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) |
a__U21#(tt,M,N) | → | mark#(N) | (22) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (23) |
a__plus#(N,0) | → | a__U11#(a__isNat(N),N) | (24) |
mark#(plus(X1,X2)) | → | mark#(X1) | (25) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (26) |
mark#(s(X)) | → | mark#(X) | (27) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (28) |
mark#(and(X1,X2)) | → | mark#(X1) | (29) |
a__plus#(N,0) | → | a__isNat#(N) | (30) |
a__U11#(tt,N) | → | mark#(N) | (31) |
mark#(U11(X1,X2)) | → | mark#(X1) | (32) |
mark#(plus(X1,X2)) | → | mark#(X2) | (33) |
mark#(isNat(X)) | → | a__isNat#(X) | (34) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNat(N)) | (35) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNat(V1),isNat(V2)) | (36) |
a__and#(tt,X) | → | mark#(X) | (37) |
a__plus#(N,s(M)) | → | a__U21#(a__and(a__isNat(M),isNat(N)),M,N) | (38) |
mark#(U21(X1,X2,X3)) | → | mark#(X1) | (39) |
a__U21#(tt,M,N) | → | mark#(M) | (40) |
a__U21#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (41) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (42) |
mark#(U21(X1,X2,X3)) | → | a__U21#(mark(X1),X2,X3) | (43) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (44) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (45) |
The dependency pairs are split into 1 component.
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (45) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (44) |
a__plus#(N,0) | → | a__isNat#(N) | (30) |
mark#(U21(X1,X2,X3)) | → | a__U21#(mark(X1),X2,X3) | (43) |
mark#(and(X1,X2)) | → | mark#(X1) | (29) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (42) |
a__U21#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (41) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (28) |
a__U21#(tt,M,N) | → | mark#(M) | (40) |
mark#(s(X)) | → | mark#(X) | (27) |
mark#(U21(X1,X2,X3)) | → | mark#(X1) | (39) |
a__plus#(N,s(M)) | → | a__U21#(a__and(a__isNat(M),isNat(N)),M,N) | (38) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (26) |
mark#(plus(X1,X2)) | → | mark#(X1) | (25) |
a__plus#(N,0) | → | a__U11#(a__isNat(N),N) | (24) |
a__and#(tt,X) | → | mark#(X) | (37) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNat(V1),isNat(V2)) | (36) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (23) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNat(N)) | (35) |
mark#(isNat(X)) | → | a__isNat#(X) | (34) |
mark#(plus(X1,X2)) | → | mark#(X2) | (33) |
mark#(U11(X1,X2)) | → | mark#(X1) | (32) |
a__U21#(tt,M,N) | → | mark#(N) | (22) |
a__U11#(tt,N) | → | mark#(N) | (31) |
[a__plus(x1, x2)] | = | x1 + x2 + 1 |
[U21(x1, x2, x3)] | = | x1 + x2 + x3 + 3 |
[U11(x1, x2)] | = | x1 + x2 + 2449 |
[s(x1)] | = | x1 + 2 |
[a__isNat#(x1)] | = | 0 |
[and(x1, x2)] | = | x1 + x2 + 0 |
[a__U11#(x1, x2)] | = | x2 + 1 |
[a__U21#(x1, x2, x3)] | = | x2 + x3 + 1 |
[a__plus#(x1, x2)] | = | x1 + x2 + 0 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 2448 |
[a__and#(x1, x2)] | = | x2 + 0 |
[a__U21(x1, x2, x3)] | = | x1 + x2 + x3 + 3 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2)] | = | x1 + x2 + 2449 |
[isNat(x1)] | = | 0 |
[plus(x1, x2)] | = | x1 + x2 + 1 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 0 |
[a__and(x1, x2)] | = | x1 + x2 + 0 |
a__U21(X1,X2,X3) | → | U21(X1,X2,X3) | (18) |
a__isNat(0) | → | tt | (4) |
mark(s(X)) | → | s(mark(X)) | (15) |
a__plus(N,s(M)) | → | a__U21(a__and(a__isNat(M),isNat(N)),M,N) | (8) |
a__U11(tt,N) | → | mark(N) | (1) |
a__and(tt,X) | → | mark(X) | (3) |
mark(0) | → | 0 | (16) |
a__isNat(X) | → | isNat(X) | (21) |
a__plus(X1,X2) | → | plus(X1,X2) | (19) |
a__U11(X1,X2) | → | U11(X1,X2) | (17) |
a__isNat(plus(V1,V2)) | → | a__and(a__isNat(V1),isNat(V2)) | (5) |
mark(U21(X1,X2,X3)) | → | a__U21(mark(X1),X2,X3) | (10) |
a__plus(N,0) | → | a__U11(a__isNat(N),N) | (7) |
a__and(X1,X2) | → | and(X1,X2) | (20) |
mark(tt) | → | tt | (14) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (12) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (9) |
mark(isNat(X)) | → | a__isNat(X) | (13) |
a__isNat(s(V1)) | → | a__isNat(V1) | (6) |
a__U21(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (2) |
a__plus#(N,0) | → | a__isNat#(N) | (30) |
mark#(U21(X1,X2,X3)) | → | a__U21#(mark(X1),X2,X3) | (43) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (42) |
a__U21#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (41) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (28) |
a__U21#(tt,M,N) | → | mark#(M) | (40) |
mark#(s(X)) | → | mark#(X) | (27) |
mark#(U21(X1,X2,X3)) | → | mark#(X1) | (39) |
a__plus#(N,s(M)) | → | a__U21#(a__and(a__isNat(M),isNat(N)),M,N) | (38) |
mark#(plus(X1,X2)) | → | mark#(X1) | (25) |
a__plus#(N,0) | → | a__U11#(a__isNat(N),N) | (24) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (23) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNat(N)) | (35) |
mark#(plus(X1,X2)) | → | mark#(X2) | (33) |
mark#(U11(X1,X2)) | → | mark#(X1) | (32) |
a__U21#(tt,M,N) | → | mark#(N) | (22) |
a__U11#(tt,N) | → | mark#(N) | (31) |
The dependency pairs are split into 1 component.
a__and#(tt,X) | → | mark#(X) | (37) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (45) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNat(V1),isNat(V2)) | (36) |
mark#(and(X1,X2)) | → | mark#(X1) | (29) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (26) |
mark#(isNat(X)) | → | a__isNat#(X) | (34) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (44) |
[a__plus(x1, x2)] | = | x1 + x2 + 70 |
[U21(x1, x2, x3)] | = | x2 + x3 + 74 |
[U11(x1, x2)] | = | x1 + 2449 |
[s(x1)] | = | x1 + 1 |
[a__isNat#(x1)] | = | x1 + 20586 |
[and(x1, x2)] | = | x1 + x2 + 2 |
[a__U11#(x1, x2)] | = | 1 |
[a__U21#(x1, x2, x3)] | = | 1 |
[a__plus#(x1, x2)] | = | 0 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 146 |
[a__and#(x1, x2)] | = | x2 + 1 |
[a__U21(x1, x2, x3)] | = | 77 |
[mark(x1)] | = | x1 + 3 |
[a__U11(x1, x2)] | = | x2 + 2453 |
[isNat(x1)] | = | x1 + 20587 |
[plus(x1, x2)] | = | x1 + x2 + 71 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 2 |
[a__and(x1, x2)] | = | 2 |
a__and#(tt,X) | → | mark#(X) | (37) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (45) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNat(V1),isNat(V2)) | (36) |
mark#(and(X1,X2)) | → | mark#(X1) | (29) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (26) |
mark#(isNat(X)) | → | a__isNat#(X) | (34) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (44) |
The dependency pairs are split into 0 components.