YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

b(a(b(b(x))))	→	b(b(b(a(b(x)))))
b(a(a(b(b(x)))))	→	b(a(b(b(a(a(b(x)))))))
b(a(a(a(b(b(x))))))	→	b(a(a(b(b(a(a(a(b(x)))))))))

Proof

1 Critical Pair Closing System

Confluence is proven using the following terminating critical-pair-closing-system R:

b(a(b(b(x))))	→	b(b(b(a(b(x)))))
b(a(a(b(b(x)))))	→	b(a(b(b(a(a(b(x)))))))
b(a(a(a(b(b(x))))))	→	b(a(a(b(b(a(a(a(b(x)))))))))

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS

b(b(a(b(x))))	→	b(a(b(b(b(x)))))
b(b(a(a(b(x)))))	→	b(a(a(b(b(a(b(x)))))))
b(b(a(a(a(b(x))))))	→	b(a(a(a(b(b(a(a(b(x)))))))))

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pairs

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

1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

1	-1
-∞	0

· x₁ +

-4	-∞
1	-∞

[b(x₁)]

-2	-2
0	0

· x₁ +

0	-∞
0	-∞

[b^#(x₁)]

2	-∞
-∞	-∞

· x₁ +

0	-∞
-∞	-∞

together with the usable rules

b(b(a(b(x))))	→	b(a(b(b(b(x)))))
b(b(a(a(b(x)))))	→	b(a(a(b(b(a(b(x)))))))
b(b(a(a(a(b(x))))))	→	b(a(a(a(b(b(a(a(b(x)))))))))

(w.r.t. the implicit argument filter of the reduction pair), the pairs

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

remain.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a(x₁)]

0	1	1
0	0	0
0	1	0

· x₁ +

0	0	0
0	0	0
1	0	0

[b(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[b^#(x₁)]

0	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

together with the usable rules

b(b(a(b(x))))	→	b(a(b(b(b(x)))))
b(b(a(a(b(x)))))	→	b(a(a(b(b(a(b(x)))))))
b(b(a(a(a(b(x))))))	→	b(a(a(a(b(b(a(a(b(x)))))))))

(w.r.t. the implicit argument filter of the reduction pair), the pairs

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

remain.

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[a(x₁)]

0	0
2	0

· x₁ +

0	0
1	0

[b(x₁)]

0	1
0	1

· x₁ +

0	0
0	0

[b^#(x₁)]

2	0
0	0

· x₁ +

0	0
0	0

together with the usable rules

b(b(a(b(x))))	→	b(a(b(b(b(x)))))
b(b(a(a(b(x)))))	→	b(a(a(b(b(a(b(x)))))))
b(b(a(a(a(b(x))))))	→	b(a(a(a(b(b(a(a(b(x)))))))))

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

1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

Tool configuration

csi

version: csi 1.2.5 [hg: unknown]
strategy: (cr -cpcs2 -cpcscert; ((( 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])!)))))!