MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could not be shown: (0) ETRS (1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms] (2) EDP (3) EDependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) EDP (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (7) EDP (8) EUsableRulesReductionPairsProof [EQUIVALENT, 42 ms] (9) EDP (10) PisEmptyProof [EQUIVALENT, 0 ms] (11) YES (12) EDP (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (14) EDP (15) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (16) EDP (17) EDependencyGraphProof [EQUIVALENT, 0 ms] (18) TRUE (19) EDP (20) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (21) EDP (22) EUsableRulesProof [EQUIVALENT, 0 ms] (23) EDP (24) EDP (25) EDP (26) EDP ---------------------------------------- (0) Obligation: Equational rewrite system: The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) ---------------------------------------- (1) EquationalDependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: The TRS P consists of the following rules: U11^1(tt, M, N) -> U12^1(tt, M, N) U12^1(tt, M, N) -> _+_^1(N, _+_(M, _*_(N, M))) U12^1(tt, M, N) -> _+_^1(M, _*_(N, M)) U12^1(tt, M, N) -> _*_^1(N, M) U21^1(tt, M, N) -> U22^1(tt, M, N) U22^1(tt, M, N) -> _+_^1(N, M) U31^1(tt, M, N) -> U32^1(tt, M, N) U32^1(tt, M, N) -> _>_^1(M, N) U41^1(tt, M, N) -> U42^1(tt, M, N) U42^1(tt, M, N) -> _>_^1(N, M) U51^1(tt, M, N) -> U52^1(tt, M, N) U52^1(tt, M, N) -> D(N, M) U61^1(tt, M', N') -> U62^1(tt, M', N') U62^1(tt, M', N') -> U63^1(equal(_>_(N', M'), true), M', N') U62^1(tt, M', N') -> EQUAL(_>_(N', M'), true) U62^1(tt, M', N') -> _>_^1(N', M') U63^1(tt, M', N') -> GCD(d(N', M'), M') U63^1(tt, M', N') -> D(N', M') U71^1(tt, M', N) -> U72^1(tt, M', N) U72^1(tt, M', N) -> U73^1(equal(_>_(M', N), true)) U72^1(tt, M', N) -> EQUAL(_>_(M', N), true) U72^1(tt, M', N) -> _>_^1(M', N) U81^1(tt, M', N) -> U82^1(tt, M', N) U82^1(tt, M', N) -> U83^1(equal(_>_(N, M'), true), M', N) U82^1(tt, M', N) -> EQUAL(_>_(N, M'), true) U82^1(tt, M', N) -> _>_^1(N, M') U83^1(tt, M', N) -> QUOT(d(N, M'), M') U83^1(tt, M', N) -> D(N, M') _*_^1(s_(N), s_(M)) -> U11^1(tt, M, N) _+_^1(s_(N), s_(M)) -> U21^1(tt, M, N) _<_^1(N, M) -> U31^1(tt, M, N) _>_^1(s_(N), s_(M)) -> U41^1(tt, M, N) D(s_(N), s_(M)) -> U51^1(tt, M, N) GCD(N', M') -> U61^1(tt, M', N') QUOT(N, M') -> U71^1(tt, M', N) QUOT(N, M') -> U81^1(tt, M', N) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (2) Obligation: The TRS P consists of the following rules: U11^1(tt, M, N) -> U12^1(tt, M, N) U12^1(tt, M, N) -> _+_^1(N, _+_(M, _*_(N, M))) U12^1(tt, M, N) -> _+_^1(M, _*_(N, M)) U12^1(tt, M, N) -> _*_^1(N, M) U21^1(tt, M, N) -> U22^1(tt, M, N) U22^1(tt, M, N) -> _+_^1(N, M) U31^1(tt, M, N) -> U32^1(tt, M, N) U32^1(tt, M, N) -> _>_^1(M, N) U41^1(tt, M, N) -> U42^1(tt, M, N) U42^1(tt, M, N) -> _>_^1(N, M) U51^1(tt, M, N) -> U52^1(tt, M, N) U52^1(tt, M, N) -> D(N, M) U61^1(tt, M', N') -> U62^1(tt, M', N') U62^1(tt, M', N') -> U63^1(equal(_>_(N', M'), true), M', N') U62^1(tt, M', N') -> EQUAL(_>_(N', M'), true) U62^1(tt, M', N') -> _>_^1(N', M') U63^1(tt, M', N') -> GCD(d(N', M'), M') U63^1(tt, M', N') -> D(N', M') U71^1(tt, M', N) -> U72^1(tt, M', N) U72^1(tt, M', N) -> U73^1(equal(_>_(M', N), true)) U72^1(tt, M', N) -> EQUAL(_>_(M', N), true) U72^1(tt, M', N) -> _>_^1(M', N) U81^1(tt, M', N) -> U82^1(tt, M', N) U82^1(tt, M', N) -> U83^1(equal(_>_(N, M'), true), M', N) U82^1(tt, M', N) -> EQUAL(_>_(N, M'), true) U82^1(tt, M', N) -> _>_^1(N, M') U83^1(tt, M', N) -> QUOT(d(N, M'), M') U83^1(tt, M', N) -> D(N, M') _*_^1(s_(N), s_(M)) -> U11^1(tt, M, N) _+_^1(s_(N), s_(M)) -> U21^1(tt, M, N) _<_^1(N, M) -> U31^1(tt, M, N) _>_^1(s_(N), s_(M)) -> U41^1(tt, M, N) D(s_(N), s_(M)) -> U51^1(tt, M, N) GCD(N', M') -> U61^1(tt, M', N') QUOT(N, M') -> U71^1(tt, M', N) QUOT(N, M') -> U81^1(tt, M', N) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (3) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 6 SCCs with 16 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: The TRS P consists of the following rules: U51^1(tt, M, N) -> U52^1(tt, M, N) D(s_(N), s_(M)) -> U51^1(tt, M, N) U52^1(tt, M, N) -> D(N, M) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (6) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) GCD(x, y) == GCD(y, x) ---------------------------------------- (7) Obligation: The TRS P consists of the following rules: U51^1(tt, M, N) -> U52^1(tt, M, N) D(s_(N), s_(M)) -> U51^1(tt, M, N) U52^1(tt, M, N) -> D(N, M) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: D(x, y) == D(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (8) EUsableRulesReductionPairsProof (EQUIVALENT) By using the improved usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding improved usable rules can be oriented non-strictly, the improved usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and improved usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: U51^1(tt, M, N) -> U52^1(tt, M, N) D(s_(N), s_(M)) -> U51^1(tt, M, N) U52^1(tt, M, N) -> D(N, M) The following rules are removed from R: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The following equations are removed from E: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(D(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U51^1(x_1, x_2, x_3)) = 3 + 2*x_1 + 2*x_2 + 2*x_3 POL(U52^1(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(s_(x_1)) = 2 + 2*x_1 POL(tt) = 0 ---------------------------------------- (9) Obligation: P is empty. R is empty. E is empty. The set E# consists of the following equations: D(x, y) == D(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (10) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: The TRS P consists of the following rules: U42^1(tt, M, N) -> _>_^1(N, M) _>_^1(s_(N), s_(M)) -> U41^1(tt, M, N) U41^1(tt, M, N) -> U42^1(tt, M, N) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (13) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) ---------------------------------------- (14) Obligation: The TRS P consists of the following rules: U42^1(tt, M, N) -> _>_^1(N, M) _>_^1(s_(N), s_(M)) -> U41^1(tt, M, N) U41^1(tt, M, N) -> U42^1(tt, M, N) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (15) EUsableRulesReductionPairsProof (EQUIVALENT) By using the improved usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding improved usable rules can be oriented non-strictly, the improved usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and improved usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. The following dependency pairs can be deleted: _>_^1(s_(N), s_(M)) -> U41^1(tt, M, N) U41^1(tt, M, N) -> U42^1(tt, M, N) The following rules are removed from R: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The following equations are removed from E: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(U41^1(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 3*x_3 POL(U42^1(x_1, x_2, x_3)) = 2*x_1 + x_2 + 3*x_3 POL(_>_^1(x_1, x_2)) = 2 + 3*x_1 + x_2 POL(s_(x_1)) = 2 + 3*x_1 POL(tt) = 1 ---------------------------------------- (16) Obligation: The TRS P consists of the following rules: U42^1(tt, M, N) -> _>_^1(N, M) R is empty. E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (17) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 1 less node. ---------------------------------------- (18) TRUE ---------------------------------------- (19) Obligation: The TRS P consists of the following rules: U82^1(tt, M', N) -> U83^1(equal(_>_(N, M'), true), M', N) QUOT(N, M') -> U81^1(tt, M', N) U81^1(tt, M', N) -> U82^1(tt, M', N) U83^1(tt, M', N) -> QUOT(d(N, M'), M') The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (20) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) ---------------------------------------- (21) Obligation: The TRS P consists of the following rules: U82^1(tt, M', N) -> U83^1(equal(_>_(N, M'), true), M', N) QUOT(N, M') -> U81^1(tt, M', N) U81^1(tt, M', N) -> U82^1(tt, M', N) U83^1(tt, M', N) -> QUOT(d(N, M'), M') The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (22) EUsableRulesProof (EQUIVALENT) We use the improved usable rules and equations processor [DA_STEIN] to delete all non-usable rules from R and all non-usable equations from E, but we lose minimality and add the following 2 Ce-rules: c(x, y) -> x c(x, y) -> y ---------------------------------------- (23) Obligation: The TRS P consists of the following rules: U82^1(tt, M', N) -> U83^1(equal(_>_(N, M'), true), M', N) QUOT(N, M') -> U81^1(tt, M', N) U81^1(tt, M', N) -> U82^1(tt, M', N) U83^1(tt, M', N) -> QUOT(d(N, M'), M') The TRS R consists of the following rules: equal(X, X) -> tt _>_(0, M) -> false U42(tt, M, N) -> _>_(N, M) U41(tt, M, N) -> U42(tt, M, N) d(s_(N), s_(M)) -> U51(tt, M, N) U51(tt, M, N) -> U52(tt, M, N) _>_(s_(N), s_(M)) -> U41(tt, M, N) _>_(N', 0) -> true U52(tt, M, N) -> d(N, M) d(0, N) -> N c(x, y) -> x c(x, y) -> y The set E consists of the following equations: d(x, y) == d(y, x) E# is empty. We have to consider all (P,E#,R,E)-chains ---------------------------------------- (24) Obligation: The TRS P consists of the following rules: U62^1(tt, M', N') -> U63^1(equal(_>_(N', M'), true), M', N') GCD(N', M') -> U61^1(tt, M', N') U61^1(tt, M', N') -> U62^1(tt, M', N') U63^1(tt, M', N') -> GCD(d(N', M'), M') The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (25) Obligation: The TRS P consists of the following rules: _+_^1(s_(N), s_(M)) -> U21^1(tt, M, N) U21^1(tt, M, N) -> U22^1(tt, M, N) U22^1(tt, M, N) -> _+_^1(N, M) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (26) Obligation: The TRS P consists of the following rules: U12^1(tt, M, N) -> _*_^1(N, M) _*_^1(s_(N), s_(M)) -> U11^1(tt, M, N) U11^1(tt, M, N) -> U12^1(tt, M, N) The TRS R consists of the following rules: 1 -> s_(0) 2 -> s_(s_(0)) 3 -> s_(s_(s_(0))) 4 -> s_(s_(s_(s_(0)))) 5 -> s_(s_(s_(s_(s_(0))))) 6 -> s_(s_(s_(s_(s_(s_(0)))))) 7 -> s_(s_(s_(s_(s_(s_(s_(0))))))) U11(tt, M, N) -> U12(tt, M, N) U12(tt, M, N) -> s_(_+_(N, _+_(M, _*_(N, M)))) U21(tt, M, N) -> U22(tt, M, N) U22(tt, M, N) -> s_(s_(_+_(N, M))) U31(tt, M, N) -> U32(tt, M, N) U32(tt, M, N) -> _>_(M, N) U41(tt, M, N) -> U42(tt, M, N) U42(tt, M, N) -> _>_(N, M) U51(tt, M, N) -> U52(tt, M, N) U52(tt, M, N) -> d(N, M) U61(tt, M', N') -> U62(tt, M', N') U62(tt, M', N') -> U63(equal(_>_(N', M'), true), M', N') U63(tt, M', N') -> gcd(d(N', M'), M') U71(tt, M', N) -> U72(tt, M', N) U72(tt, M', N) -> U73(equal(_>_(M', N), true)) U73(tt) -> 0 U81(tt, M', N) -> U82(tt, M', N) U82(tt, M', N) -> U83(equal(_>_(N, M'), true), M', N) U83(tt, M', N) -> s_(quot(d(N, M'), M')) _*_(N, 0) -> 0 _*_(s_(N), s_(M)) -> U11(tt, M, N) _+_(N, 0) -> N _+_(s_(N), s_(M)) -> U21(tt, M, N) _<_(N, M) -> U31(tt, M, N) _>_(0, M) -> false _>_(N', 0) -> true _>_(s_(N), s_(M)) -> U41(tt, M, N) d(0, N) -> N d(s_(N), s_(M)) -> U51(tt, M, N) equal(X, X) -> tt gcd(0, N) -> 0 gcd(N', M') -> U61(tt, M', N') gcd(N', N') -> N' p_(s_(N)) -> N quot(M', M') -> s_(0) quot(N, M') -> U71(tt, M', N) quot(N, M') -> U81(tt, M', N) The set E consists of the following equations: _*_(x, y) == _*_(y, x) _+_(x, y) == _+_(y, x) d(x, y) == d(y, x) gcd(x, y) == gcd(y, x) The set E# consists of the following equations: _*_^1(x, y) == _*_^1(y, x) _+_^1(x, y) == _+_^1(y, x) D(x, y) == D(y, x) GCD(x, y) == GCD(y, x) We have to consider all minimal (P,E#,R,E)-chains