Certification Problem

Input (TPDB TRS_Standard/AG01/#3.13)

The rewrite relation of the following TRS is considered.

eq(0,0)	→	true	(1)
eq(0,s(x))	→	false	(2)
eq(s(x),0)	→	false	(3)
eq(s(x),s(y))	→	eq(x,y)	(4)
or(true,y)	→	true	(5)
or(false,y)	→	y	(6)
union(empty,h)	→	h	(7)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
reach(x,y,empty,h)	→	false	(9)
reach(x,y,edge(u,v,i),h)	→	if_reach_1(eq(x,u),x,y,edge(u,v,i),h)	(10)
if_reach_1(true,x,y,edge(u,v,i),h)	→	if_reach_2(eq(y,v),x,y,edge(u,v,i),h)	(11)
if_reach_2(true,x,y,edge(u,v,i),h)	→	true	(12)
if_reach_2(false,x,y,edge(u,v,i),h)	→	or(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(13)
if_reach_1(false,x,y,edge(u,v,i),h)	→	reach(x,y,i,edge(u,v,h))	(14)

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_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(15)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	eq^#(y,v)	(16)
reach^#(x,y,edge(u,v,i),h)	→	eq^#(x,u)	(17)
eq^#(s(x),s(y))	→	eq^#(x,y)	(18)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(19)
union^#(edge(x,y,i),h)	→	union^#(i,h)	(20)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	if_reach_2^#(eq(y,v),x,y,edge(u,v,i),h)	(21)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(v,y,union(i,h),empty)	(22)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	or^#(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(23)
reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	union^#(i,h)	(25)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(v,y,union(i,h),empty)	(22)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	if_reach_2^#(eq(y,v),x,y,edge(u,v,i),h)	(21)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(19)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(15)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	1
[if_reach_1^#(x₁,...,x₅)]	=	x₂ + x₃ + x₄ + x₅ + 2
[edge(x₁, x₂, x₃)]	=	x₂ + x₃ + 89206
[if_reach_2^#(x₁,...,x₅)]	=	x₁ + x₂ + x₃ + x₄ + x₅ + 0
[eq(x₁, x₂)]	=	1
[false]	=	1
[reach(x₁,...,x₄)]	=	0
[if_reach_2(x₁,...,x₅)]	=	0
[true]	=	1
[reach^#(x₁,...,x₄)]	=	x₁ + x₂ + x₃ + x₄ + 2
[eq^#(x₁, x₂)]	=	0
[0]	=	0
[union(x₁, x₂)]	=	x₁ + x₂ + 36336
[or(x₁, x₂)]	=	0
[empty]	=	52868
[if_reach_1(x₁,...,x₅)]	=	0
[or^#(x₁, x₂)]	=	0
[union^#(x₁, x₂)]	=	0

together with the usable rules

eq(s(x),s(y))	→	eq(x,y)	(4)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
eq(0,0)	→	true	(1)
eq(s(x),0)	→	false	(3)
union(empty,h)	→	h	(7)
eq(0,s(x))	→	false	(2)

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

if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(v,y,union(i,h),empty)	(22)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	if_reach_2^#(eq(y,v),x,y,edge(u,v,i),h)	(21)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(15)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(19)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	1
[if_reach_1^#(x₁,...,x₅)]	=	x₂ + x₄ + 2
[edge(x₁, x₂, x₃)]	=	x₂ + x₃ + 2
[if_reach_2^#(x₁,...,x₅)]	=	x₁ + 0
[eq(x₁, x₂)]	=	1
[false]	=	1
[reach(x₁,...,x₄)]	=	0
[if_reach_2(x₁,...,x₅)]	=	0
[true]	=	1
[reach^#(x₁,...,x₄)]	=	x₁ + x₃ + 3
[eq^#(x₁, x₂)]	=	0
[0]	=	0
[union(x₁, x₂)]	=	x₁ + x₂ + 36336
[or(x₁, x₂)]	=	0
[empty]	=	52868
[if_reach_1(x₁,...,x₅)]	=	0
[or^#(x₁, x₂)]	=	0
[union^#(x₁, x₂)]	=	0

together with the usable rules

eq(s(x),s(y))	→	eq(x,y)	(4)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
eq(0,0)	→	true	(1)
eq(s(x),0)	→	false	(3)
union(empty,h)	→	h	(7)
eq(0,s(x))	→	false	(2)

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

reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(19)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

union^#(edge(x,y,i),h)

→

union^#(i,h)

(20)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	1
[if_reach_1^#(x₁,...,x₅)]	=	2
[edge(x₁, x₂, x₃)]	=	x₂ + x₃ + 2
[if_reach_2^#(x₁,...,x₅)]	=	x₁ + 0
[eq(x₁, x₂)]	=	1
[false]	=	1
[reach(x₁,...,x₄)]	=	0
[if_reach_2(x₁,...,x₅)]	=	0
[true]	=	1
[reach^#(x₁,...,x₄)]	=	x₃ + 3
[eq^#(x₁, x₂)]	=	0
[0]	=	0
[union(x₁, x₂)]	=	x₁ + x₂ + 1
[or(x₁, x₂)]	=	0
[empty]	=	1
[if_reach_1(x₁,...,x₅)]	=	0
[or^#(x₁, x₂)]	=	0
[union^#(x₁, x₂)]	=	x₁ + 0

together with the usable rules

eq(s(x),s(y))	→	eq(x,y)	(4)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
eq(0,0)	→	true	(1)
eq(s(x),0)	→	false	(3)
union(empty,h)	→	h	(7)
eq(0,s(x))	→	false	(2)

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

union^#(edge(x,y,i),h)

→

union^#(i,h)

(20)

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

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

→

eq^#(x,y)

(18)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[s(x₁)]	=	x₁ + 1
[if_reach_1^#(x₁,...,x₅)]	=	2
[edge(x₁, x₂, x₃)]	=	x₂ + x₃ + 2
[if_reach_2^#(x₁,...,x₅)]	=	x₁ + 0
[eq(x₁, x₂)]	=	1
[false]	=	1
[reach(x₁,...,x₄)]	=	0
[if_reach_2(x₁,...,x₅)]	=	0
[true]	=	1
[reach^#(x₁,...,x₄)]	=	x₃ + 3
[eq^#(x₁, x₂)]	=	x₁ + 0
[0]	=	0
[union(x₁, x₂)]	=	x₁ + x₂ + 1
[or(x₁, x₂)]	=	0
[empty]	=	1
[if_reach_1(x₁,...,x₅)]	=	0
[or^#(x₁, x₂)]	=	0
[union^#(x₁, x₂)]	=	0

together with the usable rules

eq(s(x),s(y))	→	eq(x,y)	(4)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
eq(0,0)	→	true	(1)
eq(s(x),0)	→	false	(3)
union(empty,h)	→	h	(7)
eq(0,s(x))	→	false	(2)

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

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

→

eq^#(x,y)

(18)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.