Certification Problem

Input (TPDB TRS_Standard/MNZ_10/8)

The rewrite relation of the following TRS is considered.

s(s(0))	→	f(s(0))	(1)
g(x)	→	h(x,x)	(2)
s(x)	→	h(x,0)	(3)
s(x)	→	h(0,x)	(4)
f(g(x))	→	g(g(f(x)))	(5)
g(s(x))	→	s(s(g(x)))	(6)
h(f(x),g(x))	→	f(s(x))	(7)
s(s(0))	→	k(0)	(8)
k(0)	→	0	(9)
s(s(s(0)))	→	k(s(0))	(10)
k(s(0))	→	s(0)	(11)
s(s(s(s(s(s(s(s(s(s(0))))))))))	→	k(s(s(0)))	(12)
k(s(s(0)))	→	s(s(s(s(s(s(s(s(0))))))))	(13)
h(k(x),g(x))	→	k(s(x))	(14)

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.

s^#(s(0))	→	f^#(s(0))	(15)
g^#(x)	→	h^#(x,x)	(16)
s^#(x)	→	h^#(x,0)	(17)
s^#(x)	→	h^#(0,x)	(18)
f^#(g(x))	→	g^#(g(f(x)))	(19)
f^#(g(x))	→	g^#(f(x))	(20)
f^#(g(x))	→	f^#(x)	(21)
g^#(s(x))	→	s^#(s(g(x)))	(22)
g^#(s(x))	→	s^#(g(x))	(23)
g^#(s(x))	→	g^#(x)	(24)
h^#(f(x),g(x))	→	f^#(s(x))	(25)
h^#(f(x),g(x))	→	s^#(x)	(26)
s^#(s(0))	→	k^#(0)	(27)
s^#(s(s(0)))	→	k^#(s(0))	(28)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(s(0))))))))	(30)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(0)))))))	(31)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(0))))))	(32)
k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
h^#(k(x),g(x))	→	k^#(s(x))	(36)
h^#(k(x),g(x))	→	s^#(x)	(37)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

f^#(g(x))	→	g^#(g(f(x)))	(19)
g^#(x)	→	h^#(x,x)	(16)
h^#(f(x),g(x))	→	f^#(s(x))	(25)
f^#(g(x))	→	g^#(f(x))	(20)
g^#(s(x))	→	s^#(s(g(x)))	(22)
s^#(s(0))	→	f^#(s(0))	(15)
f^#(g(x))	→	f^#(x)	(21)
s^#(s(s(0)))	→	k^#(s(0))	(28)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(s(0))))))))	(30)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(0)))))))	(31)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(0))))))	(32)
k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
g^#(s(x))	→	s^#(g(x))	(23)
g^#(s(x))	→	g^#(x)	(24)
h^#(f(x),g(x))	→	s^#(x)	(26)
h^#(k(x),g(x))	→	k^#(s(x))	(36)
h^#(k(x),g(x))	→	s^#(x)	(37)

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pairs

f^#(g(x))	→	g^#(g(f(x)))	(19)
g^#(x)	→	h^#(x,x)	(16)
h^#(f(x),g(x))	→	f^#(s(x))	(25)
f^#(g(x))	→	g^#(f(x))	(20)
g^#(s(x))	→	s^#(s(g(x)))	(22)
f^#(g(x))	→	f^#(x)	(21)
g^#(s(x))	→	s^#(g(x))	(23)
h^#(f(x),g(x))	→	s^#(x)	(26)
h^#(k(x),g(x))	→	k^#(s(x))	(36)
h^#(k(x),g(x))	→	s^#(x)	(37)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

s^#(s(s(0)))	→	k^#(s(0))	(28)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(s(0))))))))	(30)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(0)))))))	(31)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(0))))))	(32)
k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)

1.1.1.1.1 Reduction Pair Processor

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

[s^#(x₁)]

0	0
0	0

· x₁

[s(x₁)]

0	1
0	0

· x₁

[0]

[k^#(x₁)]

1	0
0	0

· x₁

[h(x₁, x₂)]

0	0
0	0

· x₁ +

0	1
0	0

· x₂

[f(x₁)]

0	1
0	1

· x₁

[g(x₁)]

0	1
0	0

· x₁

[k(x₁)]

0	1
0	1

· x₁

the pair

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

→

k^#(s(0))

(28)

could be deleted.

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

k^#(s(s(0)))	→	s^#(h(s(s(s(s(s(s(0)))))),0))	(38)
k^#(s(s(0)))	→	s^#(h(0,s(s(s(s(s(s(0))))))))	(39)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)

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

k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(0)))))))	(31)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(0))))))	(32)
k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)

1.1.1.1.1.1.1.1 Narrowing Processor

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

k^#(s(s(0)))	→	s^#(h(s(s(s(s(s(0))))),0))	(54)
k^#(s(s(0)))	→	s^#(h(0,s(s(s(s(s(0)))))))	(55)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)

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

k^#(s(s(0)))	→	s^#(s(s(s(s(s(0))))))	(32)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)

1.1.1.1.1.1.1.1.1.1 Narrowing Processor

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

k^#(s(s(0)))	→	s^#(h(s(s(s(s(0)))),0))	(68)
k^#(s(s(0)))	→	s^#(h(0,s(s(s(s(0))))))	(69)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

k^#(s(s(0)))	→	s^#(s(s(s(s(0)))))	(33)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)

1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

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

k^#(s(s(0)))	→	s^#(h(s(s(s(0))),0))	(80)
k^#(s(s(0)))	→	s^#(h(0,s(s(s(0)))))	(81)
k^#(s(s(0)))	→	s^#(s(h(s(s(0)),0)))	(82)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(0)))))	(83)
k^#(s(s(0)))	→	s^#(s(k(s(0))))	(84)
k^#(s(s(0)))	→	s^#(s(s(f(s(0)))))	(85)
k^#(s(s(0)))	→	s^#(s(s(h(s(0),0))))	(86)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(0)))))	(87)
k^#(s(s(0)))	→	s^#(s(s(k(0))))	(88)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,0)))))	(89)

1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

k^#(s(s(0)))	→	s^#(s(s(s(0))))	(34)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)
k^#(s(s(0)))	→	s^#(s(h(s(s(0)),0)))	(82)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(0)))))	(83)
k^#(s(s(0)))	→	s^#(s(k(s(0))))	(84)
k^#(s(s(0)))	→	s^#(s(s(f(s(0)))))	(85)
k^#(s(s(0)))	→	s^#(s(s(h(s(0),0))))	(86)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(0)))))	(87)
k^#(s(s(0)))	→	s^#(s(s(k(0))))	(88)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,0)))))	(89)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

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

k^#(s(s(0)))	→	s^#(h(s(s(0)),0))	(90)
k^#(s(s(0)))	→	s^#(h(0,s(s(0))))	(91)
k^#(s(s(0)))	→	s^#(k(s(0)))	(92)
k^#(s(s(0)))	→	s^#(s(f(s(0))))	(93)
k^#(s(s(0)))	→	s^#(s(h(s(0),0)))	(94)
k^#(s(s(0)))	→	s^#(s(h(0,s(0))))	(95)
k^#(s(s(0)))	→	s^#(s(k(0)))	(96)
k^#(s(s(0)))	→	s^#(s(s(h(0,0))))	(97)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

k^#(s(s(0)))	→	s^#(s(s(0)))	(35)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)
k^#(s(s(0)))	→	s^#(s(h(s(s(0)),0)))	(82)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(0)))))	(83)
k^#(s(s(0)))	→	s^#(s(k(s(0))))	(84)
k^#(s(s(0)))	→	s^#(s(s(f(s(0)))))	(85)
k^#(s(s(0)))	→	s^#(s(s(h(s(0),0))))	(86)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(0)))))	(87)
k^#(s(s(0)))	→	s^#(s(s(k(0))))	(88)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,0)))))	(89)
k^#(s(s(0)))	→	s^#(k(s(0)))	(92)
k^#(s(s(0)))	→	s^#(s(f(s(0))))	(93)
k^#(s(s(0)))	→	s^#(s(h(s(0),0)))	(94)
k^#(s(s(0)))	→	s^#(s(h(0,s(0))))	(95)
k^#(s(s(0)))	→	s^#(s(k(0)))	(96)
k^#(s(s(0)))	→	s^#(s(s(h(0,0))))	(97)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

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

k^#(s(s(0)))	→	s^#(f(s(0)))	(98)
k^#(s(s(0)))	→	s^#(h(s(0),0))	(99)
k^#(s(s(0)))	→	s^#(h(0,s(0)))	(100)
k^#(s(s(0)))	→	s^#(k(0))	(101)
k^#(s(s(0)))	→	s^#(s(h(0,0)))	(102)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(s(0)))),0))))	(42)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(s(0))))))))	(43)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(s(0)),0))))))	(46)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(s(0))))))))	(47)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(s(0)))))))	(48)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(k(0)))))))	(52)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(s(h(0,0))))))))	(53)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(s(0))),0))))	(58)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(s(0)))))))	(59)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(s(0),0))))))	(64)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,s(0)))))))	(65)
k^#(s(s(0)))	→	s^#(s(s(s(s(k(0))))))	(66)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)
k^#(s(s(0)))	→	s^#(s(h(s(s(0)),0)))	(82)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(0)))))	(83)
k^#(s(s(0)))	→	s^#(s(k(s(0))))	(84)
k^#(s(s(0)))	→	s^#(s(s(f(s(0)))))	(85)
k^#(s(s(0)))	→	s^#(s(s(h(s(0),0))))	(86)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(0)))))	(87)
k^#(s(s(0)))	→	s^#(s(s(k(0))))	(88)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,0)))))	(89)
k^#(s(s(0)))	→	s^#(k(s(0)))	(92)
k^#(s(s(0)))	→	s^#(s(f(s(0))))	(93)
k^#(s(s(0)))	→	s^#(s(h(s(0),0)))	(94)
k^#(s(s(0)))	→	s^#(s(h(0,s(0))))	(95)
k^#(s(s(0)))	→	s^#(s(k(0)))	(96)
k^#(s(s(0)))	→	s^#(s(s(h(0,0))))	(97)
k^#(s(s(0)))	→	s^#(f(s(0)))	(98)
k^#(s(s(0)))	→	s^#(k(0))	(101)
k^#(s(s(0)))	→	s^#(s(h(0,0)))	(102)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

k^#(s(s(0)))

→

s^#(f(s(0)))

(98)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

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

the pair

k^#(s(s(0)))

→

s^#(k(0))

(101)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

[k^#(x₁)]

0	0
0	0

· x₁

[s(x₁)]

0	0
1	0

· x₁

[0]

[s^#(x₁)]

0	1
0	0

· x₁

[h(x₁, x₂)]

0	0
1	0

· x₁ +

0	0
0	0

· x₂

[k(x₁)]

1	0
0	1

· x₁

[f(x₁)]

0	0
1	0

· x₁

[g(x₁)]

0	0
1	0

· x₁

the pairs

k^#(s(s(0)))	→	s^#(k(s(0)))	(92)
k^#(s(s(0)))	→	s^#(s(k(0)))	(96)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

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

[k^#(x₁)]

3	0
0	0

· x₁

[s(x₁)]

0	1
1	0

· x₁

[0]

[s^#(x₁)]

2	0
0	0

· x₁

[h(x₁, x₂)]

0	1
1	0

· x₁ +

0	1
1	0

· x₂

[k(x₁)]

3	0
3	0

· x₁

[f(x₁)]

0	0
0	0

· x₁

[g(x₁)]

3	3
3	3

· x₁

the pairs

k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(s(0))))),0)))	(40)
s^#(s(s(s(s(s(s(s(s(s(0))))))))))	→	k^#(s(s(0)))	(29)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(s(0))))))))	(41)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(s(0))),0)))))	(44)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(s(0))))))))	(45)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(f(s(0))))))))	(49)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(s(0),0)))))))	(50)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,s(0))))))))	(51)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(s(0)))),0)))	(56)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(s(0)))))))	(57)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(s(0)),0)))))	(60)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(s(0)))))))	(61)
k^#(s(s(0)))	→	s^#(s(s(s(k(s(0))))))	(62)
k^#(s(s(0)))	→	s^#(s(s(s(s(f(s(0)))))))	(63)
k^#(s(s(0)))	→	s^#(s(s(s(s(s(h(0,0)))))))	(67)
k^#(s(s(0)))	→	s^#(s(h(s(s(s(0))),0)))	(70)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(s(0))))))	(71)
k^#(s(s(0)))	→	s^#(s(s(h(s(s(0)),0))))	(72)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(s(0))))))	(73)
k^#(s(s(0)))	→	s^#(s(s(k(s(0)))))	(74)
k^#(s(s(0)))	→	s^#(s(s(s(f(s(0))))))	(75)
k^#(s(s(0)))	→	s^#(s(s(s(h(s(0),0)))))	(76)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,s(0))))))	(77)
k^#(s(s(0)))	→	s^#(s(s(s(k(0)))))	(78)
k^#(s(s(0)))	→	s^#(s(s(s(s(h(0,0))))))	(79)
k^#(s(s(0)))	→	s^#(s(h(s(s(0)),0)))	(82)
k^#(s(s(0)))	→	s^#(s(h(0,s(s(0)))))	(83)
k^#(s(s(0)))	→	s^#(s(k(s(0))))	(84)
k^#(s(s(0)))	→	s^#(s(s(f(s(0)))))	(85)
k^#(s(s(0)))	→	s^#(s(s(h(s(0),0))))	(86)
k^#(s(s(0)))	→	s^#(s(s(h(0,s(0)))))	(87)
k^#(s(s(0)))	→	s^#(s(s(k(0))))	(88)
k^#(s(s(0)))	→	s^#(s(s(s(h(0,0)))))	(89)
k^#(s(s(0)))	→	s^#(s(f(s(0))))	(93)
k^#(s(s(0)))	→	s^#(s(h(s(0),0)))	(94)
k^#(s(s(0)))	→	s^#(s(h(0,s(0))))	(95)
k^#(s(s(0)))	→	s^#(s(s(h(0,0))))	(97)
k^#(s(s(0)))	→	s^#(s(h(0,0)))	(102)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair
g^#(s(x)) → g^#(x) (24)

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.
1.1.1.1.2.1 Size-Change Termination

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

g^#(s(x)) → g^#(x) (24)

1 > 1

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

Certification Problem

Input (TPDB TRS_Standard/MNZ_10/8)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Reduction Pair Processor

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.2.1 Size-Change Termination