Certification Problem

Input (TPDB TRS_Equational/Mixed_AC_and_C/AC47)

The rewrite relation of the following equational 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_1(false,x,y,edge(u,v,i),h)	→	reach(x,y,i,edge(u,v,h))	(12)
if_reach_2(true,x,y,edge(u,v,i),h)	→	true	(13)
if_reach_2(false,x,y,edge(u,v,i),h)	→	or(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(14)

Associative symbols: or

Commutative symbols: eq, or

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 AC Dependency Pair Transformation

The following set of (strict) dependency pairs is constructed for the TRS.

eq^#(s(x),s(y))	→	eq^#(x,y)	(22)
union^#(edge(x,y,i),h)	→	union^#(i,h)	(23)
reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)
reach^#(x,y,edge(u,v,i),h)	→	eq^#(x,u)	(25)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	if_reach_2^#(eq(y,v),x,y,edge(u,v,i),h)	(26)
if_reach_1^#(true,x,y,edge(u,v,i),h)	→	eq^#(y,v)	(27)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(28)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	or^#(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(29)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(30)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(v,y,union(i,h),empty)	(31)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	union^#(i,h)	(32)
or^#(or(true,y),ext)	→	or^#(true,ext)	(33)
or^#(or(false,y),ext)	→	or^#(y,ext)	(34)

The extended rules of the TRS

or(or(true,y),ext)	→	or(true,ext)	(35)
or(or(false,y),ext)	→	or(y,ext)	(36)

give rise to even more dependency pairs (by sharping the root symbols of each rule). Finiteness for all DPs in combination with the equational DPs is proven as follows.

1.1 Dependency Graph Processor

The dependency pairs are split into 4 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_1^#(true,x,y,edge(u,v,i),h)	→	if_reach_2^#(eq(y,v),x,y,edge(u,v,i),h)	(26)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(30)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(v,y,union(i,h),empty)	(31)
if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(28)

1.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

together with the usable rules

union(empty,h)	→	h	(7)
union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)

(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)	(31)
if_reach_2^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,h)	(30)

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

if_reach_1^#(false,x,y,edge(u,v,i),h)	→	reach^#(x,y,i,edge(u,v,h))	(28)
reach^#(x,y,edge(u,v,i),h)	→	if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)	(24)

1.1.1.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

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

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

reach^#(x,y,edge(u,v,i),h)

→

if_reach_1^#(eq(x,u),x,y,edge(u,v,i),h)

(24)

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

eq^#(x,y)	→	eq^#(y,x)	(18)
eq^#(s(x),s(y))	→	eq^#(x,y)	(22)

1.1.2 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[eq^#(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[s(x₁)]	=	3 · x₁

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

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

→

eq^#(x,y)

(22)

and the rules

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_1(false,x,y,edge(u,v,i),h)	→	reach(x,y,i,edge(u,v,h))	(12)
if_reach_2(true,x,y,edge(u,v,i),h)	→	true	(13)
if_reach_2(false,x,y,edge(u,v,i),h)	→	or(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(14)
or(or(true,y),ext)	→	or(true,ext)	(35)
or(or(false,y),ext)	→	or(y,ext)	(36)

could be deleted.

1.1.2.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

The 3^rd component contains the pair

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

→

union^#(i,h)

(23)

1.1.3 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[union^#(x₁, x₂)]	=	3 · x₁ + 1 · x₂
[edge(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 3 · x₃

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

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

→

union^#(i,h)

(23)

and the rules

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_1(false,x,y,edge(u,v,i),h)	→	reach(x,y,i,edge(u,v,h))	(12)
if_reach_2(true,x,y,edge(u,v,i),h)	→	true	(13)
if_reach_2(false,x,y,edge(u,v,i),h)	→	or(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(14)
or(or(true,y),ext)	→	or(true,ext)	(35)
or(or(false,y),ext)	→	or(y,ext)	(36)

could be deleted.

1.1.3.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

The 4^th component contains the pair

or^#(or(false,y),ext)	→	or^#(y,ext)	(34)
or^#(or(true,y),ext)	→	or^#(true,ext)	(33)
or^#(or(x,y),z)	→	or^#(y,z)	(21)
or^#(x,y)	→	or^#(y,x)	(19)
or^#(or(x,y),z)	→	or^#(x,or(y,z))	(20)

1.1.4 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[or^#(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[or(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[true]	=	0
[false]	=	0

together with the usable rules

or(true,y)	→	true	(5)
or(false,y)	→	y	(6)
or(or(true,y),ext)	→	or(true,ext)	(35)
or(or(false,y),ext)	→	or(y,ext)	(36)
or(or(x,y),z)	→	or(x,or(y,z))	(17)
or(x,y)	→	or(y,x)	(16)

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

or^#(or(x,y),z)	→	or^#(y,z)	(21)
or^#(or(true,y),ext)	→	or^#(true,ext)	(33)
or^#(or(false,y),ext)	→	or^#(y,ext)	(34)

and the rules

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_1(false,x,y,edge(u,v,i),h)	→	reach(x,y,i,edge(u,v,h))	(12)
if_reach_2(true,x,y,edge(u,v,i),h)	→	true	(13)
if_reach_2(false,x,y,edge(u,v,i),h)	→	or(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(14)
or(or(true,y),ext)	→	or(true,ext)	(35)
or(or(false,y),ext)	→	or(y,ext)	(36)

could be deleted.

1.1.4.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.