29.92/10.38 WORST_CASE(Omega(n^1), O(n^1)) 29.92/10.40 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.92/10.40 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.92/10.40 29.92/10.40 29.92/10.40 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.92/10.40 29.92/10.40 (0) CpxTRS 29.92/10.40 (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (2) CpxTRS 29.92/10.40 (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (4) CpxWeightedTrs 29.92/10.40 (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (6) CpxTypedWeightedTrs 29.92/10.40 (7) CompletionProof [UPPER BOUND(ID), 0 ms] 29.92/10.40 (8) CpxTypedWeightedCompleteTrs 29.92/10.40 (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (10) CpxTypedWeightedCompleteTrs 29.92/10.40 (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] 29.92/10.40 (12) CpxRNTS 29.92/10.40 (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (14) CpxRNTS 29.92/10.40 (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (16) CpxRNTS 29.92/10.40 (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms] 29.92/10.40 (18) CpxRNTS 29.92/10.40 (19) IntTrsBoundProof [UPPER BOUND(ID), 5010 ms] 29.92/10.40 (20) CpxRNTS 29.92/10.40 (21) IntTrsBoundProof [UPPER BOUND(ID), 140 ms] 29.92/10.40 (22) CpxRNTS 29.92/10.40 (23) FinalProof [FINISHED, 0 ms] 29.92/10.40 (24) BOUNDS(1, n^1) 29.92/10.40 (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (26) CpxTRS 29.92/10.40 (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 29.92/10.40 (28) typed CpxTrs 29.92/10.40 (29) OrderProof [LOWER BOUND(ID), 0 ms] 29.92/10.40 (30) typed CpxTrs 29.92/10.40 (31) RewriteLemmaProof [LOWER BOUND(ID), 1099 ms] 29.92/10.40 (32) proven lower bound 29.92/10.40 (33) LowerBoundPropagationProof [FINISHED, 0 ms] 29.92/10.40 (34) BOUNDS(n^1, INF) 29.92/10.40 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (0) 29.92/10.40 Obligation: 29.92/10.40 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.92/10.40 29.92/10.40 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 S is empty. 29.92/10.40 Rewrite Strategy: FULL 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Converted rc-obligation to irc-obligation. 29.92/10.40 29.92/10.40 As the TRS does not nest defined symbols, we have rc = irc. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (2) 29.92/10.40 Obligation: 29.92/10.40 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 29.92/10.40 29.92/10.40 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 S is empty. 29.92/10.40 Rewrite Strategy: INNERMOST 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Transformed relative TRS to weighted TRS 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (4) 29.92/10.40 Obligation: 29.92/10.40 The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1). 29.92/10.40 29.92/10.40 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one [1] 29.92/10.40 dx(a) -> zero [1] 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] 29.92/10.40 29.92/10.40 Rewrite Strategy: INNERMOST 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Infered types. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (6) 29.92/10.40 Obligation: 29.92/10.40 Runtime Complexity Weighted TRS with Types. 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one [1] 29.92/10.40 dx(a) -> zero [1] 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] 29.92/10.40 29.92/10.40 The TRS has the following type information: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 Rewrite Strategy: INNERMOST 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (7) CompletionProof (UPPER BOUND(ID)) 29.92/10.40 The transformation into a RNTS is sound, since: 29.92/10.40 29.92/10.40 (a) The obligation is a constructor system where every type has a constant constructor, 29.92/10.40 29.92/10.40 (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 29.92/10.40 29.92/10.40 dx_1 29.92/10.40 29.92/10.40 (c) The following functions are completely defined: 29.92/10.40 none 29.92/10.40 29.92/10.40 Due to the following rules being added: 29.92/10.40 none 29.92/10.40 29.92/10.40 And the following fresh constants: none 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (8) 29.92/10.40 Obligation: 29.92/10.40 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 29.92/10.40 29.92/10.40 Runtime Complexity Weighted TRS with Types. 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one [1] 29.92/10.40 dx(a) -> zero [1] 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] 29.92/10.40 29.92/10.40 The TRS has the following type information: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 Rewrite Strategy: INNERMOST 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (9) NarrowingProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Narrowed the inner basic terms of all right-hand sides by a single narrowing step. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (10) 29.92/10.40 Obligation: 29.92/10.40 Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: 29.92/10.40 29.92/10.40 Runtime Complexity Weighted TRS with Types. 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one [1] 29.92/10.40 dx(a) -> zero [1] 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1] 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) [1] 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) [1] 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1] 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) [1] 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1] 29.92/10.40 29.92/10.40 The TRS has the following type information: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 Rewrite Strategy: INNERMOST 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) 29.92/10.40 Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. 29.92/10.40 The constant constructors are abstracted as follows: 29.92/10.40 29.92/10.40 one => 1 29.92/10.40 a => 0 29.92/10.40 zero => 3 29.92/10.40 two => 2 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (12) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: X >= 0, z = X 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) :|: ALPHA >= 0, z = 1 + ALPHA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + ALPHA :|: ALPHA >= 0, z = 1 + ALPHA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (13) SimplificationProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Simplified the RNTS by moving equalities from the constraints into the right-hand sides. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (14) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: z >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Found the following analysis order by SCC decomposition: 29.92/10.40 29.92/10.40 { dx } 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (16) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: z >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 Function symbols to be analyzed: {dx} 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (17) ResultPropagationProof (UPPER BOUND(ID)) 29.92/10.40 Applied inner abstraction using the recently inferred runtime/size bounds where possible. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (18) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: z >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 Function symbols to be analyzed: {dx} 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (19) IntTrsBoundProof (UPPER BOUND(ID)) 29.92/10.40 29.92/10.40 Computed SIZE bound using CoFloCo for: dx 29.92/10.40 after applying outer abstraction to obtain an ITS, 29.92/10.40 resulting in: O(n^2) with polynomial bound: 3 + 10*z + 3*z^2 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (20) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: z >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 Function symbols to be analyzed: {dx} 29.92/10.40 Previous analysis results are: 29.92/10.40 dx: runtime: ?, size: O(n^2) [3 + 10*z + 3*z^2] 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (21) IntTrsBoundProof (UPPER BOUND(ID)) 29.92/10.40 29.92/10.40 Computed RUNTIME bound using CoFloCo for: dx 29.92/10.40 after applying outer abstraction to obtain an ITS, 29.92/10.40 resulting in: O(n^1) with polynomial bound: 1 + 2*z 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (22) 29.92/10.40 Obligation: 29.92/10.40 Complexity RNTS consisting of the following rules: 29.92/10.40 29.92/10.40 dx(z) -{ 1 }-> 3 :|: z = 0 29.92/10.40 dx(z) -{ 1 }-> 1 :|: z >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 dx(z) -{ 1 }-> 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA 29.92/10.40 29.92/10.40 Function symbols to be analyzed: 29.92/10.40 Previous analysis results are: 29.92/10.40 dx: runtime: O(n^1) [1 + 2*z], size: O(n^2) [3 + 10*z + 3*z^2] 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (23) FinalProof (FINISHED) 29.92/10.40 Computed overall runtime complexity 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (24) 29.92/10.40 BOUNDS(1, n^1) 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (25) RenamingProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Renamed function symbols to avoid clashes with predefined symbol. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (26) 29.92/10.40 Obligation: 29.92/10.40 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 29.92/10.40 29.92/10.40 29.92/10.40 The TRS R consists of the following rules: 29.92/10.40 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 S is empty. 29.92/10.40 Rewrite Strategy: FULL 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (27) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 29.92/10.40 Infered types. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (28) 29.92/10.40 Obligation: 29.92/10.40 TRS: 29.92/10.40 Rules: 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 Types: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (29) OrderProof (LOWER BOUND(ID)) 29.92/10.40 Heuristically decided to analyse the following defined symbols: 29.92/10.40 dx 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (30) 29.92/10.40 Obligation: 29.92/10.40 TRS: 29.92/10.40 Rules: 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 Types: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 29.92/10.40 Generator Equations: 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0) <=> a 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(x, 1)) <=> plus(a, gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(x)) 29.92/10.40 29.92/10.40 29.92/10.40 The following defined symbols remain to be analysed: 29.92/10.40 dx 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (31) RewriteLemmaProof (LOWER BOUND(ID)) 29.92/10.40 Proved the following rewrite lemma: 29.92/10.40 dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0)) -> *3_0, rt in Omega(n4_0) 29.92/10.40 29.92/10.40 Induction Base: 29.92/10.40 dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0)) 29.92/10.40 29.92/10.40 Induction Step: 29.92/10.40 dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(n4_0, 1))) ->_R^Omega(1) 29.92/10.40 plus(dx(a), dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) ->_R^Omega(1) 29.92/10.40 plus(one, dx(gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(n4_0))) ->_IH 29.92/10.40 plus(one, *3_0) 29.92/10.40 29.92/10.40 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (32) 29.92/10.40 Obligation: 29.92/10.40 Proved the lower bound n^1 for the following obligation: 29.92/10.40 29.92/10.40 TRS: 29.92/10.40 Rules: 29.92/10.40 dx(X) -> one 29.92/10.40 dx(a) -> zero 29.92/10.40 dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) 29.92/10.40 dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA)) 29.92/10.40 dx(neg(ALPHA)) -> neg(dx(ALPHA)) 29.92/10.40 dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) 29.92/10.40 dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA) 29.92/10.40 dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) 29.92/10.40 29.92/10.40 Types: 29.92/10.40 dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 hole_one:a:zero:plus:times:minus:neg:div:two:exp:ln1_0 :: one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0 :: Nat -> one:a:zero:plus:times:minus:neg:div:two:exp:ln 29.92/10.40 29.92/10.40 29.92/10.40 Generator Equations: 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(0) <=> a 29.92/10.40 gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(+(x, 1)) <=> plus(a, gen_one:a:zero:plus:times:minus:neg:div:two:exp:ln2_0(x)) 29.92/10.40 29.92/10.40 29.92/10.40 The following defined symbols remain to be analysed: 29.92/10.40 dx 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (33) LowerBoundPropagationProof (FINISHED) 29.92/10.40 Propagated lower bound. 29.92/10.40 ---------------------------------------- 29.92/10.40 29.92/10.40 (34) 29.92/10.40 BOUNDS(n^1, INF) 30.13/10.45 EOF