5.62/2.41 YES 5.62/2.42 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 5.62/2.42 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 5.62/2.42 5.62/2.42 5.62/2.42 Termination of the given ETRS could be proven: 5.62/2.42 5.62/2.42 (0) ETRS 5.62/2.42 (1) EquationalDependencyPairsProof [EQUIVALENT, 0 ms] 5.62/2.42 (2) EDP 5.62/2.42 (3) EDependencyGraphProof [EQUIVALENT, 2 ms] 5.62/2.42 (4) AND 5.62/2.42 (5) EDP 5.62/2.42 (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.62/2.42 (7) EDP 5.62/2.42 (8) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] 5.62/2.42 (9) EDP 5.62/2.42 (10) PisEmptyProof [EQUIVALENT, 0 ms] 5.62/2.42 (11) YES 5.62/2.42 (12) EDP 5.62/2.42 (13) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.62/2.42 (14) EDP 5.62/2.42 (15) EDPPoloProof [EQUIVALENT, 0 ms] 5.62/2.42 (16) EDP 5.62/2.42 (17) PisEmptyProof [EQUIVALENT, 0 ms] 5.62/2.42 (18) YES 5.62/2.42 (19) EDP 5.62/2.42 (20) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.62/2.42 (21) EDP 5.62/2.42 (22) EUsableRulesReductionPairsProof [EQUIVALENT, 21 ms] 5.62/2.42 (23) EDP 5.62/2.42 (24) ERuleRemovalProof [EQUIVALENT, 0 ms] 5.62/2.42 (25) EDP 5.62/2.42 (26) EDPPoloProof [EQUIVALENT, 0 ms] 5.62/2.42 (27) EDP 5.62/2.42 (28) PisEmptyProof [EQUIVALENT, 0 ms] 5.62/2.42 (29) YES 5.62/2.42 (30) EDP 5.62/2.42 (31) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] 5.62/2.42 (32) EDP 5.62/2.42 (33) EDPPoloProof [EQUIVALENT, 4 ms] 5.62/2.42 (34) EDP 5.62/2.42 (35) EDPPoloProof [EQUIVALENT, 41 ms] 5.62/2.42 (36) EDP 5.62/2.42 (37) PisEmptyProof [EQUIVALENT, 0 ms] 5.62/2.42 (38) YES 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (0) 5.62/2.42 Obligation: 5.62/2.42 Equational rewrite system: 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (1) EquationalDependencyPairsProof (EQUIVALENT) 5.62/2.42 Using Dependency Pairs [AG00,DA_STEIN] we result in the following initial EDP problem: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 TIMES(x, s(y)) -> PLUS(times(x, y), x) 5.62/2.42 TIMES(x, s(y)) -> TIMES(x, y) 5.62/2.42 MINUS(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 DIV(s(x), s(y)) -> DIV(minus(x, y), s(y)) 5.62/2.42 DIV(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(plus(times(x, y), x), ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> PLUS(times(x, y), x) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (2) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 TIMES(x, s(y)) -> PLUS(times(x, y), x) 5.62/2.42 TIMES(x, s(y)) -> TIMES(x, y) 5.62/2.42 MINUS(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 DIV(s(x), s(y)) -> DIV(minus(x, y), s(y)) 5.62/2.42 DIV(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(plus(times(x, y), x), ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> PLUS(times(x, y), x) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (3) EDependencyGraphProof (EQUIVALENT) 5.62/2.42 The approximation of the Equational Dependency Graph [DA_STEIN] contains 4 SCCs with 3 less nodes. 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (4) 5.62/2.42 Complex Obligation (AND) 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (5) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 MINUS(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (6) ESharpUsableEquationsProof (EQUIVALENT) 5.62/2.42 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (7) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 MINUS(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 E# is empty. 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (8) EUsableRulesReductionPairsProof (EQUIVALENT) 5.62/2.42 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. 5.62/2.42 5.62/2.42 The following dependency pairs can be deleted: 5.62/2.42 5.62/2.42 MINUS(s(x), s(y)) -> MINUS(x, y) 5.62/2.42 The following rules are removed from R: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 The following equations are removed from E: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(MINUS(x_1, x_2)) = 3*x_1 + 3*x_2 5.62/2.42 POL(s(x_1)) = x_1 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (9) 5.62/2.42 Obligation: 5.62/2.42 P is empty. 5.62/2.42 R is empty. 5.62/2.42 E is empty. 5.62/2.42 E# is empty. 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (10) PisEmptyProof (EQUIVALENT) 5.62/2.42 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (11) 5.62/2.42 YES 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (12) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 DIV(s(x), s(y)) -> DIV(minus(x, y), s(y)) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (13) ESharpUsableEquationsProof (EQUIVALENT) 5.62/2.42 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (14) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 DIV(s(x), s(y)) -> DIV(minus(x, y), s(y)) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 E# is empty. 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (15) EDPPoloProof (EQUIVALENT) 5.62/2.42 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.62/2.42 5.62/2.42 5.62/2.42 DIV(s(x), s(y)) -> DIV(minus(x, y), s(y)) 5.62/2.42 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.62/2.42 5.62/2.42 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 There is no equation of E#. 5.62/2.42 5.62/2.42 5.62/2.42 With the implicit AFS there is no usable equation of E. 5.62/2.42 5.62/2.42 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(0) = 0 5.62/2.42 POL(DIV(x_1, x_2)) = 2*x_1 5.62/2.42 POL(minus(x_1, x_2)) = 2*x_1 5.62/2.42 POL(s(x_1)) = 2 + 2*x_1 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (16) 5.62/2.42 Obligation: 5.62/2.42 P is empty. 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 E# is empty. 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (17) PisEmptyProof (EQUIVALENT) 5.62/2.42 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (18) 5.62/2.42 YES 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (19) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (20) ESharpUsableEquationsProof (EQUIVALENT) 5.62/2.42 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (21) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (22) EUsableRulesReductionPairsProof (EQUIVALENT) 5.62/2.42 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. 5.62/2.42 5.62/2.42 No dependency pairs are removed. 5.62/2.42 5.62/2.42 The following rules are removed from R: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 The following equations are removed from E: 5.62/2.42 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(0) = 0 5.62/2.42 POL(PLUS(x_1, x_2)) = x_1 + x_2 5.62/2.42 POL(plus(x_1, x_2)) = x_1 + x_2 5.62/2.42 POL(s(x_1)) = x_1 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (23) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (24) ERuleRemovalProof (EQUIVALENT) 5.62/2.42 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. 5.62/2.42 5.62/2.42 Strictly oriented dependency pairs: 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(x, y) 5.62/2.42 PLUS(x, s(y)) -> PLUS(x, y) 5.62/2.42 5.62/2.42 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(PLUS(x_1, x_2)) = x_1 + x_2 5.62/2.42 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 5.62/2.42 POL(s(x_1)) = 2 + x_1 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (25) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (26) EDPPoloProof (EQUIVALENT) 5.62/2.42 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.62/2.42 5.62/2.42 5.62/2.42 PLUS(plus(x, s(y)), ext) -> PLUS(s(plus(x, y)), ext) 5.62/2.42 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.62/2.42 5.62/2.42 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 We had to orient the following equations of E# equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(PLUS(x_1, x_2)) = x_1 + x_2 5.62/2.42 POL(plus(x_1, x_2)) = 1 + x_1 + x_2 5.62/2.42 POL(s(x_1)) = 0 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (27) 5.62/2.42 Obligation: 5.62/2.42 P is empty. 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (28) PisEmptyProof (EQUIVALENT) 5.62/2.42 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (29) 5.62/2.42 YES 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (30) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(plus(times(x, y), x), ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(x, y) 5.62/2.42 TIMES(x, s(y)) -> TIMES(x, y) 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (31) ESharpUsableEquationsProof (EQUIVALENT) 5.62/2.42 We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: 5.62/2.42 PLUS(x, y) == PLUS(y, x) 5.62/2.42 PLUS(plus(x, y), z) == PLUS(x, plus(y, z)) 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (32) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(plus(times(x, y), x), ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(x, y) 5.62/2.42 TIMES(x, s(y)) -> TIMES(x, y) 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (33) EDPPoloProof (EQUIVALENT) 5.62/2.42 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. 5.62/2.42 5.62/2.42 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(plus(times(x, y), x), ext) 5.62/2.42 TIMES(times(x, s(y)), ext) -> TIMES(x, y) 5.62/2.42 TIMES(x, s(y)) -> TIMES(x, y) 5.62/2.42 The remaining Dependency Pairs were at least non-strictly oriented. 5.62/2.42 5.62/2.42 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.62/2.42 5.62/2.42 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 plus(x, 0) -> x 5.62/2.42 We had to orient the following equations of E# equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(0) = 0 5.62/2.42 POL(TIMES(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.62/2.42 POL(plus(x_1, x_2)) = x_1 + x_2 5.62/2.42 POL(s(x_1)) = 1 + x_1 5.62/2.42 POL(times(x_1, x_2)) = x_1 + x_1*x_2 + x_2 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (34) 5.62/2.42 Obligation: 5.62/2.42 The TRS P consists of the following rules: 5.62/2.42 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (35) EDPPoloProof (EQUIVALENT) 5.62/2.42 We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. 5.62/2.42 5.62/2.42 5.62/2.42 TIMES(times(x, 0), ext) -> TIMES(0, ext) 5.62/2.42 With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 5.62/2.42 5.62/2.42 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 plus(x, 0) -> x 5.62/2.42 We had to orient the following equations of E# equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 With the implicit AFS we had to orient the following usable equations of E equivalently. 5.62/2.42 5.62/2.42 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 Used ordering: POLO with Polynomial interpretation [POLO]: 5.62/2.42 5.62/2.42 POL(0) = 2 5.62/2.42 POL(TIMES(x_1, x_2)) = 2*x_1 + x_1*x_2 + 2*x_2 5.62/2.42 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 5.62/2.42 POL(s(x_1)) = 2 + x_1 5.62/2.42 POL(times(x_1, x_2)) = 2 + 2*x_1 + x_1*x_2 + 2*x_2 5.62/2.42 5.62/2.42 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (36) 5.62/2.42 Obligation: 5.62/2.42 P is empty. 5.62/2.42 The TRS R consists of the following rules: 5.62/2.42 5.62/2.42 plus(x, 0) -> x 5.62/2.42 plus(x, s(y)) -> s(plus(x, y)) 5.62/2.42 times(x, 0) -> 0 5.62/2.42 times(x, s(y)) -> plus(times(x, y), x) 5.62/2.42 minus(x, 0) -> x 5.62/2.42 minus(s(x), s(y)) -> minus(x, y) 5.62/2.42 div(0, s(y)) -> 0 5.62/2.42 div(s(x), s(y)) -> s(div(minus(x, y), s(y))) 5.62/2.42 plus(plus(x, s(y)), ext) -> plus(s(plus(x, y)), ext) 5.62/2.42 times(times(x, 0), ext) -> times(0, ext) 5.62/2.42 times(times(x, s(y)), ext) -> times(plus(times(x, y), x), ext) 5.62/2.42 5.62/2.42 The set E consists of the following equations: 5.62/2.42 5.62/2.42 plus(x, y) == plus(y, x) 5.62/2.42 times(x, y) == times(y, x) 5.62/2.42 plus(plus(x, y), z) == plus(x, plus(y, z)) 5.62/2.42 times(times(x, y), z) == times(x, times(y, z)) 5.62/2.42 5.62/2.42 The set E# consists of the following equations: 5.62/2.42 5.62/2.42 TIMES(times(x, y), z) == TIMES(x, times(y, z)) 5.62/2.42 TIMES(x, y) == TIMES(y, x) 5.62/2.42 5.62/2.42 We have to consider all minimal (P,E#,R,E)-chains 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (37) PisEmptyProof (EQUIVALENT) 5.62/2.42 The TRS P is empty. Hence, there is no (P,E#,R,E) chain. 5.62/2.42 ---------------------------------------- 5.62/2.42 5.62/2.42 (38) 5.62/2.42 YES 5.83/2.46 EOF