Certification Problem

Input (TPDB TRS_Standard/AProVE_07/wiehe09)

The rewrite relation of the following TRS is considered.

minus(x,0)	→	x	(1)
minus(s(x),s(y))	→	minus(x,y)	(2)
quot(0,s(y))	→	0	(3)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(4)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(5)
plus(s(x),s(y))	→	s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))	(6)
plus(s(x),x)	→	plus(if(gt(x,x),id(x),id(x)),s(x))	(7)
plus(zero,y)	→	y	(8)
plus(id(x),s(y))	→	s(plus(x,if(gt(s(y),y),y,s(y))))	(9)
id(x)	→	x	(10)
if(true,x,y)	→	x	(11)
if(false,x,y)	→	y	(12)
not(x)	→	if(x,false,true)	(13)
gt(s(x),zero)	→	true	(14)
gt(zero,y)	→	false	(15)
gt(s(x),s(y))	→	gt(x,y)	(16)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(17)
plus^#(s(x),s(y))	→	not^#(gt(x,y))	(18)
plus^#(id(x),s(y))	→	gt^#(s(y),y)	(19)
plus^#(s(x),s(y))	→	id^#(y)	(20)
plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(21)
quot^#(s(x),s(y))	→	minus^#(x,y)	(22)
plus^#(id(x),s(y))	→	if^#(gt(s(y),y),y,s(y))	(23)
plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(24)
minus^#(minus(x,y),z)	→	plus^#(y,z)	(25)
plus^#(s(x),s(y))	→	gt^#(x,y)	(26)
not^#(x)	→	if^#(x,false,true)	(27)
plus^#(s(x),s(y))	→	if^#(not(gt(x,y)),id(x),id(y))	(28)
quot^#(s(x),s(y))	→	quot^#(minus(x,y),s(y))	(29)
plus^#(s(x),x)	→	id^#(x)	(30)
minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(31)
gt^#(s(x),s(y))	→	gt^#(x,y)	(32)
plus^#(s(x),x)	→	gt^#(x,x)	(33)
plus^#(s(x),s(y))	→	if^#(gt(x,y),x,y)	(34)
minus^#(s(x),s(y))	→	minus^#(x,y)	(35)
plus^#(s(x),s(y))	→	id^#(x)	(36)
plus^#(s(x),x)	→	if^#(gt(x,x),id(x),id(x))	(37)
plus^#(s(x),s(y))	→	gt^#(x,y)	(26)
plus^#(s(x),x)	→	id^#(x)	(30)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

quot^#(s(x),s(y))

→

quot^#(minus(x,y),s(y))

(29)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 9228
[gt(x₁, x₂)]	=	1
[minus(x₁, x₂)]	=	x₁ + 9227
[plus^#(x₁, x₂)]	=	0
[false]	=	20539
[id^#(x₁)]	=	0
[true]	=	2
[not^#(x₁)]	=	0
[0]	=	1
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 14680
[quot(x₁, x₂)]	=	0
[gt^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 7830
[quot^#(x₁, x₂)]	=	x₁ + 0
[not(x₁)]	=	x₁ + 2997

together with the usable rules

minus(x,0)	→	x	(1)
minus(minus(x,y),z)	→	minus(x,plus(y,z))	(5)
minus(s(x),s(y))	→	minus(x,y)	(2)

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

quot^#(s(x),s(y))

→

quot^#(minus(x,y),s(y))

(29)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

minus^#(s(x),s(y))	→	minus^#(x,y)	(35)
minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(31)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	x₁ + 1
[gt(x₁, x₂)]	=	1
[minus(x₁, x₂)]	=	x₁ + x₂ + 9227
[plus^#(x₁, x₂)]	=	0
[false]	=	20539
[id^#(x₁)]	=	0
[true]	=	2
[not^#(x₁)]	=	0
[0]	=	0
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[quot(x₁, x₂)]	=	0
[gt^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	x₁ + 0
[plus(x₁, x₂)]	=	x₁ + x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 1
[quot^#(x₁, x₂)]	=	0
[not(x₁)]	=	x₁ + 9725

together with the usable rule

minus(x,0)

→

(1)

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

minus^#(s(x),s(y))	→	minus^#(x,y)	(35)
minus^#(minus(x,y),z)	→	minus^#(x,plus(y,z))	(31)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 3^rd component contains the pair

plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(24)
plus^#(id(x),s(y))	→	plus^#(x,if(gt(s(y),y),y,s(y)))	(21)
plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(17)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	0
[s(x₁)]	=	x₁ + 11711
[gt(x₁, x₂)]	=	max(0)
[minus(x₁, x₂)]	=	max(0)
[plus^#(x₁, x₂)]	=	max(x₁ + 33957, x₂ + 28101, 0)
[false]	=	0
[id^#(x₁)]	=	0
[true]	=	0
[not^#(x₁)]	=	0
[0]	=	0
[if(x₁, x₂, x₃)]	=	max(x₁ + 5856, x₂ + 1, x₃ + 0, 0)
[quot(x₁, x₂)]	=	max(0)
[gt^#(x₁, x₂)]	=	max(0)
[minus^#(x₁, x₂)]	=	max(0)
[plus(x₁, x₂)]	=	max(0)
[if^#(x₁, x₂, x₃)]	=	max(0)
[id(x₁)]	=	x₁ + 0
[quot^#(x₁, x₂)]	=	max(0)
[not(x₁)]	=	x₁ + 5856

together with the usable rules

gt(zero,y)	→	false	(15)
gt(s(x),s(y))	→	gt(x,y)	(16)
id(x)	→	x	(10)
gt(s(x),zero)	→	true	(14)
if(false,x,y)	→	y	(12)
if(true,x,y)	→	x	(11)
not(x)	→	if(x,false,true)	(13)

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

plus^#(s(x),x)	→	plus^#(if(gt(x,x),id(x),id(x)),s(x))	(24)
plus^#(s(x),s(y))	→	plus^#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))	(17)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

plus^#(id(x),s(y))

→

plus^#(x,if(gt(s(y),y),y,s(y)))

(21)

1.1.3.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	1
[s(x₁)]	=	4
[gt(x₁, x₂)]	=	1
[minus(x₁, x₂)]	=	x₁ + x₂ + 9227
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[id^#(x₁)]	=	0
[true]	=	1
[not^#(x₁)]	=	0
[0]	=	0
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 1
[quot(x₁, x₂)]	=	0
[gt^#(x₁, x₂)]	=	0
[minus^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₁ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 2
[quot^#(x₁, x₂)]	=	0
[not(x₁)]	=	1

together with the usable rules

gt(zero,y)	→	false	(15)
minus(x,0)	→	x	(1)
gt(s(x),s(y))	→	gt(x,y)	(16)
id(x)	→	x	(10)
gt(s(x),zero)	→	true	(14)
if(false,x,y)	→	y	(12)
if(true,x,y)	→	x	(11)

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

plus^#(id(x),s(y))

→

plus^#(x,if(gt(s(y),y),y,s(y)))

(21)

could be deleted.

1.1.3.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

gt^#(s(x),s(y))

→

gt^#(x,y)

(32)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[zero]	=	0
[s(x₁)]	=	x₁ + 1
[gt(x₁, x₂)]	=	x₁ + 1
[minus(x₁, x₂)]	=	x₁ + x₂ + 9227
[plus^#(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[id^#(x₁)]	=	0
[true]	=	1
[not^#(x₁)]	=	0
[0]	=	0
[if(x₁, x₂, x₃)]	=	x₂ + x₃ + 15620
[quot(x₁, x₂)]	=	0
[gt^#(x₁, x₂)]	=	x₁ + x₂ + 0
[minus^#(x₁, x₂)]	=	0
[plus(x₁, x₂)]	=	x₂ + 1
[if^#(x₁, x₂, x₃)]	=	0
[id(x₁)]	=	x₁ + 23637
[quot^#(x₁, x₂)]	=	0
[not(x₁)]	=	1

together with the usable rules

gt(zero,y)	→	false	(15)
minus(x,0)	→	x	(1)
gt(s(x),s(y))	→	gt(x,y)	(16)
id(x)	→	x	(10)
gt(s(x),zero)	→	true	(14)
if(false,x,y)	→	y	(12)
if(true,x,y)	→	x	(11)

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

gt^#(s(x),s(y))

→

gt^#(x,y)

(32)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.