Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/#3.45)

The rewrite relation of the following TRS is considered.

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

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(cons,app(fun,x)),app(app(map,fun),xs))	(11)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(cons,x),app(app(filter,fun),xs))	(12)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(13)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(filter,fun)	(14)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(filter,fun)	(15)
app^#(f,app(s,x))	→	app^#(f,x)	(16)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(filter2,app(fun,x)),fun),x)	(17)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(cons,x)	(18)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(19)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(21)
app^#(h,app(app(cons,x),y))	→	app^#(g,app(app(cons,x),y))	(22)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(filter2,app(fun,x))	(23)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(cons,app(fun,x))	(24)
app^#(g,app(app(cons,0),y))	→	app^#(g,y)	(25)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(26)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(filter2,app(fun,x)),fun)	(27)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(28)
app^#(h,app(app(cons,x),y))	→	app^#(h,app(g,app(app(cons,x),y)))	(29)

1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(28)
app^#(app(app(app(filter2,true),fun),x),xs)	→	app^#(app(filter,fun),xs)	(26)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(13)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(21)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)
app^#(app(app(app(filter2,false),fun),x),xs)	→	app^#(app(filter,fun),xs)	(19)

1.1.1 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	0
[s]	=	1
[false]	=	32106
[true]	=	25907
[f]	=	0
[filter2]	=	20534
[0]	=	46558
[nil]	=	22119
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	32173
[cons]	=	23535
[filter]	=	70533
[g]	=	42207
[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^#(fun,x)	(28)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(app(app(app(filter2,app(fun,x)),fun),x),xs)	(13)
app^#(app(filter,fun),app(app(cons,x),xs))	→	app^#(fun,x)	(21)
app^#(app(map,fun),app(app(cons,x),xs))	→	app^#(app(map,fun),xs)	(20)

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

→

app^#(f,x)

(16)

1.1.2 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	0
[s]	=	1
[false]	=	9119
[true]	=	28224
[f]	=	0
[filter2]	=	44735
[0]	=	20334
[nil]	=	1458
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	30374
[cons]	=	31205
[filter]	=	96331
[g]	=	40519
[app(x₁, x₂)]	=	x₁ + x₂ + 21238

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)

(16)

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^#(h,app(app(cons,x),y))

→

app^#(h,app(g,app(app(cons,x),y)))

(29)

1.1.3 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	0
[s]	=	0
[false]	=	0
[true]	=	0
[f]	=	0
[filter2]	=	37895
[0]	=	0
[nil]	=	0
[app^#(x₁, x₂)]	=	x₂ + 0
[map]	=	50174
[cons]	=	37895
[filter]	=	37895
[g]	=	37894
[app(x₁, x₂)]	=	x₁ + 0

together with the usable rules

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

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

app^#(h,app(app(cons,x),y))

→

app^#(h,app(g,app(app(cons,x),y)))

(29)

could be deleted.

1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 4^th component contains the pair

app^#(g,app(app(cons,0),y))

→

app^#(g,y)

(25)

1.1.4 Reduction Pair Processor with Usable Rules

Using the Max-polynomial interpretation

[h]	=	0
[s]	=	0
[false]	=	0
[true]	=	0
[f]	=	0
[filter2]	=	0
[0]	=	2437
[nil]	=	0
[app^#(x₁, x₂)]	=	max(x₁ + 1, x₂ + 1, 0)
[map]	=	0
[cons]	=	2
[filter]	=	0
[g]	=	1
[app(x₁, x₂)]	=	max(x₁ + 11797, x₂ + 2, 0)

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

app^#(g,app(app(cons,0),y))

→

app^#(g,y)

(25)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 0 components.