Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/08)

The rewrite relation of the following TRS is considered.

app(D,t)	→	1	(1)
app(D,constant)	→	0	(2)
app(D,app(app(+,x),y))	→	app(app(+,app(D,x)),app(D,y))	(3)
app(D,app(app(*,x),y))	→	app(app(+,app(app(,y),app(D,x))),app(app(,x),app(D,y)))	(4)
app(D,app(app(-,x),y))	→	app(app(-,app(D,x)),app(D,y))	(5)
app(app(map,f),nil)	→	nil	(6)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(7)
app(app(filter,f),nil)	→	nil	(8)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(9)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(10)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(11)

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.

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(f,x)),f),x)	(12)
app^#(D,app(app(*,x),y))	→	app^#(app(*,x),app(D,y))	(13)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(14)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(cons,x)	(15)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(cons,app(f,x))	(16)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)
app^#(D,app(app(*,x),y))	→	app^#(app(+,app(app(,y),app(D,x))),app(app(,x),app(D,y)))	(18)
app^#(D,app(app(+,x),y))	→	app^#(D,y)	(19)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(cons,x),app(app(filter,f),xs))	(20)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(21)
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(22)
app^#(D,app(app(*,x),y))	→	app^#(+,app(app(*,y),app(D,x)))	(23)
app^#(D,app(app(+,x),y))	→	app^#(app(+,app(D,x)),app(D,y))	(24)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(filter,f)	(25)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(27)
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(28)
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(29)
app^#(D,app(app(*,x),y))	→	app^#(*,y)	(30)
app^#(D,app(app(-,x),y))	→	app^#(-,app(D,x))	(31)
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(32)
app^#(D,app(app(*,x),y))	→	app^#(app(*,y),app(D,x))	(33)
app^#(D,app(app(+,x),y))	→	app^#(+,app(D,x))	(34)
app^#(D,app(app(-,x),y))	→	app^#(app(-,app(D,x)),app(D,y))	(35)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(filter2,app(f,x)),f)	(36)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(37)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(cons,app(f,x)),app(app(map,f),xs))	(38)
app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(filter,f)	(39)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(filter2,app(f,x))	(40)
app^#(D,app(app(-,x),y))	→	app^#(D,x)	(41)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

app^#(app(app(app(filter2,true),f),x),xs)	→	app^#(app(filter,f),xs)	(21)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(37)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)
app^#(app(app(app(filter2,false),f),x),xs)	→	app^#(app(filter,f),xs)	(14)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(27)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	106114
[constant]	=	1
[t]	=	32774
[false]	=	37247
[D]	=	41231
[true]	=	32106
[filter2]	=	56519
[0]	=	73341
[nil]	=	20534
[-]	=	59992
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	2466
[cons]	=	18458
[filter]	=	70168
[+]	=	23535
[*]	=	42207
[app(x₁, x₂)]	=	x₁ + x₂ + 32108

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

app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(f,x)),f),x),xs)	(37)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(app(map,f),xs)	(17)
app^#(app(filter,f),app(app(cons,x),xs))	→	app^#(f,x)	(27)
app^#(app(map,f),app(app(cons,x),xs))	→	app^#(f,x)	(26)

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

app^#(D,app(app(-,x),y))	→	app^#(D,x)	(41)
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(22)
app^#(D,app(app(+,x),y))	→	app^#(D,y)	(19)
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(32)
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(29)
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(28)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	120298
[constant]	=	1
[t]	=	48019
[false]	=	34743
[D]	=	55209
[true]	=	17067
[filter2]	=	20086
[0]	=	72280
[nil]	=	13080
[-]	=	37112
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	19158
[cons]	=	29405
[filter]	=	7749
[+]	=	38472
[*]	=	57246
[app(x₁, x₂)]	=	x₁ + x₂ + 17069

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

app^#(D,app(app(-,x),y))	→	app^#(D,x)	(41)
app^#(D,app(app(*,x),y))	→	app^#(D,y)	(22)
app^#(D,app(app(+,x),y))	→	app^#(D,y)	(19)
app^#(D,app(app(-,x),y))	→	app^#(D,y)	(32)
app^#(D,app(app(+,x),y))	→	app^#(D,x)	(29)
app^#(D,app(app(*,x),y))	→	app^#(D,x)	(28)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.