Certification Problem

Input (TPDB TRS_Standard/AG01/#3.8b)

The rewrite relation of the following TRS is considered.

le(0,y)	→	true	(1)
le(s(x),0)	→	false	(2)
le(s(x),s(y))	→	le(x,y)	(3)
minus(0,y)	→	0	(4)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(true,s(x),y)	→	0	(6)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(0,s(y))	→	0	(8)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
log(s(0))	→	0	(10)
log(s(s(x)))	→	s(log(s(quot(x,s(s(0))))))	(11)

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.

quot^#(s(x),s(y))	→	minus^#(x,y)	(12)
minus^#(s(x),y)	→	if_minus^#(le(s(x),y),s(x),y)	(13)
minus^#(s(x),y)	→	le^#(s(x),y)	(14)
if_minus^#(false,s(x),y)	→	minus^#(x,y)	(15)
log^#(s(s(x)))	→	log^#(s(quot(x,s(s(0)))))	(16)
quot^#(s(x),s(y))	→	quot^#(minus(x,y),s(y))	(17)
le^#(s(x),s(y))	→	le^#(x,y)	(18)
log^#(s(s(x)))	→	quot^#(x,s(s(0)))	(19)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

log^#(s(s(x)))

→

log^#(s(quot(x,s(s(0)))))

(16)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[le(x₁, x₂)]	=	1
[s(x₁)]	=	x₁ + 36467
[le^#(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[log^#(x₁)]	=	x₁ + 0
[true]	=	1
[log(x₁)]	=	0
[0]	=	1
[quot(x₁, x₂)]	=	x₁ + 36466
[if_minus^#(x₁, x₂, x₃)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus(x₁, x₂, x₃)]	=	x₂ + 0
[quot^#(x₁, x₂)]	=	0

together with the usable rules

minus(0,y)	→	0	(4)
quot(0,s(y))	→	0	(8)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
if_minus(true,s(x),y)	→	0	(6)
le(s(x),0)	→	false	(2)

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

log^#(s(s(x)))

→

log^#(s(quot(x,s(s(0)))))

(16)

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

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

→

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

(17)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[le(x₁, x₂)]	=	1
[s(x₁)]	=	x₁ + 1
[le^#(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[log^#(x₁)]	=	x₁ + 0
[true]	=	1
[log(x₁)]	=	0
[0]	=	1
[quot(x₁, x₂)]	=	x₁ + 36466
[if_minus^#(x₁, x₂, x₃)]	=	0
[minus^#(x₁, x₂)]	=	0
[if_minus(x₁, x₂, x₃)]	=	x₂ + 0
[quot^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

minus(0,y)	→	0	(4)
quot(0,s(y))	→	0	(8)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
if_minus(true,s(x),y)	→	0	(6)
le(s(x),0)	→	false	(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))

(17)

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

minus^#(s(x),y)	→	if_minus^#(le(s(x),y),s(x),y)	(13)
if_minus^#(false,s(x),y)	→	minus^#(x,y)	(15)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[le(x₁, x₂)]	=	1
[s(x₁)]	=	x₁ + 2
[le^#(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[log^#(x₁)]	=	x₁ + 0
[true]	=	1
[log(x₁)]	=	0
[0]	=	2
[quot(x₁, x₂)]	=	x₁ + 29283
[if_minus^#(x₁, x₂, x₃)]	=	x₁ + x₂ + 0
[minus^#(x₁, x₂)]	=	x₁ + 2
[if_minus(x₁, x₂, x₃)]	=	x₂ + 0
[quot^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

minus(0,y)	→	0	(4)
quot(0,s(y))	→	0	(8)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
if_minus(true,s(x),y)	→	0	(6)
le(s(x),0)	→	false	(2)

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

minus^#(s(x),y)	→	if_minus^#(le(s(x),y),s(x),y)	(13)
if_minus^#(false,s(x),y)	→	minus^#(x,y)	(15)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

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

→

le^#(x,y)

(18)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[le(x₁, x₂)]	=	1
[s(x₁)]	=	x₁ + 1
[le^#(x₁, x₂)]	=	x₁ + 0
[minus(x₁, x₂)]	=	x₁ + 0
[false]	=	1
[log^#(x₁)]	=	x₁ + 0
[true]	=	1
[log(x₁)]	=	0
[0]	=	1
[quot(x₁, x₂)]	=	x₁ + 29283
[if_minus^#(x₁, x₂, x₃)]	=	x₁ + 0
[minus^#(x₁, x₂)]	=	2
[if_minus(x₁, x₂, x₃)]	=	x₂ + 0
[quot^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

minus(0,y)	→	0	(4)
quot(0,s(y))	→	0	(8)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),y)	→	if_minus(le(s(x),y),s(x),y)	(5)
if_minus(false,s(x),y)	→	s(minus(x,y))	(7)
quot(s(x),s(y))	→	s(quot(minus(x,y),s(y)))	(9)
if_minus(true,s(x),y)	→	0	(6)
le(s(x),0)	→	false	(2)

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

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

→

le^#(x,y)

(18)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.