YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Termination of the given ETRS could be proven: (0) ETRS (1) EquationalDependencyPairsProof [EQUIVALENT, 147 ms] (2) EDP (3) EDependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) EDP (6) ESharpUsableEquationsProof [EQUIVALENT, 5 ms] (7) EDP (8) EUsableRulesReductionPairsProof [EQUIVALENT, 30 ms] (9) EDP (10) EDependencyGraphProof [EQUIVALENT, 0 ms] (11) TRUE (12) EDP (13) ESharpUsableEquationsProof [EQUIVALENT, 12 ms] (14) EDP (15) EDPPoloProof [EQUIVALENT, 93 ms] (16) EDP (17) EDependencyGraphProof [EQUIVALENT, 0 ms] (18) EDP (19) EDPPoloProof [EQUIVALENT, 81 ms] (20) EDP (21) EDPPoloProof [EQUIVALENT, 84 ms] (22) EDP (23) EDependencyGraphProof [EQUIVALENT, 0 ms] (24) TRUE (25) EDP (26) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (27) EDP (28) EUsableRulesReductionPairsProof [EQUIVALENT, 13 ms] (29) EDP (30) PisEmptyProof [EQUIVALENT, 0 ms] (31) YES (32) EDP (33) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (34) EDP (35) EDPPoloProof [EQUIVALENT, 263 ms] (36) EDP (37) EDependencyGraphProof [EQUIVALENT, 0 ms] (38) EDP (39) EDPPoloProof [EQUIVALENT, 240 ms] (40) EDP (41) EDependencyGraphProof [EQUIVALENT, 0 ms] (42) EDP (43) EDPPoloProof [EQUIVALENT, 78 ms] (44) EDP (45) PisEmptyProof [EQUIVALENT, 0 ms] (46) YES (47) EDP (48) ESharpUsableEquationsProof [EQUIVALENT, 0 ms] (49) EDP (50) EUsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (51) EDP (52) EDependencyGraphProof [EQUIVALENT, 0 ms] (53) 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), 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: U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) U101^1(tt, X, Y) -> ISBIN(Y) U102^1(tt, X, Y) -> 0^1(mult(X, Y)) U102^1(tt, X, Y) -> MULT(X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) U111^1(tt, X, Y) -> ISBIN(Y) U112^1(tt, X, Y) -> PLUS(0(mult(X, Y)), Y) U112^1(tt, X, Y) -> 0^1(mult(X, Y)) U112^1(tt, X, Y) -> MULT(X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) U131^1(tt, X, Y) -> ISBIN(Y) U132^1(tt, X, Y) -> 0^1(plus(X, Y)) U132^1(tt, X, Y) -> PLUS(X, Y) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) U141^1(tt, X, Y) -> ISBIN(Y) U142^1(tt, X, Y) -> PLUS(X, Y) U151^1(tt, X, Y) -> U152^1(isBin(Y), X, Y) U151^1(tt, X, Y) -> ISBIN(Y) U152^1(tt, X, Y) -> 0^1(plus(plus(X, Y), 1(z))) U152^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U152^1(tt, X, Y) -> PLUS(X, Y) U171^1(tt, A, B) -> U172^1(isBag(B), A, B) U171^1(tt, A, B) -> ISBAG(B) U172^1(tt, A, B) -> MULT(prod(A), prod(B)) U172^1(tt, A, B) -> PROD(A) U172^1(tt, A, B) -> PROD(B) U191^1(tt, A, B) -> U192^1(isBag(B), A, B) U191^1(tt, A, B) -> ISBAG(B) U192^1(tt, A, B) -> PLUS(sum(A), sum(B)) U192^1(tt, A, B) -> SUM(A) U192^1(tt, A, B) -> SUM(B) U21^1(tt, V2) -> U22^1(isBag(V2)) U21^1(tt, V2) -> ISBAG(V2) U51^1(tt, V2) -> U52^1(isBin(V2)) U51^1(tt, V2) -> ISBIN(V2) U61^1(tt, V2) -> U62^1(isBin(V2)) U61^1(tt, V2) -> ISBIN(V2) ISBAG(singl(V1)) -> U11^1(isBin(V1)) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> U21^1(isBag(V1), V2) ISBAG(union(V1, V2)) -> ISBAG(V1) ISBIN(0(V1)) -> U31^1(isBin(V1)) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(1(V1)) -> U41^1(isBin(V1)) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> U51^1(isBin(V1), V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> U61^1(isBin(V1), V2) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(prod(V1)) -> U71^1(isBag(V1)) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(sum(V1)) -> U81^1(isBag(V1)) ISBIN(sum(V1)) -> ISBAG(V1) MULT(z, X) -> U91^1(isBin(X)) MULT(z, X) -> ISBIN(X) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) MULT(0(X), Y) -> ISBIN(X) MULT(1(X), Y) -> U111^1(isBin(X), X, Y) MULT(1(X), Y) -> ISBIN(X) PLUS(z, X) -> U121^1(isBin(X), X) PLUS(z, X) -> ISBIN(X) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> ISBIN(X) PLUS(0(X), 1(Y)) -> U141^1(isBin(X), X, Y) PLUS(0(X), 1(Y)) -> ISBIN(X) PLUS(1(X), 1(Y)) -> U151^1(isBin(X), X, Y) PLUS(1(X), 1(Y)) -> ISBIN(X) PROD(singl(X)) -> U161^1(isBin(X), X) PROD(singl(X)) -> ISBIN(X) PROD(union(A, B)) -> U171^1(isBag(A), A, B) PROD(union(A, B)) -> ISBAG(A) SUM(empty) -> 0^1(z) SUM(singl(X)) -> U181^1(isBin(X), X) SUM(singl(X)) -> ISBIN(X) SUM(union(A, B)) -> U191^1(isBag(A), A, B) SUM(union(A, B)) -> ISBAG(A) MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) MULT(mult(z, X), ext) -> U91^1(isBin(X)) MULT(mult(z, X), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) MULT(mult(0(X), Y), ext) -> ISBIN(X) MULT(mult(1(X), Y), ext) -> MULT(U111(isBin(X), X, Y), ext) MULT(mult(1(X), Y), ext) -> U111^1(isBin(X), X, Y) MULT(mult(1(X), Y), ext) -> ISBIN(X) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) PLUS(plus(z, X), ext) -> U121^1(isBin(X), X) PLUS(plus(z, X), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 1(Y)), ext) -> U141^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U151(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U151^1(isBin(X), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(X) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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: U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) U101^1(tt, X, Y) -> ISBIN(Y) U102^1(tt, X, Y) -> 0^1(mult(X, Y)) U102^1(tt, X, Y) -> MULT(X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) U111^1(tt, X, Y) -> ISBIN(Y) U112^1(tt, X, Y) -> PLUS(0(mult(X, Y)), Y) U112^1(tt, X, Y) -> 0^1(mult(X, Y)) U112^1(tt, X, Y) -> MULT(X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) U131^1(tt, X, Y) -> ISBIN(Y) U132^1(tt, X, Y) -> 0^1(plus(X, Y)) U132^1(tt, X, Y) -> PLUS(X, Y) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) U141^1(tt, X, Y) -> ISBIN(Y) U142^1(tt, X, Y) -> PLUS(X, Y) U151^1(tt, X, Y) -> U152^1(isBin(Y), X, Y) U151^1(tt, X, Y) -> ISBIN(Y) U152^1(tt, X, Y) -> 0^1(plus(plus(X, Y), 1(z))) U152^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U152^1(tt, X, Y) -> PLUS(X, Y) U171^1(tt, A, B) -> U172^1(isBag(B), A, B) U171^1(tt, A, B) -> ISBAG(B) U172^1(tt, A, B) -> MULT(prod(A), prod(B)) U172^1(tt, A, B) -> PROD(A) U172^1(tt, A, B) -> PROD(B) U191^1(tt, A, B) -> U192^1(isBag(B), A, B) U191^1(tt, A, B) -> ISBAG(B) U192^1(tt, A, B) -> PLUS(sum(A), sum(B)) U192^1(tt, A, B) -> SUM(A) U192^1(tt, A, B) -> SUM(B) U21^1(tt, V2) -> U22^1(isBag(V2)) U21^1(tt, V2) -> ISBAG(V2) U51^1(tt, V2) -> U52^1(isBin(V2)) U51^1(tt, V2) -> ISBIN(V2) U61^1(tt, V2) -> U62^1(isBin(V2)) U61^1(tt, V2) -> ISBIN(V2) ISBAG(singl(V1)) -> U11^1(isBin(V1)) ISBAG(singl(V1)) -> ISBIN(V1) ISBAG(union(V1, V2)) -> U21^1(isBag(V1), V2) ISBAG(union(V1, V2)) -> ISBAG(V1) ISBIN(0(V1)) -> U31^1(isBin(V1)) ISBIN(0(V1)) -> ISBIN(V1) ISBIN(1(V1)) -> U41^1(isBin(V1)) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> U51^1(isBin(V1), V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(plus(V1, V2)) -> U61^1(isBin(V1), V2) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(prod(V1)) -> U71^1(isBag(V1)) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(sum(V1)) -> U81^1(isBag(V1)) ISBIN(sum(V1)) -> ISBAG(V1) MULT(z, X) -> U91^1(isBin(X)) MULT(z, X) -> ISBIN(X) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) MULT(0(X), Y) -> ISBIN(X) MULT(1(X), Y) -> U111^1(isBin(X), X, Y) MULT(1(X), Y) -> ISBIN(X) PLUS(z, X) -> U121^1(isBin(X), X) PLUS(z, X) -> ISBIN(X) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> ISBIN(X) PLUS(0(X), 1(Y)) -> U141^1(isBin(X), X, Y) PLUS(0(X), 1(Y)) -> ISBIN(X) PLUS(1(X), 1(Y)) -> U151^1(isBin(X), X, Y) PLUS(1(X), 1(Y)) -> ISBIN(X) PROD(singl(X)) -> U161^1(isBin(X), X) PROD(singl(X)) -> ISBIN(X) PROD(union(A, B)) -> U171^1(isBag(A), A, B) PROD(union(A, B)) -> ISBAG(A) SUM(empty) -> 0^1(z) SUM(singl(X)) -> U181^1(isBin(X), X) SUM(singl(X)) -> ISBIN(X) SUM(union(A, B)) -> U191^1(isBag(A), A, B) SUM(union(A, B)) -> ISBAG(A) MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) MULT(mult(z, X), ext) -> U91^1(isBin(X)) MULT(mult(z, X), ext) -> ISBIN(X) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) MULT(mult(0(X), Y), ext) -> ISBIN(X) MULT(mult(1(X), Y), ext) -> MULT(U111(isBin(X), X, Y), ext) MULT(mult(1(X), Y), ext) -> U111^1(isBin(X), X, Y) MULT(mult(1(X), Y), ext) -> ISBIN(X) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) PLUS(plus(z, X), ext) -> U121^1(isBin(X), X) PLUS(plus(z, X), ext) -> ISBIN(X) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> ISBIN(X) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 1(Y)), ext) -> U141^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> ISBIN(X) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U151(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U151^1(isBin(X), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> ISBIN(X) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 47 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: The TRS P consists of the following rules: U51^1(tt, V2) -> ISBIN(V2) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) U61^1(tt, V2) -> ISBIN(V2) U21^1(tt, V2) -> ISBAG(V2) ISBAG(singl(V1)) -> ISBIN(V1) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBAG(union(V1, V2)) -> U21^1(isBag(V1), V2) ISBIN(plus(V1, V2)) -> U61^1(isBin(V1), V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> U51^1(isBin(V1), V2) 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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: U51^1(tt, V2) -> ISBIN(V2) ISBIN(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) U61^1(tt, V2) -> ISBIN(V2) U21^1(tt, V2) -> ISBAG(V2) ISBAG(singl(V1)) -> ISBIN(V1) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBAG(union(V1, V2)) -> U21^1(isBag(V1), V2) ISBIN(plus(V1, V2)) -> U61^1(isBin(V1), V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> U51^1(isBin(V1), V2) 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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(plus(V1, V2)) -> ISBIN(V1) ISBIN(0(V1)) -> ISBIN(V1) ISBAG(singl(V1)) -> ISBIN(V1) ISBIN(prod(V1)) -> ISBAG(V1) ISBIN(1(V1)) -> ISBIN(V1) ISBIN(sum(V1)) -> ISBAG(V1) ISBAG(union(V1, V2)) -> U21^1(isBag(V1), V2) ISBIN(plus(V1, V2)) -> U61^1(isBin(V1), V2) ISBIN(mult(V1, V2)) -> ISBIN(V1) ISBIN(mult(V1, V2)) -> U51^1(isBin(V1), V2) 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 + 3*x_1 POL(1(x_1)) = 3 + x_1 POL(ISBAG(x_1)) = 3*x_1 POL(ISBIN(x_1)) = 2*x_1 POL(U11(x_1)) = 2*x_1 POL(U21(x_1, x_2)) = x_1 + 3*x_2 POL(U21^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1)) = x_1 POL(U51(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U51^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U61^1(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U62(x_1)) = x_1 POL(U71(x_1)) = x_1 POL(U81(x_1)) = x_1 POL(empty) = 0 POL(isBag(x_1)) = 2*x_1 POL(isBin(x_1)) = 2*x_1 POL(mult(x_1, x_2)) = 3*x_1 + 3*x_2 POL(plus(x_1, x_2)) = 3 + 3*x_1 + 3*x_2 POL(prod(x_1)) = 2*x_1 POL(singl(x_1)) = 3*x_1 POL(sum(x_1)) = 2*x_1 POL(tt) = 0 POL(union(x_1, x_2)) = 2*x_1 + 3*x_2 POL(z) = 0 ---------------------------------------- (9) Obligation: The TRS P consists of the following rules: U51^1(tt, V2) -> ISBIN(V2) U61^1(tt, V2) -> ISBIN(V2) U21^1(tt, V2) -> ISBAG(V2) The TRS R consists of the following rules: U22(tt) -> tt U81(tt) -> tt U41(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U51(tt, V2) -> U52(isBin(V2)) U21(tt, V2) -> U22(isBag(V2)) U11(tt) -> tt U52(tt) -> tt U71(tt) -> tt U31(tt) -> tt U62(tt) -> tt E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (10) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 3 less nodes. ---------------------------------------- (11) TRUE ---------------------------------------- (12) Obligation: The TRS P consists of the following rules: U152^1(tt, X, Y) -> PLUS(X, Y) U151^1(tt, X, Y) -> U152^1(isBin(Y), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U151^1(isBin(X), X, Y) U152^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> U141^1(isBin(X), X, Y) PLUS(1(X), 1(Y)) -> U151^1(isBin(X), X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 1(Y)) -> U141^1(isBin(X), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U151(isBin(X), X, Y), ext) U142^1(tt, X, Y) -> PLUS(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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: U152^1(tt, X, Y) -> PLUS(X, Y) U151^1(tt, X, Y) -> U152^1(isBin(Y), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> U151^1(isBin(X), X, Y) U152^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> U141^1(isBin(X), X, Y) PLUS(1(X), 1(Y)) -> U151^1(isBin(X), X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(0(X), 1(Y)) -> U141^1(isBin(X), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U151(isBin(X), X, Y), ext) U142^1(tt, X, Y) -> PLUS(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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. U152^1(tt, X, Y) -> PLUS(X, Y) U151^1(tt, X, Y) -> U152^1(isBin(Y), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> U151^1(isBin(X), X, Y) U152^1(tt, X, Y) -> PLUS(plus(X, Y), 1(z)) PLUS(plus(0(X), 1(Y)), ext) -> U141^1(isBin(X), X, Y) PLUS(1(X), 1(Y)) -> U151^1(isBin(X), X, Y) PLUS(0(X), 1(Y)) -> U141^1(isBin(X), X, Y) PLUS(plus(1(X), 1(Y)), ext) -> PLUS(U151(isBin(X), X, Y), ext) The remaining Dependency Pairs were at least non-strictly oriented. PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) U142^1(tt, X, Y) -> PLUS(X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U131(tt, X, Y) -> U132(isBin(Y), X, Y) U81(tt) -> tt U22(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U11(tt) -> tt U142(tt, X, Y) -> 1(plus(X, Y)) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U52(tt) -> tt 0(z) -> z U132(tt, X, Y) -> 0(plus(X, Y)) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) U71(tt) -> tt isBag(empty) -> tt isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBag(singl(V1)) -> U11(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(0(V1)) -> U31(isBin(V1)) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(sum(V1)) -> U81(isBag(V1)) isBin(z) -> tt isBin(1(V1)) -> U41(isBin(V1)) isBin(prod(V1)) -> U71(isBag(V1)) U21(tt, V2) -> U22(isBag(V2)) U31(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U121(tt, X) -> X U41(tt) -> tt U62(tt) -> tt 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)) = x_1 POL(1(x_1)) = 2 + x_1 POL(PLUS(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U11(x_1)) = 2 POL(U121(x_1, x_2)) = x_2 POL(U131(x_1, x_2, x_3)) = x_2 + x_3 POL(U131^1(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 POL(U132(x_1, x_2, x_3)) = x_2 + x_3 POL(U132^1(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 POL(U141(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U141^1(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 POL(U142(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U142^1(x_1, x_2, x_3)) = 2*x_2 + 2*x_3 POL(U151(x_1, x_2, x_3)) = 3 + x_2 + x_3 POL(U151^1(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U152(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U152^1(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U21(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1)) = 2 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = 2 POL(U61(x_1, x_2)) = 2 POL(U62(x_1)) = x_1 POL(U71(x_1)) = 2 POL(U81(x_1)) = 2 POL(empty) = 2 POL(isBag(x_1)) = 2*x_1 POL(isBin(x_1)) = 2 POL(mult(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 0 POL(singl(x_1)) = 2 POL(sum(x_1)) = 0 POL(tt) = 2 POL(union(x_1, x_2)) = 2 + 3*x_1 + 2*x_2 POL(z) = 0 ---------------------------------------- (16) Obligation: The TRS P consists of the following rules: PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) U141^1(tt, X, Y) -> U142^1(isBin(Y), X, Y) U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) U142^1(tt, X, Y) -> PLUS(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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: U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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]. The following set of Dependency Pairs of this DP problem can be strictly oriented. PLUS(plus(0(X), 0(Y)), ext) -> U131^1(isBin(X), X, Y) PLUS(plus(z, X), ext) -> PLUS(U121(isBin(X), X), ext) The remaining Dependency Pairs were at least non-strictly oriented. U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U121(tt, X) -> X U141(tt, X, Y) -> U142(isBin(Y), X, Y) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) 0(z) -> z U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U142(tt, X, Y) -> 1(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)) = x_1 POL(1(x_1)) = 1 + x_1 POL(PLUS(x_1, x_2)) = 3*x_1 + 3*x_2 POL(U11(x_1)) = 3 POL(U121(x_1, x_2)) = x_2 POL(U131(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U131^1(x_1, x_2, x_3)) = 3*x_2 + 3*x_3 POL(U132(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U132^1(x_1, x_2, x_3)) = 3*x_2 + 3*x_3 POL(U141(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U142(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U151(x_1, x_2, x_3)) = 3 + x_2 + x_3 POL(U152(x_1, x_2, x_3)) = 3 + x_2 + x_3 POL(U21(x_1, x_2)) = 3 + 3*x_2 POL(U22(x_1)) = 3 POL(U31(x_1)) = 3 POL(U41(x_1)) = 3 POL(U51(x_1, x_2)) = 3 + 3*x_2 POL(U52(x_1)) = 3 POL(U61(x_1, x_2)) = 3 + 3*x_2 POL(U62(x_1)) = 3 POL(U71(x_1)) = 3 POL(U81(x_1)) = 3 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)) = 1 + x_1 + x_2 POL(prod(x_1)) = 0 POL(singl(x_1)) = 0 POL(sum(x_1)) = 0 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (20) Obligation: The TRS P consists of the following rules: U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) PLUS(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) U132^1(tt, X, Y) -> PLUS(X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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) 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(0(X), 0(Y)) -> U131^1(isBin(X), X, Y) PLUS(plus(0(X), 1(Y)), ext) -> PLUS(U141(isBin(X), X, Y), ext) PLUS(plus(0(X), 0(Y)), ext) -> PLUS(U131(isBin(X), X, Y), ext) The remaining Dependency Pairs were at least non-strictly oriented. U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) U132^1(tt, X, Y) -> PLUS(X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U142(tt, X, Y) -> 1(plus(X, Y)) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) 0(z) -> z 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)) = 2 + x_1 POL(PLUS(x_1, x_2)) = x_1 + x_2 POL(U11(x_1)) = 3 POL(U121(x_1, x_2)) = x_2 POL(U131(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U131^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U132(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U132^1(x_1, x_2, x_3)) = x_2 + x_3 POL(U141(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U142(x_1, x_2, x_3)) = 2 + x_2 + x_3 POL(U151(x_1, x_2, x_3)) = 3 + x_2 + x_3 POL(U152(x_1, x_2, x_3)) = 3 + x_2 + x_3 POL(U21(x_1, x_2)) = 3 + 3*x_2 POL(U22(x_1)) = 3 POL(U31(x_1)) = 3 POL(U41(x_1)) = 3 POL(U51(x_1, x_2)) = 3 + 3*x_2 POL(U52(x_1)) = 3 POL(U61(x_1, x_2)) = 3 + 3*x_2 POL(U62(x_1)) = 3 POL(U71(x_1)) = 3 POL(U81(x_1)) = 3 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)) = 0 POL(singl(x_1)) = 0 POL(sum(x_1)) = 0 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (22) Obligation: The TRS P consists of the following rules: U131^1(tt, X, Y) -> U132^1(isBin(Y), X, Y) U132^1(tt, X, Y) -> PLUS(X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (23) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 2 less nodes. ---------------------------------------- (24) TRUE ---------------------------------------- (25) Obligation: The TRS P consists of the following rules: U192^1(tt, A, B) -> SUM(B) U191^1(tt, A, B) -> U192^1(isBag(B), A, B) SUM(union(A, B)) -> U191^1(isBag(A), A, B) U192^1(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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (26) 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')) ---------------------------------------- (27) Obligation: The TRS P consists of the following rules: U192^1(tt, A, B) -> SUM(B) U191^1(tt, A, B) -> U192^1(isBag(B), A, B) SUM(union(A, B)) -> U191^1(isBag(A), A, B) U192^1(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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (28) 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: U192^1(tt, A, B) -> SUM(B) U191^1(tt, A, B) -> U192^1(isBag(B), A, B) SUM(union(A, B)) -> U191^1(isBag(A), A, B) U192^1(tt, A, B) -> SUM(A) The following rules are removed from R: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U31(tt) -> tt U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 + 3*x_1 POL(1(x_1)) = 1 + 2*x_1 POL(SUM(x_1)) = x_1 POL(U11(x_1)) = 2 + 2*x_1 POL(U191^1(x_1, x_2, x_3)) = x_1 + x_2 + 3*x_3 POL(U192^1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 + 3*x_2 POL(U52(x_1)) = 2*x_1 POL(U61(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U62(x_1)) = 2*x_1 POL(U71(x_1)) = 2 + x_1 POL(U81(x_1)) = 2 + x_1 POL(empty) = 2 POL(isBag(x_1)) = 2*x_1 POL(isBin(x_1)) = 1 + x_1 POL(mult(x_1, x_2)) = 1 + 2*x_1 + 3*x_2 POL(plus(x_1, x_2)) = 3 + 2*x_1 + 3*x_2 POL(prod(x_1)) = 3 + 3*x_1 POL(singl(x_1)) = 2 + 2*x_1 POL(sum(x_1)) = 3 + 3*x_1 POL(tt) = 2 POL(union(x_1, x_2)) = 3*x_1 + 3*x_2 POL(z) = 1 ---------------------------------------- (29) Obligation: P is empty. The TRS R consists of the following rules: U22(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (30) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (31) YES ---------------------------------------- (32) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U112^1(tt, X, Y) -> MULT(X, Y) MULT(mult(1(X), Y), ext) -> U111^1(isBin(X), X, Y) MULT(mult(1(X), Y), ext) -> MULT(U111(isBin(X), X, Y), ext) U102^1(tt, X, Y) -> MULT(X, Y) U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) MULT(1(X), Y) -> U111^1(isBin(X), X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (33) 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')) ---------------------------------------- (34) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U112^1(tt, X, Y) -> MULT(X, Y) MULT(mult(1(X), Y), ext) -> U111^1(isBin(X), X, Y) MULT(mult(1(X), Y), ext) -> MULT(U111(isBin(X), X, Y), ext) U102^1(tt, X, Y) -> MULT(X, Y) U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) MULT(1(X), Y) -> U111^1(isBin(X), X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (35) 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) -> U111^1(isBin(X), X, Y) MULT(mult(1(X), Y), ext) -> MULT(U111(isBin(X), X, Y), ext) MULT(1(X), Y) -> U111^1(isBin(X), X, Y) The remaining Dependency Pairs were at least non-strictly oriented. MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U112^1(tt, X, Y) -> MULT(X, Y) U102^1(tt, X, Y) -> MULT(X, Y) U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U51(tt, V2) -> U52(isBin(V2)) U61(tt, V2) -> U62(isBin(V2)) 0(z) -> z U111(tt, X, Y) -> U112(isBin(Y), X, Y) isBag(empty) -> tt isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBag(singl(V1)) -> U11(isBin(V1)) mult(1(X), Y) -> U111(isBin(X), X, Y) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(z, X) -> U91(isBin(X)) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) U121(tt, X) -> X U102(tt, X, Y) -> 0(mult(X, Y)) isBin(z) -> tt isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(0(V1)) -> U31(isBin(V1)) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(sum(V1)) -> U81(isBag(V1)) isBin(prod(V1)) -> U71(isBag(V1)) U71(tt) -> tt plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(z, X) -> U121(isBin(X), X) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U81(tt) -> tt U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U101(tt, X, Y) -> U102(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U62(tt) -> tt U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U52(tt) -> tt U91(tt) -> z U11(tt) -> tt U31(tt) -> tt U41(tt) -> tt U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U131(tt, X, Y) -> U132(isBin(Y), X, Y) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(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. 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(U101(x_1, x_2, x_3)) = x_1*x_2*x_3 + x_1*x_3 + x_2 POL(U101^1(x_1, x_2, x_3)) = x_1*x_2 + x_1*x_2*x_3 + x_1*x_3 POL(U102(x_1, x_2, x_3)) = x_1*x_2*x_3 + x_1*x_3 + x_2 POL(U102^1(x_1, x_2, x_3)) = x_1*x_3 + x_2 + x_2*x_3 POL(U11(x_1)) = 1 POL(U111(x_1, x_2, x_3)) = x_1*x_3 + x_2 + x_2*x_3 + x_3 POL(U111^1(x_1, x_2, x_3)) = x_1*x_2 + x_1*x_2*x_3 + x_1*x_3 POL(U112(x_1, x_2, x_3)) = x_1*x_3 + x_2 + x_2*x_3 + x_3 POL(U112^1(x_1, x_2, x_3)) = x_2 + x_2*x_3 + x_3 POL(U121(x_1, x_2)) = x_1*x_2 POL(U131(x_1, x_2, x_3)) = x_2 + x_3 POL(U132(x_1, x_2, x_3)) = x_1*x_2 + x_1*x_3 POL(U141(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U142(x_1, x_2, x_3)) = 1 + x_1*x_2 + x_3 POL(U151(x_1, x_2, x_3)) = 1 + x_1 + x_1*x_3 + x_2 POL(U152(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(U21(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(U22(x_1)) = 1 POL(U31(x_1)) = x_1^2 POL(U41(x_1)) = x_1^2 POL(U51(x_1, x_2)) = 1 POL(U52(x_1)) = 1 POL(U61(x_1, x_2)) = x_1 POL(U62(x_1)) = 1 POL(U71(x_1)) = 1 POL(U81(x_1)) = 1 POL(U91(x_1)) = 0 POL(empty) = 1 POL(isBag(x_1)) = x_1^2 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)) = 0 POL(singl(x_1)) = 1 POL(sum(x_1)) = 0 POL(tt) = 1 POL(union(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(z) = 0 ---------------------------------------- (36) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U112^1(tt, X, Y) -> MULT(X, Y) U102^1(tt, X, Y) -> MULT(X, Y) U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) U111^1(tt, X, Y) -> U112^1(isBin(Y), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (37) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 2 less nodes. ---------------------------------------- (38) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) U102^1(tt, X, Y) -> MULT(X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (39) 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. U101^1(tt, X, Y) -> U102^1(isBin(Y), X, Y) The remaining Dependency Pairs were at least non-strictly oriented. MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U102^1(tt, X, Y) -> MULT(X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U11(tt) -> tt U102(tt, X, Y) -> 0(mult(X, Y)) U51(tt, V2) -> U52(isBin(V2)) U21(tt, V2) -> U22(isBag(V2)) U61(tt, V2) -> U62(isBin(V2)) U52(tt) -> tt U101(tt, X, Y) -> U102(isBin(Y), X, Y) U71(tt) -> tt U142(tt, X, Y) -> 1(plus(X, Y)) U111(tt, X, Y) -> U112(isBin(Y), X, Y) U91(tt) -> z U141(tt, X, Y) -> U142(isBin(Y), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(z, X) -> U91(isBin(X)) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) isBin(z) -> tt isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(0(V1)) -> U31(isBin(V1)) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(sum(V1)) -> U81(isBag(V1)) isBin(prod(V1)) -> U71(isBag(V1)) U41(tt) -> tt U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U62(tt) -> tt U22(tt) -> tt 0(z) -> z plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(z, X) -> U121(isBin(X), X) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) U132(tt, X, Y) -> 0(plus(X, Y)) U81(tt) -> tt isBag(empty) -> tt isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBag(singl(V1)) -> U11(isBin(V1)) U31(tt) -> tt U151(tt, X, Y) -> U152(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. 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)) = 1 + x_1 POL(1(x_1)) = 1 + x_1 POL(MULT(x_1, x_2)) = 1 + x_1 + x_1*x_2 + x_2 POL(U101(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_2 + x_2*x_3 POL(U101^1(x_1, x_2, x_3)) = 1 + x_1 + x_1*x_2*x_3 + x_2 + x_3 POL(U102(x_1, x_2, x_3)) = 1 + x_2 + x_2*x_3 + x_3 POL(U102^1(x_1, x_2, x_3)) = 1 + x_1*x_2 + x_2*x_3 + x_3 POL(U11(x_1)) = 1 POL(U111(x_1, x_2, x_3)) = 1 + x_1*x_3 + x_2 + x_2*x_3 + x_3 POL(U112(x_1, x_2, x_3)) = 1 + x_1*x_2 + x_1*x_3 + x_2*x_3 + x_3 POL(U121(x_1, x_2)) = x_1*x_2 POL(U131(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U132(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U141(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U142(x_1, x_2, x_3)) = 1 + x_2 + x_3 POL(U151(x_1, x_2, x_3)) = 1 + x_1 + x_1*x_2 + x_1*x_3 POL(U152(x_1, x_2, x_3)) = 1 + x_1 + x_1*x_2 + x_3 POL(U21(x_1, x_2)) = x_1*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = x_1^2 POL(U41(x_1)) = x_1^2 POL(U51(x_1, x_2)) = 1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = 1 POL(U62(x_1)) = 1 POL(U71(x_1)) = 1 POL(U81(x_1)) = 1 POL(U91(x_1)) = 0 POL(empty) = 1 POL(isBag(x_1)) = x_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)) = 0 POL(singl(x_1)) = 1 POL(sum(x_1)) = 0 POL(tt) = 1 POL(union(x_1, x_2)) = x_1 + x_1*x_2 + x_2 POL(z) = 0 ---------------------------------------- (40) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) U102^1(tt, X, Y) -> MULT(X, Y) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) MULT(mult(0(X), Y), ext) -> U101^1(isBin(X), X, Y) MULT(0(X), Y) -> U101^1(isBin(X), X, Y) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (41) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 1 SCC with 3 less nodes. ---------------------------------------- (42) Obligation: The TRS P consists of the following rules: MULT(mult(z, X), ext) -> MULT(U91(isBin(X)), ext) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (43) 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(z, X), ext) -> MULT(U91(isBin(X)), ext) MULT(mult(0(X), Y), ext) -> MULT(U101(isBin(X), X, Y), ext) With the implicit AFS we had to orient the following set of usable rules of R non-strictly. U141(tt, X, Y) -> U142(isBin(Y), X, Y) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), X, Y), ext) plus(z, X) -> U121(isBin(X), X) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) U102(tt, X, Y) -> 0(mult(X, Y)) U101(tt, X, Y) -> U102(isBin(Y), X, Y) U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U91(tt) -> z U142(tt, X, Y) -> 1(plus(X, Y)) U121(tt, X) -> X U111(tt, X, Y) -> U112(isBin(Y), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(z, X) -> U91(isBin(X)) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) 0(z) -> z U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(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)) = 2 POL(MULT(x_1, x_2)) = x_1 + x_2 POL(U101(x_1, x_2, x_3)) = 0 POL(U102(x_1, x_2, x_3)) = 0 POL(U11(x_1)) = 3 POL(U111(x_1, x_2, x_3)) = 2 + x_3 POL(U112(x_1, x_2, x_3)) = x_3 POL(U121(x_1, x_2)) = x_2 POL(U131(x_1, x_2, x_3)) = 0 POL(U132(x_1, x_2, x_3)) = 0 POL(U141(x_1, x_2, x_3)) = 2 POL(U142(x_1, x_2, x_3)) = 2 POL(U151(x_1, x_2, x_3)) = 3 POL(U152(x_1, x_2, x_3)) = 2 POL(U21(x_1, x_2)) = 3 + 3*x_2 POL(U22(x_1)) = 3 POL(U31(x_1)) = 3 POL(U41(x_1)) = 3 POL(U51(x_1, x_2)) = 3 + 3*x_2 POL(U52(x_1)) = 3 POL(U61(x_1, x_2)) = 3 + 3*x_2 POL(U62(x_1)) = 3 POL(U71(x_1)) = 3 POL(U81(x_1)) = 3 POL(U91(x_1)) = 0 POL(empty) = 0 POL(isBag(x_1)) = 0 POL(isBin(x_1)) = 0 POL(mult(x_1, x_2)) = 2 + x_1 + x_2 POL(plus(x_1, x_2)) = x_1 + x_2 POL(prod(x_1)) = 0 POL(singl(x_1)) = 0 POL(sum(x_1)) = 0 POL(tt) = 0 POL(union(x_1, x_2)) = 0 POL(z) = 0 ---------------------------------------- (44) 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, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (45) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,E#,R,E) chain. ---------------------------------------- (46) YES ---------------------------------------- (47) Obligation: The TRS P consists of the following rules: U171^1(tt, A, B) -> U172^1(isBag(B), A, B) U172^1(tt, A, B) -> PROD(B) U172^1(tt, A, B) -> PROD(A) PROD(union(A, B)) -> U171^1(isBag(A), A, B) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (48) 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')) ---------------------------------------- (49) Obligation: The TRS P consists of the following rules: U171^1(tt, A, B) -> U172^1(isBag(B), A, B) U172^1(tt, A, B) -> PROD(B) U172^1(tt, A, B) -> PROD(A) PROD(union(A, B)) -> U171^1(isBag(A), A, B) The TRS R consists of the following rules: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U11(tt) -> tt U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U21(tt, V2) -> U22(isBag(V2)) U22(tt) -> tt U31(tt) -> tt U41(tt) -> tt U51(tt, V2) -> U52(isBin(V2)) U52(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U62(tt) -> tt U71(tt) -> tt U81(tt) -> tt U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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 ---------------------------------------- (50) 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: PROD(union(A, B)) -> U171^1(isBag(A), A, B) The following rules are removed from R: union(X, empty) -> X union(empty, X) -> X 0(z) -> z U101(tt, X, Y) -> U102(isBin(Y), X, Y) U102(tt, X, Y) -> 0(mult(X, Y)) U111(tt, X, Y) -> U112(isBin(Y), X, Y) U112(tt, X, Y) -> plus(0(mult(X, Y)), Y) U121(tt, X) -> X U131(tt, X, Y) -> U132(isBin(Y), X, Y) U132(tt, X, Y) -> 0(plus(X, Y)) U141(tt, X, Y) -> U142(isBin(Y), X, Y) U142(tt, X, Y) -> 1(plus(X, Y)) U151(tt, X, Y) -> U152(isBin(Y), X, Y) U152(tt, X, Y) -> 0(plus(plus(X, Y), 1(z))) U161(tt, X) -> X U171(tt, A, B) -> U172(isBag(B), A, B) U172(tt, A, B) -> mult(prod(A), prod(B)) U181(tt, X) -> X U191(tt, A, B) -> U192(isBag(B), A, B) U192(tt, A, B) -> plus(sum(A), sum(B)) U51(tt, V2) -> U52(isBin(V2)) U91(tt) -> z isBag(empty) -> tt isBag(singl(V1)) -> U11(isBin(V1)) isBag(union(V1, V2)) -> U21(isBag(V1), V2) isBin(z) -> tt isBin(0(V1)) -> U31(isBin(V1)) isBin(1(V1)) -> U41(isBin(V1)) isBin(mult(V1, V2)) -> U51(isBin(V1), V2) isBin(plus(V1, V2)) -> U61(isBin(V1), V2) isBin(prod(V1)) -> U71(isBag(V1)) isBin(sum(V1)) -> U81(isBag(V1)) mult(z, X) -> U91(isBin(X)) mult(0(X), Y) -> U101(isBin(X), X, Y) mult(1(X), Y) -> U111(isBin(X), X, Y) plus(z, X) -> U121(isBin(X), X) plus(0(X), 0(Y)) -> U131(isBin(X), X, Y) plus(0(X), 1(Y)) -> U141(isBin(X), X, Y) plus(1(X), 1(Y)) -> U151(isBin(X), X, Y) prod(empty) -> 1(z) prod(singl(X)) -> U161(isBin(X), X) prod(union(A, B)) -> U171(isBag(A), A, B) sum(empty) -> 0(z) sum(singl(X)) -> U181(isBin(X), X) sum(union(A, B)) -> U191(isBag(A), A, B) mult(mult(z, X), ext) -> mult(U91(isBin(X)), ext) mult(mult(0(X), Y), ext) -> mult(U101(isBin(X), X, Y), ext) mult(mult(1(X), Y), ext) -> mult(U111(isBin(X), X, Y), ext) plus(plus(z, X), ext) -> plus(U121(isBin(X), X), ext) plus(plus(0(X), 0(Y)), ext) -> plus(U131(isBin(X), X, Y), ext) plus(plus(0(X), 1(Y)), ext) -> plus(U141(isBin(X), X, Y), ext) plus(plus(1(X), 1(Y)), ext) -> plus(U151(isBin(X), 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)) = 2 + 3*x_1 POL(1(x_1)) = 3*x_1 POL(PROD(x_1)) = x_1 POL(U11(x_1)) = 2*x_1 POL(U171^1(x_1, x_2, x_3)) = x_1 + x_2 + 3*x_3 POL(U172^1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(U21(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U22(x_1)) = x_1 POL(U31(x_1)) = 2*x_1 POL(U41(x_1)) = 2*x_1 POL(U51(x_1, x_2)) = 1 + 2*x_1 + 3*x_2 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = 2*x_1 + 3*x_2 POL(U62(x_1)) = x_1 POL(U71(x_1)) = x_1 POL(U81(x_1)) = 2*x_1 POL(empty) = 0 POL(isBag(x_1)) = 2*x_1 POL(isBin(x_1)) = 2*x_1 POL(mult(x_1, x_2)) = 2 + 3*x_1 + 3*x_2 POL(plus(x_1, x_2)) = 2 + 3*x_1 + 3*x_2 POL(prod(x_1)) = 2*x_1 POL(singl(x_1)) = 3*x_1 POL(sum(x_1)) = 2*x_1 POL(tt) = 0 POL(union(x_1, x_2)) = 3*x_1 + 3*x_2 POL(z) = 0 ---------------------------------------- (51) Obligation: The TRS P consists of the following rules: U171^1(tt, A, B) -> U172^1(isBag(B), A, B) U172^1(tt, A, B) -> PROD(B) U172^1(tt, A, B) -> PROD(A) The TRS R consists of the following rules: U22(tt) -> tt U81(tt) -> tt U41(tt) -> tt U61(tt, V2) -> U62(isBin(V2)) U21(tt, V2) -> U22(isBag(V2)) U11(tt) -> tt U52(tt) -> tt U71(tt) -> tt U31(tt) -> tt U62(tt) -> tt E is empty. E# is empty. We have to consider all minimal (P,E#,R,E)-chains ---------------------------------------- (52) EDependencyGraphProof (EQUIVALENT) The approximation of the Equational Dependency Graph [DA_STEIN] contains 0 SCCs with 3 less nodes. ---------------------------------------- (53) TRUE