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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
U11#(tt,V2) | → | activate#(V2) | (19) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (20) |
U11#(tt,V2) | → | U12#(isNat(activate(V2))) | (21) |
U31#(tt,N) | → | activate#(N) | (22) |
U41#(tt,M,N) | → | activate#(M) | (23) |
U41#(tt,M,N) | → | activate#(N) | (24) |
U41#(tt,M,N) | → | isNat#(activate(N)) | (25) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (26) |
U42#(tt,M,N) | → | activate#(M) | (27) |
U42#(tt,M,N) | → | activate#(N) | (28) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (29) |
U42#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (30) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (31) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (33) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (34) |
isNat#(n__s(V1)) | → | activate#(V1) | (35) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
isNat#(n__s(V1)) | → | U21#(isNat(activate(V1))) | (37) |
plus#(N,0) | → | isNat#(N) | (38) |
plus#(N,0) | → | U31#(isNat(N),N) | (39) |
plus#(N,s(M)) | → | isNat#(M) | (40) |
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (41) |
activate#(n__0) | → | 0# | (42) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (43) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (44) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (45) |
activate#(n__s(X)) | → | activate#(X) | (46) |
activate#(n__s(X)) | → | s#(activate(X)) | (47) |
The dependency pairs are split into 1 component.
plus#(N,0) | → | U31#(isNat(N),N) | (39) |
U31#(tt,N) | → | activate#(N) | (22) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (43) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (44) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (45) |
plus#(N,0) | → | isNat#(N) | (38) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (31) |
activate#(n__s(X)) | → | activate#(X) | (46) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (33) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (34) |
U11#(tt,V2) | → | activate#(V2) | (19) |
U11#(tt,V2) | → | isNat#(activate(V2)) | (20) |
isNat#(n__s(V1)) | → | activate#(V1) | (35) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
plus#(N,s(M)) | → | isNat#(M) | (40) |
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (41) |
U41#(tt,M,N) | → | activate#(M) | (23) |
U41#(tt,M,N) | → | activate#(N) | (24) |
U41#(tt,M,N) | → | isNat#(activate(N)) | (25) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (26) |
U42#(tt,M,N) | → | activate#(M) | (27) |
U42#(tt,M,N) | → | activate#(N) | (28) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (29) |
[U12(x1)] | = | 6 · x1 + 13 |
[U11#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U42(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 1 · x3 + 6 |
[U11(x1, x2)] | = | -∞ · x1 + 4 · x2 + -∞ |
[isNat(x1)] | = | 0 · x1 + 1 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[U41#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 6 |
[isNat#(x1)] | = | 0 · x1 + -∞ |
[U42#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + -∞ |
[U21(x1)] | = | 1 · x1 + 8 |
[s(x1)] | = | 0 · x1 + 6 |
[tt] | = | 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 1 · x3 + 6 |
[U31#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[0] | = | 4 |
[n__plus(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[n__0] | = | 4 |
[plus(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[n__s(x1)] | = | 0 · x1 + 6 |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
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) |
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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (44) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
isNat#(n__plus(V1,V2)) | → | isNat#(activate(V1)) | (33) |
[U12(x1)] | = | 6 · x1 + -16 |
[U11#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U42(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 0 · x3 + 5 |
[U11(x1, x2)] | = | -∞ · x1 + 0 · x2 + -16 |
[isNat(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[U41#(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 0 · x3 + 0 |
[isNat#(x1)] | = | 0 · x1 + 0 |
[U42#(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 0 · x3 + -16 |
[U21(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 3 |
[tt] | = | 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 0 · x3 + 5 |
[U31#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 2 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[plus#(x1, x2)] | = | 0 · x1 + 2 · x2 + -∞ |
[n__0] | = | 0 |
[plus(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[n__s(x1)] | = | 0 · x1 + 3 |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + -8 |
[activate(x1)] | = | 0 · x1 + -∞ |
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) |
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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (43) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (31) |
plus#(N,s(M)) | → | isNat#(M) | (40) |
U41#(tt,M,N) | → | activate#(M) | (23) |
U42#(tt,M,N) | → | activate#(M) | (27) |
[U12(x1)] | = | 0 · x1 + 0 |
[U11#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U42(x1, x2, x3)] | = | -∞ · x1 + 1 · x2 + 1 · x3 + 1 |
[U11(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + 0 |
[U41#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 0 |
[isNat#(x1)] | = | 0 · x1 + 0 |
[U42#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 0 |
[U21(x1)] | = | -∞ · x1 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[tt] | = | 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 1 · x2 + 1 · x3 + 1 |
[U31#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 1 |
[plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[n__0] | = | 0 |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 1 |
[n__s(x1)] | = | 0 · x1 + -∞ |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + 0 |
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) |
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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (45) |
isNat#(n__plus(V1,V2)) | → | U11#(isNat(activate(V1)),activate(V2)) | (34) |
The dependency pairs are split into 3 components.
plus#(N,s(M)) | → | U41#(isNat(M),M,N) | (41) |
U41#(tt,M,N) | → | U42#(isNat(activate(N)),activate(M),activate(N)) | (26) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (29) |
[U12(x1)] | = | 0 · x1 + 0 |
[U42(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 1 · x3 + 1 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNat(x1)] | = | 0 · x1 + 1 |
[U41#(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 4 |
[U42#(x1, x2, x3)] | = | 2 · x1 + 4 · x2 + 0 · x3 + 2 |
[U21(x1)] | = | 0 · x1 + 1 |
[s(x1)] | = | 1 · x1 + 1 |
[tt] | = | 0 |
[U41(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 1 · x3 + 1 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[plus#(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[n__0] | = | 0 |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[n__s(x1)] | = | 1 · x1 + 1 |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[activate(x1)] | = | 1 · x1 + 0 |
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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
U42#(tt,M,N) | → | plus#(activate(N),activate(M)) | (29) |
The dependency pairs are split into 0 components.
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
[U12(x1)] | = | 1/2 · x1 + 3/2 |
[U42(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 2 · x3 + 3/2 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNat(x1)] | = | 1 · x1 + 2 |
[isNat#(x1)] | = | 1 · x1 + 0 |
[U21(x1)] | = | 5/2 · x1 + 1 |
[s(x1)] | = | 1 · x1 + 1/2 |
[tt] | = | 1 |
[U41(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 2 · x3 + 2 |
[0] | = | 0 |
[n__plus(x1, x2)] | = | 2 · x1 + 2 · x2 + 1 |
[n__0] | = | 0 |
[plus(x1, x2)] | = | 2 · x1 + 2 · x2 + 1 |
[n__s(x1)] | = | 1 · x1 + 1/2 |
[U31(x1, x2)] | = | 0 · x1 + 2 · x2 + 1 |
[activate(x1)] | = | 1 · x1 + 0 |
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) |
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(activate(X1),activate(X2)) | (16) |
activate(n__s(X)) | → | s(activate(X)) | (17) |
activate(X) | → | X | (18) |
isNat#(n__s(V1)) | → | isNat#(activate(V1)) | (36) |
There are no pairs anymore.
activate#(n__s(X)) | → | activate#(X) | (46) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
activate#(n__s(X)) | → | activate#(X) | (46) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.