Certification Problem

Input (TPDB TRS_Standard/AProVE_04/IJCAR_26a)

The rewrite relation of the following TRS is considered.

p(0)	→	0	(1)
p(s(x))	→	x	(2)
plus(x,0)	→	x	(3)
plus(0,y)	→	y	(4)
plus(s(x),y)	→	s(plus(x,y))	(5)
plus(s(x),y)	→	s(plus(p(s(x)),y))	(6)
plus(x,s(y))	→	s(plus(x,p(s(y))))	(7)
times(0,y)	→	0	(8)
times(s(0),y)	→	y	(9)
times(s(x),y)	→	plus(y,times(x,y))	(10)
div(0,y)	→	0	(11)
div(x,y)	→	quot(x,y,y)	(12)
quot(0,s(y),z)	→	0	(13)
quot(s(x),s(y),z)	→	quot(x,y,z)	(14)
quot(x,0,s(z))	→	s(div(x,s(z)))	(15)
div(div(x,y),z)	→	div(x,times(y,z))	(16)
eq(0,0)	→	true	(17)
eq(s(x),0)	→	false	(18)
eq(0,s(y))	→	false	(19)
eq(s(x),s(y))	→	eq(x,y)	(20)
divides(y,x)	→	eq(x,times(div(x,y),y))	(21)
prime(s(s(x)))	→	pr(s(s(x)),s(x))	(22)
pr(x,s(0))	→	true	(23)
pr(x,s(s(y)))	→	if(divides(s(s(y)),x),x,s(y))	(24)
if(true,x,y)	→	false	(25)
if(false,x,y)	→	pr(x,y)	(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.

plus^#(s(x),y)	→	plus^#(x,y)	(27)
plus^#(s(x),y)	→	plus^#(p(s(x)),y)	(28)
plus^#(s(x),y)	→	p^#(s(x))	(29)
plus^#(x,s(y))	→	plus^#(x,p(s(y)))	(30)
plus^#(x,s(y))	→	p^#(s(y))	(31)
times^#(s(x),y)	→	plus^#(y,times(x,y))	(32)
times^#(s(x),y)	→	times^#(x,y)	(33)
div^#(x,y)	→	quot^#(x,y,y)	(34)
quot^#(s(x),s(y),z)	→	quot^#(x,y,z)	(35)
quot^#(x,0,s(z))	→	div^#(x,s(z))	(36)
div^#(div(x,y),z)	→	div^#(x,times(y,z))	(37)
div^#(div(x,y),z)	→	times^#(y,z)	(38)
eq^#(s(x),s(y))	→	eq^#(x,y)	(39)
divides^#(y,x)	→	eq^#(x,times(div(x,y),y))	(40)
divides^#(y,x)	→	times^#(div(x,y),y)	(41)
divides^#(y,x)	→	div^#(x,y)	(42)
prime^#(s(s(x)))	→	pr^#(s(s(x)),s(x))	(43)
pr^#(x,s(s(y)))	→	if^#(divides(s(s(y)),x),x,s(y))	(44)
pr^#(x,s(s(y)))	→	divides^#(s(s(y)),x)	(45)
if^#(false,x,y)	→	pr^#(x,y)	(46)

1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

The 1^st component contains the pair

pr^#(x,s(s(y))) → if^#(divides(s(s(y)),x),x,s(y)) (44)

if^#(false,x,y) → pr^#(x,y) (46)

1.1.1 Size-Change Termination

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

if^#(false,x,y) → pr^#(x,y) (46)

2 ≥ 1

3 ≥ 2

pr^#(x,s(s(y))) → if^#(divides(s(s(y)),x),x,s(y)) (44)

1 ≥ 2

2 > 3

As there is no critical graph in the transitive closure, there are no infinite chains.
The 2^nd component contains the pair

div^#(x,y) → quot^#(x,y,y) (34)

quot^#(s(x),s(y),z) → quot^#(x,y,z) (35)

quot^#(x,0,s(z)) → div^#(x,s(z)) (36)

div^#(div(x,y),z) → div^#(x,times(y,z)) (37)

1.1.2 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(div) = 1 weight(div) = 1
in combination with the following argument filter

π(div^#) = 1

π(quot^#) = 1

π(s) = 1

π(div) = [1]

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
div^#(div(x,y),z) → div^#(x,times(y,z)) (37)
could be deleted.
1.1.2.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(s) = 0 weight(s) = 1
in combination with the following argument filter

π(div^#) = 1

π(quot^#) = 1

π(s) = [1]

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
quot^#(s(x),s(y),z) → quot^#(x,y,z) (35)
could be deleted.
1.1.2.1.1 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) = 2

prec(s) = 0 weight(s) = 1

in combination with the following argument filter

π(div^#) = 2

π(quot^#) = 2

π(0) = []

π(s) = []

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair
quot^#(x,0,s(z)) → div^#(x,s(z)) (36)
could be deleted.
1.1.2.1.1.1 Size-Change Termination

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

div^#(x,y) → quot^#(x,y,y) (34)

1 ≥ 1

2 ≥ 2

2 ≥ 3

As there is no critical graph in the transitive closure, there are no infinite chains.
The 3^rd component contains the pair
times^#(s(x),y) → times^#(x,y) (33)

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

[s(x₁)] = 1 · x₁

[times^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

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

times^#(s(x),y) → times^#(x,y) (33)

1 > 1

2 ≥ 2

As there is no critical graph in the transitive closure, there are no infinite chains.
The 4^th component contains the pair

plus^#(s(x),y) → plus^#(p(s(x)),y) (28)

plus^#(s(x),y) → plus^#(x,y) (27)

plus^#(x,s(y)) → plus^#(x,p(s(y))) (30)

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

[p(x₁)] = 1 · x₁

[s(x₁)] = 1 · x₁

[plus^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

together with the usable rule
p(s(x)) → x (2)
(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.4.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.4.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[p(x₁)] = 1 · x₁

[s(x₁)] = 1 + 2 · x₁

[plus^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

the pair
plus^#(s(x),y) → plus^#(x,y) (27)
and the rule
p(s(x)) → x (2)
could be deleted.
1.1.4.1.1.1 Dependency Graph Processor
The dependency pairs are split into 0 components.
The 5^th component contains the pair
eq^#(s(x),s(y)) → eq^#(x,y) (39)

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

[s(x₁)] = 1 · x₁

[eq^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

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

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

eq^#(s(x),s(y)) → eq^#(x,y) (39)

1 > 1

2 > 2

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

[s(x₁)]	=	1 · x₁
[times^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

[p(x₁)]	=	1 · x₁
[s(x₁)]	=	1 · x₁
[plus^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

π(div^#)	=	1
π(quot^#)	=	1
π(s)	=	1
π(div)	=	[1]

π(div^#)	=	1
π(quot^#)	=	1
π(s)	=	[1]

prec(0)	=	1		weight(0)	=	2
prec(s)	=	0		weight(s)	=	1

π(div^#)	=	2
π(quot^#)	=	2
π(0)	=	[]
π(s)	=	[]

Certification Problem

Input (TPDB TRS_Standard/AProVE_04/IJCAR_26a)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Size-Change Termination

1.1.2 Reduction Pair Processor with Usable Rules

1.1.2.1 Reduction Pair Processor with Usable Rules

1.1.2.1.1 Reduction Pair Processor with Usable Rules

1.1.2.1.1.1 Size-Change Termination

1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.3.1 Size-Change Termination

1.1.4 Monotonic Reduction Pair Processor with Usable Rules

1.1.4.1 Switch to Innermost Termination

1.1.4.1.1 Monotonic Reduction Pair Processor

1.1.4.1.1.1 Dependency Graph Processor

1.1.5 Monotonic Reduction Pair Processor with Usable Rules

1.1.5.1 Size-Change Termination