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) |
and(tt,X) | → | activate(X) | (3) |
isNat(n__0) | → | tt | (4) |
isNat(n__plus(V1,V2)) | → | and(isNat(activate(V1)),n__isNat(activate(V2))) | (5) |
isNat(n__s(V1)) | → | isNat(activate(V1)) | (6) |
plus(N,0) | → | U11(isNat(N),N) | (7) |
plus(N,s(M)) | → | U21(and(isNat(M),n__isNat(N)),M,N) | (8) |
0 | → | n__0 | (9) |
plus(X1,X2) | → | n__plus(X1,X2) | (10) |
isNat(X) | → | n__isNat(X) | (11) |
s(X) | → | n__s(X) | (12) |
activate(n__0) | → | 0 | (13) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (14) |
activate(n__isNat(X)) | → | isNat(X) | (15) |
activate(n__s(X)) | → | s(activate(X)) | (16) |
activate(X) | → | X | (17) |
U11#(tt,N) | → | activate#(N) | (18) |
U21#(tt,M,N) | → | activate#(M) | (19) |
U21#(tt,M,N) | → | activate#(N) | (20) |
U21#(tt,M,N) | → | plus#(activate(N),activate(M)) | (21) |
U21#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (22) |
and#(tt,X) | → | activate#(X) | (23) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (24) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (25) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (26) |
isNat#(n__plus(V1,V2)) | → | and#(isNat(activate(V1)),n__isNat(activate(V2))) | (27) |
isNat#(n__s(V1)) | → | activate#(V1) | (28) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (29) |
plus#(N,0) | → | isNat#(N) | (30) |
plus#(N,0) | → | U11#(isNat(N),N) | (31) |
plus#(N,s(M)) | → | isNat#(M) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNat(N)) | (33) |
plus#(N,s(M)) | → | U21#(and(isNat(M),n__isNat(N)),M,N) | (34) |
activate#(n__0) | → | 0# | (35) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (36) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (37) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (38) |
activate#(n__isNat(X)) | → | isNat#(X) | (39) |
activate#(n__s(X)) | → | activate#(X) | (40) |
activate#(n__s(X)) | → | s#(activate(X)) | (41) |
The dependency pairs are split into 1 component.
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (29) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (24) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (36) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (37) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (38) |
plus#(N,0) | → | isNat#(N) | (30) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (25) |
activate#(n__isNat(X)) | → | isNat#(X) | (39) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (26) |
isNat#(n__plus(V1,V2)) | → | and#(isNat(activate(V1)),n__isNat(activate(V2))) | (27) |
and#(tt,X) | → | activate#(X) | (23) |
activate#(n__s(X)) | → | activate#(X) | (40) |
isNat#(n__s(V1)) | → | activate#(V1) | (28) |
plus#(N,0) | → | U11#(isNat(N),N) | (31) |
U11#(tt,N) | → | activate#(N) | (18) |
plus#(N,s(M)) | → | isNat#(M) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNat(N)) | (33) |
plus#(N,s(M)) | → | U21#(and(isNat(M),n__isNat(N)),M,N) | (34) |
U21#(tt,M,N) | → | activate#(M) | (19) |
U21#(tt,M,N) | → | activate#(N) | (20) |
U21#(tt,M,N) | → | plus#(activate(N),activate(M)) | (21) |
[and#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[plus(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[U21#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[U11(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[U21(x1, x2, x3)] | = | -∞ · x1 + 4 · x2 + 0 · x3 + 0 |
[plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[isNat#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[n__isNat(x1)] | = | 0 · x1 + 0 |
[tt] | = | 0 |
[n__0] | = | 0 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[0] | = | 0 |
[n__s(x1)] | = | 0 · x1 + 0 |
[n__plus(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[U11#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
U11(tt,N) | → | activate(N) | (1) |
U21(tt,M,N) | → | s(plus(activate(N),activate(M))) | (2) |
and(tt,X) | → | activate(X) | (3) |
isNat(n__0) | → | tt | (4) |
isNat(n__plus(V1,V2)) | → | and(isNat(activate(V1)),n__isNat(activate(V2))) | (5) |
isNat(n__s(V1)) | → | isNat(activate(V1)) | (6) |
plus(N,0) | → | U11(isNat(N),N) | (7) |
plus(N,s(M)) | → | U21(and(isNat(M),n__isNat(N)),M,N) | (8) |
0 | → | n__0 | (9) |
plus(X1,X2) | → | n__plus(X1,X2) | (10) |
isNat(X) | → | n__isNat(X) | (11) |
s(X) | → | n__s(X) | (12) |
activate(n__0) | → | 0 | (13) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (14) |
activate(n__isNat(X)) | → | isNat(X) | (15) |
activate(n__s(X)) | → | s(activate(X)) | (16) |
activate(X) | → | X | (17) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (24) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (36) |
[and#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[plus(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[U21#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 0 |
[isNat(x1)] | = | 0 · x1 + -∞ |
[U11(x1, x2)] | = | 0 · x1 + 2 · x2 + 1 |
[U21(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 2 · x3 + 0 |
[plus#(x1, x2)] | = | 2 · x1 + 0 · x2 + -∞ |
[isNat#(x1)] | = | 0 · x1 + -∞ |
[s(x1)] | = | 0 · x1 + 0 |
[n__isNat(x1)] | = | 0 · x1 + -∞ |
[tt] | = | 0 |
[n__0] | = | 0 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[0] | = | 0 |
[n__s(x1)] | = | 0 · x1 + 0 |
[n__plus(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[U11#(x1, x2)] | = | 0 · x1 + 2 · x2 + -∞ |
[and(x1, x2)] | = | -∞ · x1 + 2 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
U11(tt,N) | → | activate(N) | (1) |
U21(tt,M,N) | → | s(plus(activate(N),activate(M))) | (2) |
and(tt,X) | → | activate(X) | (3) |
isNat(n__0) | → | tt | (4) |
isNat(n__plus(V1,V2)) | → | and(isNat(activate(V1)),n__isNat(activate(V2))) | (5) |
isNat(n__s(V1)) | → | isNat(activate(V1)) | (6) |
plus(N,0) | → | U11(isNat(N),N) | (7) |
plus(N,s(M)) | → | U21(and(isNat(M),n__isNat(N)),M,N) | (8) |
0 | → | n__0 | (9) |
plus(X1,X2) | → | n__plus(X1,X2) | (10) |
isNat(X) | → | n__isNat(X) | (11) |
s(X) | → | n__s(X) | (12) |
activate(n__0) | → | 0 | (13) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (14) |
activate(n__isNat(X)) | → | isNat(X) | (15) |
activate(n__s(X)) | → | s(activate(X)) | (16) |
activate(X) | → | X | (17) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (37) |
plus#(N,0) | → | isNat#(N) | (30) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (25) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (26) |
isNat#(n__plus(V1,V2)) | → | and#(isNat(activate(V1)),n__isNat(activate(V2))) | (27) |
U11#(tt,N) | → | activate#(N) | (18) |
U21#(tt,M,N) | → | activate#(N) | (20) |
The dependency pairs are split into 1 component.
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (29) |
isNat#(n__s(V1)) | → | activate#(V1) | (28) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (38) |
plus#(N,s(M)) | → | isNat#(M) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNat(N)) | (33) |
and#(tt,X) | → | activate#(X) | (23) |
activate#(n__isNat(X)) | → | isNat#(X) | (39) |
activate#(n__s(X)) | → | activate#(X) | (40) |
plus#(N,s(M)) | → | U21#(and(isNat(M),n__isNat(N)),M,N) | (34) |
U21#(tt,M,N) | → | activate#(M) | (19) |
U21#(tt,M,N) | → | plus#(activate(N),activate(M)) | (21) |
[and#(x1, x2)] | = | 4 · x1 + 4 · x2 + 1 |
[plus(x1, x2)] | = | 4 · x1 + 4 · x2 + 1 |
[U21#(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 4 · x3 + 3 |
[isNat(x1)] | = | 1 · x1 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[U21(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 4 · x3 + 3 |
[plus#(x1, x2)] | = | 4 · x1 + 4 · x2 + 2 |
[isNat#(x1)] | = | 4 · x1 + 0 |
[s(x1)] | = | 1 · x1 + 1 |
[n__isNat(x1)] | = | 1 · x1 + 0 |
[tt] | = | 1 |
[n__0] | = | 2 |
[activate#(x1)] | = | 4 · x1 + 0 |
[0] | = | 2 |
[n__s(x1)] | = | 1 · x1 + 1 |
[n__plus(x1, x2)] | = | 4 · x1 + 4 · x2 + 1 |
[and(x1, x2)] | = | 0 · x1 + 2 · x2 + 1 |
[activate(x1)] | = | 1 · x1 + 0 |
U11(tt,N) | → | activate(N) | (1) |
U21(tt,M,N) | → | s(plus(activate(N),activate(M))) | (2) |
and(tt,X) | → | activate(X) | (3) |
isNat(n__0) | → | tt | (4) |
isNat(n__plus(V1,V2)) | → | and(isNat(activate(V1)),n__isNat(activate(V2))) | (5) |
isNat(n__s(V1)) | → | isNat(activate(V1)) | (6) |
plus(N,0) | → | U11(isNat(N),N) | (7) |
plus(N,s(M)) | → | U21(and(isNat(M),n__isNat(N)),M,N) | (8) |
0 | → | n__0 | (9) |
plus(X1,X2) | → | n__plus(X1,X2) | (10) |
isNat(X) | → | n__isNat(X) | (11) |
s(X) | → | n__s(X) | (12) |
activate(n__0) | → | 0 | (13) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (14) |
activate(n__isNat(X)) | → | isNat(X) | (15) |
activate(n__s(X)) | → | s(activate(X)) | (16) |
activate(X) | → | X | (17) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (29) |
isNat#(n__s(V1)) | → | activate#(V1) | (28) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (38) |
plus#(N,s(M)) | → | isNat#(M) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNat(N)) | (33) |
and#(tt,X) | → | activate#(X) | (23) |
activate#(n__s(X)) | → | activate#(X) | (40) |
plus#(N,s(M)) | → | U21#(and(isNat(M),n__isNat(N)),M,N) | (34) |
U21#(tt,M,N) | → | activate#(M) | (19) |
U21#(tt,M,N) | → | plus#(activate(N),activate(M)) | (21) |
The dependency pairs are split into 0 components.