Certification Problem

Input (TPDB TRS_Standard/AG01/#3.5)

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(x,0)	→	x	(4)
minus(s(x),s(y))	→	minus(x,y)	(5)
mod(0,y)	→	0	(6)
mod(s(x),0)	→	0	(7)
mod(s(x),s(y))	→	if_mod(le(y,x),s(x),s(y))	(8)
if_mod(true,s(x),s(y))	→	mod(minus(x,y),s(y))	(9)
if_mod(false,s(x),s(y))	→	s(x)	(10)

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.

if_mod^#(true,s(x),s(y))	→	minus^#(x,y)	(11)
mod^#(s(x),s(y))	→	le^#(y,x)	(12)
mod^#(s(x),s(y))	→	if_mod^#(le(y,x),s(x),s(y))	(13)
minus^#(s(x),s(y))	→	minus^#(x,y)	(14)
if_mod^#(true,s(x),s(y))	→	mod^#(minus(x,y),s(y))	(15)
le^#(s(x),s(y))	→	le^#(x,y)	(16)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

if_mod^#(true,s(x),s(y))	→	mod^#(minus(x,y),s(y))	(15)
mod^#(s(x),s(y))	→	if_mod^#(le(y,x),s(x),s(y))	(13)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[le(x₁, x₂)]	=	1
[s(x₁)]	=	x₁ + 33956
[le^#(x₁, x₂)]	=	0
[minus(x₁, x₂)]	=	x₁ + 33954
[mod(x₁, x₂)]	=	0
[false]	=	0
[true]	=	1
[if_mod(x₁, x₂, x₃)]	=	0
[0]	=	1
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	x₁ + x₂ + 2
[if_mod^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0

together with the usable rules

minus(x,0)	→	x	(4)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),s(y))	→	minus(x,y)	(5)
le(s(x),0)	→	false	(2)

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

if_mod^#(true,s(x),s(y))	→	mod^#(minus(x,y),s(y))	(15)
mod^#(s(x),s(y))	→	if_mod^#(le(y,x),s(x),s(y))	(13)

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

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

→

le^#(x,y)

(16)

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)]	=	x₁ + x₂ + 0
[minus(x₁, x₂)]	=	x₁ + 33954
[mod(x₁, x₂)]	=	0
[false]	=	0
[true]	=	1
[if_mod(x₁, x₂, x₃)]	=	0
[0]	=	1
[minus^#(x₁, x₂)]	=	0
[mod^#(x₁, x₂)]	=	x₁ + 2
[if_mod^#(x₁, x₂, x₃)]	=	x₁ + 0

together with the usable rules

minus(x,0)	→	x	(4)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),s(y))	→	minus(x,y)	(5)
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)

(16)

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),s(y))

→

minus^#(x,y)

(14)

1.1.3 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 + 33954
[mod(x₁, x₂)]	=	0
[false]	=	0
[true]	=	1
[if_mod(x₁, x₂, x₃)]	=	0
[0]	=	1
[minus^#(x₁, x₂)]	=	x₁ + x₂ + 0
[mod^#(x₁, x₂)]	=	x₁ + 2
[if_mod^#(x₁, x₂, x₃)]	=	x₁ + 0

together with the usable rules

minus(x,0)	→	x	(4)
le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
minus(s(x),s(y))	→	minus(x,y)	(5)
le(s(x),0)	→	false	(2)

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

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

→

minus^#(x,y)

(14)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.