YES proof of Transformed_CSR_04_PEANO_complete_Z.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) QTRSRRRProof [EQUIVALENT, 151 ms] (2) QTRS (3) QTRSRRRProof [EQUIVALENT, 0 ms] (4) QTRS (5) RisEmptyProof [EQUIVALENT, 0 ms] (6) YES ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U13(tt) -> tt U21(tt, V1) -> U22(isNat(activate(V1))) U22(tt) -> tt U31(tt, N) -> activate(N) U41(tt, M, N) -> s(plus(activate(N), activate(M))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1, X2) -> n__and(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. ---------------------------------------- (1) QTRSRRRProof (EQUIVALENT) Used ordering: U11/3(YES,YES,YES) tt/0) U12/2(YES,YES) isNat/1)YES( activate/1)YES( U13/1)YES( U21/2(YES,YES) U22/1)YES( U31/2(YES,YES) U41/3(YES,YES,YES) s/1(YES) plus/2(YES,YES) and/2(YES,YES) n__0/0) n__plus/2(YES,YES) isNatKind/1)YES( n__isNatKind/1)YES( n__s/1(YES) 0/0) n__and/2(YES,YES) Quasi precedence: [U41_3, plus_2, n__plus_2] > U11_3 > U12_2 [U41_3, plus_2, n__plus_2] > U31_2 [U41_3, plus_2, n__plus_2] > [s_1, n__s_1] > U21_2 [U41_3, plus_2, n__plus_2] > [s_1, n__s_1] > [and_2, n__and_2] [n__0, 0] > tt Status: U11_3: multiset status tt: multiset status U12_2: [2,1] U21_2: multiset status U31_2: multiset status U41_3: [3,2,1] s_1: multiset status plus_2: [1,2] and_2: [1,2] n__0: multiset status n__plus_2: [1,2] n__s_1: multiset status 0: multiset status n__and_2: [1,2] With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U11(tt, V1, V2) -> U12(isNat(activate(V1)), activate(V2)) U12(tt, V2) -> U13(isNat(activate(V2))) U21(tt, V1) -> U22(isNat(activate(V1))) U31(tt, N) -> activate(N) U41(tt, M, N) -> s(plus(activate(N), activate(M))) and(tt, X) -> activate(X) isNat(n__0) -> tt isNat(n__plus(V1, V2)) -> U11(and(isNatKind(activate(V1)), n__isNatKind(activate(V2))), activate(V1), activate(V2)) isNat(n__s(V1)) -> U21(isNatKind(activate(V1)), activate(V1)) isNatKind(n__0) -> tt isNatKind(n__plus(V1, V2)) -> and(isNatKind(activate(V1)), n__isNatKind(activate(V2))) isNatKind(n__s(V1)) -> isNatKind(activate(V1)) plus(N, 0) -> U31(and(isNat(N), n__isNatKind(N)), N) plus(N, s(M)) -> U41(and(and(isNat(M), n__isNatKind(M)), n__and(isNat(N), n__isNatKind(N))), M, N) ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: U13(tt) -> tt U22(tt) -> tt 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1, X2) -> n__and(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X Q is empty. ---------------------------------------- (3) QTRSRRRProof (EQUIVALENT) Used ordering: Knuth-Bendix order [KBO] with precedence:U22_1 > tt > activate_1 > and_2 > n__and_2 > s_1 > plus_2 > isNatKind_1 > n__s_1 > n__isNatKind_1 > n__plus_2 > n__0 > 0 > U13_1 and weight map: tt=1 0=2 n__0=1 U13_1=1 U22_1=0 isNatKind_1=2 n__isNatKind_1=1 s_1=1 n__s_1=1 activate_1=1 plus_2=1 n__plus_2=0 and_2=0 n__and_2=0 The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: U13(tt) -> tt U22(tt) -> tt 0 -> n__0 plus(X1, X2) -> n__plus(X1, X2) isNatKind(X) -> n__isNatKind(X) s(X) -> n__s(X) and(X1, X2) -> n__and(X1, X2) activate(n__0) -> 0 activate(n__plus(X1, X2)) -> plus(X1, X2) activate(n__isNatKind(X)) -> isNatKind(X) activate(n__s(X)) -> s(X) activate(n__and(X1, X2)) -> and(X1, X2) activate(X) -> X ---------------------------------------- (4) Obligation: Q restricted rewrite system: R is empty. Q is empty. ---------------------------------------- (5) RisEmptyProof (EQUIVALENT) The TRS R is empty. Hence, termination is trivially proven. ---------------------------------------- (6) YES