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