Certification Problem

Input (TPDB TRS_Standard/AProVE_04/Liveness6.4)

The rewrite relation of the following TRS is considered.

top1(free(x),y)	→	top2(check(new(x)),y)	(1)
top1(free(x),y)	→	top2(new(x),check(y))	(2)
top1(free(x),y)	→	top2(check(x),new(y))	(3)
top1(free(x),y)	→	top2(x,check(new(y)))	(4)
top2(x,free(y))	→	top1(check(new(x)),y)	(5)
top2(x,free(y))	→	top1(new(x),check(y))	(6)
top2(x,free(y))	→	top1(check(x),new(y))	(7)
top2(x,free(y))	→	top1(x,check(new(y)))	(8)
new(free(x))	→	free(new(x))	(9)
old(free(x))	→	free(old(x))	(10)
new(serve)	→	free(serve)	(11)
old(serve)	→	free(serve)	(12)
check(free(x))	→	free(check(x))	(13)
check(new(x))	→	new(check(x))	(14)
check(old(x))	→	old(check(x))	(15)
check(old(x))	→	old(x)	(16)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

[top1(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[free(x₁)]	=	1 · x₁
[top2(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[check(x₁)]	=	1 · x₁
[new(x₁)]	=	1 · x₁
[old(x₁)]	=	2 · x₁
[serve]	=	2

all of the following rules can be deleted.

old(serve)

→

free(serve)

(12)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[top1(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[free(x₁)]	=	2 + 1 · x₁
[top2(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[check(x₁)]	=	1 · x₁
[new(x₁)]	=	2 + 1 · x₁
[old(x₁)]	=	2 · x₁
[serve]	=	0

all of the following rules can be deleted.

old(free(x))

→

free(old(x))

(10)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

top1^#(free(x),y)	→	top2^#(check(new(x)),y)	(17)
top1^#(free(x),y)	→	check^#(new(x))	(18)
top1^#(free(x),y)	→	new^#(x)	(19)
top1^#(free(x),y)	→	top2^#(new(x),check(y))	(20)
top1^#(free(x),y)	→	check^#(y)	(21)
top1^#(free(x),y)	→	top2^#(check(x),new(y))	(22)
top1^#(free(x),y)	→	check^#(x)	(23)
top1^#(free(x),y)	→	new^#(y)	(24)
top1^#(free(x),y)	→	top2^#(x,check(new(y)))	(25)
top1^#(free(x),y)	→	check^#(new(y))	(26)
top2^#(x,free(y))	→	top1^#(check(new(x)),y)	(27)
top2^#(x,free(y))	→	check^#(new(x))	(28)
top2^#(x,free(y))	→	new^#(x)	(29)
top2^#(x,free(y))	→	top1^#(new(x),check(y))	(30)
top2^#(x,free(y))	→	check^#(y)	(31)
top2^#(x,free(y))	→	top1^#(check(x),new(y))	(32)
top2^#(x,free(y))	→	check^#(x)	(33)
top2^#(x,free(y))	→	new^#(y)	(34)
top2^#(x,free(y))	→	top1^#(x,check(new(y)))	(35)
top2^#(x,free(y))	→	check^#(new(y))	(36)
new^#(free(x))	→	new^#(x)	(37)
check^#(free(x))	→	check^#(x)	(38)
check^#(new(x))	→	new^#(check(x))	(39)
check^#(new(x))	→	check^#(x)	(40)
check^#(old(x))	→	check^#(x)	(41)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

top2^#(x,free(y))	→	top1^#(check(new(x)),y)	(27)
top1^#(free(x),y)	→	top2^#(check(new(x)),y)	(17)
top2^#(x,free(y))	→	top1^#(new(x),check(y))	(30)
top1^#(free(x),y)	→	top2^#(new(x),check(y))	(20)
top2^#(x,free(y))	→	top1^#(check(x),new(y))	(32)
top1^#(free(x),y)	→	top2^#(check(x),new(y))	(22)
top2^#(x,free(y))	→	top1^#(x,check(new(y)))	(35)
top1^#(free(x),y)	→	top2^#(x,check(new(y)))	(25)

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[new(x₁)]	=	1 · x₁
[free(x₁)]	=	1 · x₁
[serve]	=	0
[check(x₁)]	=	1 · x₁
[old(x₁)]	=	1 · x₁
[top1^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top2^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

new(free(x))	→	free(new(x))	(9)
new(serve)	→	free(serve)	(11)
check(free(x))	→	free(check(x))	(13)
check(new(x))	→	new(check(x))	(14)
check(old(x))	→	old(check(x))	(15)
check(old(x))	→	old(x)	(16)

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

top2^#(x0,free(y1))	→	top1^#(new(check(x0)),y1)	(42)
top2^#(free(x0),free(y1))	→	top1^#(check(free(new(x0))),y1)	(43)
top2^#(serve,free(y1))	→	top1^#(check(free(serve)),y1)	(44)

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

top2^#(free(x0),free(y1))	→	top1^#(free(new(x0)),check(y1))	(45)
top2^#(serve,free(y1))	→	top1^#(free(serve),check(y1))	(46)

1.1.1.1.1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 2 to get the following set of pairs

top1^#(free(y0),free(x0))	→	top2^#(new(y0),free(check(x0)))	(47)
top1^#(free(y0),new(x0))	→	top2^#(new(y0),new(check(x0)))	(48)
top1^#(free(y0),old(x0))	→	top2^#(new(y0),old(check(x0)))	(49)
top1^#(free(y0),old(x0))	→	top2^#(new(y0),old(x0))	(50)

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

top2^#(x,free(y))	→	top1^#(check(x),new(y))	(32)
top1^#(free(x),y)	→	top2^#(check(new(x)),y)	(17)
top2^#(x,free(y))	→	top1^#(x,check(new(y)))	(35)
top1^#(free(x),y)	→	top2^#(check(x),new(y))	(22)
top2^#(x0,free(y1))	→	top1^#(new(check(x0)),y1)	(42)
top1^#(free(x),y)	→	top2^#(x,check(new(y)))	(25)
top2^#(free(x0),free(y1))	→	top1^#(check(free(new(x0))),y1)	(43)
top1^#(free(y0),free(x0))	→	top2^#(new(y0),free(check(x0)))	(47)
top2^#(free(x0),free(y1))	→	top1^#(free(new(x0)),check(y1))	(45)
top1^#(free(y0),new(x0))	→	top2^#(new(y0),new(check(x0)))	(48)
top2^#(serve,free(y1))	→	top1^#(check(free(serve)),y1)	(44)
top2^#(serve,free(y1))	→	top1^#(free(serve),check(y1))	(46)

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

top2^#(free(x0),free(y1))	→	top1^#(free(check(x0)),new(y1))	(51)
top2^#(new(x0),free(y1))	→	top1^#(new(check(x0)),new(y1))	(52)
top2^#(old(x0),free(y1))	→	top1^#(old(check(x0)),new(y1))	(53)
top2^#(old(x0),free(y1))	→	top1^#(old(x0),new(y1))	(54)

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

top2^#(x,free(y))	→	top1^#(x,check(new(y)))	(35)
top1^#(free(x),y)	→	top2^#(check(new(x)),y)	(17)
top2^#(x0,free(y1))	→	top1^#(new(check(x0)),y1)	(42)
top1^#(free(x),y)	→	top2^#(check(x),new(y))	(22)
top2^#(free(x0),free(y1))	→	top1^#(check(free(new(x0))),y1)	(43)
top1^#(free(x),y)	→	top2^#(x,check(new(y)))	(25)
top2^#(serve,free(y1))	→	top1^#(check(free(serve)),y1)	(44)
top1^#(free(y0),free(x0))	→	top2^#(new(y0),free(check(x0)))	(47)
top2^#(free(x0),free(y1))	→	top1^#(free(new(x0)),check(y1))	(45)
top1^#(free(y0),new(x0))	→	top2^#(new(y0),new(check(x0)))	(48)
top2^#(free(x0),free(y1))	→	top1^#(free(check(x0)),new(y1))	(51)
top2^#(new(x0),free(y1))	→	top1^#(new(check(x0)),new(y1))	(52)
top2^#(serve,free(y1))	→	top1^#(free(serve),check(y1))	(46)

1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 2 to get the following set of pairs

top1^#(free(y0),free(x0))	→	top2^#(check(y0),free(new(x0)))	(55)
top1^#(free(y0),serve)	→	top2^#(check(y0),free(serve))	(56)

1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position 2 to get the following set of pairs

top1^#(free(y0),x0)	→	top2^#(y0,new(check(x0)))	(57)
top1^#(free(y0),free(x0))	→	top2^#(y0,check(free(new(x0))))	(58)
top1^#(free(y0),serve)	→	top2^#(y0,check(free(serve)))	(59)

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

[top1^#(x₁, x₂)]	=	x₁
[top2^#(x₁, x₂)]	=	x₁
[new(x₁)]	=	x₁
[free(x₁)]	=	x₁
[serve]	=	1
[check(x₁)]	=	-2
[old(x₁)]	=	-2

the pair

top2^#(serve,free(y1))

→

top1^#(check(free(serve)),y1)

(44)

could be deleted.

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

[top1^#(x₁, x₂)]	=	2 · x₂
[top2^#(x₁, x₂)]	=	2 · x₂
[new(x₁)]	=	x₁
[free(x₁)]	=	x₁
[serve]	=	2
[check(x₁)]	=	-2
[old(x₁)]	=	-2

the pair

top1^#(free(y0),serve)

→

top2^#(y0,check(free(serve)))

(59)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Semantic Labeling Processor

The following interpretations form a model of the rules.

As carrier we take the set {0,1}. Symbols are labeled by the interpretation of their arguments using the interpretations (modulo 2):

[old(x₁)]	=	1
[new(x₁)]	=	1x₁
[top2^#(x₁, x₂)]	=	0
[serve]	=	0
[check(x₁)]	=	1
[free(x₁)]	=	1x₁
[top1^#(x₁, x₂)]	=	0

We obtain the set of labeled pairs

top2^#₀₀(x,free₀(y))	→	top1^#₀₁(x,check₀(new₀(y)))	(60)
top1^#₀₀(free₀(x),y)	→	top2^#₁₀(check₀(new₀(x)),y)	(61)
top1^#₀₁(free₀(x),y)	→	top2^#₁₁(check₀(new₀(x)),y)	(62)
top1^#₁₀(free₁(x),y)	→	top2^#₁₀(check₁(new₁(x)),y)	(63)
top1^#₁₁(free₁(x),y)	→	top2^#₁₁(check₁(new₁(x)),y)	(64)
top2^#₀₁(x,free₁(y))	→	top1^#₀₁(x,check₁(new₁(y)))	(65)
top2^#₁₀(x,free₀(y))	→	top1^#₁₁(x,check₀(new₀(y)))	(66)
top2^#₁₁(x,free₁(y))	→	top1^#₁₁(x,check₁(new₁(y)))	(67)
top1^#₀₀(free₀(y0),free₀(x0))	→	top2^#₀₁(new₀(y0),free₁(check₀(x0)))	(68)
top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(new₀(y0),free₁(check₁(x0)))	(69)
top1^#₁₀(free₁(y0),free₀(x0))	→	top2^#₁₁(new₁(y0),free₁(check₀(x0)))	(70)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(new₁(y0),free₁(check₁(x0)))	(71)
top1^#₀₀(free₀(y0),new₀(x0))	→	top2^#₀₁(new₀(y0),new₁(check₀(x0)))	(72)
top1^#₀₁(free₀(y0),new₁(x0))	→	top2^#₀₁(new₀(y0),new₁(check₁(x0)))	(73)
top1^#₁₀(free₁(y0),new₀(x0))	→	top2^#₁₁(new₁(y0),new₁(check₀(x0)))	(74)
top1^#₁₁(free₁(y0),new₁(x0))	→	top2^#₁₁(new₁(y0),new₁(check₁(x0)))	(75)
top1^#₀₀(free₀(y0),free₀(x0))	→	top2^#₁₀(check₀(y0),free₀(new₀(x0)))	(76)
top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₁₁(check₀(y0),free₁(new₁(x0)))	(77)
top1^#₁₀(free₁(y0),free₀(x0))	→	top2^#₁₀(check₁(y0),free₀(new₀(x0)))	(78)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(check₁(y0),free₁(new₁(x0)))	(79)
top1^#₀₀(free₀(y0),x0)	→	top2^#₀₁(y0,new₁(check₀(x0)))	(80)
top1^#₀₁(free₀(y0),x0)	→	top2^#₀₁(y0,new₁(check₁(x0)))	(81)
top1^#₁₀(free₁(y0),x0)	→	top2^#₁₁(y0,new₁(check₀(x0)))	(82)
top1^#₁₁(free₁(y0),x0)	→	top2^#₁₁(y0,new₁(check₁(x0)))	(83)
top1^#₀₀(free₀(y0),free₀(x0))	→	top2^#₀₁(y0,check₀(free₀(new₀(x0))))	(84)
top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(y0,check₁(free₁(new₁(x0))))	(85)
top1^#₁₀(free₁(y0),free₀(x0))	→	top2^#₁₁(y0,check₀(free₀(new₀(x0))))	(86)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(y0,check₁(free₁(new₁(x0))))	(87)
top2^#₀₀(x0,free₀(y1))	→	top1^#₁₀(new₁(check₀(x0)),y1)	(88)
top2^#₀₁(x0,free₁(y1))	→	top1^#₁₁(new₁(check₀(x0)),y1)	(89)
top2^#₁₀(x0,free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),y1)	(90)
top2^#₁₁(x0,free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),y1)	(91)
top2^#₀₀(free₀(x0),free₀(y1))	→	top1^#₁₀(check₀(free₀(new₀(x0))),y1)	(92)
top2^#₀₁(free₀(x0),free₁(y1))	→	top1^#₁₁(check₀(free₀(new₀(x0))),y1)	(93)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(check₁(free₁(new₁(x0))),y1)	(94)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(check₁(free₁(new₁(x0))),y1)	(95)
top2^#₀₀(free₀(x0),free₀(y1))	→	top1^#₀₁(free₀(new₀(x0)),check₀(y1))	(96)
top2^#₀₁(free₀(x0),free₁(y1))	→	top1^#₀₁(free₀(new₀(x0)),check₁(y1))	(97)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₁(free₁(new₁(x0)),check₀(y1))	(98)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(new₁(x0)),check₁(y1))	(99)
top2^#₀₀(free₀(x0),free₀(y1))	→	top1^#₁₀(free₁(check₀(x0)),new₀(y1))	(100)
top2^#₀₁(free₀(x0),free₁(y1))	→	top1^#₁₁(free₁(check₀(x0)),new₁(y1))	(101)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(free₁(check₁(x0)),new₀(y1))	(102)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(check₁(x0)),new₁(y1))	(103)
top2^#₀₀(new₀(x0),free₀(y1))	→	top1^#₁₀(new₁(check₀(x0)),new₀(y1))	(104)
top2^#₀₁(new₀(x0),free₁(y1))	→	top1^#₁₁(new₁(check₀(x0)),new₁(y1))	(105)
top2^#₁₀(new₁(x0),free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),new₀(y1))	(106)
top2^#₁₁(new₁(x0),free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),new₁(y1))	(107)
top1^#₀₀(free₀(y0),serve)	→	top2^#₁₀(check₀(y0),free₀(serve))	(108)
top1^#₁₀(free₁(y0),serve)	→	top2^#₁₀(check₁(y0),free₀(serve))	(109)
top2^#₀₀(serve,free₀(y1))	→	top1^#₀₁(free₀(serve),check₀(y1))	(110)
top2^#₀₁(serve,free₁(y1))	→	top1^#₀₁(free₀(serve),check₁(y1))	(111)

and the set of labeled rules:

new₀(free₀(x))	→	free₀(new₀(x))	(112)
new₁(free₁(x))	→	free₁(new₁(x))	(113)
new₀(serve)	→	free₀(serve)	(114)
check₀(free₀(x))	→	free₁(check₀(x))	(115)
check₁(free₁(x))	→	free₁(check₁(x))	(116)
check₀(new₀(x))	→	new₁(check₀(x))	(117)
check₁(new₁(x))	→	new₁(check₁(x))	(118)
check₁(old₀(x))	→	old₁(check₀(x))	(119)
check₁(old₁(x))	→	old₁(check₁(x))	(120)
check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)

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 3 components.

The 1^st component contains the pair

top2^#₁₀(x0,free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),y1)	(90)
top1^#₁₀(free₁(x),y)	→	top2^#₁₀(check₁(new₁(x)),y)	(63)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(check₁(free₁(new₁(x0))),y1)	(94)
top1^#₁₀(free₁(y0),free₀(x0))	→	top2^#₁₀(check₁(y0),free₀(new₀(x0)))	(78)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(free₁(check₁(x0)),new₀(y1))	(102)
top2^#₁₀(new₁(x0),free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),new₀(y1))	(106)
top1^#₁₀(free₁(y0),serve)	→	top2^#₁₀(check₁(y0),free₀(serve))	(109)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[new₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 · x₁
[serve]	=	0
[check₀(x₁)]	=	1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 + 1 · x₁
[old₁(x₁)]	=	1 · x₁
[top2^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top1^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the rule

check₁(old₀(x))

→

old₁(check₀(x))

(119)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[check₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 · x₁
[new₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[serve]	=	0
[top2^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top1^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

check₁(free₁(x))	→	free₁(check₁(x))	(116)
check₁(new₁(x))	→	new₁(check₁(x))	(118)
check₁(old₁(x))	→	old₁(check₁(x))	(120)
check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)
new₁(free₁(x))	→	free₁(new₁(x))	(113)
new₀(free₀(x))	→	free₀(new₀(x))	(112)
new₀(serve)	→	free₀(serve)	(114)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[check₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[new₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 + 1 · x₁
[free₀(x₁)]	=	1 + 1 · x₁
[serve]	=	0
[top2^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top1^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the pairs

top2^#₁₀(x0,free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),y1)	(90)
top1^#₁₀(free₁(x),y)	→	top2^#₁₀(check₁(new₁(x)),y)	(63)
top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(check₁(free₁(new₁(x0))),y1)	(94)

and no rules could be deleted.

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 1 component.

The 1^st component contains the pair

top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(free₁(check₁(x0)),new₀(y1))	(102)
top1^#₁₀(free₁(y0),free₀(x0))	→	top2^#₁₀(check₁(y0),free₀(new₀(x0)))	(78)
top2^#₁₀(new₁(x0),free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),new₀(y1))	(106)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[check₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[new₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[serve]	=	0
[top2^#₁₀(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[top1^#₁₀(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the pairs

top2^#₁₀(free₁(x0),free₀(y1))	→	top1^#₁₀(free₁(check₁(x0)),new₀(y1))	(102)
top2^#₁₀(new₁(x0),free₀(y1))	→	top1^#₁₀(new₁(check₁(x0)),new₀(y1))	(106)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[check₁(x₁)]	=	1 + 1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[new₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[serve]	=	0
[top1^#₁₀(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[top2^#₁₀(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂

together with the usable rules

check₁(free₁(x))	→	free₁(check₁(x))	(116)
check₁(new₁(x))	→	new₁(check₁(x))	(118)
check₁(old₁(x))	→	old₁(check₁(x))	(120)
check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)
new₀(free₀(x))	→	free₀(new₀(x))	(112)
new₀(serve)	→	free₀(serve)	(114)
new₁(free₁(x))	→	free₁(new₁(x))	(113)

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

top1^#₁₀(free₁(y0),free₀(x0))

→

top2^#₁₀(check₁(y0),free₀(new₀(x0)))

(78)

and the rules

check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(new₀(y0),free₁(check₁(x0)))	(69)
top2^#₀₁(x,free₁(y))	→	top1^#₀₁(x,check₁(new₁(y)))	(65)
top1^#₀₁(free₀(y0),new₁(x0))	→	top2^#₀₁(new₀(y0),new₁(check₁(x0)))	(73)
top2^#₀₁(free₀(x0),free₁(y1))	→	top1^#₀₁(free₀(new₀(x0)),check₁(y1))	(97)
top1^#₀₁(free₀(y0),x0)	→	top2^#₀₁(y0,new₁(check₁(x0)))	(81)
top2^#₀₁(serve,free₁(y1))	→	top1^#₀₁(free₀(serve),check₁(y1))	(111)
top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(y0,check₁(free₁(new₁(x0))))	(85)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[new₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 · x₁
[serve]	=	0
[check₀(x₁)]	=	1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 + 1 · x₁
[old₁(x₁)]	=	1 · x₁
[top1^#₀₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top2^#₀₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the rule

check₁(old₀(x))

→

old₁(check₀(x))

(119)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[new₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[serve]	=	0
[top1^#₀₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top2^#₀₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

new₁(free₁(x))	→	free₁(new₁(x))	(113)
check₁(free₁(x))	→	free₁(check₁(x))	(116)
check₁(new₁(x))	→	new₁(check₁(x))	(118)
check₁(old₁(x))	→	old₁(check₁(x))	(120)
check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)
new₀(free₀(x))	→	free₀(new₀(x))	(112)
new₀(serve)	→	free₀(serve)	(114)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[new₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[new₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 · x₁
[serve]	=	0
[top1^#₀₁(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[top2^#₀₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the pairs

top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(new₀(y0),free₁(check₁(x0)))	(69)
top1^#₀₁(free₀(y0),new₁(x0))	→	top2^#₀₁(new₀(y0),new₁(check₁(x0)))	(73)
top1^#₀₁(free₀(y0),x0)	→	top2^#₀₁(y0,new₁(check₁(x0)))	(81)
top1^#₀₁(free₀(y0),free₁(x0))	→	top2^#₀₁(y0,check₁(free₁(new₁(x0))))	(85)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

top2^#₁₁(x,free₁(y))	→	top1^#₁₁(x,check₁(new₁(y)))	(67)
top1^#₁₁(free₁(x),y)	→	top2^#₁₁(check₁(new₁(x)),y)	(64)
top2^#₁₁(x0,free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),y1)	(91)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(new₁(y0),free₁(check₁(x0)))	(71)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(check₁(free₁(new₁(x0))),y1)	(95)
top1^#₁₁(free₁(y0),new₁(x0))	→	top2^#₁₁(new₁(y0),new₁(check₁(x0)))	(75)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(new₁(x0)),check₁(y1))	(99)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(check₁(y0),free₁(new₁(x0)))	(79)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(check₁(x0)),new₁(y1))	(103)
top1^#₁₁(free₁(y0),x0)	→	top2^#₁₁(y0,new₁(check₁(x0)))	(83)
top2^#₁₁(new₁(x0),free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),new₁(y1))	(107)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(y0,check₁(free₁(new₁(x0))))	(87)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[new₁(x₁)]	=	1 + 1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 + 1 · x₁
[old₁(x₁)]	=	1 · x₁
[check₀(x₁)]	=	1 · x₁
[free₀(x₁)]	=	1 + 1 · x₁
[new₀(x₁)]	=	1 + 1 · x₁
[top2^#₁₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top1^#₁₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

new₁(free₁(x))	→	free₁(new₁(x))	(113)
check₁(free₁(x))	→	free₁(check₁(x))	(116)
check₁(new₁(x))	→	new₁(check₁(x))	(118)
check₁(old₀(x))	→	old₁(check₀(x))	(119)
check₁(old₁(x))	→	old₁(check₁(x))	(120)
check₁(old₀(x))	→	old₀(x)	(121)
check₁(old₁(x))	→	old₁(x)	(122)
check₀(free₀(x))	→	free₁(check₀(x))	(115)
check₀(new₀(x))	→	new₁(check₀(x))	(117)

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

check₁(old₀(x))	→	old₁(check₀(x))	(119)
check₀(free₀(x))	→	free₁(check₀(x))	(115)
check₀(new₀(x))	→	new₁(check₀(x))	(117)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[new₁(x₁)]	=	1 · x₁
[free₁(x₁)]	=	1 + 1 · x₁
[check₁(x₁)]	=	1 · x₁
[old₁(x₁)]	=	1 · x₁
[old₀(x₁)]	=	1 · x₁
[top2^#₁₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[top1^#₁₁(x₁, x₂)]	=	1 · x₁ + 1 · x₂

the pairs

top2^#₁₁(x,free₁(y))	→	top1^#₁₁(x,check₁(new₁(y)))	(67)
top1^#₁₁(free₁(x),y)	→	top2^#₁₁(check₁(new₁(x)),y)	(64)
top2^#₁₁(x0,free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),y1)	(91)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(new₁(y0),free₁(check₁(x0)))	(71)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(check₁(free₁(new₁(x0))),y1)	(95)
top1^#₁₁(free₁(y0),new₁(x0))	→	top2^#₁₁(new₁(y0),new₁(check₁(x0)))	(75)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(new₁(x0)),check₁(y1))	(99)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(check₁(y0),free₁(new₁(x0)))	(79)
top2^#₁₁(free₁(x0),free₁(y1))	→	top1^#₁₁(free₁(check₁(x0)),new₁(y1))	(103)
top1^#₁₁(free₁(y0),x0)	→	top2^#₁₁(y0,new₁(check₁(x0)))	(83)
top2^#₁₁(new₁(x0),free₁(y1))	→	top1^#₁₁(new₁(check₁(x0)),new₁(y1))	(107)
top1^#₁₁(free₁(y0),free₁(x0))	→	top2^#₁₁(y0,check₁(free₁(new₁(x0))))	(87)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

check^#(new(x)) → check^#(x) (40)

check^#(free(x)) → check^#(x) (38)

check^#(old(x)) → check^#(x) (41)

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

[new(x₁)] = 1 · x₁

[free(x₁)] = 1 · x₁

[old(x₁)] = 1 · x₁

[check^#(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.

check^#(new(x)) → check^#(x) (40)

1 > 1

check^#(free(x)) → check^#(x) (38)

1 > 1

check^#(old(x)) → check^#(x) (41)

1 > 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 3^rd component contains the pair
new^#(free(x)) → new^#(x) (37)

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

[free(x₁)] = 1 · x₁

[new^#(x₁)] = 1 · x₁

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

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

new^#(free(x)) → new^#(x) (37)

1 > 1

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

Certification Problem

Input (TPDB TRS_Standard/AProVE_04/Liveness6.4)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

1.1 Rule Removal

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Narrowing Processor

1.1.1.1.1.1.1 Narrowing 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 Narrowing Processor

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 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Semantic Labeling 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 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 P is empty

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.3.1.1 P is empty

1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.2.1 Size-Change Termination

1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.3.1 Size-Change Termination