59 Transformed a relative TRS into a decreasing-loop problem. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (8) 622.13/291.59 Obligation: 622.13/291.59 Analyzing the following TRS for decreasing loops: 622.13/291.59 622.13/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 S is empty. 622.13/291.59 Rewrite Strategy: FULL 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (9) RelTrsToTrsProof (UPPER BOUND(ID)) 622.13/291.59 transformed relative TRS to TRS 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (10) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(g(x))) -> a 622.13/291.59 622.13/291.59 S is empty. 622.13/291.59 Rewrite Strategy: FULL 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (11) NonCtorToCtorProof (UPPER BOUND(ID)) 622.13/291.59 transformed non-ctor to ctor-system 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (12) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(c_g(x))) -> a 622.13/291.59 622.13/291.59 The (relative) TRS S consists of the following rules: 622.13/291.59 622.13/291.59 g(x0) -> c_g(x0) 622.13/291.59 622.13/291.59 Rewrite Strategy: FULL 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (13) RcToIrcProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Converted rc-obligation to irc-obligation. 622.13/291.59 622.13/291.59 As the TRS is a non-duplicating overlay system, we have rc = irc. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (14) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) 622.13/291.59 f(x, y) -> x 622.13/291.59 g(x) -> i(x) 622.13/291.59 f(x, i(x)) -> f(x, x) 622.13/291.59 f(i(x), i(c_g(x))) -> a 622.13/291.59 622.13/291.59 The (relative) TRS S consists of the following rules: 622.13/291.59 622.13/291.59 g(x0) -> c_g(x0) 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (15) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Transformed relative TRS to weighted TRS 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (16) 622.13/291.59 Obligation: 622.13/291.59 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). 622.13/291.59 622.13/291.59 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) [1] 622.13/291.59 f(x, y) -> x [1] 622.13/291.59 g(x) -> i(x) [1] 622.13/291.59 f(x, i(x)) -> f(x, x) [1] 622.13/291.59 f(i(x), i(c_g(x))) -> a [1] 622.13/291.59 g(x0) -> c_g(x0) [0] 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (17) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Infered types. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (18) 622.13/291.59 Obligation: 622.13/291.59 Runtime Complexity Weighted TRS with Types. 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) [1] 622.13/291.59 f(x, y) -> x [1] 622.13/291.59 g(x) -> i(x) [1] 622.13/291.59 f(x, i(x)) -> f(x, x) [1] 622.13/291.59 f(i(x), i(c_g(x))) -> a [1] 622.13/291.59 g(x0) -> c_g(x0) [0] 622.13/291.59 622.13/291.59 The TRS has the following type information: 622.13/291.59 f :: i:c_g:a -> i:c_g:a -> i:c_g:a 622.13/291.59 i :: i:c_g:a -> i:c_g:a 622.13/291.59 g :: i:c_g:a -> i:c_g:a 622.13/291.59 c_g :: i:c_g:a -> i:c_g:a 622.13/291.59 a :: i:c_g:a 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (19) CompletionProof (UPPER BOUND(ID)) 622.13/291.59 The transformation into a RNTS is sound, since: 622.13/291.59 622.13/291.59 (a) The obligation is a constructor system where every type has a constant constructor, 622.13/291.59 622.13/291.59 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 622.13/291.59 622.13/291.59 f_2 622.13/291.59 622.13/291.59 (c) The following functions are completely defined: 622.13/291.59 622.13/291.59 g_1 622.13/291.59 622.13/291.59 Due to the following rules being added: 622.13/291.59 622.13/291.59 g(v0) -> a [0] 622.13/291.59 622.13/291.59 And the following fresh constants: none 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (20) 622.13/291.59 Obligation: 622.13/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 622.13/291.59 622.13/291.59 Runtime Complexity Weighted TRS with Types. 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) [1] 622.13/291.59 f(x, y) -> x [1] 622.13/291.59 g(x) -> i(x) [1] 622.13/291.59 f(x, i(x)) -> f(x, x) [1] 622.13/291.59 f(i(x), i(c_g(x))) -> a [1] 622.13/291.59 g(x0) -> c_g(x0) [0] 622.13/291.59 g(v0) -> a [0] 622.13/291.59 622.13/291.59 The TRS has the following type information: 622.13/291.59 f :: i:c_g:a -> i:c_g:a -> i:c_g:a 622.13/291.59 i :: i:c_g:a -> i:c_g:a 622.13/291.59 g :: i:c_g:a -> i:c_g:a 622.13/291.59 c_g :: i:c_g:a -> i:c_g:a 622.13/291.59 a :: i:c_g:a 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (21) NarrowingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (22) 622.13/291.59 Obligation: 622.13/291.59 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 622.13/291.59 622.13/291.59 Runtime Complexity Weighted TRS with Types. 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(i(x))) [2] 622.13/291.59 f(x, x) -> f(i(x), g(c_g(x))) [1] 622.13/291.59 f(x, x) -> f(i(x), g(a)) [1] 622.13/291.59 f(x, y) -> x [1] 622.13/291.59 g(x) -> i(x) [1] 622.13/291.59 f(x, i(x)) -> f(x, x) [1] 622.13/291.59 f(i(x), i(c_g(x))) -> a [1] 622.13/291.59 g(x0) -> c_g(x0) [0] 622.13/291.59 g(v0) -> a [0] 622.13/291.59 622.13/291.59 The TRS has the following type information: 622.13/291.59 f :: i:c_g:a -> i:c_g:a -> i:c_g:a 622.13/291.59 i :: i:c_g:a -> i:c_g:a 622.13/291.59 g :: i:c_g:a -> i:c_g:a 622.13/291.59 c_g :: i:c_g:a -> i:c_g:a 622.13/291.59 a :: i:c_g:a 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (23) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 622.13/291.59 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 622.13/291.59 The constant constructors are abstracted as follows: 622.13/291.59 622.13/291.59 a => 0 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (24) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y 622.13/291.59 f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, g(0)) :|: z' = x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 2 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, g(1 + x)) :|: z' = x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x) 622.13/291.59 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 622.13/291.59 g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x 622.13/291.59 g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (25) InliningProof (UPPER BOUND(ID)) 622.13/291.59 Inlined the following terminating rules on right-hand sides where appropriate: 622.13/291.59 622.13/291.59 g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x 622.13/291.59 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (26) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y 622.13/291.59 f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 2 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 1 + x = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, 0) :|: z' = x, x >= 0, z = x, v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 1 + x = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + x, 1 + x') :|: z' = x, x >= 0, z = x, x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 1 + x = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, 1 + x0) :|: z' = x, x >= 0, z = x, 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x) 622.13/291.59 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 622.13/291.59 g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x 622.13/291.59 g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (27) SimplificationProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (28) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (29) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Found the following analysis order by SCC decomposition: 622.13/291.59 622.13/291.59 { g } 622.13/291.59 { f } 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (30) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {g}, {f} 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (31) ResultPropagationProof (UPPER BOUND(ID)) 622.13/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (32) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {g}, {f} 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (33) IntTrsBoundProof (UPPER BOUND(ID)) 622.13/291.59 622.13/291.59 Computed SIZE bound using CoFloCo for: g 622.13/291.59 after applying outer abstraction to obtain an ITS, 622.13/291.59 resulting in: O(n^1) with polynomial bound: 1 + z 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (34) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {g}, {f} 622.13/291.59 Previous analysis results are: 622.13/291.59 g: runtime: ?, size: O(n^1) [1 + z] 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (35) IntTrsBoundProof (UPPER BOUND(ID)) 622.13/291.59 622.13/291.59 Computed RUNTIME bound using CoFloCo for: g 622.13/291.59 after applying outer abstraction to obtain an ITS, 622.13/291.59 resulting in: O(1) with polynomial bound: 1 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (36) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {f} 622.13/291.59 Previous analysis results are: 622.13/291.59 g: runtime: O(1) [1], size: O(n^1) [1 + z] 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (37) ResultPropagationProof (UPPER BOUND(ID)) 622.13/291.59 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (38) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {f} 622.13/291.59 Previous analysis results are: 622.13/291.59 g: runtime: O(1) [1], size: O(n^1) [1 + z] 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (39) IntTrsBoundProof (UPPER BOUND(ID)) 622.13/291.59 622.13/291.59 Computed SIZE bound using CoFloCo for: f 622.13/291.59 after applying outer abstraction to obtain an ITS, 622.13/291.59 resulting in: INF with polynomial bound: ? 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (40) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {f} 622.13/291.59 Previous analysis results are: 622.13/291.59 g: runtime: O(1) [1], size: O(n^1) [1 + z] 622.13/291.59 f: runtime: ?, size: INF 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (41) IntTrsBoundProof (UPPER BOUND(ID)) 622.13/291.59 622.13/291.59 Computed RUNTIME bound using CoFloCo for: f 622.13/291.59 after applying outer abstraction to obtain an ITS, 622.13/291.59 resulting in: INF with polynomial bound: ? 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (42) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> z :|: z >= 0, z' >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(z' - 1, z' - 1) :|: z' - 1 >= 0, z = z' - 1 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 1 + z' = v0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 0) :|: z' >= 0, z = z', v0 >= 0, 0 = v0 622.13/291.59 f(z, z') -{ 3 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 1 + z' = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x') :|: z' >= 0, z = z', x' >= 0, 0 = x' 622.13/291.59 f(z, z') -{ 2 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 1 + z' = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> f(1 + z', 1 + x0) :|: z' >= 0, z = z', 0 = x0, x0 >= 0 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 1 + (1 + (z - 1)) 622.13/291.59 g(z) -{ 0 }-> 0 :|: z >= 0 622.13/291.59 g(z) -{ 1 }-> 1 + z :|: z >= 0 622.13/291.59 g(z) -{ 0 }-> 1 + z :|: z >= 0 622.13/291.59 622.13/291.59 Function symbols to be analyzed: {f} 622.13/291.59 Previous analysis results are: 622.13/291.59 g: runtime: O(1) [1], size: O(n^1) [1 + z] 622.13/291.59 f: runtime: INF, size: INF 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (43) CompletionProof (UPPER BOUND(ID)) 622.13/291.59 The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: 622.13/291.59 622.13/291.59 g(v0) -> null_g [0] 622.13/291.59 622.13/291.59 And the following fresh constants: null_g 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (44) 622.13/291.59 Obligation: 622.13/291.59 Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: 622.13/291.59 622.13/291.59 Runtime Complexity Weighted TRS with Types. 622.13/291.59 The TRS R consists of the following rules: 622.13/291.59 622.13/291.59 f(x, x) -> f(i(x), g(g(x))) [1] 622.13/291.59 f(x, y) -> x [1] 622.13/291.59 g(x) -> i(x) [1] 622.13/291.59 f(x, i(x)) -> f(x, x) [1] 622.13/291.59 f(i(x), i(c_g(x))) -> a [1] 622.13/291.59 g(x0) -> c_g(x0) [0] 622.13/291.59 g(v0) -> null_g [0] 622.13/291.59 622.13/291.59 The TRS has the following type information: 622.13/291.59 f :: i:c_g:a:null_g -> i:c_g:a:null_g -> i:c_g:a:null_g 622.13/291.59 i :: i:c_g:a:null_g -> i:c_g:a:null_g 622.13/291.59 g :: i:c_g:a:null_g -> i:c_g:a:null_g 622.13/291.59 c_g :: i:c_g:a:null_g -> i:c_g:a:null_g 622.13/291.59 a :: i:c_g:a:null_g 622.13/291.59 null_g :: i:c_g:a:null_g 622.13/291.59 622.13/291.59 Rewrite Strategy: INNERMOST 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 622.13/291.59 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 622.13/291.59 The constant constructors are abstracted as follows: 622.13/291.59 622.13/291.59 a => 0 622.13/291.59 null_g => 0 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (46) 622.13/291.59 Obligation: 622.13/291.59 Complexity RNTS consisting of the following rules: 622.13/291.59 622.13/291.59 f(z, z') -{ 1 }-> x :|: x >= 0, y >= 0, z = x, z' = y 622.13/291.59 f(z, z') -{ 1 }-> f(x, x) :|: z' = 1 + x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 1 }-> f(1 + x, g(g(x))) :|: z' = x, x >= 0, z = x 622.13/291.59 f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 1 + (1 + x) 622.13/291.59 g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 622.13/291.59 g(z) -{ 1 }-> 1 + x :|: x >= 0, z = x 622.13/291.59 g(z) -{ 0 }-> 1 + x0 :|: z = x0, x0 >= 0 622.13/291.59 622.13/291.59 Only complete derivations are relevant for the runtime complexity. 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (47) CpxTrsToCdtProof (UPPER BOUND(ID)) 622.13/291.59 Converted Cpx (relative) TRS to CDT 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (48) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 f(z0, z0) -> f(i(z0), g(g(z0))) 622.13/291.59 f(z0, z1) -> z0 622.13/291.59 f(z0, i(z0)) -> f(z0, z0) 622.13/291.59 f(i(z0), i(c_g(z0))) -> a 622.13/291.59 Tuples: 622.13/291.59 G(z0) -> c 622.13/291.59 G(z0) -> c1 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0)) 622.13/291.59 F(z0, z1) -> c3 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(i(z0), i(c_g(z0))) -> c5 622.13/291.59 S tuples: 622.13/291.59 G(z0) -> c1 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0)) 622.13/291.59 F(z0, z1) -> c3 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(i(z0), i(c_g(z0))) -> c5 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: f_2, g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: G_1, F_2 622.13/291.59 622.13/291.59 Compound Symbols: c, c1, c2_3, c3, c4_1, c5 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 4 trailing nodes: 622.13/291.59 G(z0) -> c 622.13/291.59 F(i(z0), i(c_g(z0))) -> c5 622.13/291.59 G(z0) -> c1 622.13/291.59 F(z0, z1) -> c3 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (50) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 f(z0, z0) -> f(i(z0), g(g(z0))) 622.13/291.59 f(z0, z1) -> z0 622.13/291.59 f(z0, i(z0)) -> f(z0, z0) 622.13/291.59 f(i(z0), i(c_g(z0))) -> a 622.13/291.59 Tuples: 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0)) 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 S tuples: 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0))), G(g(z0)), G(z0)) 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: f_2, g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c2_3, c4_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 2 trailing tuple parts 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (52) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 f(z0, z0) -> f(i(z0), g(g(z0))) 622.13/291.59 f(z0, z1) -> z0 622.13/291.59 f(z0, i(z0)) -> f(z0, z0) 622.13/291.59 f(i(z0), i(c_g(z0))) -> a 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: f_2, g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 The following rules are not usable and were removed: 622.13/291.59 f(z0, z0) -> f(i(z0), g(g(z0))) 622.13/291.59 f(z0, z1) -> z0 622.13/291.59 f(z0, i(z0)) -> f(z0, z0) 622.13/291.59 f(i(z0), i(c_g(z0))) -> a 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (54) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(g(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(g(z0)))) by 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(g(x0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (56) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(g(x0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(g(x0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 1 trailing nodes: 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(g(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (58) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(g(x0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(g(x0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use narrowing to replace F(x0, x0) -> c2(F(i(x0), i(g(x0)))) by 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (60) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 1 trailing nodes: 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(c_g(z0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (62) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(c_g(z0)))) by 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(c_g(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (64) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(c_g(x0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(c_g(x0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 2 trailing nodes: 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(c_g(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(c_g(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (66) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), g(i(z0)))) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use narrowing to replace F(z0, z0) -> c2(F(i(z0), g(i(z0)))) by 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(i(x0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), i(i(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (68) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(i(x0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(i(x0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Removed 1 trailing nodes: 622.13/291.59 F(x0, x0) -> c2(F(i(x0), c_g(i(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (70) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols: g_1 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 The following rules are not usable and were removed: 622.13/291.59 g(z0) -> c_g(z0) 622.13/291.59 g(z0) -> i(z0) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (72) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules:none 622.13/291.59 Tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, i(z0)) -> c4(F(z0, z0)) 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols:none 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (73) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use instantiation to replace F(z0, i(z0)) -> c4(F(z0, z0)) by 622.13/291.59 F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (74) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules:none 622.13/291.59 Tuples: 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) 622.13/291.59 S tuples: 622.13/291.59 F(z0, z0) -> c2(F(i(z0), i(i(z0)))) 622.13/291.59 F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols:none 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c2_1, c4_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (75) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use instantiation to replace F(z0, z0) -> c2(F(i(z0), i(i(z0)))) by 622.13/291.59 F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (76) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules:none 622.13/291.59 Tuples: 622.13/291.59 F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) 622.13/291.59 F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) 622.13/291.59 S tuples: 622.13/291.59 F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) 622.13/291.59 F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols:none 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (77) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use instantiation to replace F(i(x0), i(i(x0))) -> c4(F(i(x0), i(x0))) by 622.13/291.59 F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0)))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (78) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules:none 622.13/291.59 Tuples: 622.13/291.59 F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) 622.13/291.59 F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0)))) 622.13/291.59 S tuples: 622.13/291.59 F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) 622.13/291.59 F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0)))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols:none 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c2_1, c4_1 622.13/291.59 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (79) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) 622.13/291.59 Use instantiation to replace F(i(x0), i(x0)) -> c2(F(i(i(x0)), i(i(i(x0))))) by 622.13/291.59 F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0)))))) 622.13/291.59 622.13/291.59 ---------------------------------------- 622.13/291.59 622.13/291.59 (80) 622.13/291.59 Obligation: 622.13/291.59 Complexity Dependency Tuples Problem 622.13/291.59 622.13/291.59 Rules:none 622.13/291.59 Tuples: 622.13/291.59 F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0)))) 622.13/291.59 F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0)))))) 622.13/291.59 S tuples: 622.13/291.59 F(i(i(x0)), i(i(i(x0)))) -> c4(F(i(i(x0)), i(i(x0)))) 622.13/291.59 F(i(i(x0)), i(i(x0))) -> c2(F(i(i(i(x0))), i(i(i(i(x0)))))) 622.13/291.59 K tuples:none 622.13/291.59 Defined Rule Symbols:none 622.13/291.59 622.13/291.59 Defined Pair Symbols: F_2 622.13/291.59 622.13/291.59 Compound Symbols: c4_1, c2_1 622.13/291.59 622.13/291.62 EOF