The rewrite relation of the following TRS is considered.
app(app(\,x),x) | → | e | (1) |
app(app(\,e),x) | → | x | (2) |
app(app(\,x),app(app(.,x),y)) | → | y | (3) |
app(app(\,app(app(/,x),y)),x) | → | y | (4) |
app(app(/,x),x) | → | e | (5) |
app(app(/,x),e) | → | x | (6) |
app(app(/,app(app(.,y),x)),x) | → | y | (7) |
app(app(/,x),app(app(\,y),x)) | → | y | (8) |
app(app(.,e),x) | → | x | (9) |
app(app(.,x),e) | → | x | (10) |
app(app(.,x),app(app(\,x),y)) | → | y | (11) |
app(app(.,app(app(/,y),x)),x) | → | y | (12) |
app(app(map,f),nil) | → | nil | (13) |
app(app(map,f),app(app(cons,x),xs)) | → | app(app(cons,app(f,x)),app(app(map,f),xs)) | (14) |
app(app(filter,f),nil) | → | nil | (15) |
app(app(filter,f),app(app(cons,x),xs)) | → | app(app(app(app(filter2,app(f,x)),f),x),xs) | (16) |
app(app(app(app(filter2,true),f),x),xs) | → | app(app(cons,x),app(app(filter,f),xs)) | (17) |
app(app(app(app(filter2,false),f),x),xs) | → | app(app(filter,f),xs) | (18) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(app(map,f),xs) | (19) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(f,x) | (20) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(cons,app(f,x)) | (21) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(app(cons,app(f,x)),app(app(map,f),xs)) | (22) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(f,x) | (23) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(filter2,app(f,x)) | (24) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(app(filter2,app(f,x)),f) | (25) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(app(app(filter2,app(f,x)),f),x) | (26) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(app(app(app(filter2,app(f,x)),f),x),xs) | (27) |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(filter,f) | (28) |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(app(filter,f),xs) | (29) |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(cons,x) | (30) |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(app(cons,x),app(app(filter,f),xs)) | (31) |
app#(app(app(app(filter2,false),f),x),xs) | → | app#(filter,f) | (32) |
app#(app(app(app(filter2,false),f),x),xs) | → | app#(app(filter,f),xs) | (33) |
The dependency pairs are split into 1 component.
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(app(app(app(filter2,app(f,x)),f),x),xs) | (27) |
app#(app(app(app(filter2,false),f),x),xs) | → | app#(app(filter,f),xs) | (33) |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(f,x) | (23) |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(app(filter,f),xs) | (29) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(f,x) | (20) |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(app(map,f),xs) | (19) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(app(app(app(filter2,app(f,x)),f),x),xs) | (27) |
2 | > | 2 | |
app#(app(app(app(filter2,false),f),x),xs) | → | app#(app(filter,f),xs) | (33) |
2 | ≥ | 2 | |
app#(app(filter,f),app(app(cons,x),xs)) | → | app#(f,x) | (23) |
2 | > | 2 | |
1 | > | 1 | |
app#(app(app(app(filter2,true),f),x),xs) | → | app#(app(filter,f),xs) | (29) |
2 | ≥ | 2 | |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(f,x) | (20) |
2 | > | 2 | |
1 | > | 1 | |
app#(app(map,f),app(app(cons,x),xs)) | → | app#(app(map,f),xs) | (19) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.