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