Certification Problem

Input (TPDB TRS_Standard/Secret_07_TRS/secret1)

The rewrite relation of the following TRS is considered.

D(t)	→	s(h)	(1)
D(constant)	→	h	(2)
D(b(x,y))	→	b(D(x),D(y))	(3)
D(c(x,y))	→	b(c(y,D(x)),c(x,D(y)))	(4)
D(m(x,y))	→	m(D(x),D(y))	(5)
D(opp(x))	→	opp(D(x))	(6)
D(div(x,y))	→	m(div(D(x),y),div(c(x,D(y)),pow(y,2)))	(7)
D(ln(x))	→	div(D(x),x)	(8)
D(pow(x,y))	→	b(c(c(y,pow(x,m(y,1))),D(x)),c(c(pow(x,y),ln(x)),D(y)))	(9)
b(h,x)	→	x	(10)
b(x,h)	→	x	(11)
b(s(x),s(y))	→	s(s(b(x,y)))	(12)
b(b(x,y),z)	→	b(x,b(y,z))	(13)

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.

b^#(b(x,y),z)	→	b^#(x,b(y,z))	(14)
D^#(pow(x,y))	→	D^#(y)	(15)
D^#(b(x,y))	→	D^#(x)	(16)
D^#(div(x,y))	→	D^#(y)	(17)
D^#(m(x,y))	→	D^#(y)	(18)
D^#(b(x,y))	→	b^#(D(x),D(y))	(19)
D^#(b(x,y))	→	D^#(y)	(20)
b^#(s(x),s(y))	→	b^#(x,y)	(21)
D^#(m(x,y))	→	D^#(x)	(22)
D^#(pow(x,y))	→	D^#(x)	(23)
D^#(c(x,y))	→	D^#(x)	(24)
D^#(c(x,y))	→	b^#(c(y,D(x)),c(x,D(y)))	(25)
b^#(b(x,y),z)	→	b^#(y,z)	(26)
D^#(opp(x))	→	D^#(x)	(27)
D^#(div(x,y))	→	D^#(x)	(28)
D^#(ln(x))	→	D^#(x)	(29)
D^#(pow(x,y))	→	b^#(c(c(y,pow(x,m(y,1))),D(x)),c(c(pow(x,y),ln(x)),D(y)))	(30)
D^#(c(x,y))	→	D^#(y)	(31)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

D^#(c(x,y))	→	D^#(y)	(31)
D^#(b(x,y))	→	D^#(y)	(20)
D^#(ln(x))	→	D^#(x)	(29)
D^#(div(x,y))	→	D^#(x)	(28)
D^#(opp(x))	→	D^#(x)	(27)
D^#(m(x,y))	→	D^#(y)	(18)
D^#(div(x,y))	→	D^#(y)	(17)
D^#(b(x,y))	→	D^#(x)	(16)
D^#(c(x,y))	→	D^#(x)	(24)
D^#(pow(x,y))	→	D^#(y)	(15)
D^#(pow(x,y))	→	D^#(x)	(23)
D^#(m(x,y))	→	D^#(x)	(22)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	0
[1]	=	0
[ln(x₁)]	=	x₁ + 1
[s(x₁)]	=	0
[constant]	=	0
[b(x₁, x₂)]	=	x₁ + x₂ + 1
[pow(x₁, x₂)]	=	x₁ + x₂ + 1
[t]	=	0
[div(x₁, x₂)]	=	x₁ + x₂ + 1
[c(x₁, x₂)]	=	x₁ + x₂ + 1
[D(x₁)]	=	0
[D^#(x₁)]	=	x₁ + 0
[opp(x₁)]	=	x₁ + 1
[2]	=	0
[b^#(x₁, x₂)]	=	0
[m(x₁, x₂)]	=	x₁ + x₂ + 1

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

D^#(c(x,y))	→	D^#(y)	(31)
D^#(b(x,y))	→	D^#(y)	(20)
D^#(ln(x))	→	D^#(x)	(29)
D^#(div(x,y))	→	D^#(x)	(28)
D^#(opp(x))	→	D^#(x)	(27)
D^#(m(x,y))	→	D^#(y)	(18)
D^#(div(x,y))	→	D^#(y)	(17)
D^#(b(x,y))	→	D^#(x)	(16)
D^#(c(x,y))	→	D^#(x)	(24)
D^#(pow(x,y))	→	D^#(y)	(15)
D^#(pow(x,y))	→	D^#(x)	(23)
D^#(m(x,y))	→	D^#(x)	(22)

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

b^#(b(x,y),z)	→	b^#(y,z)	(26)
b^#(s(x),s(y))	→	b^#(x,y)	(21)
b^#(b(x,y),z)	→	b^#(x,b(y,z))	(14)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	1
[1]	=	0
[ln(x₁)]	=	1
[s(x₁)]	=	x₁ + 1
[constant]	=	0
[b(x₁, x₂)]	=	x₁ + x₂ + 1
[pow(x₁, x₂)]	=	1
[t]	=	0
[div(x₁, x₂)]	=	1
[c(x₁, x₂)]	=	1
[D(x₁)]	=	0
[D^#(x₁)]	=	0
[opp(x₁)]	=	1
[2]	=	0
[b^#(x₁, x₂)]	=	x₁ + 0
[m(x₁, x₂)]	=	1

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

b^#(b(x,y),z)	→	b^#(y,z)	(26)
b^#(s(x),s(y))	→	b^#(x,y)	(21)
b^#(b(x,y),z)	→	b^#(x,b(y,z))	(14)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.