Certification Problem

Input (TPDB TRS_Standard/Zantema_05/jw41)

The rewrite relation of the following TRS is considered.

f(f(a,x),a) → f(f(f(x,f(a,a)),a),a) (1)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol f in combination with the following symbol map which also determines the applicative arities of these symbols.

a is mapped to a, a1(x1), a2(x1, x2)


There are no uncurry rules.
No rules have to be added for the eta-expansion.

Uncurrying the rules and adding the uncurrying rules yields the new set of rules
a2(x,a) → f(f(f(x,a1(a)),a),a) (4)
f(a,y1) → a1(y1) (2)
f(a1(x0),y1) → a2(x0,y1) (3)

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

10

Hence, it suffices to show innermost termination in the following.

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a2#(x,a) → f#(f(f(x,a1(a)),a),a) (5)
a2#(x,a) → f#(f(x,a1(a)),a) (6)
a2#(x,a) → f#(x,a1(a)) (7)
f#(a1(x0),y1) → a2#(x0,y1) (8)

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(a) = 1 weight(a) = 1
prec(a1) = 0 weight(a1) = 1
in combination with the following argument filter

π(a2#) = 2
π(a) = []
π(f#) = 2
π(a1) = []

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
a2#(x,a) → f#(x,a1(a)) (7)
could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[a2#(x1, x2)] =
1
0
+
1 0
0 0
· x1 +
0 1
0 0
· x2
[a] =
1
0
[f#(x1, x2)] =
0
0
+
0 1
0 0
· x1 +
0 1
0 0
· x2
[f(x1, x2)] =
0
0
+
0 0
1 0
· x1 +
0 0
1 0
· x2
[a1(x1)] =
0
1
+
0 0
1 0
· x1
[a2(x1, x2)] =
0
0
+
0 0
0 0
· x1 +
0 0
1 0
· x2
the pair
a2#(x,a) → f#(f(x,a1(a)),a) (6)
could be deleted.

1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers
[a2#(x1, x2)] =
0
0
+
0 0
0 0
· x1 +
0 1
0 0
· x2
[a] =
1
0
[f#(x1, x2)] =
0
0
+
0 1
0 0
· x1 +
0 1
0 0
· x2
[f(x1, x2)] =
0
0
+
0 0
1 0
· x1 +
0 0
0 0
· x2
[a1(x1)] =
0
1
+
0 0
0 0
· x1
[a2(x1, x2)] =
0
0
+
0 0
0 0
· x1 +
0 0
0 0
· x2
the pair
f#(a1(x0),y1) → a2#(x0,y1) (8)
could be deleted.

1.1.1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

a2#(x,a) → f#(f(f(x,a1(a)),a),a) (5)
2 ≥ 2

As there is no critical graph in the transitive closure, there are no infinite chains.