Certification Problem

Input (TPDB TRS_Innermost/Applicative_AG01_innermost/#4.36)

The rewrite relation of the following TRS is considered.

app(app(eq,0),0)	→	true	(1)
app(app(eq,0),app(s,m))	→	false	(2)
app(app(eq,app(s,n)),0)	→	false	(3)
app(app(eq,app(s,n)),app(s,m))	→	app(app(eq,n),m)	(4)
app(app(le,0),m)	→	true	(5)
app(app(le,app(s,n)),0)	→	false	(6)
app(app(le,app(s,n)),app(s,m))	→	app(app(le,n),m)	(7)
app(min,app(app(cons,0),nil))	→	0	(8)
app(min,app(app(cons,app(s,n)),nil))	→	app(s,n)	(9)
app(min,app(app(cons,n),app(app(cons,m),x)))	→	app(app(if_min,app(app(le,n),m)),app(app(cons,n),app(app(cons,m),x)))	(10)
app(app(if_min,true),app(app(cons,n),app(app(cons,m),x)))	→	app(min,app(app(cons,n),x))	(11)
app(app(if_min,false),app(app(cons,n),app(app(cons,m),x)))	→	app(min,app(app(cons,m),x))	(12)
app(app(app(replace,n),m),nil)	→	nil	(13)
app(app(app(replace,n),m),app(app(cons,k),x))	→	app(app(app(app(if_replace,app(app(eq,n),k)),n),m),app(app(cons,k),x))	(14)
app(app(app(app(if_replace,true),n),m),app(app(cons,k),x))	→	app(app(cons,m),x)	(15)
app(app(app(app(if_replace,false),n),m),app(app(cons,k),x))	→	app(app(cons,k),app(app(app(replace,n),m),x))	(16)
app(sort,nil)	→	nil	(17)
app(sort,app(app(cons,n),x))	→	app(app(cons,app(min,app(app(cons,n),x))),app(sort,app(app(app(replace,app(min,app(app(cons,n),x))),n),x)))	(18)
app(app(map,f),nil)	→	nil	(19)
app(app(map,f),app(app(cons,x),xs))	→	app(app(cons,app(f,x)),app(app(map,f),xs))	(20)
app(app(filter,f),nil)	→	nil	(21)
app(app(filter,f),app(app(cons,x),xs))	→	app(app(app(app(filter2,app(f,x)),f),x),xs)	(22)
app(app(app(app(filter2,true),f),x),xs)	→	app(app(cons,x),app(app(filter,f),xs))	(23)
app(app(app(app(filter2,false),f),x),xs)	→	app(app(filter,f),xs)	(24)

The evaluation strategy is innermost.

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
le	is mapped to	le,	le1(x₁),	le2(x₁, x₂)
min	is mapped to	min,	min1(x₁)
cons	is mapped to	cons,	cons1(x₁),	cons2(x₁, x₂)
nil	is mapped to	nil
if_min	is mapped to	if_min,	if_min1(x₁),	if_min2(x₁, x₂)
replace	is mapped to	replace,	replace1(x₁),	replace2(x₁, x₂),	replace3(x₁, x₂, x₃)
if_replace	is mapped to	if_replace,	if_replace1(x₁),	if_replace2(x₁, x₂),	if_replace3(x₁, x₂, x₃),	if_replace4(x₁,...,x₄)
sort	is mapped to	sort,	sort1(x₁)
map	is mapped to	map,	map1(x₁),	map2(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	(51)
eq2(0,s1(m))	→	false	(52)
eq2(s1(n),0)	→	false	(53)
eq2(s1(n),s1(m))	→	eq2(n,m)	(54)
le2(0,m)	→	true	(55)
le2(s1(n),0)	→	false	(56)
le2(s1(n),s1(m))	→	le2(n,m)	(57)
min1(cons2(0,nil))	→	0	(58)
min1(cons2(s1(n),nil))	→	s1(n)	(59)
min1(cons2(n,cons2(m,x)))	→	if_min2(le2(n,m),cons2(n,cons2(m,x)))	(60)
if_min2(true,cons2(n,cons2(m,x)))	→	min1(cons2(n,x))	(61)
if_min2(false,cons2(n,cons2(m,x)))	→	min1(cons2(m,x))	(62)
replace3(n,m,nil)	→	nil	(63)
replace3(n,m,cons2(k,x))	→	if_replace4(eq2(n,k),n,m,cons2(k,x))	(64)
if_replace4(true,n,m,cons2(k,x))	→	cons2(m,x)	(65)
if_replace4(false,n,m,cons2(k,x))	→	cons2(k,replace3(n,m,x))	(66)
sort1(nil)	→	nil	(67)
sort1(cons2(n,x))	→	cons2(min1(cons2(n,x)),sort1(replace3(min1(cons2(n,x)),n,x)))	(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)	(25)
app(eq1(x0),y1)	→	eq2(x0,y1)	(26)
app(s,y1)	→	s1(y1)	(27)
app(le,y1)	→	le1(y1)	(28)
app(le1(x0),y1)	→	le2(x0,y1)	(29)
app(min,y1)	→	min1(y1)	(30)
app(cons,y1)	→	cons1(y1)	(31)
app(cons1(x0),y1)	→	cons2(x0,y1)	(32)
app(if_min,y1)	→	if_min1(y1)	(33)
app(if_min1(x0),y1)	→	if_min2(x0,y1)	(34)
app(replace,y1)	→	replace1(y1)	(35)
app(replace1(x0),y1)	→	replace2(x0,y1)	(36)
app(replace2(x0,x1),y1)	→	replace3(x0,x1,y1)	(37)
app(if_replace,y1)	→	if_replace1(y1)	(38)
app(if_replace1(x0),y1)	→	if_replace2(x0,y1)	(39)
app(if_replace2(x0,x1),y1)	→	if_replace3(x0,x1,y1)	(40)
app(if_replace3(x0,x1,x2),y1)	→	if_replace4(x0,x1,x2,y1)	(41)
app(sort,y1)	→	sort1(y1)	(42)
app(map,y1)	→	map1(y1)	(43)
app(map1(x0),y1)	→	map2(x0,y1)	(44)
app(filter,y1)	→	filter1(y1)	(45)
app(filter1(x0),y1)	→	filter3(x0,y1)	(46)
app(filter2,y1)	→	filter21(y1)	(47)
app(filter21(x0),y1)	→	filter22(x0,y1)	(48)
app(filter22(x0,x1),y1)	→	filter23(x0,x1,y1)	(49)
app(filter23(x0,x1,x2),y1)	→	filter24(x0,x1,x2,y1)	(50)

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(n),s1(m))	→	eq2^#(n,m)	(75)
le2^#(s1(n),s1(m))	→	le2^#(n,m)	(76)
min1^#(cons2(n,cons2(m,x)))	→	if_min2^#(le2(n,m),cons2(n,cons2(m,x)))	(77)
min1^#(cons2(n,cons2(m,x)))	→	le2^#(n,m)	(78)
if_min2^#(true,cons2(n,cons2(m,x)))	→	min1^#(cons2(n,x))	(79)
if_min2^#(false,cons2(n,cons2(m,x)))	→	min1^#(cons2(m,x))	(80)
replace3^#(n,m,cons2(k,x))	→	if_replace4^#(eq2(n,k),n,m,cons2(k,x))	(81)
replace3^#(n,m,cons2(k,x))	→	eq2^#(n,k)	(82)
if_replace4^#(false,n,m,cons2(k,x))	→	replace3^#(n,m,x)	(83)
sort1^#(cons2(n,x))	→	min1^#(cons2(n,x))	(84)
sort1^#(cons2(n,x))	→	sort1^#(replace3(min1(cons2(n,x)),n,x))	(85)
sort1^#(cons2(n,x))	→	replace3^#(min1(cons2(n,x)),n,x)	(86)
map2^#(f,cons2(x,xs))	→	app^#(f,x)	(87)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(88)
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(89)
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(90)
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(91)
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(92)
app^#(eq1(x0),y1)	→	eq2^#(x0,y1)	(93)
app^#(le1(x0),y1)	→	le2^#(x0,y1)	(94)
app^#(min,y1)	→	min1^#(y1)	(95)
app^#(if_min1(x0),y1)	→	if_min2^#(x0,y1)	(96)
app^#(replace2(x0,x1),y1)	→	replace3^#(x0,x1,y1)	(97)
app^#(if_replace3(x0,x1,x2),y1)	→	if_replace4^#(x0,x1,x2,y1)	(98)
app^#(sort,y1)	→	sort1^#(y1)	(99)
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(100)
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(101)
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(102)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 6 components.

The 1^st component contains the pair

app^#(map1(x0),y1)	→	map2^#(x0,y1)	(100)
map2^#(f,cons2(x,xs))	→	app^#(f,x)	(87)
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(101)
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(89)
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(91)
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(90)
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(102)
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(92)
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(88)

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)	(87)

1	≥	1
2	>	2
map2^#(f,cons2(x,xs))	→	map2^#(f,xs)	(88)

1	≥	1
2	>	2
filter3^#(f,cons2(x,xs))	→	app^#(f,x)	(90)

1	≥	1
2	>	2
app^#(map1(x0),y1)	→	map2^#(x0,y1)	(100)

1	>	1
2	≥	2
filter3^#(f,cons2(x,xs))	→	filter24^#(app(f,x),f,x,xs)	(89)

1	≥	2
2	>	3
2	>	4
app^#(filter1(x0),y1)	→	filter3^#(x0,y1)	(101)

1	>	1
2	≥	2
app^#(filter23(x0,x1,x2),y1)	→	filter24^#(x0,x1,x2,y1)	(102)

1	>	1
1	>	2
1	>	3
2	≥	4
filter24^#(true,f,x,xs)	→	filter3^#(f,xs)	(91)

2	≥	1
4	≥	2
filter24^#(false,f,x,xs)	→	filter3^#(f,xs)	(92)

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

sort1^#(cons2(n,x))

→

sort1^#(replace3(min1(cons2(n,x)),n,x))

(85)

1.1.1.1.2 Usable Rules Processor

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

min1(cons2(0,nil))	→	0	(58)
min1(cons2(s1(n),nil))	→	s1(n)	(59)
min1(cons2(n,cons2(m,x)))	→	if_min2(le2(n,m),cons2(n,cons2(m,x)))	(60)
if_min2(true,cons2(n,cons2(m,x)))	→	min1(cons2(n,x))	(61)
if_min2(false,cons2(n,cons2(m,x)))	→	min1(cons2(m,x))	(62)
replace3(n,m,nil)	→	nil	(63)
replace3(n,m,cons2(k,x))	→	if_replace4(eq2(n,k),n,m,cons2(k,x))	(64)
eq2(0,0)	→	true	(51)
eq2(0,s1(m))	→	false	(52)
eq2(s1(n),0)	→	false	(53)
eq2(s1(n),s1(m))	→	eq2(n,m)	(54)
if_replace4(true,n,m,cons2(k,x))	→	cons2(m,x)	(65)
if_replace4(false,n,m,cons2(k,x))	→	cons2(k,replace3(n,m,x))	(66)
le2(0,m)	→	true	(55)
le2(s1(n),0)	→	false	(56)
le2(s1(n),s1(m))	→	le2(n,m)	(57)

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))

le2(0,x0)

le2(s1(x0),0)

le2(s1(x0),s1(x1))

min1(cons2(0,nil))

min1(cons2(s1(x0),nil))

min1(cons2(x0,cons2(x1,x2)))

if_min2(true,cons2(x0,cons2(x1,x2)))

if_min2(false,cons2(x0,cons2(x1,x2)))

replace3(x0,x1,nil)

replace3(x0,x1,cons2(x2,x3))

if_replace4(true,x0,x1,cons2(x2,x3))

if_replace4(false,x0,x1,cons2(x2,x3))

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[sort1^#(x₁)]	=	2 + 2 · x₁
[replace3(x₁, x₂, x₃)]	=	1 + 2 · x₂ + x₃
[min1(x₁)]	=	-1 + x₁
[cons2(x₁, x₂)]	=	2 + 2 · x₁ + x₂
[0]	=	2
[nil]	=	1
[s1(x₁)]	=	2
[if_min2(x₁, x₂)]	=	-2 + 2 · x₁ + x₂
[le2(x₁, x₂)]	=	2 + x₁ + 2 · x₂
[true]	=	0
[false]	=	0
[if_replace4(x₁,...,x₄)]	=	1 + 2 · x₃ + x₄
[eq2(x₁, x₂)]	=	0

together with the usable rules

replace3(n,m,nil)	→	nil	(63)
replace3(n,m,cons2(k,x))	→	if_replace4(eq2(n,k),n,m,cons2(k,x))	(64)
if_replace4(true,n,m,cons2(k,x))	→	cons2(m,x)	(65)
if_replace4(false,n,m,cons2(k,x))	→	cons2(k,replace3(n,m,x))	(66)

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

sort1^#(cons2(n,x))

→

sort1^#(replace3(min1(cons2(n,x)),n,x))

(85)

could be deleted.

1.1.1.1.2.1.1.1 P is empty

There are no pairs anymore.

The 3^rd component contains the pair

replace3^#(n,m,cons2(k,x))	→	if_replace4^#(eq2(n,k),n,m,cons2(k,x))	(81)
if_replace4^#(false,n,m,cons2(k,x))	→	replace3^#(n,m,x)	(83)

1.1.1.1.3 Usable Rules Processor

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

eq2(0,0)	→	true	(51)
eq2(0,s1(m))	→	false	(52)
eq2(s1(n),0)	→	false	(53)
eq2(s1(n),s1(m))	→	eq2(n,m)	(54)

1.1.1.1.3.1 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))

1.1.1.1.3.1.1 Size-Change Termination

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

if_replace4^#(false,n,m,cons2(k,x))	→	replace3^#(n,m,x)	(83)

2	≥	1
3	≥	2
4	>	3
replace3^#(n,m,cons2(k,x))	→	if_replace4^#(eq2(n,k),n,m,cons2(k,x))	(81)

1	≥	2
2	≥	3
3	≥	4

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

The 4^th component contains the pair

min1^#(cons2(n,cons2(m,x)))	→	if_min2^#(le2(n,m),cons2(n,cons2(m,x)))	(77)
if_min2^#(true,cons2(n,cons2(m,x)))	→	min1^#(cons2(n,x))	(79)
if_min2^#(false,cons2(n,cons2(m,x)))	→	min1^#(cons2(m,x))	(80)

1.1.1.1.4 Usable Rules Processor

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

le2(0,m)	→	true	(55)
le2(s1(n),0)	→	false	(56)
le2(s1(n),s1(m))	→	le2(n,m)	(57)

1.1.1.1.4.1 Innermost Lhss Removal Processor

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

le2(0,x0)

le2(s1(x0),0)

le2(s1(x0),s1(x1))

1.1.1.1.4.1.1 Reduction Pair Processor with Usable Rules

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions

prec(cons2)

weight(cons2)

in combination with the following argument filter

π(min1^#)	=	1
π(cons2)	=	[2]
π(if_min2^#)	=	2

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

if_min2^#(true,cons2(n,cons2(m,x)))	→	min1^#(cons2(n,x))	(79)
if_min2^#(false,cons2(n,cons2(m,x)))	→	min1^#(cons2(m,x))	(80)

could be deleted.

1.1.1.1.4.1.1.1 Size-Change Termination

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

min1^#(cons2(n,cons2(m,x)))	→	if_min2^#(le2(n,m),cons2(n,cons2(m,x)))	(77)

1	≥	2

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

The 5^th component contains the pair
eq2^#(s1(n),s1(m)) → eq2^#(n,m) (75)

1.1.1.1.5 Usable Rules Processor

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

There are no rules.

1.1.1.1.5.1 Innermost Lhss Removal Processor

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

There are no lhss.

1.1.1.1.5.1.1 Size-Change Termination

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

eq2^#(s1(n),s1(m)) → eq2^#(n,m) (75)

1 > 1

2 > 2

As there is no critical graph in the transitive closure, there are no infinite chains.
The 6^th component contains the pair
le2^#(s1(n),s1(m)) → le2^#(n,m) (76)

1.1.1.1.6 Usable Rules Processor

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

There are no rules.

1.1.1.1.6.1 Innermost Lhss Removal Processor

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

There are no lhss.

1.1.1.1.6.1.1 Size-Change Termination

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

le2^#(s1(n),s1(m)) → le2^#(n,m) (76)

1 > 1

2 > 2

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

Certification Problem

Input (TPDB TRS_Innermost/Applicative_AG01_innermost/#4.36)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Uncurrying

1.1 Switch to Innermost Termination

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Size-Change Termination

1.1.1.1.2 Usable Rules Processor

1.1.1.1.2.1 Innermost Lhss Removal Processor

1.1.1.1.2.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.2.1.1.1 P is empty

1.1.1.1.3 Usable Rules Processor

1.1.1.1.3.1 Innermost Lhss Removal Processor

1.1.1.1.3.1.1 Size-Change Termination

1.1.1.1.4 Usable Rules Processor

1.1.1.1.4.1 Innermost Lhss Removal Processor

1.1.1.1.4.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.4.1.1.1 Size-Change Termination

1.1.1.1.5 Usable Rules Processor

1.1.1.1.5.1 Innermost Lhss Removal Processor

1.1.1.1.5.1.1 Size-Change Termination

1.1.1.1.6 Usable Rules Processor

1.1.1.1.6.1 Innermost Lhss Removal Processor

1.1.1.1.6.1.1 Size-Change Termination