The rewrite relation of the following TRS is considered.
| U11(tt,N,X,XS) | → | U12(splitAt(activate(N),activate(XS)),activate(X)) | (1) |
| U12(pair(YS,ZS),X) | → | pair(cons(activate(X),YS),ZS) | (2) |
| afterNth(N,XS) | → | snd(splitAt(N,XS)) | (3) |
| and(tt,X) | → | activate(X) | (4) |
| fst(pair(X,Y)) | → | X | (5) |
| head(cons(N,XS)) | → | N | (6) |
| natsFrom(N) | → | cons(N,n__natsFrom(n__s(N))) | (7) |
| sel(N,XS) | → | head(afterNth(N,XS)) | (8) |
| snd(pair(X,Y)) | → | Y | (9) |
| splitAt(0,XS) | → | pair(nil,XS) | (10) |
| splitAt(s(N),cons(X,XS)) | → | U11(tt,N,X,activate(XS)) | (11) |
| tail(cons(N,XS)) | → | activate(XS) | (12) |
| take(N,XS) | → | fst(splitAt(N,XS)) | (13) |
| natsFrom(X) | → | n__natsFrom(X) | (14) |
| s(X) | → | n__s(X) | (15) |
| activate(n__natsFrom(X)) | → | natsFrom(activate(X)) | (16) |
| activate(n__s(X)) | → | s(activate(X)) | (17) |
| activate(X) | → | X | (18) |
| U11#(tt,N,X,XS) | → | activate#(X) | (19) |
| U11#(tt,N,X,XS) | → | activate#(XS) | (20) |
| U11#(tt,N,X,XS) | → | activate#(N) | (21) |
| U11#(tt,N,X,XS) | → | splitAt#(activate(N),activate(XS)) | (22) |
| U11#(tt,N,X,XS) | → | U12#(splitAt(activate(N),activate(XS)),activate(X)) | (23) |
| U12#(pair(YS,ZS),X) | → | activate#(X) | (24) |
| afterNth#(N,XS) | → | splitAt#(N,XS) | (25) |
| afterNth#(N,XS) | → | snd#(splitAt(N,XS)) | (26) |
| and#(tt,X) | → | activate#(X) | (27) |
| sel#(N,XS) | → | afterNth#(N,XS) | (28) |
| sel#(N,XS) | → | head#(afterNth(N,XS)) | (29) |
| splitAt#(s(N),cons(X,XS)) | → | activate#(XS) | (30) |
| splitAt#(s(N),cons(X,XS)) | → | U11#(tt,N,X,activate(XS)) | (31) |
| tail#(cons(N,XS)) | → | activate#(XS) | (32) |
| take#(N,XS) | → | splitAt#(N,XS) | (33) |
| take#(N,XS) | → | fst#(splitAt(N,XS)) | (34) |
| activate#(n__natsFrom(X)) | → | activate#(X) | (35) |
| activate#(n__natsFrom(X)) | → | natsFrom#(activate(X)) | (36) |
| activate#(n__s(X)) | → | activate#(X) | (37) |
| activate#(n__s(X)) | → | s#(activate(X)) | (38) |
The dependency pairs are split into 2 components.
| splitAt#(s(N),cons(X,XS)) | → | U11#(tt,N,X,activate(XS)) | (31) |
| U11#(tt,N,X,XS) | → | splitAt#(activate(N),activate(XS)) | (22) |
| π(splitAt#) | = | { 1 } |
| π(U11#) | = | { 2, 2 } |
| π(s) | = | { 1, 1, 1 } |
| π(n__natsFrom) | = | { 1 } |
| π(n__s) | = | { 1, 1, 1 } |
| π(natsFrom) | = | { 1 } |
| π(cons) | = | { 1 } |
| π(pair) | = | { 1 } |
| π(U12) | = | { 2 } |
| π(activate) | = | { 1 } |
| splitAt#(s(N),cons(X,XS)) | → | U11#(tt,N,X,activate(XS)) | (31) |
| U11#(tt,N,X,XS) | → | splitAt#(activate(N),activate(XS)) | (22) |
There are no pairs anymore.
| activate#(n__natsFrom(X)) | → | activate#(X) | (35) |
| activate#(n__s(X)) | → | activate#(X) | (37) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| activate#(n__natsFrom(X)) | → | activate#(X) | (35) |
| 1 | > | 1 | |
| activate#(n__s(X)) | → | activate#(X) | (37) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.