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,V2) | → | U32(isNat(activate(V2))) | (4) |
U32(tt) | → | tt | (5) |
U41(tt,N) | → | activate(N) | (6) |
U51(tt,M,N) | → | U52(isNat(activate(N)),activate(M),activate(N)) | (7) |
U52(tt,M,N) | → | s(plus(activate(N),activate(M))) | (8) |
U61(tt) | → | 0 | (9) |
U71(tt,M,N) | → | U72(isNat(activate(N)),activate(M),activate(N)) | (10) |
U72(tt,M,N) | → | plus(x(activate(N),activate(M)),activate(N)) | (11) |
isNat(n__0) | → | tt | (12) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (13) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (14) |
isNat(n__x(V1,V2)) | → | U31(isNat(activate(V1)),activate(V2)) | (15) |
plus(N,0) | → | U41(isNat(N),N) | (16) |
plus(N,s(M)) | → | U51(isNat(M),M,N) | (17) |
x(N,0) | → | U61(isNat(N)) | (18) |
x(N,s(M)) | → | U71(isNat(M),M,N) | (19) |
0 | → | n__0 | (20) |
plus(X1,X2) | → | n__plus(X1,X2) | (21) |
s(X) | → | n__s(X) | (22) |
x(X1,X2) | → | n__x(X1,X2) | (23) |
activate(n__0) | → | 0 | (24) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (25) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
activate(n__x(X1,X2)) | → | x(activate(X1),activate(X2)) | (27) |
activate(X) | → | X | (28) |
U11#(tt,V2) | → | U12#(isNat(activate(V2))) | (29) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (30) |
U11#(tt,V2) | → | activate#(V2) | (31) |
U31#(tt,V2) | → | U32#(isNat(activate(V2))) | (32) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (33) |
U31#(tt,V2) | → | activate#(V2) | (34) |
U41#(tt,N) | → | activate#(N) | (35) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (36) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (37) |
U51#(tt,M,N) | → | activate#(N) | (38) |
U51#(tt,M,N) | → | activate#(M) | (39) |
U52#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (40) |
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
U52#(tt,M,N) | → | activate#(N) | (42) |
U52#(tt,M,N) | → | activate#(M) | (43) |
U61#(tt) | → | 0# | (44) |
U71#(tt,M,N) | → | U72#(isNat(activate(N)),activate(M),activate(N)) | (45) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (46) |
U71#(tt,M,N) | → | activate#(N) | (47) |
U71#(tt,M,N) | → | activate#(M) | (48) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (49) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (50) |
U72#(tt,M,N) | → | activate#(N) | (51) |
U72#(tt,M,N) | → | activate#(M) | (52) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (53) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (54) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (56) |
isNat#(n__s(V1)) | → | U21#(isNat(activate(V1))) | (57) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (58) |
isNat#(n__s(V1)) | → | activate#(V1) | (59) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (60) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (61) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (62) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (63) |
plus#(N,0) | → | U41#(isNat(N),N) | (64) |
plus#(N,0) | → | isNat#(N) | (65) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (66) |
plus#(N,s(M)) | → | isNat#(M) | (67) |
x#(N,0) | → | U61#(isNat(N)) | (68) |
x#(N,0) | → | isNat#(N) | (69) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (70) |
x#(N,s(M)) | → | isNat#(M) | (71) |
activate#(n__0) | → | 0# | (72) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (73) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (74) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (75) |
activate#(n__s(X)) | → | s#(activate(X)) | (76) |
activate#(n__s(X)) | → | activate#(X) | (77) |
activate#(n__x(X1,X2)) | → | x#(activate(X1),activate(X2)) | (78) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (79) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (80) |
The dependency pairs are split into 1 component.
U11#(tt,V2) | → | isNat#(activate(V2)) | (30) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (53) |
U11#(tt,V2) | → | activate#(V2) | (31) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (73) |
plus#(N,0) | → | U41#(isNat(N),N) | (64) |
U41#(tt,N) | → | activate#(N) | (35) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (74) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (75) |
activate#(n__s(X)) | → | activate#(X) | (77) |
activate#(n__x(X1,X2)) | → | x#(activate(X1),activate(X2)) | (78) |
x#(N,0) | → | isNat#(N) | (69) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (54) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (79) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (80) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (56) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (58) |
isNat#(n__s(V1)) | → | activate#(V1) | (59) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (60) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (33) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (61) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (62) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (63) |
U31#(tt,V2) | → | activate#(V2) | (34) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (70) |
U71#(tt,M,N) | → | U72#(isNat(activate(N)),activate(M),activate(N)) | (45) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (49) |
plus#(N,0) | → | isNat#(N) | (65) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (66) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (36) |
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
plus#(N,s(M)) | → | isNat#(M) | (67) |
U52#(tt,M,N) | → | activate#(N) | (42) |
U52#(tt,M,N) | → | activate#(M) | (43) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (37) |
U51#(tt,M,N) | → | activate#(N) | (38) |
U51#(tt,M,N) | → | activate#(M) | (39) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (50) |
x#(N,s(M)) | → | isNat#(M) | (71) |
U72#(tt,M,N) | → | activate#(N) | (51) |
U72#(tt,M,N) | → | activate#(M) | (52) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (46) |
U71#(tt,M,N) | → | activate#(N) | (47) |
U71#(tt,M,N) | → | activate#(M) | (48) |
prec(U11#) | = | 1 | stat(U11#) | = | mul | |
prec(tt) | = | 3 | stat(tt) | = | mul | |
prec(isNat#) | = | 0 | stat(isNat#) | = | mul | |
prec(n__plus) | = | 5 | stat(n__plus) | = | mul | |
prec(isNat) | = | 3 | stat(isNat) | = | mul | |
prec(plus#) | = | 2 | stat(plus#) | = | mul | |
prec(0) | = | 6 | stat(0) | = | mul | |
prec(n__s) | = | 4 | stat(n__s) | = | mul | |
prec(n__x) | = | 9 | stat(n__x) | = | mul | |
prec(x#) | = | 9 | stat(x#) | = | mul | |
prec(U31#) | = | 7 | stat(U31#) | = | mul | |
prec(s) | = | 4 | stat(s) | = | mul | |
prec(U71#) | = | 9 | stat(U71#) | = | mul | |
prec(U72#) | = | 9 | stat(U72#) | = | mul | |
prec(x) | = | 9 | stat(x) | = | mul | |
prec(U51#) | = | 2 | stat(U51#) | = | mul | |
prec(U52#) | = | 2 | stat(U52#) | = | mul | |
prec(n__0) | = | 6 | stat(n__0) | = | mul | |
prec(plus) | = | 5 | stat(plus) | = | mul | |
prec(U71) | = | 9 | stat(U71) | = | mul | |
prec(U72) | = | 9 | stat(U72) | = | mul | |
prec(U61) | = | 8 | stat(U61) | = | mul | |
prec(U51) | = | 5 | stat(U51) | = | mul | |
prec(U32) | = | 3 | stat(U32) | = | mul | |
prec(U52) | = | 5 | stat(U52) | = | mul |
π(U11#) | = | [2] |
π(tt) | = | [] |
π(isNat#) | = | [1] |
π(activate) | = | 1 |
π(n__plus) | = | [1,2] |
π(isNat) | = | [] |
π(activate#) | = | 1 |
π(plus#) | = | [1,2] |
π(0) | = | [] |
π(U41#) | = | 2 |
π(n__s) | = | [1] |
π(n__x) | = | [1,2] |
π(x#) | = | [1,2] |
π(U31#) | = | [1,2] |
π(s) | = | [1] |
π(U71#) | = | [1,2,3] |
π(U72#) | = | [1,2,3] |
π(x) | = | [1,2] |
π(U51#) | = | [1,2,3] |
π(U52#) | = | [2,3] |
π(n__0) | = | [] |
π(plus) | = | [1,2] |
π(U41) | = | 2 |
π(U71) | = | [1,2,3] |
π(U72) | = | [1,2,3] |
π(U11) | = | 1 |
π(U21) | = | 1 |
π(U31) | = | 1 |
π(U61) | = | [1] |
π(U51) | = | [1,2,3] |
π(U32) | = | [] |
π(U12) | = | 1 |
π(U52) | = | [1,2,3] |
U11#(tt,V2) | → | isNat#(activate(V2)) | (30) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (53) |
U11#(tt,V2) | → | activate#(V2) | (31) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (73) |
plus#(N,0) | → | U41#(isNat(N),N) | (64) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (74) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (75) |
activate#(n__s(X)) | → | activate#(X) | (77) |
x#(N,0) | → | isNat#(N) | (69) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (54) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (79) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (80) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (56) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (58) |
isNat#(n__s(V1)) | → | activate#(V1) | (59) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (60) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (33) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (61) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (62) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (63) |
U31#(tt,V2) | → | activate#(V2) | (34) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (70) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (49) |
plus#(N,0) | → | isNat#(N) | (65) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (66) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (36) |
plus#(N,s(M)) | → | isNat#(M) | (67) |
U52#(tt,M,N) | → | activate#(N) | (42) |
U52#(tt,M,N) | → | activate#(M) | (43) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (37) |
U51#(tt,M,N) | → | activate#(N) | (38) |
U51#(tt,M,N) | → | activate#(M) | (39) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (50) |
x#(N,s(M)) | → | isNat#(M) | (71) |
U72#(tt,M,N) | → | activate#(N) | (51) |
U72#(tt,M,N) | → | activate#(M) | (52) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (46) |
U71#(tt,M,N) | → | activate#(N) | (47) |
U71#(tt,M,N) | → | activate#(M) | (48) |
The dependency pairs are split into 0 components.