YES
Confluence Proof
Confluence Proof
by csi
Input
The rewrite relation of the following TRS is considered.
|
f(g(x),g(y)) |
→ |
f(g(x),h(y)) |
|
f(h(x),g(y)) |
→ |
f(g(x),g(y)) |
|
f(g(x),h(y)) |
→ |
f(x,y) |
|
f(h(x),h(y)) |
→ |
f(y,x) |
|
f(x,y) |
→ |
f(y,x) |
|
g(x) |
→ |
h(x) |
|
h(x) |
→ |
g(x) |
Proof
1 Decreasing Diagrams
1.1 Relative Termination Proof
The duplicating rules (R) terminate relative to the other rules (S).
1.1.1 R is empty
There are no rules in the TRS R. Hence, R/S is relative terminating.
1.2 Rule Labeling
Confluence is proven, because all critical peaks can be joined decreasingly
using the following rule labeling function (rules that are not shown have label 0).
-
f(g(x),g(y))→f(g(x),h(y)) ↦ 0
-
f(h(x),g(y))→f(g(x),g(y)) ↦ 4
-
f(g(x),h(y))→f(x,y) ↦ 2
-
f(h(x),h(y))→f(y,x) ↦ 1
-
f(x,y)→f(y,x) ↦ 0
-
g(x)→h(x) ↦ 0
-
h(x)→g(x) ↦ 4
All critical pairs are joinable:
-
f(g(x182),h(x183))→f(h(x183),g(x182))←f(g(x183),g(x182))
-
f(g(x182),h(x183))→f(g(x182),g(x183))←f(g(x183),g(x182))
-
f(g(x184),g(x185))→f(g(x185),g(x184))←f(g(x185),h(x184))
-
f(g(x184),g(x185))→f(h(x184),g(x185))←f(g(x185),h(x184))
-
f(x186,x187)←f(g(x186),h(x187))←f(h(x187),g(x186))
-
f(x186,x187)←f(h(x187),h(x186))←f(h(x187),g(x186))
-
f(x189,x188)→f(x188,x189)←f(h(x189),h(x188))
-
f(g(y),g(x))→f(g(x),g(y))←f(g(x),h(y))
-
f(g(y),g(x))→f(h(y),g(x))←f(g(x),h(y))
-
f(g(y),h(x))→f(h(x),g(y))←f(g(x),g(y))
-
f(g(y),h(x))→f(g(y),g(x))←f(g(x),g(y))
-
f(h(y),g(x))→f(g(x),h(y))→f(x,y)
-
f(h(y),g(x))→f(h(y),h(x))→f(x,y)
-
f(h(y),h(x))→f(x,y)←f(y,x)
-
f(h(x),g(y))→f(g(x),g(y))←f(g(x),h(y))
-
f(h(x),g(y))→f(h(x),h(y))←f(g(x),h(y))
-
f(g(x),h(y))
-
f(h(x),h(y))→f(h(x),g(y))←f(g(x),g(y))
-
f(h(x),h(y))→f(g(x),h(y))←f(g(x),g(y))
-
f(h(x),h(y))→f(y,x)←f(x,y)
-
f(g(x),g(y))
-
f(g(x),g(y))→f(g(x),h(y))→f(x,y)
-
f(g(x),h(y))→f(x,y)←f(y,x)
-
f(h(x),g(y))→f(g(y),h(x))→f(y,x)
-
f(h(x),g(y))→f(h(x),h(y))→f(y,x)
Tool configuration
csi
- version: csi 1.2.5 [hg: unknown]
- strategy:
(if left-linear then (cr -dup;(( lpo -quasi || (matrix -dim 1 -ib 3 -ob 4 | matrix -dim 2 -ib 2 -ob 2 | matrix -dim 3 -ib 1 -ob 2 | arctic -dim 2 -ib 2 -ob 2) || (if duplicating then fail else (bounds -rt || bounds -rt -qc))[1] || poly -ib 2 -ob 4 -nl2 -heuristic 1 || fail )[5]*);shift -lstar);(rule_labeling | rule_labeling -left)?;decreasing else fail)!