Certification Problem

Input (TPDB TRS_Standard/MNZ_10/nrvsq)

The rewrite relation of the following TRS is considered.

f(g(x))	→	g(g(f(x)))	(1)
g(s(x))	→	s(s(g(x)))	(2)
s(x)	→	h(0,x)	(3)
s(x)	→	h(x,0)	(4)
f(0)	→	0	(5)
s(s(s(0)))	→	f(s(0))	(6)
f(s(0))	→	s(0)	(7)
h(f(x),g(x))	→	f(s(x))	(8)
g(x)	→	h(h(h(h(x,x),x),x),x)	(9)
f(s(s(x)))	→	h(f(x),g(h(x,x)))	(10)
s(0)	→	r(0)	(11)
s(s(s(0)))	→	r(s(0))	(12)
r(s(0))	→	s(0)	(13)
g(x)	→	r(x)	(14)
s(0)	→	p(0)	(15)
s(s(0))	→	p(s(0))	(16)
p(s(0))	→	0	(17)
s(s(s(s(s(0)))))	→	p(s(s(0)))	(18)
p(s(s(0)))	→	s(s(s(0)))	(19)
h(p(x),g(x))	→	p(s(x))	(20)
s(0)	→	k(0)	(21)
s(s(p(p(a))))	→	s(k(p(a)))	(22)
s(k(p(a)))	→	p(p(a))	(23)
g(x)	→	k(x)	(24)
a	→	0	(25)
s(h(r(k(p(x))),r(x)))	→	h(r(r(p(x))),k(x))	(26)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

f^#(g(x))	→	g^#(g(f(x)))	(27)
f^#(g(x))	→	g^#(f(x))	(28)
f^#(g(x))	→	f^#(x)	(29)
g^#(s(x))	→	s^#(s(g(x)))	(30)
g^#(s(x))	→	s^#(g(x))	(31)
g^#(s(x))	→	g^#(x)	(32)
s^#(x)	→	h^#(0,x)	(33)
s^#(x)	→	h^#(x,0)	(34)
s^#(s(s(0)))	→	f^#(s(0))	(35)
h^#(f(x),g(x))	→	f^#(s(x))	(36)
h^#(f(x),g(x))	→	s^#(x)	(37)
g^#(x)	→	h^#(h(h(h(x,x),x),x),x)	(38)
g^#(x)	→	h^#(h(h(x,x),x),x)	(39)
g^#(x)	→	h^#(h(x,x),x)	(40)
g^#(x)	→	h^#(x,x)	(41)
f^#(s(s(x)))	→	h^#(f(x),g(h(x,x)))	(42)
f^#(s(s(x)))	→	f^#(x)	(43)
f^#(s(s(x)))	→	g^#(h(x,x))	(44)
f^#(s(s(x)))	→	h^#(x,x)	(45)
s^#(0)	→	r^#(0)	(46)
s^#(s(s(0)))	→	r^#(s(0))	(47)
g^#(x)	→	r^#(x)	(48)
s^#(0)	→	p^#(0)	(49)
s^#(s(0))	→	p^#(s(0))	(50)
s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
p^#(s(s(0)))	→	s^#(s(s(0)))	(52)
h^#(p(x),g(x))	→	p^#(s(x))	(53)
h^#(p(x),g(x))	→	s^#(x)	(54)
s^#(s(p(p(a))))	→	s^#(k(p(a)))	(55)
s^#(k(p(a)))	→	p^#(p(a))	(56)
s^#(h(r(k(p(x))),r(x)))	→	h^#(r(r(p(x))),k(x))	(57)
s^#(h(r(k(p(x))),r(x)))	→	r^#(r(p(x)))	(58)
s^#(h(r(k(p(x))),r(x)))	→	r^#(p(x))	(59)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

g^#(s(x))	→	s^#(s(g(x)))	(30)
s^#(s(s(0)))	→	f^#(s(0))	(35)
f^#(g(x))	→	g^#(g(f(x)))	(27)
g^#(s(x))	→	s^#(g(x))	(31)
s^#(s(0))	→	p^#(s(0))	(50)
p^#(s(s(0)))	→	s^#(s(s(0)))	(52)
s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
s^#(s(p(p(a))))	→	s^#(k(p(a)))	(55)
s^#(k(p(a)))	→	p^#(p(a))	(56)
g^#(s(x))	→	g^#(x)	(32)
g^#(x)	→	h^#(h(h(h(x,x),x),x),x)	(38)
h^#(f(x),g(x))	→	f^#(s(x))	(36)
f^#(g(x))	→	g^#(f(x))	(28)
g^#(x)	→	h^#(h(h(x,x),x),x)	(39)
h^#(f(x),g(x))	→	s^#(x)	(37)
h^#(p(x),g(x))	→	p^#(s(x))	(53)
h^#(p(x),g(x))	→	s^#(x)	(54)
g^#(x)	→	h^#(h(x,x),x)	(40)
g^#(x)	→	h^#(x,x)	(41)
f^#(g(x))	→	f^#(x)	(29)
f^#(s(s(x)))	→	h^#(f(x),g(h(x,x)))	(42)
f^#(s(s(x)))	→	f^#(x)	(43)
f^#(s(s(x)))	→	g^#(h(x,x))	(44)
f^#(s(s(x)))	→	h^#(x,x)	(45)

1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 1 to get the following set of pairs

s^#(k(p(a)))

→

p^#(p(0))

(60)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

s^#(s(s(0)))	→	f^#(s(0))	(35)
f^#(g(x))	→	g^#(g(f(x)))	(27)
g^#(s(x))	→	s^#(s(g(x)))	(30)
s^#(s(0))	→	p^#(s(0))	(50)
p^#(s(s(0)))	→	s^#(s(s(0)))	(52)
s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
g^#(s(x))	→	s^#(g(x))	(31)
g^#(s(x))	→	g^#(x)	(32)
g^#(x)	→	h^#(h(h(h(x,x),x),x),x)	(38)
h^#(f(x),g(x))	→	f^#(s(x))	(36)
f^#(g(x))	→	g^#(f(x))	(28)
g^#(x)	→	h^#(h(h(x,x),x),x)	(39)
h^#(f(x),g(x))	→	s^#(x)	(37)
h^#(p(x),g(x))	→	p^#(s(x))	(53)
h^#(p(x),g(x))	→	s^#(x)	(54)
g^#(x)	→	h^#(h(x,x),x)	(40)
g^#(x)	→	h^#(x,x)	(41)
f^#(g(x))	→	f^#(x)	(29)
f^#(s(s(x)))	→	h^#(f(x),g(h(x,x)))	(42)
f^#(s(s(x)))	→	f^#(x)	(43)
f^#(s(s(x)))	→	g^#(h(x,x))	(44)
f^#(s(s(x)))	→	h^#(x,x)	(45)

1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[f^#(x₁)]	=	2 · x₁
[g^#(x₁)]	=	x₁
[h^#(x₁, x₂)]	=	x₁
[p^#(x₁)]	=	-2
[s^#(x₁)]	=	-2
[s(x₁)]	=	x₁
[h(x₁, x₂)]	=	x₁
[0]	=	0
[r(x₁)]	=	-2
[p(x₁)]	=	x₁
[k(x₁)]	=	x₁
[g(x₁)]	=	2 + x₁
[f(x₁)]	=	2 · x₁
[a]	=	2

the pairs

f^#(g(x))	→	g^#(g(f(x)))	(27)
f^#(g(x))	→	g^#(f(x))	(28)
f^#(g(x))	→	f^#(x)	(29)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[f^#(x₁)]	=	1 + x₁
[g^#(x₁)]	=	1 + x₁
[h^#(x₁, x₂)]	=	-1 + x₂
[p^#(x₁)]	=	1
[s^#(x₁)]	=	1
[s(x₁)]	=	x₁
[h(x₁, x₂)]	=	x₁
[0]	=	0
[r(x₁)]	=	0
[p(x₁)]	=	-2
[k(x₁)]	=	x₁
[g(x₁)]	=	2 + x₁
[f(x₁)]	=	2 · x₁
[a]	=	0

together with the usable rules

s(x)	→	h(0,x)	(3)
s(x)	→	h(x,0)	(4)
s(0)	→	r(0)	(11)
s(0)	→	p(0)	(15)
s(0)	→	k(0)	(21)
g(s(x))	→	s(s(g(x)))	(2)
s(s(s(0)))	→	f(s(0))	(6)
f(g(x))	→	g(g(f(x)))	(1)
g(x)	→	h(h(h(h(x,x),x),x),x)	(9)
h(f(x),g(x))	→	f(s(x))	(8)
f(s(s(x)))	→	h(f(x),g(h(x,x)))	(10)
h(p(x),g(x))	→	p(s(x))	(20)
p(s(s(0)))	→	s(s(s(0)))	(19)
s(s(0))	→	p(s(0))	(16)
s(s(s(s(s(0)))))	→	p(s(s(0)))	(18)
s(s(p(p(a))))	→	s(k(p(a)))	(22)
s(k(p(a)))	→	p(p(a))	(23)
g(x)	→	r(x)	(14)
g(x)	→	k(x)	(24)
s(s(s(0)))	→	r(s(0))	(12)
s(h(r(k(p(x))),r(x)))	→	h(r(r(p(x))),k(x))	(26)
f(0)	→	0	(5)
f(s(0))	→	s(0)	(7)
p(s(0))	→	0	(17)
r(s(0))	→	s(0)	(13)

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

g^#(x)	→	h^#(h(h(h(x,x),x),x),x)	(38)
g^#(x)	→	h^#(h(h(x,x),x),x)	(39)
g^#(x)	→	h^#(h(x,x),x)	(40)
g^#(x)	→	h^#(x,x)	(41)
f^#(s(s(x)))	→	h^#(x,x)	(45)

could be deleted.

1.1.1.1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 1 to get the following set of pairs

s^#(s(s(0)))	→	f^#(h(0,0))	(61)
s^#(s(s(0)))	→	f^#(r(0))	(62)
s^#(s(s(0)))	→	f^#(p(0))	(63)
s^#(s(s(0)))	→	f^#(k(0))	(64)

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

f^#(s(s(x)))	→	h^#(f(x),g(h(x,x)))	(42)
h^#(f(x),g(x))	→	f^#(s(x))	(36)
f^#(s(s(x)))	→	f^#(x)	(43)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

[f^#(x₁)]

-∞

-∞	-∞	0
-∞	-∞	-∞
-∞	-∞	-∞

· x₁

[s(x₁)]

-∞	-∞	-∞
1	-∞	0
0	0	-∞

· x₁

[h^#(x₁, x₂)]

-∞

0	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

0	1	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₂

[f(x₁)]

0	-∞	-∞
1	-∞	-∞
1	-∞	-∞

· x₁

[g(x₁)]

0	-∞	-∞
0	0	0
0	0	0

· x₁

[h(x₁, x₂)]

-∞	-∞	-∞
-∞	-∞	-∞
-∞	-∞	-∞

· x₁ +

-∞	-∞	-∞
0	-∞	-∞
0	-∞	-∞

· x₂

[0]

-∞

[p(x₁)]

0	-∞	-∞
0	-∞	-∞
1	-∞	-∞

· x₁

[a]

[k(x₁)]

-∞

-∞	-∞	-∞
-∞	-∞	0
-∞	-∞	-∞

· x₁

[r(x₁)]

-∞	-∞	-∞
0	0	-∞
-∞	-∞	-∞

· x₁

the pair

h^#(f(x),g(x))

→

f^#(s(x))

(36)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
f^#(s(s(x))) → f^#(x) (43)

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

[s(x₁)] = 1 · x₁

[f^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

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

f^#(s(s(x))) → f^#(x) (43)

1 > 1

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

The 2^nd component contains the pair
g^#(s(x)) → g^#(x) (32)

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

[s(x₁)] = 1 · x₁

[g^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1.1.1.1.1.2.1 Forward Instantiation Processor
We instantiate the pair to the following set of pairs
g^#(s(s(y_0))) → g^#(s(y_0)) (65)

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

g^#(s(s(y_0))) → g^#(s(y_0)) (65)

1 > 1

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

The 3^rd component contains the pair

p^#(s(s(0)))	→	s^#(s(s(0)))	(52)
s^#(s(0))	→	p^#(s(0))	(50)
s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)

1.1.1.1.1.1.1.1.3 Narrowing Processor

We consider all narrowings of the pair below position 1 to get the following set of pairs

p^#(s(s(0)))	→	s^#(h(0,s(0)))	(66)
p^#(s(s(0)))	→	s^#(h(s(0),0))	(67)
p^#(s(s(0)))	→	s^#(p(s(0)))	(68)
p^#(s(s(0)))	→	s^#(s(h(0,0)))	(69)
p^#(s(s(0)))	→	s^#(s(r(0)))	(70)
p^#(s(s(0)))	→	s^#(s(p(0)))	(71)
p^#(s(s(0)))	→	s^#(s(k(0)))	(72)

1.1.1.1.1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

p^#(s(s(0)))	→	s^#(p(s(0)))	(68)
s^#(s(0))	→	p^#(s(0))	(50)
p^#(s(s(0)))	→	s^#(s(h(0,0)))	(69)
s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
p^#(s(s(0)))	→	s^#(s(r(0)))	(70)
p^#(s(s(0)))	→	s^#(s(p(0)))	(71)
p^#(s(s(0)))	→	s^#(s(k(0)))	(72)

1.1.1.1.1.1.1.1.3.1.1 Narrowing Processor

We consider all narrowings of the pair below position 1 to get the following set of pairs

s^#(s(0))	→	p^#(h(0,0))	(73)
s^#(s(0))	→	p^#(r(0))	(74)
s^#(s(0))	→	p^#(p(0))	(75)
s^#(s(0))	→	p^#(k(0))	(76)

1.1.1.1.1.1.1.1.3.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
p^#(s(s(0)))	→	s^#(p(s(0)))	(68)
p^#(s(s(0)))	→	s^#(s(h(0,0)))	(69)
p^#(s(s(0)))	→	s^#(s(r(0)))	(70)
p^#(s(s(0)))	→	s^#(s(p(0)))	(71)
p^#(s(s(0)))	→	s^#(s(k(0)))	(72)

1.1.1.1.1.1.1.1.3.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 2 with strict dimension 1 over the integers

[s^#(x₁)]

1	0
0	0

· x₁

[s(x₁)]

0	1
0	0

· x₁

[0]

[p^#(x₁)]

0	0
0	0

· x₁

[p(x₁)]

1	0
0	1

· x₁

[h(x₁, x₂)]

0	1
0	0

· x₁ +

0	0
0	0

· x₂

[r(x₁)]

0	0
0	0

· x₁

[k(x₁)]

0	0
0	0

· x₁

[g(x₁)]

0	0
0	0

· x₁

[f(x₁)]

0	0
0	0

· x₁

[a]

the pairs

p^#(s(s(0)))	→	s^#(p(s(0)))	(68)
p^#(s(s(0)))	→	s^#(s(p(0)))	(71)
p^#(s(s(0)))	→	s^#(s(k(0)))	(72)

could be deleted.

1.1.1.1.1.1.1.1.3.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

prec(0)	=	1	weight(0)	=	4
prec(p^#)	=	3	weight(p^#)	=	3
prec(h)	=	2	weight(h)	=	1
prec(r)	=	0	weight(r)	=	2

in combination with the following argument filter

π(s^#)	=	1
π(s)	=	1
π(0)	=	[]
π(p^#)	=	[]
π(h)	=	[]
π(r)	=	[]

together with the usable rules

s(x)	→	h(0,x)	(3)
s(x)	→	h(x,0)	(4)

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

s^#(s(s(s(s(0)))))	→	p^#(s(s(0)))	(51)
p^#(s(s(0)))	→	s^#(s(h(0,0)))	(69)
p^#(s(s(0)))	→	s^#(s(r(0)))	(70)

could be deleted.

1.1.1.1.1.1.1.1.3.1.1.1.1.1.1 P is empty

There are no pairs anymore.

Certification Problem

Input (TPDB TRS_Standard/MNZ_10/nrvsq)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Narrowing Processor

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination

1.1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.2.1 Forward Instantiation Processor

1.1.1.1.1.1.1.1.2.1.1 Size-Change Termination

1.1.1.1.1.1.1.1.3 Narrowing Processor

1.1.1.1.1.1.1.1.3.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.3.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.3.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.3.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.3.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.3.1.1.1.1.1.1 P is empty