4.01/2.05 YES 4.01/2.06 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 4.01/2.06 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.01/2.06 4.01/2.06 4.01/2.06 Termination of the given ETRS could be proven: 4.01/2.06 4.01/2.06 (0) ETRS 4.01/2.06 (1) EquationalDependencyPairsProof [EQUIVALENT, 14 ms] 4.01/2.06 (2) EDP 4.01/2.06 (3) EDependencyGraphProof [EQUIVALENT, 0 ms] 4.01/2.06 (4) AND 4.01/2.06 (5) EDP 4.01/2.06 (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 4.01/2.06 (7) EDP 4.01/2.06 (8) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 4.01/2.06 (9) EDP 4.01/2.06 (10) PisEmptyProof [EQUIVALENT, 0 ms] 4.01/2.06 (11) YES 4.01/2.06 (12) EDP 4.01/2.06 (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 4.01/2.06 (14) EDP 4.01/2.06 (15) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 4.01/2.06 (16) EDP 4.01/2.06 (17) PisEmptyProof [EQUIVALENT, 0 ms] 4.01/2.06 (18) YES 4.01/2.06 (19) EDP 4.01/2.06 (20) EDPPoloProof [EQUIVALENT, 0 ms] 4.01/2.06 (21) EDP 4.01/2.06 (22) PisEmptyProof [EQUIVALENT, 0 ms] 4.01/2.06 (23) YES 4.01/2.06 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (0) 4.01/2.06 Obligation: 4.01/2.06 Equational rewrite system: 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (1) EquationalDependencyPairsProof (EQUIVALENT) 4.01/2.06 Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 LE(s(x), s(y)) -> LE(x, y) 4.01/2.06 MINUS(x, s(y)) -> PRED(minus(x, y)) 4.01/2.06 MINUS(x, s(y)) -> MINUS(x, y) 4.01/2.06 GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y)) 4.01/2.06 GCD(s(x), s(y)) -> LE(y, x) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y)) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> MINUS(x, y) 4.01/2.06 IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x)) 4.01/2.06 IF_GCD(false, s(x), s(y)) -> MINUS(y, x) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (2) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 LE(s(x), s(y)) -> LE(x, y) 4.01/2.06 MINUS(x, s(y)) -> PRED(minus(x, y)) 4.01/2.06 MINUS(x, s(y)) -> MINUS(x, y) 4.01/2.06 GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y)) 4.01/2.06 GCD(s(x), s(y)) -> LE(y, x) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y)) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> MINUS(x, y) 4.01/2.06 IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x)) 4.01/2.06 IF_GCD(false, s(x), s(y)) -> MINUS(y, x) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (3) EDependencyGraphProof (EQUIVALENT) 4.01/2.06 The approximation of the Equational Dependency Graph [DA_STEIN] contains 3 SCCs with 4 less nodes. 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (4) 4.01/2.06 Complex Obligation (AND) 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (5) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 MINUS(x, s(y)) -> MINUS(x, y) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (6) ESharpUsableEquationsProof (EQUIVALENT) 4.01/2.06 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (7) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 MINUS(x, s(y)) -> MINUS(x, y) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 E# is empty. 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (8) EUsableRulesReductionPairsProof (EQUIVALENT) 4.01/2.06 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. 4.01/2.06 4.01/2.06 The following dependency pairs can be deleted: 4.01/2.06 4.01/2.06 MINUS(x, s(y)) -> MINUS(x, y) 4.01/2.06 The following rules are removed from R: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 The following equations are removed from E: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 Used ordering: POLO with Polynomial interpretation [POLO]: 4.01/2.06 4.01/2.06 POL(MINUS(x_1, x_2)) = x_1 + 3*x_2 4.01/2.06 POL(s(x_1)) = 3*x_1 4.01/2.06 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (9) 4.01/2.06 Obligation: 4.01/2.06 P is empty. 4.01/2.06 R is empty. 4.01/2.06 E is empty. 4.01/2.06 E# is empty. 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (10) PisEmptyProof (EQUIVALENT) 4.01/2.06 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (11) 4.01/2.06 YES 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (12) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 LE(s(x), s(y)) -> LE(x, y) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (13) ESharpUsableEquationsProof (EQUIVALENT) 4.01/2.06 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (14) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 LE(s(x), s(y)) -> LE(x, y) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 E# is empty. 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (15) EUsableRulesReductionPairsProof (EQUIVALENT) 4.01/2.06 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. 4.01/2.06 4.01/2.06 The following dependency pairs can be deleted: 4.01/2.06 4.01/2.06 LE(s(x), s(y)) -> LE(x, y) 4.01/2.06 The following rules are removed from R: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 The following equations are removed from E: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 Used ordering: POLO with Polynomial interpretation [POLO]: 4.01/2.06 4.01/2.06 POL(LE(x_1, x_2)) = 3*x_1 + 3*x_2 4.01/2.06 POL(s(x_1)) = x_1 4.01/2.06 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (16) 4.01/2.06 Obligation: 4.01/2.06 P is empty. 4.01/2.06 R is empty. 4.01/2.06 E is empty. 4.01/2.06 E# is empty. 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (17) PisEmptyProof (EQUIVALENT) 4.01/2.06 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (18) 4.01/2.06 YES 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (19) 4.01/2.06 Obligation: 4.01/2.06 The TRS P consists of the following rules: 4.01/2.06 4.01/2.06 IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x)) 4.01/2.06 GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y)) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y)) 4.01/2.06 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (20) EDPPoloProof (EQUIVALENT) 4.01/2.06 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 4.01/2.06 4.01/2.06 4.01/2.06 IF_GCD(false, s(x), s(y)) -> GCD(minus(y, x), s(x)) 4.01/2.06 GCD(s(x), s(y)) -> IF_GCD(le(y, x), s(x), s(y)) 4.01/2.06 IF_GCD(true, s(x), s(y)) -> GCD(minus(x, y), s(y)) 4.01/2.06 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 4.01/2.06 4.01/2.06 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 minus(x, 0) -> x 4.01/2.06 pred(s(x)) -> x 4.01/2.06 We had to orient the following equations of E# equivalently. 4.01/2.06 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 With the implicit AFS there is no usable equation of E. 4.01/2.06 4.01/2.06 4.01/2.06 Used ordering: POLO with Polynomial interpretation [POLO]: 4.01/2.06 4.01/2.06 POL(0) = 0 4.01/2.06 POL(GCD(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 4.01/2.06 POL(IF_GCD(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 4.01/2.06 POL(false) = 0 4.01/2.06 POL(le(x_1, x_2)) = 0 4.01/2.06 POL(minus(x_1, x_2)) = x_1 4.01/2.06 POL(pred(x_1)) = x_1 4.01/2.06 POL(s(x_1)) = 2 + x_1 4.01/2.06 POL(true) = 0 4.01/2.06 4.01/2.06 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (21) 4.01/2.06 Obligation: 4.01/2.06 P is empty. 4.01/2.06 The TRS R consists of the following rules: 4.01/2.06 4.01/2.06 le(0, y) -> true 4.01/2.06 le(s(x), 0) -> false 4.01/2.06 le(s(x), s(y)) -> le(x, y) 4.01/2.06 pred(s(x)) -> x 4.01/2.06 minus(x, 0) -> x 4.01/2.06 minus(x, s(y)) -> pred(minus(x, y)) 4.01/2.06 gcd(0, y) -> y 4.01/2.06 gcd(s(x), 0) -> s(x) 4.01/2.06 gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) 4.01/2.06 if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y)) 4.01/2.06 if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x)) 4.01/2.06 4.01/2.06 The set E consists of the following equations: 4.01/2.06 4.01/2.06 gcd(x, y) == gcd(y, x) 4.01/2.06 4.01/2.06 The set E# consists of the following equations: 4.01/2.06 4.01/2.06 GCD(x, y) == GCD(y, x) 4.01/2.06 4.01/2.06 We have to consider all minimal (P,E#,R,E)-chains 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (22) PisEmptyProof (EQUIVALENT) 4.01/2.06 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 4.01/2.06 ---------------------------------------- 4.01/2.06 4.01/2.06 (23) 4.01/2.06 YES 4.18/2.07 EOF