5.69/2.48 YES 5.69/2.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.69/2.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.69/2.49 5.69/2.49 5.69/2.49 Termination of the given ETRS could be proven: 5.69/2.49 5.69/2.49 (0) ETRS 5.69/2.49 (1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms] 5.69/2.49 (2) EDP 5.69/2.49 (3) EDependencyGraphProof [EQUIVALENT, 0 ms] 5.69/2.49 (4) AND 5.69/2.49 (5) EDP 5.69/2.49 (6) ESharpUsableEquationsProof [EQUIVALENT, 4 ms] 5.69/2.49 (7) EDP 5.69/2.49 (8) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 5.69/2.49 (9) EDP 5.69/2.49 (10) ERuleRemovalProof [EQUIVALENT, 9 ms] 5.69/2.49 (11) EDP 5.69/2.49 (12) EDPPoloProof [EQUIVALENT, 0 ms] 5.69/2.49 (13) EDP 5.69/2.49 (14) PisEmptyProof [EQUIVALENT, 0 ms] 5.69/2.49 (15) YES 5.69/2.49 (16) EDP 5.69/2.49 (17) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.69/2.49 (18) EDP 5.69/2.49 (19) EDPPoloProof [EQUIVALENT, 3 ms] 5.69/2.49 (20) EDP 5.69/2.49 (21) EDPPoloProof [EQUIVALENT, 26 ms] 5.69/2.49 (22) EDP 5.69/2.49 (23) PisEmptyProof [EQUIVALENT, 0 ms] 5.69/2.49 (24) YES 5.69/2.49 (25) EDP 5.69/2.49 (26) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.69/2.49 (27) EDP 5.69/2.49 (28) EDPPoloProof [EQUIVALENT, 22 ms] 5.69/2.49 (29) EDP 5.69/2.49 (30) EDPPoloProof [EQUIVALENT, 0 ms] 5.69/2.49 (31) EDP 5.69/2.49 (32) PisEmptyProof [EQUIVALENT, 0 ms] 5.69/2.49 (33) YES 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (0) 5.69/2.49 Obligation: 5.69/2.49 Equational rewrite system: 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (1) EquationalDependencyPairsProof (EQUIVALENT) 5.69/2.49 Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 TIMES(s(x), y) -> PLUS(y, times(x, y)) 5.69/2.49 TIMES(s(x), y) -> TIMES(x, y) 5.69/2.49 DIV(x, y) -> QUOT(x, y, y) 5.69/2.49 QUOT(s(x), s(y), z) -> QUOT(x, y, z) 5.69/2.49 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.49 DIV(div(x, y), z) -> DIV(x, times(y, z)) 5.69/2.49 DIV(div(x, y), z) -> TIMES(y, z) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(plus(y, times(x, y)), ext) 5.69/2.49 TIMES(times(s(x), y), ext) -> PLUS(y, times(x, y)) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(x, y) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (2) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 TIMES(s(x), y) -> PLUS(y, times(x, y)) 5.69/2.49 TIMES(s(x), y) -> TIMES(x, y) 5.69/2.49 DIV(x, y) -> QUOT(x, y, y) 5.69/2.49 QUOT(s(x), s(y), z) -> QUOT(x, y, z) 5.69/2.49 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.49 DIV(div(x, y), z) -> DIV(x, times(y, z)) 5.69/2.49 DIV(div(x, y), z) -> TIMES(y, z) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(plus(y, times(x, y)), ext) 5.69/2.49 TIMES(times(s(x), y), ext) -> PLUS(y, times(x, y)) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(x, y) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (3) EDependencyGraphProof (EQUIVALENT) 5.69/2.49 The approximation of the Equational Dependency Graph [DA_STEIN] contains 3 SCCs with 3 less nodes. 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (4) 5.69/2.49 Complex Obligation (AND) 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (5) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (6) ESharpUsableEquationsProof (EQUIVALENT) 5.69/2.49 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (7) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (8) EUsableRulesReductionPairsProof (EQUIVALENT) 5.69/2.49 By using the usable rules and equations with reduction pair processor [DA_STEIN] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules can be oriented non-strictly, the usable equations and the esharp equations can be oriented equivalently. All non-usable rules and equations are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. 5.69/2.49 5.69/2.49 No dependency pairs are removed. 5.69/2.49 5.69/2.49 The following rules are removed from R: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 The following equations are removed from E: 5.69/2.49 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.49 5.69/2.49 POL(0) = 0 5.69/2.49 POL(PLUS(x_1, x_2)) = x_1 + x_2 5.69/2.49 POL(plus(x_1, x_2)) = x_1 + x_2 5.69/2.49 POL(s(x_1)) = x_1 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (9) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (10) ERuleRemovalProof (EQUIVALENT) 5.69/2.49 By using the rule removal processor [DA_STEIN] with the following polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this EDP problem can be strictly oriented. 5.69/2.49 5.69/2.49 Strictly oriented dependency pairs: 5.69/2.49 5.69/2.49 PLUS(s(x), y) -> PLUS(x, y) 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(x, y) 5.69/2.49 5.69/2.49 5.69/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.49 5.69/2.49 POL(PLUS(x_1, x_2)) = 2*x_1 + 2*x_2 5.69/2.49 POL(plus(x_1, x_2)) = x_1 + x_2 5.69/2.49 POL(s(x_1)) = 1 + x_1 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (11) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (12) EDPPoloProof (EQUIVALENT) 5.69/2.49 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.69/2.49 5.69/2.49 5.69/2.49 PLUS(plus(s(x), y), ext) -> PLUS(s(plus(x, y)), ext) 5.69/2.49 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 We had to orient the following equations of E# equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.49 5.69/2.49 POL(PLUS(x_1, x_2)) = x_1 + x_2 5.69/2.49 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 5.69/2.49 POL(s(x_1)) = 0 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (13) 5.69/2.49 Obligation: 5.69/2.49 P is empty. 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (14) PisEmptyProof (EQUIVALENT) 5.69/2.49 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (15) 5.69/2.49 YES 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (16) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(plus(y, times(x, y)), ext) 5.69/2.49 TIMES(s(x), y) -> TIMES(x, y) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(x, y) 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (17) ESharpUsableEquationsProof (EQUIVALENT) 5.69/2.49 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (18) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(plus(y, times(x, y)), ext) 5.69/2.49 TIMES(s(x), y) -> TIMES(x, y) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(x, y) 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (19) EDPPoloProof (EQUIVALENT) 5.69/2.49 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. 5.69/2.49 5.69/2.49 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(plus(y, times(x, y)), ext) 5.69/2.49 TIMES(s(x), y) -> TIMES(x, y) 5.69/2.49 TIMES(times(s(x), y), ext) -> TIMES(x, y) 5.69/2.49 The remaining Dependency Pairs were at least non-strictly oriented. 5.69/2.49 5.69/2.49 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(0, y) -> 0 5.69/2.49 We had to orient the following equations of E# equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.49 5.69/2.49 POL(0) = 0 5.69/2.49 POL(TIMES(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.69/2.49 POL(plus(x_1, x_2)) = x_1 + x_2 5.69/2.49 POL(s(x_1)) = 1 + x_1 5.69/2.49 POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (20) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (21) EDPPoloProof (EQUIVALENT) 5.69/2.49 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.69/2.49 5.69/2.49 5.69/2.49 TIMES(times(0, y), ext) -> TIMES(0, ext) 5.69/2.49 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(0, y) -> 0 5.69/2.49 We had to orient the following equations of E# equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.69/2.49 5.69/2.49 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.49 5.69/2.49 POL(0) = 1 5.69/2.49 POL(TIMES(x_1, x_2)) = 2*x_1 + x_1*x_2 + 2*x_2 5.69/2.49 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 5.69/2.49 POL(s(x_1)) = 2 + x_1 5.69/2.49 POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 5.69/2.49 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (22) 5.69/2.49 Obligation: 5.69/2.49 P is empty. 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (23) PisEmptyProof (EQUIVALENT) 5.69/2.49 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (24) 5.69/2.49 YES 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (25) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 QUOT(s(x), s(y), z) -> QUOT(x, y, z) 5.69/2.49 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.49 DIV(x, y) -> QUOT(x, y, y) 5.69/2.49 DIV(div(x, y), z) -> DIV(x, times(y, z)) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.49 5.69/2.49 The set E consists of the following equations: 5.69/2.49 5.69/2.49 plus(x, y) == plus(y, x) 5.69/2.49 times(x, y) == times(y, x) 5.69/2.49 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.49 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.49 5.69/2.49 The set E# consists of the following equations: 5.69/2.49 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (26) ESharpUsableEquationsProof (EQUIVALENT) 5.69/2.49 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.69/2.49 PLUS(x, y) == PLUS(y, x) 5.69/2.49 TIMES(x, y) == TIMES(y, x) 5.69/2.49 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.69/2.49 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.69/2.49 5.69/2.49 ---------------------------------------- 5.69/2.49 5.69/2.49 (27) 5.69/2.49 Obligation: 5.69/2.49 The TRS P consists of the following rules: 5.69/2.49 5.69/2.49 QUOT(s(x), s(y), z) -> QUOT(x, y, z) 5.69/2.49 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.49 DIV(x, y) -> QUOT(x, y, y) 5.69/2.49 DIV(div(x, y), z) -> DIV(x, times(y, z)) 5.69/2.49 5.69/2.49 The TRS R consists of the following rules: 5.69/2.49 5.69/2.49 plus(x, 0) -> x 5.69/2.49 plus(0, y) -> y 5.69/2.49 plus(s(x), y) -> s(plus(x, y)) 5.69/2.49 times(0, y) -> 0 5.69/2.49 times(s(0), y) -> y 5.69/2.49 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.49 div(0, y) -> 0 5.69/2.49 div(x, y) -> quot(x, y, y) 5.69/2.49 quot(0, s(y), z) -> 0 5.69/2.49 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.49 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.49 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.49 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.49 times(times(0, y), ext) -> times(0, ext) 5.69/2.49 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.50 5.69/2.50 The set E consists of the following equations: 5.69/2.50 5.69/2.50 plus(x, y) == plus(y, x) 5.69/2.50 times(x, y) == times(y, x) 5.69/2.50 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.50 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.50 5.69/2.50 E# is empty. 5.69/2.50 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (28) EDPPoloProof (EQUIVALENT) 5.69/2.50 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. The following set of Dependency Pairs of this DP problem can be strictly oriented. 5.69/2.50 5.69/2.50 5.69/2.50 QUOT(s(x), s(y), z) -> QUOT(x, y, z) 5.69/2.50 DIV(div(x, y), z) -> DIV(x, times(y, z)) 5.69/2.50 The remaining Dependency Pairs were at least non-strictly oriented. 5.69/2.50 5.69/2.50 5.69/2.50 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.50 DIV(x, y) -> QUOT(x, y, y) 5.69/2.50 With the implicit AFS there is no usable rule of R. 5.69/2.50 5.69/2.50 5.69/2.50 There is no equation of E#. 5.69/2.50 5.69/2.50 5.69/2.50 With the implicit AFS there is no usable equation of E. 5.69/2.50 5.69/2.50 5.69/2.50 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.50 5.69/2.50 POL(0) = 3 5.69/2.50 POL(DIV(x_1, x_2)) = 3 + 2*x_1 5.69/2.50 POL(QUOT(x_1, x_2, x_3)) = 3 + 2*x_1 5.69/2.50 POL(div(x_1, x_2)) = 3 + 3*x_1 + 3*x_2 5.69/2.50 POL(plus(x_1, x_2)) = 2 + 3*x_1 + 2*x_2 5.69/2.50 POL(s(x_1)) = 3 + 3*x_1 5.69/2.50 POL(times(x_1, x_2)) = 3 + 2*x_1 + 3*x_2 5.69/2.50 5.69/2.50 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (29) 5.69/2.50 Obligation: 5.69/2.50 The TRS P consists of the following rules: 5.69/2.50 5.69/2.50 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.50 DIV(x, y) -> QUOT(x, y, y) 5.69/2.50 5.69/2.50 The TRS R consists of the following rules: 5.69/2.50 5.69/2.50 plus(x, 0) -> x 5.69/2.50 plus(0, y) -> y 5.69/2.50 plus(s(x), y) -> s(plus(x, y)) 5.69/2.50 times(0, y) -> 0 5.69/2.50 times(s(0), y) -> y 5.69/2.50 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.50 div(0, y) -> 0 5.69/2.50 div(x, y) -> quot(x, y, y) 5.69/2.50 quot(0, s(y), z) -> 0 5.69/2.50 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.50 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.50 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.50 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.50 times(times(0, y), ext) -> times(0, ext) 5.69/2.50 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.50 5.69/2.50 The set E consists of the following equations: 5.69/2.50 5.69/2.50 plus(x, y) == plus(y, x) 5.69/2.50 times(x, y) == times(y, x) 5.69/2.50 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.50 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.50 5.69/2.50 E# is empty. 5.69/2.50 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (30) EDPPoloProof (EQUIVALENT) 5.69/2.50 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.69/2.50 5.69/2.50 5.69/2.50 QUOT(x, 0, s(z)) -> DIV(x, s(z)) 5.69/2.50 DIV(x, y) -> QUOT(x, y, y) 5.69/2.50 With the implicit AFS there is no usable rule of R. 5.69/2.50 5.69/2.50 5.69/2.50 There is no equation of E#. 5.69/2.50 5.69/2.50 5.69/2.50 With the implicit AFS there is no usable equation of E. 5.69/2.50 5.69/2.50 5.69/2.50 Used ordering: POLO with Polynomial interpretation [POLO]: 5.69/2.50 5.69/2.50 POL(0) = 3 5.69/2.50 POL(DIV(x_1, x_2)) = 2 + 2*x_2 5.69/2.50 POL(QUOT(x_1, x_2, x_3)) = 1 + 2*x_2 5.69/2.50 POL(s(x_1)) = 1 5.69/2.50 5.69/2.50 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (31) 5.69/2.50 Obligation: 5.69/2.50 P is empty. 5.69/2.50 The TRS R consists of the following rules: 5.69/2.50 5.69/2.50 plus(x, 0) -> x 5.69/2.50 plus(0, y) -> y 5.69/2.50 plus(s(x), y) -> s(plus(x, y)) 5.69/2.50 times(0, y) -> 0 5.69/2.50 times(s(0), y) -> y 5.69/2.50 times(s(x), y) -> plus(y, times(x, y)) 5.69/2.50 div(0, y) -> 0 5.69/2.50 div(x, y) -> quot(x, y, y) 5.69/2.50 quot(0, s(y), z) -> 0 5.69/2.50 quot(s(x), s(y), z) -> quot(x, y, z) 5.69/2.50 quot(x, 0, s(z)) -> s(div(x, s(z))) 5.69/2.50 div(div(x, y), z) -> div(x, times(y, z)) 5.69/2.50 plus(plus(s(x), y), ext) -> plus(s(plus(x, y)), ext) 5.69/2.50 times(times(0, y), ext) -> times(0, ext) 5.69/2.50 times(times(s(x), y), ext) -> times(plus(y, times(x, y)), ext) 5.69/2.50 5.69/2.50 The set E consists of the following equations: 5.69/2.50 5.69/2.50 plus(x, y) == plus(y, x) 5.69/2.50 times(x, y) == times(y, x) 5.69/2.50 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.69/2.50 times(times(x, y), z) == times(x, times(y, z)) 5.69/2.50 5.69/2.50 E# is empty. 5.69/2.50 We have to consider all minimal (P,E#,R,E)-chains 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (32) PisEmptyProof (EQUIVALENT) 5.69/2.50 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.69/2.50 ---------------------------------------- 5.69/2.50 5.69/2.50 (33) 5.69/2.50 YES 5.91/2.52 EOF