Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex5_Zan97_GM)

The rewrite relation of the following TRS is considered.

a__f(X)	→	a__if(mark(X),c,f(true))	(1)
a__if(true,X,Y)	→	mark(X)	(2)
a__if(false,X,Y)	→	mark(Y)	(3)
mark(f(X))	→	a__f(mark(X))	(4)
mark(if(X1,X2,X3))	→	a__if(mark(X1),mark(X2),X3)	(5)
mark(c)	→	c	(6)
mark(true)	→	true	(7)
mark(false)	→	false	(8)
a__f(X)	→	f(X)	(9)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(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.

mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)
a__f^#(X)	→	a__if^#(mark(X),c,f(true))	(12)
mark^#(f(X))	→	mark^#(X)	(13)
a__f^#(X)	→	mark^#(X)	(14)
a__if^#(false,X,Y)	→	mark^#(Y)	(15)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
a__if^#(true,X,Y)	→	mark^#(X)	(17)
mark^#(f(X))	→	a__f^#(mark(X))	(18)
mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)
mark^#(f(X))	→	a__f^#(mark(X))	(18)
a__if^#(true,X,Y)	→	mark^#(X)	(17)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
mark^#(f(X))	→	mark^#(X)	(13)
a__f^#(X)	→	a__if^#(mark(X),c,f(true))	(12)
a__if^#(false,X,Y)	→	mark^#(Y)	(15)
a__f^#(X)	→	mark^#(X)	(14)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__f(x₁)]	=	x₁ + 10451
[false]	=	1
[c]	=	0
[true]	=	0
[f(x₁)]	=	x₁ + 10451
[mark^#(x₁)]	=	x₁ + 0
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[mark(x₁)]	=	x₁ + 0
[a__if^#(x₁, x₂, x₃)]	=	x₂ + x₃ + 0
[a__f^#(x₁)]	=	x₁ + 10451
[a__if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0

together with the usable rules

mark(f(X))	→	a__f(mark(X))	(4)
mark(false)	→	false	(8)
a__f(X)	→	a__if(mark(X),c,f(true))	(1)
a__if(false,X,Y)	→	mark(Y)	(3)
mark(if(X1,X2,X3))	→	a__if(mark(X1),mark(X2),X3)	(5)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(10)
mark(true)	→	true	(7)
a__f(X)	→	f(X)	(9)
mark(c)	→	c	(6)
a__if(true,X,Y)	→	mark(X)	(2)

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

mark^#(f(X))	→	mark^#(X)	(13)
a__f^#(X)	→	mark^#(X)	(14)

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

mark^#(f(X))	→	a__f^#(mark(X))	(18)
a__f^#(X)	→	a__if^#(mark(X),c,f(true))	(12)
a__if^#(false,X,Y)	→	mark^#(Y)	(15)
mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)
a__if^#(true,X,Y)	→	mark^#(X)	(17)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__f(x₁)]	=	x₁ + 19781
[false]	=	1
[c]	=	0
[true]	=	0
[f(x₁)]	=	x₁ + 19781
[mark^#(x₁)]	=	x₁ + 0
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[mark(x₁)]	=	x₁ + 0
[a__if^#(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[a__f^#(x₁)]	=	x₁ + 19781
[a__if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0

together with the usable rules

mark(f(X))	→	a__f(mark(X))	(4)
mark(false)	→	false	(8)
a__f(X)	→	a__if(mark(X),c,f(true))	(1)
a__if(false,X,Y)	→	mark(Y)	(3)
mark(if(X1,X2,X3))	→	a__if(mark(X1),mark(X2),X3)	(5)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(10)
mark(true)	→	true	(7)
a__f(X)	→	f(X)	(9)
mark(c)	→	c	(6)
a__if(true,X,Y)	→	mark(X)	(2)

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

a__if^#(false,X,Y)

→

mark^#(Y)

(15)

could be deleted.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(f(X))	→	a__f^#(mark(X))	(18)
a__f^#(X)	→	a__if^#(mark(X),c,f(true))	(12)
mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)
a__if^#(true,X,Y)	→	mark^#(X)	(17)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__f(x₁)]	=	x₁ + 2
[false]	=	1
[c]	=	0
[true]	=	0
[f(x₁)]	=	x₁ + 2
[mark^#(x₁)]	=	x₁ + 0
[if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0
[mark(x₁)]	=	x₁ + 0
[a__if^#(x₁, x₂, x₃)]	=	x₁ + x₂ + 0
[a__f^#(x₁)]	=	x₁ + 1
[a__if(x₁, x₂, x₃)]	=	x₁ + x₂ + x₃ + 0

together with the usable rules

mark(f(X))	→	a__f(mark(X))	(4)
mark(false)	→	false	(8)
a__f(X)	→	a__if(mark(X),c,f(true))	(1)
a__if(false,X,Y)	→	mark(Y)	(3)
mark(if(X1,X2,X3))	→	a__if(mark(X1),mark(X2),X3)	(5)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(10)
mark(true)	→	true	(7)
a__f(X)	→	f(X)	(9)
mark(c)	→	c	(6)
a__if(true,X,Y)	→	mark(X)	(2)

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

mark^#(f(X))	→	a__f^#(mark(X))	(18)
a__f^#(X)	→	a__if^#(mark(X),c,f(true))	(12)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)
a__if^#(true,X,Y)	→	mark^#(X)	(17)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[a__f(x₁)]	=	x₁ + 5857
[false]	=	1
[c]	=	4681
[true]	=	0
[f(x₁)]	=	x₁ + 5857
[mark^#(x₁)]	=	x₁ + 30613
[if(x₁, x₂, x₃)]	=	max(x₁ + 4, x₂ + 2, x₃ + 0, 0)
[mark(x₁)]	=	x₁ + 0
[a__if^#(x₁, x₂, x₃)]	=	max(x₁ + 30616, x₂ + 30614, 0)
[a__f^#(x₁)]	=	0
[a__if(x₁, x₂, x₃)]	=	max(x₁ + 4, x₂ + 2, x₃ + 0, 0)

together with the usable rules

mark(f(X))	→	a__f(mark(X))	(4)
mark(false)	→	false	(8)
a__f(X)	→	a__if(mark(X),c,f(true))	(1)
a__if(false,X,Y)	→	mark(Y)	(3)
mark(if(X1,X2,X3))	→	a__if(mark(X1),mark(X2),X3)	(5)
a__if(X1,X2,X3)	→	if(X1,X2,X3)	(10)
mark(true)	→	true	(7)
a__f(X)	→	f(X)	(9)
mark(c)	→	c	(6)
a__if(true,X,Y)	→	mark(X)	(2)

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

mark^#(if(X1,X2,X3))	→	mark^#(X2)	(19)
mark^#(if(X1,X2,X3))	→	mark^#(X1)	(16)
mark^#(if(X1,X2,X3))	→	a__if^#(mark(X1),mark(X2),X3)	(11)
a__if^#(true,X,Y)	→	mark^#(X)	(17)

could be deleted.

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.