The rewrite relation of the following TRS is considered.
U11(tt,V2) | → | U12(isNat(activate(V2))) | (1) |
U12(tt) | → | tt | (2) |
U21(tt) | → | tt | (3) |
U31(tt,N) | → | activate(N) | (4) |
U41(tt,M,N) | → | U42(isNat(activate(N)),activate(M),activate(N)) | (5) |
U42(tt,M,N) | → | s(plus(activate(N),activate(M))) | (6) |
isNat(n__0) | → | tt | (7) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (8) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (9) |
plus(N,0) | → | U31(isNat(N),N) | (10) |
plus(N,s(M)) | → | U41(isNat(M),M,N) | (11) |
0 | → | n__0 | (12) |
plus(X1,X2) | → | n__plus(X1,X2) | (13) |
s(X) | → | n__s(X) | (14) |
activate(n__0) | → | 0 | (15) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (16) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
U11#(tt,V2) | → | U12#(isNat(activate(V2))) | (19) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (20) |
U11#(tt,V2) | → | activate#(V2) | (21) |
U31#(tt,N) | → | activate#(N) | (22) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (23) |
U41#(tt,M,N) | → | isNat#(activate(N)) | (24) |
U41#(tt,M,N) | → | activate#(N) | (25) |
U41#(tt,M,N) | → | activate#(M) | (26) |
U42#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (27) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (28) |
U42#(tt,M,N) | → | activate#(N) | (29) |
U42#(tt,M,N) | → | activate#(M) | (30) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (31) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (32) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (33) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (34) |
isNat#(n__s(V1)) | → | U21#(isNat(activate(V1))) | (35) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
isNat#(n__s(V1)) | → | activate#(V1) | (37) |
plus#(N,0) | → | U31#(isNat(N),N) | (38) |
plus#(N,0) | → | isNat#(N) | (39) |
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (40) |
plus#(N,s(M)) | → | isNat#(M) | (41) |
activate#(n__0) | → | 0# | (42) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (43) |
activate#(n__s(X)) | → | s#(X) | (44) |
The dependency pairs are split into 1 component.
U11#(tt,V2) | → | isNat#(activate(V2)) | (20) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (31) |
U11#(tt,V2) | → | activate#(V2) | (21) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (43) |
plus#(N,0) | → | U31#(isNat(N),N) | (38) |
U31#(tt,N) | → | activate#(N) | (22) |
plus#(N,0) | → | isNat#(N) | (39) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (32) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (33) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (34) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
isNat#(n__s(V1)) | → | activate#(V1) | (37) |
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (40) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (23) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (28) |
plus#(N,s(M)) | → | isNat#(M) | (41) |
U42#(tt,M,N) | → | activate#(N) | (29) |
U42#(tt,M,N) | → | activate#(M) | (30) |
U41#(tt,M,N) | → | isNat#(activate(N)) | (24) |
U41#(tt,M,N) | → | activate#(N) | (25) |
U41#(tt,M,N) | → | activate#(M) | (26) |
[isNat#(x1)] | = | 1 + 2 · x1 |
[plus#(x1, x2)] | = | 2 + 2 · x1 + 2 · x2 |
[U11#(x1, x2)] | = | 1 + 2 · x2 |
[U31#(x1, x2)] | = | 2 + 2 · x2 |
[U41#(x1, x2, x3)] | = | 2 + 2 · x2 + 2 · x3 |
[U42#(x1, x2, x3)] | = | 2 + 2 · x2 + 2 · x3 |
[activate(x1)] | = | x1 |
[n__0] | = | 0 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 1 + x1 + 2 · x2 |
[plus(x1, x2)] | = | 1 + x1 + 2 · x2 |
[U31(x1, x2)] | = | x2 |
[isNat(x1)] | = | -2 |
[tt] | = | 0 |
[n__s(x1)] | = | x1 |
[s(x1)] | = | x1 |
[U11(x1, x2)] | = | -2 |
[U12(x1)] | = | -2 |
[U21(x1)] | = | 0 |
[U41(x1, x2, x3)] | = | 1 + 2 · x2 + x3 |
[U42(x1, x2, x3)] | = | 1 + 2 · x2 + x3 |
[activate#(x1)] | = | 1 + 2 · x1 |
activate(n__0) | → | 0 | (15) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (16) |
plus(N,0) | → | U31(isNat(N),N) | (10) |
U31(tt,N) | → | activate(N) | (4) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (8) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (9) |
plus(X1,X2) | → | n__plus(X1,X2) | (13) |
plus(N,s(M)) | → | U41(isNat(M),M,N) | (11) |
U21(tt) | → | tt | (3) |
U11(tt,V2) | → | U12(isNat(activate(V2))) | (1) |
U12(tt) | → | tt | (2) |
U41(tt,M,N) | → | U42(isNat(activate(N)),activate(M),activate(N)) | (5) |
U42(tt,M,N) | → | s(plus(activate(N),activate(M))) | (6) |
s(X) | → | n__s(X) | (14) |
0 | → | n__0 | (12) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (31) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (43) |
U31#(tt,N) | → | activate#(N) | (22) |
plus#(N,0) | → | isNat#(N) | (39) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (32) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (33) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (34) |
plus#(N,s(M)) | → | isNat#(M) | (41) |
U42#(tt,M,N) | → | activate#(N) | (29) |
U42#(tt,M,N) | → | activate#(M) | (30) |
U41#(tt,M,N) | → | isNat#(activate(N)) | (24) |
U41#(tt,M,N) | → | activate#(N) | (25) |
U41#(tt,M,N) | → | activate#(M) | (26) |
The dependency pairs are split into 2 components.
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (40) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (23) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (28) |
[U41#(x1, x2, x3)] | = | x2 |
[U42#(x1, x2, x3)] | = | -2 + x1 + x2 |
[plus#(x1, x2)] | = | -1 + x2 |
[U11(x1, x2)] | = | -2 + 2 · x1 |
[U12(x1)] | = | x1 |
[U21(x1)] | = | 2 |
[U31(x1, x2)] | = | 1 + x2 |
[U41(x1, x2, x3)] | = | 2 · x1 + x2 + 2 · x3 |
[U42(x1, x2, x3)] | = | 2 · x1 + x2 + 2 · x3 |
[isNat(x1)] | = | 2 |
[n__0] | = | 0 |
[tt] | = | 2 |
[n__plus(x1, x2)] | = | 2 + 2 · x1 + x2 |
[activate(x1)] | = | x1 |
[n__s(x1)] | = | 2 + x1 |
[0] | = | 0 |
[plus(x1, x2)] | = | 2 + 2 · x1 + x2 |
[s(x1)] | = | 2 + x1 |
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (40) |
The dependency pairs are split into 0 components.
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
[isNat#(x1)] | = | -2 + 2 · x1 |
[activate(x1)] | = | x1 |
[n__0] | = | 0 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 2 · x1 + 2 · x2 |
[plus(x1, x2)] | = | 2 · x1 + 2 · x2 |
[U31(x1, x2)] | = | x2 |
[isNat(x1)] | = | -2 |
[tt] | = | 0 |
[n__s(x1)] | = | 2 + x1 |
[s(x1)] | = | 2 + x1 |
[U11(x1, x2)] | = | -2 |
[U12(x1)] | = | -2 |
[U21(x1)] | = | 0 |
[U41(x1, x2, x3)] | = | 2 + 2 · x2 + 2 · x3 |
[U42(x1, x2, x3)] | = | 2 + 2 · x2 + 2 · x3 |
activate(n__0) | → | 0 | (15) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (16) |
plus(N,0) | → | U31(isNat(N),N) | (10) |
U31(tt,N) | → | activate(N) | (4) |
activate(n__s(X)) | → | s(X) | (17) |
activate(X) | → | X | (18) |
plus(X1,X2) | → | n__plus(X1,X2) | (13) |
plus(N,s(M)) | → | U41(isNat(M),M,N) | (11) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (8) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (9) |
U21(tt) | → | tt | (3) |
U11(tt,V2) | → | U12(isNat(activate(V2))) | (1) |
U12(tt) | → | tt | (2) |
U41(tt,M,N) | → | U42(isNat(activate(N)),activate(M),activate(N)) | (5) |
U42(tt,M,N) | → | s(plus(activate(N),activate(M))) | (6) |
s(X) | → | n__s(X) | (14) |
0 | → | n__0 | (12) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
There are no pairs anymore.