YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could be proven: (0) ETRS (1) EquationalDependencyPairsProof [EQUIVALENT, 129 ms] (2) EDP (3) EDependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) EDP (6) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (7) EDP (8) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (9) EDP (10) PisEmptyProof [EQUIVALENT, 0 ms] (11) YES (12) EDP (13) ESharpUsableEquationsProof [EQUIVALENT, 13 ms] (14) EDP (15) EDPPoloProof [EQUIVALENT, 89 ms] (16) EDP (17) EDependencyGraphProof [EQUIVALENT, 0 ms] (18) EDP (19) EDPPoloProof [EQUIVALENT, 9 ms] (20) EDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES (23) EDP (24) ESharpUsableEquationsProof [EQUIVALENT, 6 ms] (25) EDP (26) EDPPoloProof [EQUIVALENT, 772 ms] (27) EDP (28) EDependencyGraphProof [EQUIVALENT, 0 ms] (29) EDP (30) EDPPoloProof [EQUIVALENT, 751 ms] (31) EDP (32) EDependencyGraphProof [EQUIVALENT, 0 ms] (33) EDP (34) EDPPoloProof [EQUIVALENT, 60 ms] (35) EDP (36) PisEmptyProof [EQUIVALENT, 0 ms] (37) YES (38) EDP (39) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (40) EDP (41) EUsableRulesReductionPairsProof [EQUIVALENT, 7 ms] (42) EDP (43) PisEmptyProof [EQUIVALENT, 0 ms] (44) YES (45) EDP (46) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (47) EDP (48) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (49) EDP (50) EDependencyGraphProof [EQUIVALENT, 0 ms] (51) TRUE ---------------------------------------- (0) Obligation: Equational rewrite system: The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) ---------------------------------------- (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: U111^1(tt, A, B) -> PLUS(sum(A), sum(B)) U111^1(tt, A, B) -> SUM(A) U111^1(tt, A, B) -> SUM(B) U21^1(tt, X, Y) -> 0^1(mult(X, Y)) U21^1(tt, X, Y) -> MULT(X, Y) U31^1(tt, X, Y) -> PLUS(0(mult(X, Y)), Y) U31^1(tt, X, Y) -> 0^1(mult(X, Y)) U31^1(tt, X, Y) -> MULT(X, Y) U51^1(tt, X, Y) -> 0^1(plus(X, Y)) U51^1(tt, X, Y) -> PLUS(X, Y) U61^1(tt, X, Y) -> PLUS(X, Y) U71^1(tt, X, Y) -> 0^1(plus(plus(X, Y), 1(z))) U71^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U71^1(tt, X, Y) -> PLUS(X, Y) U91^1(tt, A, B) -> MULT(prod(A), prod(B)) U91^1(tt, A, B) -> PROD(A) U91^1(tt, A, B) -> PROD(B) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> AND(isBag(V1), isBag(V2)) ISBAG(union(V1, V2)) -> ISBAG(V1) ISBAG(union(V1, V2)) -> ISBAG(V2) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> AND(isBin(V1), isBin(V2)) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> ISBIN(V2) ISBIN(plus(V1, V2)) -> AND(isBin(V1), isBin(V2)) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> ISBIN(V2) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(sum(V1)) -> ISBAG(V1) MULT(z, X) -> U11^1(isBin(X)) MULT(z, X) -> ISBIN(X) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> AND(isBin(X), isBin(Y)) MULT(0(X), Y) -> ISBIN(X) MULT(0(X), Y) -> ISBIN(Y) MULT(1(X), Y) -> U31^1(and(isBin(X), isBin(Y)), X, Y) MULT(1(X), Y) -> AND(isBin(X), isBin(Y)) MULT(1(X), Y) -> ISBIN(X) MULT(1(X), Y) -> ISBIN(Y) PLUS(z, X) -> U41^1(isBin(X), X) PLUS(z, X) -> ISBIN(X) PLUS(0(X), 0(Y)) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(0(X), 0(Y)) -> AND(isBin(X), isBin(Y)) PLUS(0(X), 0(Y)) -> ISBIN(X) PLUS(0(X), 0(Y)) -> ISBIN(Y) PLUS(0(X), 1(Y)) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(0(X), 1(Y)) -> AND(isBin(X), isBin(Y)) PLUS(0(X), 1(Y)) -> ISBIN(X) PLUS(0(X), 1(Y)) -> ISBIN(Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(1(X), 1(Y)) -> AND(isBin(X), isBin(Y)) PLUS(1(X), 1(Y)) -> ISBIN(X) PLUS(1(X), 1(Y)) -> ISBIN(Y) PROD(singl(X)) -> U81^1(isBin(X), X) PROD(singl(X)) -> ISBIN(X) PROD(union(A, B)) -> U91^1(and(isBag(A), isBag(B)), A, B) PROD(union(A, B)) -> AND(isBag(A), isBag(B)) PROD(union(A, B)) -> ISBAG(A) PROD(union(A, B)) -> ISBAG(B) SUM(empty) -> 0^1(z) SUM(singl(X)) -> U101^1(isBin(X), X) SUM(singl(X)) -> ISBIN(X) SUM(union(A, B)) -> U111^1(and(isBag(A), isBag(B)), A, B) SUM(union(A, B)) -> AND(isBag(A), isBag(B)) SUM(union(A, B)) -> ISBAG(A) SUM(union(A, B)) -> ISBAG(B) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) MULT(mult(z, X), ext) -> U11^1(isBin(X)) MULT(mult(z, X), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(mult(0(X), Y), ext) -> AND(isBin(X), isBin(Y)) MULT(mult(0(X), Y), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> ISBIN(Y) MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(1(X), Y), ext) -> U31^1(and(isBin(X), isBin(Y)), X, Y) MULT(mult(1(X), Y), ext) -> AND(isBin(X), isBin(Y)) MULT(mult(1(X), Y), ext) -> ISBIN(X) MULT(mult(1(X), Y), ext) -> ISBIN(Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(z, X), ext) -> U41^1(isBin(X), X) PLUS(plus(z, X), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U51(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U61(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(0(X), 1(Y)), ext) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(Y) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (2) Obligation: The TRS P consists of the following rules: U111^1(tt, A, B) -> PLUS(sum(A), sum(B)) U111^1(tt, A, B) -> SUM(A) U111^1(tt, A, B) -> SUM(B) U21^1(tt, X, Y) -> 0^1(mult(X, Y)) U21^1(tt, X, Y) -> MULT(X, Y) U31^1(tt, X, Y) -> PLUS(0(mult(X, Y)), Y) U31^1(tt, X, Y) -> 0^1(mult(X, Y)) U31^1(tt, X, Y) -> MULT(X, Y) U51^1(tt, X, Y) -> 0^1(plus(X, Y)) U51^1(tt, X, Y) -> PLUS(X, Y) U61^1(tt, X, Y) -> PLUS(X, Y) U71^1(tt, X, Y) -> 0^1(plus(plus(X, Y), 1(z))) U71^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U71^1(tt, X, Y) -> PLUS(X, Y) U91^1(tt, A, B) -> MULT(prod(A), prod(B)) U91^1(tt, A, B) -> PROD(A) U91^1(tt, A, B) -> PROD(B) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> AND(isBag(V1), isBag(V2)) ISBAG(union(V1, V2)) -> ISBAG(V1) ISBAG(union(V1, V2)) -> ISBAG(V2) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> AND(isBin(V1), isBin(V2)) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> ISBIN(V2) ISBIN(plus(V1, V2)) -> AND(isBin(V1), isBin(V2)) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> ISBIN(V2) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(sum(V1)) -> ISBAG(V1) MULT(z, X) -> U11^1(isBin(X)) MULT(z, X) -> ISBIN(X) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> AND(isBin(X), isBin(Y)) MULT(0(X), Y) -> ISBIN(X) MULT(0(X), Y) -> ISBIN(Y) MULT(1(X), Y) -> U31^1(and(isBin(X), isBin(Y)), X, Y) MULT(1(X), Y) -> AND(isBin(X), isBin(Y)) MULT(1(X), Y) -> ISBIN(X) MULT(1(X), Y) -> ISBIN(Y) PLUS(z, X) -> U41^1(isBin(X), X) PLUS(z, X) -> ISBIN(X) PLUS(0(X), 0(Y)) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(0(X), 0(Y)) -> AND(isBin(X), isBin(Y)) PLUS(0(X), 0(Y)) -> ISBIN(X) PLUS(0(X), 0(Y)) -> ISBIN(Y) PLUS(0(X), 1(Y)) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(0(X), 1(Y)) -> AND(isBin(X), isBin(Y)) PLUS(0(X), 1(Y)) -> ISBIN(X) PLUS(0(X), 1(Y)) -> ISBIN(Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(1(X), 1(Y)) -> AND(isBin(X), isBin(Y)) PLUS(1(X), 1(Y)) -> ISBIN(X) PLUS(1(X), 1(Y)) -> ISBIN(Y) PROD(singl(X)) -> U81^1(isBin(X), X) PROD(singl(X)) -> ISBIN(X) PROD(union(A, B)) -> U91^1(and(isBag(A), isBag(B)), A, B) PROD(union(A, B)) -> AND(isBag(A), isBag(B)) PROD(union(A, B)) -> ISBAG(A) PROD(union(A, B)) -> ISBAG(B) SUM(empty) -> 0^1(z) SUM(singl(X)) -> U101^1(isBin(X), X) SUM(singl(X)) -> ISBIN(X) SUM(union(A, B)) -> U111^1(and(isBag(A), isBag(B)), A, B) SUM(union(A, B)) -> AND(isBag(A), isBag(B)) SUM(union(A, B)) -> ISBAG(A) SUM(union(A, B)) -> ISBAG(B) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) MULT(mult(z, X), ext) -> U11^1(isBin(X)) MULT(mult(z, X), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(mult(0(X), Y), ext) -> AND(isBin(X), isBin(Y)) MULT(mult(0(X), Y), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> ISBIN(Y) MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(1(X), Y), ext) -> U31^1(and(isBin(X), isBin(Y)), X, Y) MULT(mult(1(X), Y), ext) -> AND(isBin(X), isBin(Y)) MULT(mult(1(X), Y), ext) -> ISBIN(X) MULT(mult(1(X), Y), ext) -> ISBIN(Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(z, X), ext) -> U41^1(isBin(X), X) PLUS(plus(z, X), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U51(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U61(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(0(X), 1(Y)), ext) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(Y) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> AND(isBin(X), isBin(Y)) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (3) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 5 SCCs with 59 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: The TRS P consists of the following rules: ISBIN(1(V1)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBIN(plus(V1, V2)) -> ISBIN(V2) ISBAG(union(V1, V2)) -> ISBAG(V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> ISBIN(V2) ISBIN(prod(V1)) -> ISBAG(V1) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> ISBAG(V1) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) 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]: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) ---------------------------------------- (7) Obligation: The TRS P consists of the following rules: ISBIN(1(V1)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBIN(plus(V1, V2)) -> ISBIN(V2) ISBAG(union(V1, V2)) -> ISBAG(V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> ISBIN(V2) ISBIN(prod(V1)) -> ISBAG(V1) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> ISBAG(V1) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (8) EUsableRulesReductionPairsProof (EQUIVALENT) 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. The following dependency pairs can be deleted: ISBIN(1(V1)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBIN(plus(V1, V2)) -> ISBIN(V2) ISBAG(union(V1, V2)) -> ISBAG(V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> ISBIN(V2) ISBIN(prod(V1)) -> ISBAG(V1) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> ISBAG(V1) The following rules are removed from R: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The following equations are removed from E: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 3*x_1 POL(1(x_1)) = 3*x_1 POL(ISBAG(x_1)) = 2*x_1 POL(ISBIN(x_1)) = x_1 POL(mult(x_1, x_2)) = 3*x_1 + 3*x_2 POL(plus(x_1, x_2)) = 3*x_1 + 3*x_2 POL(prod(x_1)) = 3*x_1 POL(singl(x_1)) = 3*x_1 POL(sum(x_1)) = 3*x_1 POL(union(x_1, x_2)) = 3*x_1 + 3*x_2 ---------------------------------------- (9) Obligation: P is empty. R is empty. E is empty. E# is empty. 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: PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U61(and(isBin(X), isBin(Y)), X, Y), ext) U61^1(tt, X, Y) -> PLUS(X, Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) U71^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 1(Y)) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U51(and(isBin(X), isBin(Y)), X, Y), ext) U71^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) PLUS(plus(0(X), 1(Y)), ext) -> U61^1(and(isBin(X), isBin(Y)), X, Y) U51^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 0(Y)) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U51^1(and(isBin(X), isBin(Y)), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) 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]: MULT(x, y) == MULT(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) ---------------------------------------- (14) Obligation: The TRS P consists of the following rules: PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U61(and(isBin(X), isBin(Y)), X, Y), ext) U61^1(tt, X, Y) -> PLUS(X, Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) U71^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 1(Y)) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U51(and(isBin(X), isBin(Y)), X, Y), ext) U71^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) PLUS(plus(0(X), 1(Y)), ext) -> U61^1(and(isBin(X), isBin(Y)), X, Y) U51^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 0(Y)) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U51^1(and(isBin(X), isBin(Y)), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (15) EDPPoloProof (EQUIVALENT) 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. PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U61(and(isBin(X), isBin(Y)), X, Y), ext) U61^1(tt, X, Y) -> PLUS(X, Y) U71^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 1(Y)) -> U61^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U51(and(isBin(X), isBin(Y)), X, Y), ext) U71^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) PLUS(plus(0(X), 1(Y)), ext) -> U61^1(and(isBin(X), isBin(Y)), X, Y) U51^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 0(Y)) -> U51^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U51^1(and(isBin(X), isBin(Y)), X, Y) The remaining Dependency Pairs were at least non-strictly oriented. PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) U61(tt, X, Y) -> 1(plus(X, Y)) 0(z) -> z U41(tt, X) -> X U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U51(tt, X, Y) -> 0(plus(X, Y)) We had to orient the following equations of E# equivalently. PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. plus(plus(x, y), z') == plus(x, plus(y, z')) plus(x, y) == plus(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(PLUS(x_1, x_2)) = x_1 + x_2 POL(U41(x_1, x_2)) = x_2 POL(U51(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U51^1(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U61(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U61^1(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U71(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U71^1(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(and(x_1, x_2)) = 0 POL(empty) = 0 POL(isBag(x_1)) = 0 POL(isBin(x_1)) = 0 POL(mult(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 3*x_1 POL(singl(x_1)) = 0 POL(sum(x_1)) = 0 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (16) Obligation: The TRS P consists of the following rules: PLUS(plus(1(X), 1(Y)), ext) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(1(X), 1(Y)) -> U71^1(and(isBin(X), isBin(Y)), X, Y) PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (17) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 2 less nodes. ---------------------------------------- (18) Obligation: The TRS P consists of the following rules: PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (19) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. PLUS(plus(z, X), ext) -> PLUS(U41(isBin(X), X), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U71(and(isBin(X), isBin(Y)), X, Y), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U61(tt, X, Y) -> 1(plus(X, Y)) U41(tt, X) -> X 0(z) -> z plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) U51(tt, X, Y) -> 0(plus(X, Y)) We had to orient the following equations of E# equivalently. PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) With the implicit AFS we had to orient the following usable equations of E equivalently. plus(plus(x, y), z') == plus(x, plus(y, z')) plus(x, y) == plus(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 1 POL(PLUS(x_1, x_2)) = x_1 + x_2 POL(U41(x_1, x_2)) = 1 + x_2 POL(U51(x_1, x_2, x_3)) = 0 POL(U61(x_1, x_2, x_3)) = 1 POL(U71(x_1, x_2, x_3)) = 0 POL(and(x_1, x_2)) = 0 POL(empty) = 0 POL(isBag(x_1)) = 0 POL(isBin(x_1)) = 0 POL(mult(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = 2 + x_1 + x_2 POL(prod(x_1)) = 0 POL(singl(x_1)) = 0 POL(sum(x_1)) = 3*x_1 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (20) Obligation: P is empty. The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) PLUS(x, y) == PLUS(y, x) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) U31^1(tt, X, Y) -> MULT(X, Y) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) MULT(mult(1(X), Y), ext) -> U31^1(and(isBin(X), isBin(Y)), X, Y) U21^1(tt, X, Y) -> MULT(X, Y) MULT(1(X), Y) -> U31^1(and(isBin(X), isBin(Y)), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (24) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) ---------------------------------------- (25) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) U31^1(tt, X, Y) -> MULT(X, Y) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) MULT(mult(1(X), Y), ext) -> U31^1(and(isBin(X), isBin(Y)), X, Y) U21^1(tt, X, Y) -> MULT(X, Y) MULT(1(X), Y) -> U31^1(and(isBin(X), isBin(Y)), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (26) EDPPoloProof (EQUIVALENT) 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. MULT(mult(1(X), Y), ext) -> U31^1(and(isBin(X), isBin(Y)), X, Y) MULT(1(X), Y) -> U31^1(and(isBin(X), isBin(Y)), X, Y) The remaining Dependency Pairs were at least non-strictly oriented. MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) U31^1(tt, X, Y) -> MULT(X, Y) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) U21^1(tt, X, Y) -> MULT(X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. isBag(singl(V1)) -> isBin(V1) isBag(empty) -> tt isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) and(tt, X) -> X mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) mult(z, X) -> U11(isBin(X)) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) 0(z) -> z U51(tt, X, Y) -> 0(plus(X, Y)) U41(tt, X) -> X isBin(sum(V1)) -> isBag(V1) isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(prod(V1)) -> isBag(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(z) -> tt isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) U11(tt) -> z U61(tt, X, Y) -> 1(plus(X, Y)) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) We had to orient the following equations of E# equivalently. MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) With the implicit AFS we had to orient the following usable equations of E equivalently. mult(x, y) == mult(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) plus(x, y) == plus(y, x) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = x_1 POL(1(x_1)) = 1 + x_1 POL(MULT(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(U11(x_1)) = 0 POL(U21(x_1, x_2, x_3)) = x_2 + x_2*x_3 + x_3 POL(U21^1(x_1, x_2, x_3)) = x_2 + x_2*x_3 + x_3 POL(U31(x_1, x_2, x_3)) = x_1 + x_1*x_2 + x_1*x_2*x_3 + x_1*x_3 + x_3 POL(U31^1(x_1, x_2, x_3)) = x_2 + x_2*x_3 + x_3 POL(U41(x_1, x_2)) = x_2 POL(U51(x_1, x_2, x_3)) = x_2 + x_3 POL(U61(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U71(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(and(x_1, x_2)) = x_1*x_2 POL(empty) = 1 POL(isBag(x_1)) = 1 POL(isBin(x_1)) = 1 POL(mult(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 1 + x_1 + x_1^2 POL(singl(x_1)) = 1 + x_1 + x_1^2 POL(sum(x_1)) = 1 + x_1 + x_1^2 POL(tt) = 1 POL(union(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(z) = 0 ---------------------------------------- (27) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) U31^1(tt, X, Y) -> MULT(X, Y) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) U21^1(tt, X, Y) -> MULT(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (28) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 1 less node. ---------------------------------------- (29) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) U21^1(tt, X, Y) -> MULT(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (30) EDPPoloProof (EQUIVALENT) 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. MULT(mult(0(X), Y), ext) -> U21^1(and(isBin(X), isBin(Y)), X, Y) MULT(0(X), Y) -> U21^1(and(isBin(X), isBin(Y)), X, Y) The remaining Dependency Pairs were at least non-strictly oriented. MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) U21^1(tt, X, Y) -> MULT(X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. 0(z) -> z U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) isBin(sum(V1)) -> isBag(V1) isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(prod(V1)) -> isBag(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(z) -> tt isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) U51(tt, X, Y) -> 0(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) isBag(singl(V1)) -> isBin(V1) isBag(empty) -> tt isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) U61(tt, X, Y) -> 1(plus(X, Y)) U21(tt, X, Y) -> 0(mult(X, Y)) U41(tt, X) -> X U11(tt) -> z and(tt, X) -> X mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) mult(z, X) -> U11(isBin(X)) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) We had to orient the following equations of E# equivalently. MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) With the implicit AFS we had to orient the following usable equations of E equivalently. plus(plus(x, y), z') == plus(x, plus(y, z')) plus(x, y) == plus(y, x) mult(x, y) == mult(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(MULT(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(U11(x_1)) = 0 POL(U21(x_1, x_2, x_3)) = x_1 + x_1*x_2*x_3 + x_2 + x_3 POL(U21^1(x_1, x_2, x_3)) = x_1*x_2*x_3 + x_2 + x_3 POL(U31(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_2 + x_2*x_3 + x_3 POL(U41(x_1, x_2)) = x_1*x_2 POL(U51(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U61(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U71(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(and(x_1, x_2)) = x_1*x_2 POL(empty) = 1 POL(isBag(x_1)) = 1 POL(isBin(x_1)) = 1 POL(mult(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 1 + x_1 + x_1^2 POL(singl(x_1)) = 1 + x_1 + x_1^2 POL(sum(x_1)) = 1 + x_1 + x_1^2 POL(tt) = 1 POL(union(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(z) = 0 ---------------------------------------- (31) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) U21^1(tt, X, Y) -> MULT(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (32) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 1 less node. ---------------------------------------- (33) Obligation: The TRS P consists of the following rules: MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (34) EDPPoloProof (EQUIVALENT) We use the reduction pair processor [DA_STEIN] with a polynomial ordering [POLO]. All Dependency Pairs of this DP problem can be strictly oriented. MULT(mult(1(X), Y), ext) -> MULT(U31(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(0(X), Y), ext) -> MULT(U21(and(isBin(X), isBin(Y)), X, Y), ext) MULT(mult(z, X), ext) -> MULT(U11(isBin(X)), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) mult(z, X) -> U11(isBin(X)) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) U41(tt, X) -> X U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U51(tt, X, Y) -> 0(plus(X, Y)) U11(tt) -> z U21(tt, X, Y) -> 0(mult(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) 0(z) -> z We had to orient the following equations of E# equivalently. MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) With the implicit AFS we had to orient the following usable equations of E equivalently. plus(plus(x, y), z') == plus(x, plus(y, z')) plus(x, y) == plus(y, x) mult(x, y) == mult(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 0 POL(1(x_1)) = 3 POL(MULT(x_1, x_2)) = x_1 + x_2 POL(U11(x_1)) = 0 POL(U21(x_1, x_2, x_3)) = 0 POL(U31(x_1, x_2, x_3)) = x_3 POL(U41(x_1, x_2)) = x_2 POL(U51(x_1, x_2, x_3)) = 0 POL(U61(x_1, x_2, x_3)) = 3 POL(U71(x_1, x_2, x_3)) = 2 POL(and(x_1, x_2)) = 0 POL(empty) = 0 POL(isBag(x_1)) = 0 POL(isBin(x_1)) = 0 POL(mult(x_1, x_2)) = 3 + x_1 + x_2 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 3*x_1 POL(singl(x_1)) = 0 POL(sum(x_1)) = 0 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (35) Obligation: P is empty. The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (36) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (37) YES ---------------------------------------- (38) Obligation: The TRS P consists of the following rules: U91^1(tt, A, B) -> PROD(A) PROD(union(A, B)) -> U91^1(and(isBag(A), isBag(B)), A, B) U91^1(tt, A, B) -> PROD(B) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (39) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) ---------------------------------------- (40) Obligation: The TRS P consists of the following rules: U91^1(tt, A, B) -> PROD(A) PROD(union(A, B)) -> U91^1(and(isBag(A), isBag(B)), A, B) U91^1(tt, A, B) -> PROD(B) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (41) EUsableRulesReductionPairsProof (EQUIVALENT) 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. The following dependency pairs can be deleted: U91^1(tt, A, B) -> PROD(A) PROD(union(A, B)) -> U91^1(and(isBag(A), isBag(B)), A, B) U91^1(tt, A, B) -> PROD(B) The following rules are removed from R: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The following equations are removed from E: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 3*x_1 POL(1(x_1)) = x_1 POL(PROD(x_1)) = 2 + 2*x_1 POL(U91^1(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(and(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(empty) = 2 POL(isBag(x_1)) = x_1 POL(isBin(x_1)) = 2*x_1 POL(mult(x_1, x_2)) = 2 + 3*x_1 + 2*x_2 POL(plus(x_1, x_2)) = 2 + 3*x_1 + 2*x_2 POL(prod(x_1)) = 3*x_1 POL(singl(x_1)) = 3*x_1 POL(sum(x_1)) = 3*x_1 POL(tt) = 2 POL(union(x_1, x_2)) = 3 + 3*x_1 + 3*x_2 POL(z) = 2 ---------------------------------------- (42) Obligation: P is empty. R is empty. E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (43) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: The TRS P consists of the following rules: SUM(union(A, B)) -> U111^1(and(isBag(A), isBag(B)), A, B) U111^1(tt, A, B) -> SUM(B) U111^1(tt, A, B) -> SUM(A) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) The set E# consists of the following equations: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (46) ESharpUsableEquationsProof (EQUIVALENT) We can delete the following equations of E# with the esharp usable equations processor[DA_STEIN]: MULT(x, y) == MULT(y, x) PLUS(x, y) == PLUS(y, x) UNION(x, y) == UNION(y, x) MULT(mult(x, y), z') == MULT(x, mult(y, z')) PLUS(plus(x, y), z') == PLUS(x, plus(y, z')) UNION(union(x, y), z') == UNION(x, union(y, z')) ---------------------------------------- (47) Obligation: The TRS P consists of the following rules: SUM(union(A, B)) -> U111^1(and(isBag(A), isBag(B)), A, B) U111^1(tt, A, B) -> SUM(B) U111^1(tt, A, B) -> SUM(A) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) and(tt, X) -> X isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The set E consists of the following equations: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (48) EUsableRulesReductionPairsProof (EQUIVALENT) 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. The following dependency pairs can be deleted: SUM(union(A, B)) -> U111^1(and(isBag(A), isBag(B)), A, B) The following rules are removed from R: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X) -> X U11(tt) -> z U111(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, X, Y) -> 0(mult(X, Y)) U31(tt, X, Y) -> plus(0(mult(X, Y)), Y) U41(tt, X) -> X U51(tt, X, Y) -> 0(plus(X, Y)) U61(tt, X, Y) -> 1(plus(X, Y)) U71(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U81(tt, X) -> X U91(tt, A, B) -> mult(prod(A), prod(B)) isBag(empty) -> tt isBag(singl(V1)) -> isBin(V1) isBag(union(V1, V2)) -> and(isBag(V1), isBag(V2)) isBin(z) -> tt isBin(0(V1)) -> isBin(V1) isBin(1(V1)) -> isBin(V1) isBin(mult(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(plus(V1, V2)) -> and(isBin(V1), isBin(V2)) isBin(prod(V1)) -> isBag(V1) isBin(sum(V1)) -> isBag(V1) mult(z, X) -> U11(isBin(X)) mult(0(X), Y) -> U21(and(isBin(X), isBin(Y)), X, Y) mult(1(X), Y) -> U31(and(isBin(X), isBin(Y)), X, Y) plus(z, X) -> U41(isBin(X), X) plus(0(X), 0(Y)) -> U51(and(isBin(X), isBin(Y)), X, Y) plus(0(X), 1(Y)) -> U61(and(isBin(X), isBin(Y)), X, Y) plus(1(X), 1(Y)) -> U71(and(isBin(X), isBin(Y)), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U81(isBin(X), X) prod(union(A, B)) -> U91(and(isBag(A), isBag(B)), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U101(isBin(X), X) sum(union(A, B)) -> U111(and(isBag(A), isBag(B)), A, B) mult(mult(z, X), ext) -> mult(U11(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U21(and(isBin(X), isBin(Y)), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U31(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(z, X), ext) -> plus(U41(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U51(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U61(and(isBin(X), isBin(Y)), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U71(and(isBin(X), isBin(Y)), X, Y), ext) The following equations are removed from E: mult(x, y) == mult(y, x) plus(x, y) == plus(y, x) union(x, y) == union(y, x) mult(mult(x, y), z') == mult(x, mult(y, z')) plus(plus(x, y), z') == plus(x, plus(y, z')) union(union(x, y), z') == union(x, union(y, z')) Used ordering: POLO with Polynomial interpretation [POLO]: POL(0(x_1)) = 3*x_1 POL(1(x_1)) = 3*x_1 POL(SUM(x_1)) = 2*x_1 POL(U111^1(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3 POL(and(x_1, x_2)) = 2*x_1 + x_2 POL(empty) = 0 POL(isBag(x_1)) = 2 + 2*x_1 POL(isBin(x_1)) = 2*x_1 POL(mult(x_1, x_2)) = 3 + 3*x_1 + 2*x_2 POL(plus(x_1, x_2)) = 2 + 3*x_1 + 2*x_2 POL(prod(x_1)) = 1 + 3*x_1 POL(singl(x_1)) = 3 + 3*x_1 POL(sum(x_1)) = 2 + 3*x_1 POL(tt) = 0 POL(union(x_1, x_2)) = 3 + 3*x_1 + 3*x_2 POL(z) = 0 ---------------------------------------- (49) Obligation: The TRS P consists of the following rules: U111^1(tt, A, B) -> SUM(B) U111^1(tt, A, B) -> SUM(A) The TRS R consists of the following rules: and(tt, X) -> X E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (50) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 2 less nodes. ---------------------------------------- (51) TRUE