Certification Problem

Input (TPDB TRS_Standard/Applicative_first_order_05/#3.13)

The rewrite relation of the following TRS is considered.

app(app(eq,0),0)	→	true	(1)
app(app(eq,0),app(s,x))	→	false	(2)
app(app(eq,app(s,x)),0)	→	false	(3)
app(app(eq,app(s,x)),app(s,y))	→	app(app(eq,x),y)	(4)
app(app(or,true),y)	→	true	(5)
app(app(or,false),y)	→	y	(6)
app(app(union,empty),h)	→	h	(7)
app(app(union,app(app(app(edge,x),y),i)),h)	→	app(app(app(edge,x),y),app(app(union,i),h))	(8)
app(app(app(app(reach,x),y),empty),h)	→	false	(9)
app(app(app(app(reach,x),y),app(app(app(edge,u),v),i)),h)	→	app(app(app(app(app(if_reach_1,app(app(eq,x),u)),x),y),app(app(app(edge,u),v),i)),h)	(10)
app(app(app(app(app(if_reach_1,true),x),y),app(app(app(edge,u),v),i)),h)	→	app(app(app(app(app(if_reach_2,app(app(eq,y),v)),x),y),app(app(app(edge,u),v),i)),h)	(11)
app(app(app(app(app(if_reach_1,false),x),y),app(app(app(edge,u),v),i)),h)	→	app(app(app(app(reach,x),y),i),app(app(app(edge,u),v),h))	(12)
app(app(app(app(app(if_reach_2,true),x),y),app(app(app(edge,u),v),i)),h)	→	true	(13)
app(app(app(app(app(if_reach_2,false),x),y),app(app(app(edge,u),v),i)),h)	→	app(app(or,app(app(app(app(reach,x),y),i),h)),app(app(app(app(reach,v),y),app(app(union,i),h)),empty))	(14)
app(app(map,f),nil)	→	nil	(15)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(16)
app(app(filter,f),nil)	→	nil	(17)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(18)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(19)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(20)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

We uncurry the binary symbol app in combination with the following symbol map which also determines the applicative arities of these symbols.

eq	is mapped to	eq,	eq1(x₁),	eq2(x₁, x₂)
0	is mapped to	0
true	is mapped to	true
s	is mapped to	s,	s1(x₁)
false	is mapped to	false
or	is mapped to	or,	or1(x₁),	or2(x₁, x₂)
union	is mapped to	union,	union1(x₁),	union2(x₁, x₂)
empty	is mapped to	empty
edge	is mapped to	edge,	edge1(x₁),	edge2(x₁, x₂),	edge3(x₁, x₂, x₃)
reach	is mapped to	reach,	reach1(x₁),	reach2(x₁, x₂),	reach3(x₁, x₂, x₃),	reach4(x₁,...,x₄)
if_reach_1	is mapped to	if_reach_1,	if_reach_11(x₁),	if_reach_12(x₁, x₂),	if_reach_13(x₁, x₂, x₃),	if_reach_14(x₁,...,x₄),	if_reach_15(x₁,...,x₅)
if_reach_2	is mapped to	if_reach_2,	if_reach_21(x₁),	if_reach_22(x₁, x₂),	if_reach_23(x₁, x₂, x₃),	if_reach_24(x₁,...,x₄),	if_reach_25(x₁,...,x₅)
map	is mapped to	map,	map1(x₁),	map2(x₁, x₂)
nil	is mapped to	nil
cons	is mapped to	cons,	cons1(x₁),	cons2(x₁, x₂)
filter	is mapped to	filter,	filter1(x₁),	filter3(x₁, x₂)
filter2	is mapped to	filter2,	filter21(x₁),	filter22(x₁, x₂),	filter23(x₁, x₂, x₃),	filter24(x₁,...,x₄)

There are no uncurry rules.
No rules have to be added for the eta-expansion.

Uncurrying the rules and adding the uncurrying rules yields the new set of rules

eq2(0,0)	→	true	(55)
eq2(0,s1(x))	→	false	(56)
eq2(s1(x),0)	→	false	(57)
eq2(s1(x),s1(y))	→	eq2(x,y)	(58)
or2(true,y)	→	true	(59)
or2(false,y)	→	y	(60)
union2(empty,h)	→	h	(61)
union2(edge3(x,y,i),h)	→	edge3(x,y,union2(i,h))	(62)
reach4(x,y,empty,h)	→	false	(63)
reach4(x,y,edge3(u,v,i),h)	→	if_reach_15(eq2(x,u),x,y,edge3(u,v,i),h)	(64)
if_reach_15(true,x,y,edge3(u,v,i),h)	→	if_reach_25(eq2(y,v),x,y,edge3(u,v,i),h)	(65)
if_reach_15(false,x,y,edge3(u,v,i),h)	→	reach4(x,y,i,edge3(u,v,h))	(66)
if_reach_25(true,x,y,edge3(u,v,i),h)	→	true	(67)
if_reach_25(false,x,y,edge3(u,v,i),h)	→	or2(reach4(x,y,i,h),reach4(v,y,union2(i,h),empty))	(68)
map2(f,nil)	→	nil	(69)
map2(f,cons2(x,xs))	→	cons2(app(f,x),map2(f,xs))	(70)
filter3(f,nil)	→	nil	(71)
filter3(f,cons2(x,xs))	→	filter24(app(f,x),f,x,xs)	(72)
filter24(true,f,x,xs)	→	cons2(x,filter3(f,xs))	(73)
filter24(false,f,x,xs)	→	filter3(f,xs)	(74)
app(eq,y1)	→	eq1(y1)	(21)
app(eq1(x0),y1)	→	eq2(x0,y1)	(22)
app(s,y1)	→	s1(y1)	(23)
app(or,y1)	→	or1(y1)	(24)
app(or1(x0),y1)	→	or2(x0,y1)	(25)
app(union,y1)	→	union1(y1)	(26)
app(union1(x0),y1)	→	union2(x0,y1)	(27)
app(edge,y1)	→	edge1(y1)	(28)
app(edge1(x0),y1)	→	edge2(x0,y1)	(29)
app(edge2(x0,x1),y1)	→	edge3(x0,x1,y1)	(30)
app(reach,y1)	→	reach1(y1)	(31)
app(reach1(x0),y1)	→	reach2(x0,y1)	(32)
app(reach2(x0,x1),y1)	→	reach3(x0,x1,y1)	(33)
app(reach3(x0,x1,x2),y1)	→	reach4(x0,x1,x2,y1)	(34)
app(if_reach_1,y1)	→	if_reach_11(y1)	(35)
app(if_reach_11(x0),y1)	→	if_reach_12(x0,y1)	(36)
app(if_reach_12(x0,x1),y1)	→	if_reach_13(x0,x1,y1)	(37)
app(if_reach_13(x0,x1,x2),y1)	→	if_reach_14(x0,x1,x2,y1)	(38)
app(if_reach_14(x0,x1,x2,x3),y1)	→	if_reach_15(x0,x1,x2,x3,y1)	(39)
app(if_reach_2,y1)	→	if_reach_21(y1)	(40)
app(if_reach_21(x0),y1)	→	if_reach_22(x0,y1)	(41)
app(if_reach_22(x0,x1),y1)	→	if_reach_23(x0,x1,y1)	(42)
app(if_reach_23(x0,x1,x2),y1)	→	if_reach_24(x0,x1,x2,y1)	(43)
app(if_reach_24(x0,x1,x2,x3),y1)	→	if_reach_25(x0,x1,x2,x3,y1)	(44)
app(map,y1)	→	map1(y1)	(45)
app(map1(x0),y1)	→	map2(x0,y1)	(46)
app(cons,y1)	→	cons1(y1)	(47)
app(cons1(x0),y1)	→	cons2(x0,y1)	(48)
app(filter,y1)	→	filter1(y1)	(49)
app(filter1(x0),y1)	→	filter3(x0,y1)	(50)
app(filter2,y1)	→	filter21(y1)	(51)
app(filter21(x0),y1)	→	filter22(x0,y1)	(52)
app(filter22(x0,x1),y1)	→	filter23(x0,x1,y1)	(53)
app(filter23(x0,x1,x2),y1)	→	filter24(x0,x1,x2,y1)	(54)

1.1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

Hence, it suffices to show innermost termination in the following.

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

eq2^#(s1(x),s1(y))	→	eq2^#(x,y)	(75)
union2^#(edge3(x,y,i),h)	→	union2^#(i,h)	(76)
reach4^#(x,y,edge3(u,v,i),h)	→	if_reach_15^#(eq2(x,u),x,y,edge3(u,v,i),h)	(77)
reach4^#(x,y,edge3(u,v,i),h)	→	eq2^#(x,u)	(78)
if_reach_15^#(true,x,y,edge3(u,v,i),h)	→	if_reach_25^#(eq2(y,v),x,y,edge3(u,v,i),h)	(79)
if_reach_15^#(true,x,y,edge3(u,v,i),h)	→	eq2^#(y,v)	(80)
if_reach_15^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(x,y,i,edge3(u,v,h))	(81)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	or2^#(reach4(x,y,i,h),reach4(v,y,union2(i,h),empty))	(82)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(x,y,i,h)	(83)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(v,y,union2(i,h),empty)	(84)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	union2^#(i,h)	(85)
map2^#(f,cons2(x,xs))	→	app^#(f,x)	(86)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(87)
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(88)
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(89)
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(90)
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(91)
app^#(eq1(x0),y1)	→	eq2^#(x0,y1)	(92)
app^#(or1(x0),y1)	→	or2^#(x0,y1)	(93)
app^#(union1(x0),y1)	→	union2^#(x0,y1)	(94)
app^#(reach3(x0,x1,x2),y1)	→	reach4^#(x0,x1,x2,y1)	(95)
app^#(if_reach_14(x0,x1,x2,x3),y1)	→	if_reach_15^#(x0,x1,x2,x3,y1)	(96)
app^#(if_reach_24(x0,x1,x2,x3),y1)	→	if_reach_25^#(x0,x1,x2,x3,y1)	(97)
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(98)
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(99)
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(100)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

app^#(map1(x0),y1)	→	map2^#(x0,y1)	(98)
map2^#(f,cons2(x,xs))	→	app^#(f,x)	(86)
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(99)
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(88)
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(90)
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(89)
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(100)
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(91)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(87)

1.1.1.1.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

map2^#(f,cons2(x,xs))	→	app^#(f,x)	(86)

1	≥	1
2	>	2
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(87)

1	≥	1
2	>	2
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(89)

1	≥	1
2	>	2
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(98)

1	>	1
2	≥	2
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(88)

1	≥	2
2	>	3
2	>	4
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(99)

1	>	1
2	≥	2
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(100)

1	>	1
1	>	2
1	>	3
2	≥	4
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(90)

2	≥	1
4	≥	2
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(91)

2	≥	1
4	≥	2

As there is no critical graph in the transitive closure, there are no infinite chains.

The 2^nd component contains the pair

reach4^#(x,y,edge3(u,v,i),h)	→	if_reach_15^#(eq2(x,u),x,y,edge3(u,v,i),h)	(77)
if_reach_15^#(true,x,y,edge3(u,v,i),h)	→	if_reach_25^#(eq2(y,v),x,y,edge3(u,v,i),h)	(79)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(x,y,i,h)	(83)
if_reach_25^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(v,y,union2(i,h),empty)	(84)
if_reach_15^#(false,x,y,edge3(u,v,i),h)	→	reach4^#(x,y,i,edge3(u,v,h))	(81)

1.1.1.1.2 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

union2(empty,h)	→	h	(61)
union2(edge3(x,y,i),h)	→	edge3(x,y,union2(i,h))	(62)
eq2(0,0)	→	true	(55)
eq2(0,s1(x))	→	false	(56)
eq2(s1(x),0)	→	false	(57)
eq2(s1(x),s1(y))	→	eq2(x,y)	(58)

1.1.1.1.2.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

eq2(0,0)

eq2(0,s1(x0))

eq2(s1(x0),0)

eq2(s1(x0),s1(x1))

union2(empty,x0)

union2(edge3(x0,x1,x2),x3)

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[if_reach_15^#(x₁,...,x₅)]	=	-1 + 2 · x₃ + x₄ + x₅
[if_reach_25^#(x₁,...,x₅)]	=	-2 + 2 · x₃ + x₄ + x₅
[eq2(x₁, x₂)]	=	0
[0]	=	0
[true]	=	2
[s1(x₁)]	=	2 · x₁
[false]	=	0
[reach4^#(x₁,...,x₄)]	=	-1 + 2 · x₂ + x₃ + x₄
[union2(x₁, x₂)]	=	x₁ + x₂
[empty]	=	0
[edge3(x₁, x₂, x₃)]	=	2 + x₃

together with the usable rules

union2(empty,h)	→	h	(61)
union2(edge3(x,y,i),h)	→	edge3(x,y,union2(i,h))	(62)

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

if_reach_15^#(true,x,y,edge3(u,v,i),h)

→

if_reach_25^#(eq2(y,v),x,y,edge3(u,v,i),h)

(79)

could be deleted.

1.1.1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.