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) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (29) |
U52#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (30) |
activate#(n__x(X1,X2)) | → | x#(activate(X1),activate(X2)) | (31) |
x#(N,s(M)) | → | isNat#(M) | (32) |
U52#(tt,M,N) | → | activate#(M) | (33) |
U11#(tt,V2) | → | activate#(V2) | (34) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (35) |
activate#(n__s(X)) | → | activate#(X) | (36) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (37) |
U31#(tt,V2) | → | activate#(V2) | (38) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (39) |
plus#(N,0) | → | U41#(isNat(N),N) | (40) |
U61#(tt) | → | 0# | (41) |
x#(N,0) | → | isNat#(N) | (42) |
U11#(tt,V2) | → | U12#(isNat(activate(V2))) | (43) |
U31#(tt,V2) | → | U32#(isNat(activate(V2))) | (44) |
isNat#(n__s(V1)) | → | U21#(isNat(activate(V1))) | (45) |
U72#(tt,M,N) | → | activate#(N) | (46) |
U72#(tt,M,N) | → | activate#(N) | (46) |
U71#(tt,M,N) | → | U72#(isNat(activate(N)),activate(M),activate(N)) | (47) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U52#(tt,M,N) | → | activate#(N) | (49) |
activate#(n__0) | → | 0# | (50) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (51) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (52) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (53) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (54) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (55) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (56) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (57) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (58) |
plus#(N,0) | → | isNat#(N) | (59) |
U72#(tt,M,N) | → | activate#(M) | (60) |
plus#(N,s(M)) | → | isNat#(M) | (61) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (62) |
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (63) |
U71#(tt,M,N) | → | activate#(N) | (64) |
U51#(tt,M,N) | → | activate#(M) | (65) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (66) |
U71#(tt,M,N) | → | activate#(M) | (67) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (68) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (69) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (70) |
U71#(tt,M,N) | → | activate#(N) | (64) |
activate#(n__s(X)) | → | s#(activate(X)) | (71) |
U51#(tt,M,N) | → | activate#(N) | (48) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (72) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (73) |
U41#(tt,N) | → | activate#(N) | (74) |
x#(N,0) | → | U61#(isNat(N)) | (75) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (76) |
isNat#(n__s(V1)) | → | activate#(V1) | (77) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (78) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (79) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (80) |
The dependency pairs are split into 1 component.
activate#(n__plus(X1,X2)) | → | activate#(X2) | (80) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (79) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (56) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (54) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (55) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (78) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (53) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (52) |
isNat#(n__s(V1)) | → | activate#(V1) | (77) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (51) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (76) |
U52#(tt,M,N) | → | activate#(N) | (49) |
U41#(tt,N) | → | activate#(N) | (74) |
U51#(tt,M,N) | → | activate#(N) | (48) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (73) |
U71#(tt,M,N) | → | U72#(isNat(activate(N)),activate(M),activate(N)) | (47) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (72) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U71#(tt,M,N) | → | activate#(N) | (64) |
U72#(tt,M,N) | → | activate#(N) | (46) |
U72#(tt,M,N) | → | activate#(N) | (46) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (69) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (70) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (68) |
x#(N,0) | → | isNat#(N) | (42) |
U71#(tt,M,N) | → | activate#(M) | (67) |
plus#(N,0) | → | U41#(isNat(N),N) | (40) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (66) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (39) |
U51#(tt,M,N) | → | activate#(M) | (65) |
U31#(tt,V2) | → | activate#(V2) | (38) |
U71#(tt,M,N) | → | activate#(N) | (64) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (37) |
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (63) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (62) |
activate#(n__s(X)) | → | activate#(X) | (36) |
plus#(N,s(M)) | → | isNat#(M) | (61) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (35) |
U11#(tt,V2) | → | activate#(V2) | (34) |
U72#(tt,M,N) | → | activate#(M) | (60) |
U52#(tt,M,N) | → | activate#(M) | (33) |
plus#(N,0) | → | isNat#(N) | (59) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (58) |
x#(N,s(M)) | → | isNat#(M) | (32) |
activate#(n__x(X1,X2)) | → | x#(activate(X1),activate(X2)) | (31) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (29) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (57) |
[0#] | = | 0 |
[U72#(x1, x2, x3)] | = | max(x2 + 5, x3 + 4, 0) |
[U32#(x1)] | = | 0 |
[U21(x1)] | = | x1 + 1 |
[U11(x1, x2)] | = | max(x1 + 3, 0) |
[s(x1)] | = | x1 + 0 |
[isNat#(x1)] | = | x1 + 1 |
[activate(x1)] | = | x1 + 0 |
[U71(x1, x2, x3)] | = | max(x2 + 4, x3 + 3, 0) |
[plus#(x1, x2)] | = | max(x1 + 1, x2 + 3, 0) |
[activate#(x1)] | = | x1 + 1 |
[U72(x1, x2, x3)] | = | max(x2 + 4, x3 + 3, 0) |
[U52#(x1, x2, x3)] | = | max(x2 + 3, x3 + 1, 0) |
[U12(x1)] | = | 1 |
[x(x1, x2)] | = | max(x1 + 3, x2 + 4, 0) |
[n__s(x1)] | = | x1 + 0 |
[U12#(x1)] | = | 0 |
[0] | = | 2 |
[x#(x1, x2)] | = | max(x1 + 4, x2 + 5, 0) |
[s#(x1)] | = | 0 |
[n__plus(x1, x2)] | = | max(x1 + 0, x2 + 2, 0) |
[U32(x1)] | = | 6 |
[n__0] | = | 2 |
[isNat(x1)] | = | x1 + 2 |
[n__x(x1, x2)] | = | max(x1 + 3, x2 + 4, 0) |
[U52(x1, x2, x3)] | = | max(x2 + 2, x3 + 0, 0) |
[plus(x1, x2)] | = | max(x1 + 0, x2 + 2, 0) |
[U61(x1)] | = | 4 |
[U51#(x1, x2, x3)] | = | max(x2 + 3, x3 + 1, 0) |
[U11#(x1, x2)] | = | max(x2 + 2, 0) |
[U31(x1, x2)] | = | max(x1 + 5, x2 + 6, 0) |
[U41#(x1, x2)] | = | max(x2 + 1, 0) |
[U21#(x1)] | = | 0 |
[tt] | = | 0 |
[U71#(x1, x2, x3)] | = | max(x2 + 5, x3 + 4, 0) |
[U51(x1, x2, x3)] | = | max(x2 + 2, x3 + 0, 0) |
[U41(x1, x2)] | = | max(x2 + 0, 0) |
[U31#(x1, x2)] | = | max(x2 + 3, 0) |
[U61#(x1)] | = | 0 |
x(N,0) | → | U61(isNat(N)) | (18) |
U52(tt,M,N) | → | s(plus(activate(N),activate(M))) | (8) |
plus(N,0) | → | U41(isNat(N),N) | (16) |
plus(X1,X2) | → | n__plus(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
x(N,s(M)) | → | U71(isNat(M),M,N) | (19) |
plus(N,s(M)) | → | U51(isNat(M),M,N) | (17) |
activate(n__x(X1,X2)) | → | x(activate(X1),activate(X2)) | (27) |
s(X) | → | n__s(X) | (22) |
activate(X) | → | X | (28) |
U71(tt,M,N) | → | U72(isNat(activate(N)),activate(M),activate(N)) | (10) |
U51(tt,M,N) | → | U52(isNat(activate(N)),activate(M),activate(N)) | (7) |
0 | → | n__0 | (20) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (25) |
x(X1,X2) | → | n__x(X1,X2) | (23) |
activate(n__0) | → | 0 | (24) |
U72(tt,M,N) | → | plus(x(activate(N),activate(M)),activate(N)) | (11) |
U61(tt) | → | 0 | (9) |
U41(tt,N) | → | activate(N) | (6) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (80) |
isNat#(n__x(V1,V2)) | → | activate#(V2) | (79) |
U71#(tt,M,N) | → | isNat#(activate(N)) | (56) |
U31#(tt,V2) | → | isNat#(activate(V2)) | (54) |
isNat#(n__x(V1,V2)) | → | activate#(V1) | (55) |
activate#(n__x(X1,X2)) | → | activate#(X1) | (53) |
activate#(n__x(X1,X2)) | → | activate#(X2) | (51) |
U71#(tt,M,N) | → | activate#(N) | (64) |
U72#(tt,M,N) | → | activate#(N) | (46) |
U72#(tt,M,N) | → | activate#(N) | (46) |
x#(N,0) | → | isNat#(N) | (42) |
U71#(tt,M,N) | → | activate#(M) | (67) |
isNat#(n__x(V1,V2)) | → | isNat#(activate(V1)) | (39) |
U51#(tt,M,N) | → | activate#(M) | (65) |
U31#(tt,V2) | → | activate#(V2) | (38) |
U71#(tt,M,N) | → | activate#(N) | (64) |
isNat#(n__x(V1,V2)) | → | U31#(isNat(activate(V1)),activate(V2)) | (37) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (62) |
plus#(N,s(M)) | → | isNat#(M) | (61) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (35) |
U11#(tt,V2) | → | activate#(V2) | (34) |
U72#(tt,M,N) | → | activate#(M) | (60) |
U52#(tt,M,N) | → | activate#(M) | (33) |
x#(N,s(M)) | → | isNat#(M) | (32) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (29) |
The dependency pairs are split into 1 component.
U52#(tt,M,N) | → | activate#(N) | (49) |
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (63) |
plus#(N,0) | → | isNat#(N) | (59) |
plus#(N,0) | → | U41#(isNat(N),N) | (40) |
activate#(n__s(X)) | → | activate#(X) | (36) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (66) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (69) |
activate#(n__x(X1,X2)) | → | x#(activate(X1),activate(X2)) | (31) |
U71#(tt,M,N) | → | U72#(isNat(activate(N)),activate(M),activate(N)) | (47) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (57) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (70) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (68) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (72) |
isNat#(n__s(V1)) | → | activate#(V1) | (77) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (52) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (76) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (58) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (78) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (73) |
U41#(tt,N) | → | activate#(N) | (74) |
π(U21) | = | 1 |
π(isNat#) | = | 1 |
π(activate) | = | 1 |
π(activate#) | = | 1 |
π(U12) | = | 1 |
π(U12#) | = | 1 |
π(U32) | = | 1 |
π(U61) | = | 1 |
π(U11#) | = | 1 |
π(U41#) | = | 2 |
π(U21#) | = | 1 |
π(U41) | = | 2 |
π(U31#) | = | 2 |
π(U61#) | = | 1 |
prec(0#) | = | 0 | status(0#) | = | [] | list-extension(0#) | = | Lex | ||
prec(U72#) | = | 7 | status(U72#) | = | [3, 2, 1] | list-extension(U72#) | = | Lex | ||
prec(U32#) | = | 0 | status(U32#) | = | [] | list-extension(U32#) | = | Lex | ||
prec(U11) | = | 1 | status(U11) | = | [] | list-extension(U11) | = | Lex | ||
prec(s) | = | 3 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(U71) | = | 7 | status(U71) | = | [3, 2, 1] | list-extension(U71) | = | Lex | ||
prec(plus#) | = | 4 | status(plus#) | = | [1, 2] | list-extension(plus#) | = | Lex | ||
prec(U72) | = | 7 | status(U72) | = | [3, 2, 1] | list-extension(U72) | = | Lex | ||
prec(U52#) | = | 4 | status(U52#) | = | [3, 2] | list-extension(U52#) | = | Lex | ||
prec(x) | = | 7 | status(x) | = | [1, 2] | list-extension(x) | = | Lex | ||
prec(n__s) | = | 3 | status(n__s) | = | [1] | list-extension(n__s) | = | Lex | ||
prec(0) | = | 1 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(x#) | = | 7 | status(x#) | = | [1, 2] | list-extension(x#) | = | Lex | ||
prec(s#) | = | 0 | status(s#) | = | [] | list-extension(s#) | = | Lex | ||
prec(n__plus) | = | 5 | status(n__plus) | = | [2, 1] | list-extension(n__plus) | = | Lex | ||
prec(n__0) | = | 1 | status(n__0) | = | [] | list-extension(n__0) | = | Lex | ||
prec(isNat) | = | 1 | status(isNat) | = | [] | list-extension(isNat) | = | Lex | ||
prec(n__x) | = | 7 | status(n__x) | = | [1, 2] | list-extension(n__x) | = | Lex | ||
prec(U52) | = | 5 | status(U52) | = | [2, 3, 1] | list-extension(U52) | = | Lex | ||
prec(plus) | = | 5 | status(plus) | = | [2, 1] | list-extension(plus) | = | Lex | ||
prec(U51#) | = | 4 | status(U51#) | = | [3, 2] | list-extension(U51#) | = | Lex | ||
prec(U31) | = | 1 | status(U31) | = | [] | list-extension(U31) | = | Lex | ||
prec(tt) | = | 1 | status(tt) | = | [] | list-extension(tt) | = | Lex | ||
prec(U71#) | = | 7 | status(U71#) | = | [3, 2, 1] | list-extension(U71#) | = | Lex | ||
prec(U51) | = | 5 | status(U51) | = | [2, 3, 1] | list-extension(U51) | = | Lex |
[0#] | = | 0 |
[U72#(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U32#(x1)] | = | 0 |
[U11(x1, x2)] | = | max(0) |
[s(x1)] | = | x1 + 0 |
[U71(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[plus#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U72(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U52#(x1, x2, x3)] | = | max(x2 + 0, x3 + 0, 0) |
[x(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[n__s(x1)] | = | x1 + 0 |
[0] | = | 0 |
[x#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[s#(x1)] | = | 0 |
[n__plus(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[n__0] | = | 0 |
[isNat(x1)] | = | 0 |
[n__x(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U52(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[plus(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U51#(x1, x2, x3)] | = | max(x2 + 0, x3 + 0, 0) |
[U31(x1, x2)] | = | max(0) |
[tt] | = | 0 |
[U71#(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U51(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
x(N,0) | → | U61(isNat(N)) | (18) |
U31(tt,V2) | → | U32(isNat(activate(V2))) | (4) |
isNat(n__x(V1,V2)) | → | U31(isNat(activate(V1)),activate(V2)) | (15) |
U52(tt,M,N) | → | s(plus(activate(N),activate(M))) | (8) |
U11(tt,V2) | → | U12(isNat(activate(V2))) | (1) |
U21(tt) | → | tt | (3) |
plus(N,0) | → | U41(isNat(N),N) | (16) |
plus(X1,X2) | → | n__plus(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
x(N,s(M)) | → | U71(isNat(M),M,N) | (19) |
plus(N,s(M)) | → | U51(isNat(M),M,N) | (17) |
activate(n__x(X1,X2)) | → | x(activate(X1),activate(X2)) | (27) |
s(X) | → | n__s(X) | (22) |
activate(X) | → | X | (28) |
U32(tt) | → | tt | (5) |
U71(tt,M,N) | → | U72(isNat(activate(N)),activate(M),activate(N)) | (10) |
U51(tt,M,N) | → | U52(isNat(activate(N)),activate(M),activate(N)) | (7) |
0 | → | n__0 | (20) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (25) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (14) |
isNat(n__0) | → | tt | (12) |
x(X1,X2) | → | n__x(X1,X2) | (23) |
activate(n__0) | → | 0 | (24) |
U72(tt,M,N) | → | plus(x(activate(N),activate(M)),activate(N)) | (11) |
U61(tt) | → | 0 | (9) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (13) |
U41(tt,N) | → | activate(N) | (6) |
U12(tt) | → | tt | (2) |
U52#(tt,M,N) | → | activate#(N) | (49) |
plus#(N,0) | → | isNat#(N) | (59) |
plus#(N,0) | → | U41#(isNat(N),N) | (40) |
activate#(n__s(X)) | → | activate#(X) | (36) |
x#(N,s(M)) | → | U71#(isNat(M),M,N) | (66) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U51#(tt,M,N) | → | activate#(N) | (48) |
U51#(tt,M,N) | → | isNat#(activate(N)) | (57) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (68) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (72) |
isNat#(n__s(V1)) | → | activate#(V1) | (77) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (52) |
U72#(tt,M,N) | → | x#(activate(N),activate(M)) | (76) |
U72#(tt,M,N) | → | plus#(x(activate(N),activate(M)),activate(N)) | (58) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (78) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (73) |
The dependency pairs are split into 1 component.
U52#(tt,M,N) | → | plus#(activate(N),activate(M)) | (63) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (69) |
U51#(tt,M,N) | → | U52#(isNat(activate(N)),activate(M),activate(N)) | (70) |
π(U21) | = | 1 |
π(isNat#) | = | 1 |
π(activate) | = | 1 |
π(activate#) | = | 1 |
π(U12) | = | 1 |
π(U12#) | = | 1 |
π(U32) | = | 1 |
π(U61) | = | 1 |
π(U11#) | = | 1 |
π(U41#) | = | 2 |
π(U21#) | = | 1 |
π(U41) | = | 2 |
π(U31#) | = | 2 |
π(U61#) | = | 1 |
prec(0#) | = | 0 | status(0#) | = | [] | list-extension(0#) | = | Lex | ||
prec(U72#) | = | 7 | status(U72#) | = | [3, 2, 1] | list-extension(U72#) | = | Lex | ||
prec(U32#) | = | 0 | status(U32#) | = | [] | list-extension(U32#) | = | Lex | ||
prec(U11) | = | 1 | status(U11) | = | [] | list-extension(U11) | = | Lex | ||
prec(s) | = | 3 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(U71) | = | 7 | status(U71) | = | [3, 2, 1] | list-extension(U71) | = | Lex | ||
prec(plus#) | = | 6 | status(plus#) | = | [1, 2] | list-extension(plus#) | = | Lex | ||
prec(U72) | = | 7 | status(U72) | = | [3, 2, 1] | list-extension(U72) | = | Lex | ||
prec(U52#) | = | 6 | status(U52#) | = | [3, 2] | list-extension(U52#) | = | Lex | ||
prec(x) | = | 7 | status(x) | = | [1, 2] | list-extension(x) | = | Lex | ||
prec(n__s) | = | 3 | status(n__s) | = | [1] | list-extension(n__s) | = | Lex | ||
prec(0) | = | 1 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(x#) | = | 7 | status(x#) | = | [1, 2] | list-extension(x#) | = | Lex | ||
prec(s#) | = | 0 | status(s#) | = | [] | list-extension(s#) | = | Lex | ||
prec(n__plus) | = | 5 | status(n__plus) | = | [2, 1] | list-extension(n__plus) | = | Lex | ||
prec(n__0) | = | 1 | status(n__0) | = | [] | list-extension(n__0) | = | Lex | ||
prec(isNat) | = | 1 | status(isNat) | = | [] | list-extension(isNat) | = | Lex | ||
prec(n__x) | = | 7 | status(n__x) | = | [1, 2] | list-extension(n__x) | = | Lex | ||
prec(U52) | = | 5 | status(U52) | = | [2, 3, 1] | list-extension(U52) | = | Lex | ||
prec(plus) | = | 5 | status(plus) | = | [2, 1] | list-extension(plus) | = | Lex | ||
prec(U51#) | = | 6 | status(U51#) | = | [3, 2] | list-extension(U51#) | = | Lex | ||
prec(U31) | = | 1 | status(U31) | = | [] | list-extension(U31) | = | Lex | ||
prec(tt) | = | 1 | status(tt) | = | [] | list-extension(tt) | = | Lex | ||
prec(U71#) | = | 7 | status(U71#) | = | [3, 2, 1] | list-extension(U71#) | = | Lex | ||
prec(U51) | = | 5 | status(U51) | = | [2, 3, 1] | list-extension(U51) | = | Lex |
[0#] | = | 0 |
[U72#(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U32#(x1)] | = | 0 |
[U11(x1, x2)] | = | max(0) |
[s(x1)] | = | x1 + 0 |
[U71(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[plus#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U72(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U52#(x1, x2, x3)] | = | max(x2 + 0, x3 + 0, 0) |
[x(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[n__s(x1)] | = | x1 + 0 |
[0] | = | 0 |
[x#(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[s#(x1)] | = | 0 |
[n__plus(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[n__0] | = | 0 |
[isNat(x1)] | = | 0 |
[n__x(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U52(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[plus(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[U51#(x1, x2, x3)] | = | max(x2 + 0, x3 + 0, 0) |
[U31(x1, x2)] | = | max(0) |
[tt] | = | 0 |
[U71#(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U51(x1, x2, x3)] | = | max(x1 + 0, x2 + 0, x3 + 0, 0) |
x(N,0) | → | U61(isNat(N)) | (18) |
U31(tt,V2) | → | U32(isNat(activate(V2))) | (4) |
isNat(n__x(V1,V2)) | → | U31(isNat(activate(V1)),activate(V2)) | (15) |
U52(tt,M,N) | → | s(plus(activate(N),activate(M))) | (8) |
U11(tt,V2) | → | U12(isNat(activate(V2))) | (1) |
U21(tt) | → | tt | (3) |
plus(N,0) | → | U41(isNat(N),N) | (16) |
plus(X1,X2) | → | n__plus(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
x(N,s(M)) | → | U71(isNat(M),M,N) | (19) |
plus(N,s(M)) | → | U51(isNat(M),M,N) | (17) |
activate(n__x(X1,X2)) | → | x(activate(X1),activate(X2)) | (27) |
s(X) | → | n__s(X) | (22) |
activate(X) | → | X | (28) |
U32(tt) | → | tt | (5) |
U71(tt,M,N) | → | U72(isNat(activate(N)),activate(M),activate(N)) | (10) |
U51(tt,M,N) | → | U52(isNat(activate(N)),activate(M),activate(N)) | (7) |
0 | → | n__0 | (20) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (25) |
isNat(n__s(V1)) | → | U21(isNat(activate(V1))) | (14) |
isNat(n__0) | → | tt | (12) |
x(X1,X2) | → | n__x(X1,X2) | (23) |
activate(n__0) | → | 0 | (24) |
U72(tt,M,N) | → | plus(x(activate(N),activate(M)),activate(N)) | (11) |
U61(tt) | → | 0 | (9) |
isNat(n__plus(V1,V2)) | → | U11(isNat(activate(V1)),activate(V2)) | (13) |
U41(tt,N) | → | activate(N) | (6) |
U12(tt) | → | tt | (2) |
plus#(N,s(M)) | → | U51#(isNat(M),M,N) | (69) |
The dependency pairs are split into 0 components.