The rewrite relation of the following TRS is considered.
| a__U11(tt,N) | → | mark(N) | (1) |
| a__U21(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (2) |
| a__and(tt,X) | → | mark(X) | (3) |
| a__isNat(0) | → | tt | (4) |
| a__isNat(plus(V1,V2)) | → | a__and(a__isNat(V1),isNat(V2)) | (5) |
| a__isNat(s(V1)) | → | a__isNat(V1) | (6) |
| a__plus(N,0) | → | a__U11(a__isNat(N),N) | (7) |
| a__plus(N,s(M)) | → | a__U21(a__and(a__isNat(M),isNat(N)),M,N) | (8) |
| mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (9) |
| mark(U21(X1,X2,X3)) | → | a__U21(mark(X1),X2,X3) | (10) |
| mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (11) |
| mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (12) |
| mark(isNat(X)) | → | a__isNat(X) | (13) |
| mark(tt) | → | tt | (14) |
| mark(s(X)) | → | s(mark(X)) | (15) |
| mark(0) | → | 0 | (16) |
| a__U11(X1,X2) | → | U11(X1,X2) | (17) |
| a__U21(X1,X2,X3) | → | U21(X1,X2,X3) | (18) |
| a__plus(X1,X2) | → | plus(X1,X2) | (19) |
| a__and(X1,X2) | → | and(X1,X2) | (20) |
| a__isNat(X) | → | isNat(X) | (21) |
| prec(a__U11) | = | 1 | stat(a__U11) | = | lex | |
| prec(tt) | = | 4 | stat(tt) | = | mul | |
| prec(a__U21) | = | 3 | stat(a__U21) | = | lex | |
| prec(s) | = | 0 | stat(s) | = | mul | |
| prec(a__plus) | = | 3 | stat(a__plus) | = | lex | |
| prec(a__and) | = | 2 | stat(a__and) | = | mul | |
| prec(a__isNat) | = | 2 | stat(a__isNat) | = | mul | |
| prec(0) | = | 5 | stat(0) | = | mul | |
| prec(plus) | = | 3 | stat(plus) | = | lex | |
| prec(isNat) | = | 2 | stat(isNat) | = | mul | |
| prec(U11) | = | 1 | stat(U11) | = | lex | |
| prec(U21) | = | 3 | stat(U21) | = | lex | |
| prec(and) | = | 2 | stat(and) | = | mul |
| π(a__U11) | = | [2,1] |
| π(tt) | = | [] |
| π(mark) | = | 1 |
| π(a__U21) | = | [3,2,1] |
| π(s) | = | [1] |
| π(a__plus) | = | [1,2] |
| π(a__and) | = | [1,2] |
| π(a__isNat) | = | [1] |
| π(0) | = | [] |
| π(plus) | = | [1,2] |
| π(isNat) | = | [1] |
| π(U11) | = | [2,1] |
| π(U21) | = | [3,2,1] |
| π(and) | = | [1,2] |
| a__U11(tt,N) | → | mark(N) | (1) |
| a__U21(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (2) |
| a__and(tt,X) | → | mark(X) | (3) |
| a__isNat(0) | → | tt | (4) |
| a__isNat(plus(V1,V2)) | → | a__and(a__isNat(V1),isNat(V2)) | (5) |
| a__isNat(s(V1)) | → | a__isNat(V1) | (6) |
| a__plus(N,0) | → | a__U11(a__isNat(N),N) | (7) |
| a__plus(N,s(M)) | → | a__U21(a__and(a__isNat(M),isNat(N)),M,N) | (8) |
| prec(tt) | = | 8 | weight(tt) | = | 1 | ||||
| prec(0) | = | 12 | weight(0) | = | 1 | ||||
| prec(mark) | = | 13 | weight(mark) | = | 0 | ||||
| prec(isNat) | = | 3 | weight(isNat) | = | 1 | ||||
| prec(a__isNat) | = | 5 | weight(a__isNat) | = | 1 | ||||
| prec(s) | = | 11 | weight(s) | = | 1 | ||||
| prec(U11) | = | 1 | weight(U11) | = | 0 | ||||
| prec(a__U11) | = | 4 | weight(a__U11) | = | 0 | ||||
| prec(U21) | = | 6 | weight(U21) | = | 0 | ||||
| prec(a__U21) | = | 7 | weight(a__U21) | = | 0 | ||||
| prec(plus) | = | 0 | weight(plus) | = | 0 | ||||
| prec(a__plus) | = | 2 | weight(a__plus) | = | 0 | ||||
| prec(and) | = | 9 | weight(and) | = | 0 | ||||
| prec(a__and) | = | 10 | weight(a__and) | = | 0 |
| mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (9) |
| mark(U21(X1,X2,X3)) | → | a__U21(mark(X1),X2,X3) | (10) |
| mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (11) |
| mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (12) |
| mark(isNat(X)) | → | a__isNat(X) | (13) |
| mark(tt) | → | tt | (14) |
| mark(s(X)) | → | s(mark(X)) | (15) |
| mark(0) | → | 0 | (16) |
| a__U11(X1,X2) | → | U11(X1,X2) | (17) |
| a__U21(X1,X2,X3) | → | U21(X1,X2,X3) | (18) |
| a__plus(X1,X2) | → | plus(X1,X2) | (19) |
| a__and(X1,X2) | → | and(X1,X2) | (20) |
| a__isNat(X) | → | isNat(X) | (21) |
There are no rules in the TRS. Hence, it is terminating.