Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/#3.48)

The rewrite relation of the following TRS is considered.

app(f,0)	→	true	(1)
app(f,1)	→	false	(2)
app(f,app(s,x))	→	app(f,x)	(3)
app(app(app(if,true),app(s,x)),app(s,y))	→	app(s,x)	(4)
app(app(app(if,false),app(s,x)),app(s,y))	→	app(s,y)	(5)
app(app(g,x),app(c,y))	→	app(c,app(app(g,x),y))	(6)
app(app(g,x),app(c,y))	→	app(app(g,x),app(app(app(if,app(f,x)),app(c,app(app(g,app(s,x)),y))),app(c,y)))	(7)
app(app(map,fun),nil)	→	nil	(8)
app(app(map,fun),app(app(cons,x),xs))	→	app(app(cons,app(fun,x)),app(app(map,fun),xs))	(9)
app(app(filter,fun),nil)	→	nil	(10)
app(app(filter,fun),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(fun,x)),fun),x),xs)	(11)
app(app(app(app(filter2,true),fun),x),xs)	→	app(app(cons,x),app(app(filter,fun),xs))	(12)
app(app(app(app(filter2,false),fun),x),xs)	→	app(app(filter,fun),xs)	(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.

app^#(app(g,x),app(c,y))	→	app^#(app(g,x),y)	(14)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(15)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(17)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(18)
app^#(app(g,x),app(c,y))	→	app^#(app(g,app(s,x)),y)	(19)
app^#(app(g,x),app(c,y))	→	app^#(c,app(app(g,app(s,x)),y))	(20)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(21)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(22)
app^#(app(g,x),app(c,y))	→	app^#(g,app(s,x))	(23)
app^#(app(g,x),app(c,y))	→	app^#(app(if,app(f,x)),app(c,app(app(g,app(s,x)),y)))	(24)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(26)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(27)
app^#(app(g,x),app(c,y))	→	app^#(f,x)	(28)
app^#(f,app(s,x))	→	app^#(f,x)	(29)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(30)
app^#(app(g,x),app(c,y))	→	app^#(app(app(if,app(f,x)),app(c,app(app(g,app(s,x)),y))),app(c,y))	(31)
app^#(app(g,x),app(c,y))	→	app^#(s,x)	(32)
app^#(app(g,x),app(c,y))	→	app^#(if,app(f,x))	(33)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(34)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(35)
app^#(app(g,x),app(c,y))	→	app^#(c,app(app(g,x),y))	(36)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(37)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(38)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(39)
app^#(app(g,x),app(c,y))	→	app^#(app(g,x),app(app(app(if,app(f,x)),app(c,app(app(g,app(s,x)),y))),app(c,y)))	(40)

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(37)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(35)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(34)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(17)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	1
[s]	=	34651
[false]	=	61216
[c]	=	50743
[true]	=	93107
[f]	=	52268
[filter2]	=	7177
[0]	=	31892
[if]	=	31098
[nil]	=	15823
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	21258
[cons]	=	49419
[filter]	=	86286
[g]	=	41828
[app(x₁, x₂)]	=	x₁ + x₂ + 8946

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

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(37)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(17)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(16)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(25)

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^#(app(g,x),app(c,y))	→	app^#(app(g,app(s,x)),y)	(19)
app^#(app(g,x),app(c,y))	→	app^#(app(g,x),y)	(14)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	1
[s]	=	43596
[false]	=	95227
[c]	=	53145
[true]	=	4
[f]	=	1
[filter2]	=	1
[0]	=	1
[if]	=	1
[nil]	=	13485
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	30203
[cons]	=	19335
[filter]	=	95231
[g]	=	1
[app(x₁, x₂)]	=	x₁ + x₂ + 1

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

app^#(app(g,x),app(c,y))	→	app^#(app(g,app(s,x)),y)	(19)
app^#(app(g,x),app(c,y))	→	app^#(app(g,x),y)	(14)

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

app^#(f,app(s,x))

→

app^#(f,x)

(29)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[1]	=	1
[s]	=	51519
[false]	=	46044
[c]	=	20684
[true]	=	47147
[f]	=	46041
[filter2]	=	35987
[0]	=	1103
[if]	=	21313
[nil]	=	13485
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	29260
[cons]	=	47121
[filter]	=	82034
[g]	=	1
[app(x₁, x₂)]	=	x₁ + x₂ + 1

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

app^#(f,app(s,x))

→

app^#(f,x)

(29)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.