Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove3)

The rewrite relation of the following TRS is considered.

p(0(x1))	→	s(s(0(s(s(p(x1))))))	(1)
p(s(0(x1)))	→	0(x1)	(2)
p(s(s(x1)))	→	s(p(s(x1)))	(3)
f(s(x1))	→	g(q(i(x1)))	(4)
g(x1)	→	f(p(p(x1)))	(5)
q(i(x1))	→	q(s(x1))	(6)
q(s(x1))	→	s(s(x1))	(7)
i(x1)	→	s(x1)	(8)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

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

0(p(x1))	→	p(s(s(0(s(s(x1))))))	(9)
0(s(p(x1)))	→	0(x1)	(10)
s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
g(x1)	→	p(p(f(x1)))	(13)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

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.

0^#(p(x1))	→	s^#(s(0(s(s(x1)))))	(16)
0^#(p(x1))	→	s^#(0(s(s(x1))))	(17)
0^#(p(x1))	→	0^#(s(s(x1)))	(18)
0^#(p(x1))	→	s^#(s(x1))	(19)
0^#(p(x1))	→	s^#(x1)	(20)
0^#(s(p(x1)))	→	0^#(x1)	(21)
s^#(s(p(x1)))	→	s^#(p(s(x1)))	(22)
s^#(s(p(x1)))	→	s^#(x1)	(23)
s^#(f(x1))	→	i^#(q(g(x1)))	(24)
s^#(f(x1))	→	g^#(x1)	(25)
i^#(q(x1))	→	s^#(q(x1))	(26)
s^#(q(x1))	→	s^#(s(x1))	(27)
s^#(q(x1))	→	s^#(x1)	(28)
i^#(x1)	→	s^#(x1)	(29)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

0^#(s(p(x1)))	→	0^#(x1)	(21)
0^#(p(x1))	→	0^#(s(s(x1)))	(18)

1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

s(s(p(x1)))	→	s(p(s(x1)))	(11)
s(f(x1))	→	i(q(g(x1)))	(12)
i(q(x1))	→	s(q(x1))	(14)
s(q(x1))	→	s(s(x1))	(15)
i(x1)	→	s(x1)	(8)
g(x1)	→	p(p(f(x1)))	(13)

1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

s(s(p(x0)))

s(f(x0))

g(x0)

s(q(x0))

i(x0)

1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

[0^#(x₁)]

-∞

0	0	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[s(x₁)]

-∞	0	-∞
-∞	0	-∞
0	0	-∞

· x₁

[p(x₁)]

0	0	0
0	0	-∞
0	1	0

· x₁

[f(x₁)]

0	-∞	-∞
0	-∞	-∞
0	-∞	-∞

· x₁

[i(x₁)]

0	0	0
0	0	0
0	0	0

· x₁

[q(x₁)]

-∞

-∞	-∞	-∞
-∞	0	-∞
-∞	-∞	-∞

· x₁

[g(x₁)]

1	0	0
0	-∞	-∞
1	0	0

· x₁

the pair

0^#(p(x1))

→

0^#(s(s(x1)))

(18)

could be deleted.

1.1.1.1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

s(s(p(x0)))

s(f(x0))

s(q(x0))

1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

0^#(s(p(x1)))	→	0^#(x1)	(21)

1	>	1

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

The 2^nd component contains the pair

s^#(f(x1)) → i^#(q(g(x1))) (24)

i^#(q(x1)) → s^#(q(x1)) (26)

s^#(q(x1)) → s^#(s(x1)) (27)

s^#(s(p(x1))) → s^#(x1) (23)

s^#(q(x1)) → s^#(x1) (28)

i^#(x1) → s^#(x1) (29)

1.1.1.1.2 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

s(s(p(x1))) → s(p(s(x1))) (11)

s(f(x1)) → i(q(g(x1))) (12)

i(q(x1)) → s(q(x1)) (14)

s(q(x1)) → s(s(x1)) (15)

i(x1) → s(x1) (8)

g(x1) → p(p(f(x1))) (13)

1.1.1.1.2.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

s(s(p(x0)))

s(f(x0))

g(x0)

s(q(x0))

i(x0)

1.1.1.1.2.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[i^#(x₁)] = 2 + 2 · x₁

[q(x₁)] = 2 + x₁

[s^#(x₁)] = 2 + 2 · x₁

[i(x₁)] = 2 + x₁

[g(x₁)] = 0

[p(x₁)] = -2 + x₁

[f(x₁)] = 2

[s(x₁)] = 2 + x₁

the pair
s^#(q(x1)) → s^#(x1) (28)
could be deleted.
1.1.1.1.2.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[i^#(x₁)] = 2 · x₁

[q(x₁)] = 2 + 2 · x₁

[s^#(x₁)] = 2 · x₁

[i(x₁)] = 2 + x₁

[g(x₁)] = 0

[p(x₁)] = -1 + x₁

[f(x₁)] = 2

[s(x₁)] = 2 + x₁

the pair
s^#(s(p(x1))) → s^#(x1) (23)
could be deleted.
1.1.1.1.2.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.
- The 1^st component contains the pair
  
  i^#(x1) → s^#(x1) (29)
  
  s^#(f(x1)) → i^#(q(g(x1))) (24)
  
  1.1.1.1.2.1.1.1.1.1 Usable Rules Processor
  
  We restrict the rewrite rules to the following usable rules of the DP problem.
  
  g(x1) → p(p(f(x1))) (13)
  
  1.1.1.1.2.1.1.1.1.1.1 Innermost Lhss Removal Processor
  
  We restrict the innermost strategy to the following left hand sides.
  
  g(x0)
  
  1.1.1.1.2.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
  Using the linear polynomial interpretation over the naturals
  
  [i^#(x₁)] = 1 + 2 · x₁
  
  [q(x₁)] = 1
  
  [g(x₁)] = 2 + 2 · x₁
  
  [p(x₁)] = -2
  
  [f(x₁)] = 2
  
  [s^#(x₁)] = 2 · x₁
  
  having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs
  
  i^#(x1) → s^#(x1) (29)
  
  s^#(f(x1)) → i^#(q(g(x1))) (24)
  
  could be deleted.
  1.1.1.1.2.1.1.1.1.1.1.1.1 P is empty
  
  There are no pairs anymore.

[i^#(x₁)]	=	2 + 2 · x₁
[q(x₁)]	=	2 + x₁
[s^#(x₁)]	=	2 + 2 · x₁
[i(x₁)]	=	2 + x₁
[g(x₁)]	=	0
[p(x₁)]	=	-2 + x₁
[f(x₁)]	=	2
[s(x₁)]	=	2 + x₁

[i^#(x₁)]	=	2 · x₁
[q(x₁)]	=	2 + 2 · x₁
[s^#(x₁)]	=	2 · x₁
[i(x₁)]	=	2 + x₁
[g(x₁)]	=	0
[p(x₁)]	=	-1 + x₁
[f(x₁)]	=	2
[s(x₁)]	=	2 + x₁

[i^#(x₁)]	=	1 + 2 · x₁
[q(x₁)]	=	1
[g(x₁)]	=	2 + 2 · x₁
[p(x₁)]	=	-2
[f(x₁)]	=	2
[s^#(x₁)]	=	2 · x₁

Certification Problem

Input (TPDB SRS_Standard/Secret_05_SRS/aprove3)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 String Reversal

1.1 Switch to Innermost Termination

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Usable Rules Processor

1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1 Usable Rules Processor

1.1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.1.1.1.1.1.1 Size-Change Termination

1.1.1.1.2 Usable Rules Processor

1.1.1.1.2.1 Innermost Lhss Removal Processor

1.1.1.1.2.1.1 Reduction Pair Processor

1.1.1.1.2.1.1.1 Reduction Pair Processor

1.1.1.1.2.1.1.1.1 Dependency Graph Processor

1.1.1.1.2.1.1.1.1.1 Usable Rules Processor

1.1.1.1.2.1.1.1.1.1.1 Innermost Lhss Removal Processor

1.1.1.1.2.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1.1.1.1.1.1 P is empty