The rewrite relation of the following TRS is considered.
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
U13(tt) | → | tt | (3) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
U22(tt) | → | tt | (5) |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
isNat(n__0) | → | tt | (9) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNatKind(n__0) | → | tt | (12) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
0 | → | n__0 | (17) |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
s(X) | → | n__s(X) | (20) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
activate(X) | → | X | (29) |
prec(U11) | = | 3 | stat(U11) | = | mul | |
prec(tt) | = | 1 | stat(tt) | = | mul | |
prec(U12) | = | 2 | stat(U12) | = | lex | |
prec(isNat) | = | 2 | stat(isNat) | = | lex | |
prec(U21) | = | 2 | stat(U21) | = | lex | |
prec(U22) | = | 0 | stat(U22) | = | lex | |
prec(U31) | = | 4 | stat(U31) | = | mul | |
prec(U41) | = | 6 | stat(U41) | = | lex | |
prec(s) | = | 5 | stat(s) | = | mul | |
prec(plus) | = | 6 | stat(plus) | = | lex | |
prec(and) | = | 5 | stat(and) | = | mul | |
prec(n__0) | = | 7 | stat(n__0) | = | mul | |
prec(n__plus) | = | 6 | stat(n__plus) | = | lex | |
prec(isNatKind) | = | 5 | stat(isNatKind) | = | mul | |
prec(n__isNatKind) | = | 5 | stat(n__isNatKind) | = | mul | |
prec(n__s) | = | 5 | stat(n__s) | = | mul | |
prec(0) | = | 7 | stat(0) | = | mul | |
prec(n__and) | = | 5 | stat(n__and) | = | mul | |
prec(n__isNat) | = | 2 | stat(n__isNat) | = | lex |
π(U11) | = | [1,2,3] |
π(tt) | = | [] |
π(U12) | = | [2,1] |
π(isNat) | = | [1] |
π(activate) | = | 1 |
π(U13) | = | 1 |
π(U21) | = | [2,1] |
π(U22) | = | [1] |
π(U31) | = | [1,2] |
π(U41) | = | [3,2,1] |
π(s) | = | [1] |
π(plus) | = | [1,2] |
π(and) | = | [1,2] |
π(n__0) | = | [] |
π(n__plus) | = | [1,2] |
π(isNatKind) | = | [1] |
π(n__isNatKind) | = | [1] |
π(n__s) | = | [1] |
π(0) | = | [] |
π(n__and) | = | [1,2] |
π(n__isNat) | = | [1] |
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
U22(tt) | → | tt | (5) |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
isNat(n__0) | → | tt | (9) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNatKind(n__0) | → | tt | (12) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
prec(tt) | = | 0 | weight(tt) | = | 1 | ||||
prec(0) | = | 12 | weight(0) | = | 1 | ||||
prec(n__0) | = | 1 | weight(n__0) | = | 1 | ||||
prec(U13) | = | 4 | weight(U13) | = | 1 | ||||
prec(isNatKind) | = | 5 | weight(isNatKind) | = | 1 | ||||
prec(n__isNatKind) | = | 3 | weight(n__isNatKind) | = | 1 | ||||
prec(s) | = | 11 | weight(s) | = | 1 | ||||
prec(n__s) | = | 6 | weight(n__s) | = | 1 | ||||
prec(isNat) | = | 10 | weight(isNat) | = | 1 | ||||
prec(n__isNat) | = | 2 | weight(n__isNat) | = | 1 | ||||
prec(activate) | = | 14 | weight(activate) | = | 0 | ||||
prec(plus) | = | 13 | weight(plus) | = | 0 | ||||
prec(n__plus) | = | 7 | weight(n__plus) | = | 0 | ||||
prec(and) | = | 9 | weight(and) | = | 0 | ||||
prec(n__and) | = | 8 | weight(n__and) | = | 0 |
U13(tt) | → | tt | (3) |
0 | → | n__0 | (17) |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
s(X) | → | n__s(X) | (20) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
activate(X) | → | X | (29) |
There are no rules in the TRS. Hence, it is terminating.