YES Confluence Proof

Confluence Proof

by csi

Input

The rewrite relation of the following TRS is considered.

a(s(x))	→	s(a(x))
b(a(b(s(x))))	→	a(b(s(a(x))))
b(a(b(b(x))))	→	c(s(x))
c(s(x))	→	a(b(a(b(x))))
a(b(a(a(x))))	→	b(a(b(a(x))))

Proof

1 Locally confluent and terminating

Confluence is proven by showing local confluence and termination.

1.1 Rule Removal

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

[a(x₁)]

1	0	1
0	1	0
0	0	1

· x₁ +

1	0	0
0	0	0
0	0	0

[c(x₁)]

1	1	1
0	1	0
0	1	0

· x₁ +

1	0	0
1	0	0
0	0	0

[s(x₁)]

1	0	1
0	0	1
0	1	1

· x₁ +

1	0	0
0	0	0
1	0	0

[b(x₁)]

1	1	0
0	0	1
0	0	1

· x₁ +

0	0	0
1	0	0
0	0	0

the rules

b(a(b(b(x))))	→	c(s(x))
c(s(x))	→	a(b(a(b(x))))
a(b(a(a(x))))	→	b(a(b(a(x))))

remain.

1.1.1 String Reversal

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

b(b(a(b(x))))	→	s(c(x))
s(c(x))	→	b(a(b(a(x))))
a(a(b(a(x))))	→	a(b(a(b(x))))

1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

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

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[b^#(x₁)]	=	2 · x₁ + 1
[a(x₁)]	=	1 · x₁ + 1
[c(x₁)]	=	1 · x₁ + 1
[s^#(x₁)]	=	2 · x₁ + 5
[s(x₁)]	=	1 · x₁ + 3
[a^#(x₁)]	=	2 · x₁ + 0
[b(x₁)]	=	1 · x₁ + 1

together with the usable rules

b(b(a(b(x))))	→	s(c(x))
s(c(x))	→	b(a(b(a(x))))
a(a(b(a(x))))	→	a(b(a(b(x))))

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

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

remain.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pairs

s^#(c(x)) → b^#(a(b(a(x))))

b^#(b(a(b(x)))) → s^#(c(x))

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

prec(s^#) = 3 weight(s^#) = 1

prec(b^#) = 1 weight(b^#) = 1

prec(c) = 0 weight(c) = 0

prec(b) = 7 weight(b) = 1

prec(a) = 2 weight(a) = 1

prec(s) = 6 weight(s) = 1

in combination with the following argument filter

π(s^#) = []

π(b^#) = 1

π(c) = 1

π(b) = []

π(a) = []

π(s) = [1]

together with the usable rule
a(a(b(a(x)))) → a(b(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.
The 2^nd component contains the pair
a^#(a(b(a(x)))) → a^#(b(a(b(x))))

1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[a(x₁)] = 3 · x₁ + 1

[c(x₁)] = 0 · x₁ + 0

[s(x₁)] = 0 · x₁ + 0

[a^#(x₁)] = 1 · x₁ + 2

[b(x₁)] = 0 · x₁ + 0

together with the usable rules

b(b(a(b(x)))) → s(c(x))

s(c(x)) → b(a(b(a(x))))

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

There are no pairs anymore.

1.2 Local Confluence Proof

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

Tool configuration

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])!)))))!

prec(s^#)	=	3	weight(s^#)	=	1
prec(b^#)	=	1	weight(b^#)	=	1
prec(c)	=	0	weight(c)	=	0
prec(b)	=	7	weight(b)	=	1
prec(a)	=	2	weight(a)	=	1
prec(s)	=	6	weight(s)	=	1

[a(x₁)]	=	3 · x₁ + 1
[c(x₁)]	=	0 · x₁ + 0
[s(x₁)]	=	0 · x₁ + 0
[a^#(x₁)]	=	1 · x₁ + 2
[b(x₁)]	=	0 · x₁ + 0

π(s^#)	=	[]
π(b^#)	=	1
π(c)	=	1
π(b)	=	[]
π(a)	=	[]
π(s)	=	[1]

Confluence Proof

Input

Proof

1 Locally confluent and terminating

1.1 Rule Removal

1.1.1 String Reversal

1.1.1.1 Dependency Pair Transformation

1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 P is empty

1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.2.1 P is empty

1.2 Local Confluence Proof

Tool configuration