The rewrite relation of the following TRS is considered.
| +(0,y) | → | y | (1) |
| +(s(x),y) | → | s(+(x,y)) | (2) |
| ++(nil,ys) | → | ys | (3) |
| ++(:(x,xs),ys) | → | :(x,++(xs,ys)) | (4) |
| sum(:(x,nil)) | → | :(x,nil) | (5) |
| sum(:(x,:(y,xs))) | → | sum(:(+(x,y),xs)) | (6) |
| sum(++(xs,:(x,:(y,ys)))) | → | sum(++(xs,sum(:(x,:(y,ys))))) | (7) |
| -(x,0) | → | x | (8) |
| -(0,s(y)) | → | 0 | (9) |
| -(s(x),s(y)) | → | -(x,y) | (10) |
| quot(0,s(y)) | → | 0 | (11) |
| quot(s(x),s(y)) | → | s(quot(-(x,y),s(y))) | (12) |
| length(nil) | → | 0 | (13) |
| length(:(x,xs)) | → | s(length(xs)) | (14) |
| hd(:(x,xs)) | → | x | (15) |
| avg(xs) | → | quot(hd(sum(xs)),length(xs)) | (16) |
| +#(s(x),y) | → | +#(x,y) | (17) |
| ++#(:(x,xs),ys) | → | ++#(xs,ys) | (18) |
| sum#(:(x,:(y,xs))) | → | +#(x,y) | (19) |
| sum#(:(x,:(y,xs))) | → | sum#(:(+(x,y),xs)) | (20) |
| sum#(++(xs,:(x,:(y,ys)))) | → | sum#(:(x,:(y,ys))) | (21) |
| sum#(++(xs,:(x,:(y,ys)))) | → | ++#(xs,sum(:(x,:(y,ys)))) | (22) |
| sum#(++(xs,:(x,:(y,ys)))) | → | sum#(++(xs,sum(:(x,:(y,ys))))) | (23) |
| -#(s(x),s(y)) | → | -#(x,y) | (24) |
| quot#(s(x),s(y)) | → | -#(x,y) | (25) |
| quot#(s(x),s(y)) | → | quot#(-(x,y),s(y)) | (26) |
| length#(:(x,xs)) | → | length#(xs) | (27) |
| avg#(xs) | → | length#(xs) | (28) |
| avg#(xs) | → | sum#(xs) | (29) |
| avg#(xs) | → | hd#(sum(xs)) | (30) |
| avg#(xs) | → | quot#(hd(sum(xs)),length(xs)) | (31) |
The dependency pairs are split into 7 components.
| sum#(++(xs,:(x,:(y,ys)))) | → | sum#(++(xs,sum(:(x,:(y,ys))))) | (23) |
| [++(x1, x2)] | = | 1 · x1 + 2 · x2 + 6 |
| [+(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
| [nil] | = | 0 |
| [:(x1, x2)] | = | 0 · x1 + 1 · x2 + 1 |
| [0] | = | 0 |
| [sum#(x1)] | = | 1 · x1 + 6 |
| [sum(x1)] | = | 0 · x1 + 1 |
| [s(x1)] | = | 0 · x1 + 0 |
| sum(:(x,:(y,xs))) | → | sum(:(+(x,y),xs)) | (6) |
| sum(:(x,nil)) | → | :(x,nil) | (5) |
| ++(nil,ys) | → | ys | (3) |
| ++(:(x,xs),ys) | → | :(x,++(xs,ys)) | (4) |
| sum#(++(xs,:(x,:(y,ys)))) | → | sum#(++(xs,sum(:(x,:(y,ys))))) | (23) |
There are no pairs anymore.
| sum#(:(x,:(y,xs))) | → | sum#(:(+(x,y),xs)) | (20) |
| prec(sum#) | = | 0 | stat(sum#) | = | lex | |
| prec(:) | = | 0 | stat(:) | = | lex | |
| prec(s) | = | 0 | stat(s) | = | lex | |
| prec(+) | = | 0 | stat(+) | = | lex | |
| prec(0) | = | 0 | stat(0) | = | lex |
| π(sum#) | = | 1 |
| π(:) | = | [2] |
| π(s) | = | 1 |
| π(+) | = | 2 |
| π(0) | = | [] |
| sum#(:(x,:(y,xs))) | → | sum#(:(+(x,y),xs)) | (20) |
There are no pairs anymore.
| +#(s(x),y) | → | +#(x,y) | (17) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| +#(s(x),y) | → | +#(x,y) | (17) |
| 2 | ≥ | 2 | |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| ++#(:(x,xs),ys) | → | ++#(xs,ys) | (18) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| ++#(:(x,xs),ys) | → | ++#(xs,ys) | (18) |
| 2 | ≥ | 2 | |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| quot#(s(x),s(y)) | → | quot#(-(x,y),s(y)) | (26) |
| π(quot#) | = | { 1 } |
| π(-) | = | { 1 } |
| quot#(s(x),s(y)) | → | quot#(-(x,y),s(y)) | (26) |
There are no pairs anymore.
| -#(s(x),s(y)) | → | -#(x,y) | (24) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| -#(s(x),s(y)) | → | -#(x,y) | (24) |
| 2 | > | 2 | |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
| length#(:(x,xs)) | → | length#(xs) | (27) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
| length#(:(x,xs)) | → | length#(xs) | (27) |
| 1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.