YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

Confluence is proven by showing local confluence and termination.

The following set of initial dependency pairs has been identified.

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

a^#(b(x))	→	a^#(c(a(x)))
a^#(b(x))	→	a^#(x)

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers

[b(x₁)]

-∞	0
0	1

· x₁ +

0	-∞
1	-∞

[a^#(x₁)]

0	2
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

[a(x₁)]

-∞	0
0	1

· x₁ +

0	-∞
1	-∞

[c(x₁)]

-∞	0
0	-∞

· x₁ +

0	-∞
0	-∞

together with the usable rules

(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

There are no pairs anymore.

All critical pairs are joinable which can be seen by computing normal forms of all critical pairs.

csi

version: csi 1.2.5 [hg: unknown]
strategy: (cr -kb;((( matrix -dim 1 -ib 3 -ob 5 | matrix -dim 2 -ib 2 -ob 3 | matrix -dim 3 -ib 1 -ob 1 | matrix -dim 3 -ib 1 -ob 3 | fail)[2]*);((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])! || (rev;((dp;edg[0.5]?;(sccs | (sc || sct || {ur?;( (matrix -dp -ur -dim 1 -ib 3 -ob 5 | matrix -dp -ur -dim 2 -ib 2 -ob 3 | matrix -dp -ur -dim 3 -ib 1 -ob 1 | matrix -dp -ur -dim 3 -ib 1 -ob 3) || (kbo -ur -af | lpo -ur -af) || ( arctic -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || ( arctic -bz -dp -ur -dim 2 -ib 2 -ob 2[2] | fail) || fail) }restore || fail;(bounds -dp -rfc -qc || bounds -dp -all -rfc -qc || bounds -rfc -qc)[1] || fail ))*[6])! || (( kbo || (lpo | fail;(ref;lpo)) || fail;(bounds -rfc -qc) || fail)*[7])!)))))!