Certification Problem

Input (TPDB TRS_Standard/CiME_04/log2)

The rewrite relation of the following TRS is considered.

0(#)	→	#	(1)
+(#,x)	→	x	(2)
+(x,#)	→	x	(3)
+(0(x),0(y))	→	0(+(x,y))	(4)
+(0(x),1(y))	→	1(+(x,y))	(5)
+(1(x),0(y))	→	1(+(x,y))	(6)
+(1(x),1(y))	→	0(+(+(x,y),1(#)))	(7)
+(+(x,y),z)	→	+(x,+(y,z))	(8)
-(#,x)	→	#	(9)
-(x,#)	→	x	(10)
-(0(x),0(y))	→	0(-(x,y))	(11)
-(0(x),1(y))	→	1(-(-(x,y),1(#)))	(12)
-(1(x),0(y))	→	1(-(x,y))	(13)
-(1(x),1(y))	→	0(-(x,y))	(14)
not(true)	→	false	(15)
not(false)	→	true	(16)
if(true,x,y)	→	x	(17)
if(false,x,y)	→	y	(18)
ge(0(x),0(y))	→	ge(x,y)	(19)
ge(0(x),1(y))	→	not(ge(y,x))	(20)
ge(1(x),0(y))	→	ge(x,y)	(21)
ge(1(x),1(y))	→	ge(x,y)	(22)
ge(x,#)	→	true	(23)
ge(#,0(x))	→	ge(#,x)	(24)
ge(#,1(x))	→	false	(25)
log(x)	→	-(log'(x),1(#))	(26)
log'(#)	→	#	(27)
log'(1(x))	→	+(log'(x),1(#))	(28)
log'(0(x))	→	if(ge(x,1(#)),+(log'(x),1(#)),#)	(29)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0(x₁)]	=	2 · x₁
[#]	=	0
[+(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[1(x₁)]	=	2 · x₁
[-(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[not(x₁)]	=	1 · x₁
[true]	=	0
[false]	=	0
[if(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[ge(x₁, x₂)]	=	1 · x₁ + 2 · x₂
[log(x₁)]	=	2 + 2 · x₁
[log'(x₁)]	=	2 · x₁

all of the following rules can be deleted.

log(x)

→

-(log'(x),1(#))

(26)

1.1 Rule Removal

Using the linear polynomial interpretation over the naturals

[0(x₁)]	=	2 · x₁
[#]	=	0
[+(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[1(x₁)]	=	2 · x₁
[-(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[not(x₁)]	=	1 · x₁
[true]	=	0
[false]	=	0
[if(x₁, x₂, x₃)]	=	2 · x₁ + 1 · x₂ + 1 · x₃
[ge(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[log'(x₁)]	=	1 + 2 · x₁

all of the following rules can be deleted.

log'(#)

→

(27)

1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

+^#(0(x),0(y))	→	0^#(+(x,y))	(30)
+^#(0(x),0(y))	→	+^#(x,y)	(31)
+^#(0(x),1(y))	→	+^#(x,y)	(32)
+^#(1(x),0(y))	→	+^#(x,y)	(33)
+^#(1(x),1(y))	→	0^#(+(+(x,y),1(#)))	(34)
+^#(1(x),1(y))	→	+^#(+(x,y),1(#))	(35)
+^#(1(x),1(y))	→	+^#(x,y)	(36)
+^#(+(x,y),z)	→	+^#(x,+(y,z))	(37)
+^#(+(x,y),z)	→	+^#(y,z)	(38)
-^#(0(x),0(y))	→	0^#(-(x,y))	(39)
-^#(0(x),0(y))	→	-^#(x,y)	(40)
-^#(0(x),1(y))	→	-^#(-(x,y),1(#))	(41)
-^#(0(x),1(y))	→	-^#(x,y)	(42)
-^#(1(x),0(y))	→	-^#(x,y)	(43)
-^#(1(x),1(y))	→	0^#(-(x,y))	(44)
-^#(1(x),1(y))	→	-^#(x,y)	(45)
ge^#(0(x),0(y))	→	ge^#(x,y)	(46)
ge^#(0(x),1(y))	→	not^#(ge(y,x))	(47)
ge^#(0(x),1(y))	→	ge^#(y,x)	(48)
ge^#(1(x),0(y))	→	ge^#(x,y)	(49)
ge^#(1(x),1(y))	→	ge^#(x,y)	(50)
ge^#(#,0(x))	→	ge^#(#,x)	(51)
log'^#(1(x))	→	+^#(log'(x),1(#))	(52)
log'^#(1(x))	→	log'^#(x)	(53)
log'^#(0(x))	→	if^#(ge(x,1(#)),+(log'(x),1(#)),#)	(54)
log'^#(0(x))	→	ge^#(x,1(#))	(55)
log'^#(0(x))	→	+^#(log'(x),1(#))	(56)
log'^#(0(x))	→	log'^#(x)	(57)

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 5 components.

The 1^st component contains the pair

log'^#(0(x)) → log'^#(x) (57)

log'^#(1(x)) → log'^#(x) (53)

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[0(x₁)] = 1 · x₁

[1(x₁)] = 1 · x₁

[log'^#(x₁)] = 1 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1.1.1 Size-Change Termination

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

log'^#(0(x)) → log'^#(x) (57)

1 > 1

log'^#(1(x)) → log'^#(x) (53)

1 > 1

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

The 2^nd component contains the pair

ge^#(0(x),1(y))	→	ge^#(y,x)	(48)
ge^#(0(x),0(y))	→	ge^#(x,y)	(46)
ge^#(1(x),0(y))	→	ge^#(x,y)	(49)
ge^#(1(x),1(y))	→	ge^#(x,y)	(50)

1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[0(x₁)]	=	1 · x₁
[1(x₁)]	=	1 · x₁
[ge^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

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

1.1.1.1.2.1 Size-Change Termination

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

ge^#(0(x),1(y))	→	ge^#(y,x)	(48)

2	>	1
1	>	2
ge^#(0(x),0(y))	→	ge^#(x,y)	(46)

1	>	1
2	>	2
ge^#(1(x),0(y))	→	ge^#(x,y)	(49)

1	>	1
2	>	2
ge^#(1(x),1(y))	→	ge^#(x,y)	(50)

1	>	1
2	>	2

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

The 3^rd component contains the pair

+^#(0(x),1(y))	→	+^#(x,y)	(32)
+^#(0(x),0(y))	→	+^#(x,y)	(31)
+^#(1(x),0(y))	→	+^#(x,y)	(33)
+^#(1(x),1(y))	→	+^#(+(x,y),1(#))	(35)
+^#(1(x),1(y))	→	+^#(x,y)	(36)
+^#(+(x,y),z)	→	+^#(x,+(y,z))	(37)
+^#(+(x,y),z)	→	+^#(y,z)	(38)

1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[+(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[#]	=	0
[0(x₁)]	=	1 · x₁
[1(x₁)]	=	1 · x₁
[+^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

+(#,x)	→	x	(2)
+(x,#)	→	x	(3)
+(0(x),0(y))	→	0(+(x,y))	(4)
+(0(x),1(y))	→	1(+(x,y))	(5)
+(1(x),0(y))	→	1(+(x,y))	(6)
+(1(x),1(y))	→	0(+(+(x,y),1(#)))	(7)
+(+(x,y),z)	→	+(x,+(y,z))	(8)
0(#)	→	#	(1)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.3.1 Monotonic Reduction Pair Processor

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

prec(#)	=	4	weight(#)	=	2
prec(0)	=	1	weight(0)	=	1
prec(1)	=	2	weight(1)	=	3
prec(+)	=	3	weight(+)	=	0
prec(+^#)	=	0	weight(+^#)	=	0

the pairs

+^#(0(x),1(y))	→	+^#(x,y)	(32)
+^#(0(x),0(y))	→	+^#(x,y)	(31)
+^#(1(x),0(y))	→	+^#(x,y)	(33)
+^#(1(x),1(y))	→	+^#(+(x,y),1(#))	(35)
+^#(1(x),1(y))	→	+^#(x,y)	(36)
+^#(+(x,y),z)	→	+^#(x,+(y,z))	(37)
+^#(+(x,y),z)	→	+^#(y,z)	(38)

and the rules

+(#,x)	→	x	(2)
+(x,#)	→	x	(3)
+(0(x),0(y))	→	0(+(x,y))	(4)
+(0(x),1(y))	→	1(+(x,y))	(5)
+(1(x),0(y))	→	1(+(x,y))	(6)
+(1(x),1(y))	→	0(+(+(x,y),1(#)))	(7)
+(+(x,y),z)	→	+(x,+(y,z))	(8)
0(#)	→	#	(1)

could be deleted.

1.1.1.1.3.1.1 P is empty

There are no pairs anymore.

The 4^th component contains the pair

-^#(0(x),1(y))	→	-^#(-(x,y),1(#))	(41)
-^#(0(x),1(y))	→	-^#(x,y)	(42)
-^#(0(x),0(y))	→	-^#(x,y)	(40)
-^#(1(x),0(y))	→	-^#(x,y)	(43)
-^#(1(x),1(y))	→	-^#(x,y)	(45)

1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[-(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[#]	=	0
[0(x₁)]	=	1 · x₁
[1(x₁)]	=	1 · x₁
[-^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

-(#,x)	→	#	(9)
-(x,#)	→	x	(10)
-(0(x),0(y))	→	0(-(x,y))	(11)
-(0(x),1(y))	→	1(-(-(x,y),1(#)))	(12)
-(1(x),0(y))	→	1(-(x,y))	(13)
-(1(x),1(y))	→	0(-(x,y))	(14)
0(#)	→	#	(1)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.4.1 Monotonic Reduction Pair Processor

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

prec(#)	=	0	weight(#)	=	2
prec(0)	=	2	weight(0)	=	5
prec(1)	=	1	weight(1)	=	3
prec(-)	=	4	weight(-)	=	0
prec(-^#)	=	3	weight(-^#)	=	0

the pairs

-^#(0(x),1(y))	→	-^#(-(x,y),1(#))	(41)
-^#(0(x),1(y))	→	-^#(x,y)	(42)
-^#(0(x),0(y))	→	-^#(x,y)	(40)
-^#(1(x),0(y))	→	-^#(x,y)	(43)
-^#(1(x),1(y))	→	-^#(x,y)	(45)

and the rules

-(#,x)	→	#	(9)
-(x,#)	→	x	(10)
-(0(x),0(y))	→	0(-(x,y))	(11)
-(0(x),1(y))	→	1(-(-(x,y),1(#)))	(12)
-(1(x),0(y))	→	1(-(x,y))	(13)
-(1(x),1(y))	→	0(-(x,y))	(14)
0(#)	→	#	(1)

could be deleted.

1.1.1.1.4.1.1 P is empty

There are no pairs anymore.

The 5^th component contains the pair
ge^#(#,0(x)) → ge^#(#,x) (51)

1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[#] = 0

[0(x₁)] = 1 · x₁

[ge^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.1.5.1 Size-Change Termination

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

ge^#(#,0(x)) → ge^#(#,x) (51)

1 ≥ 1

2 > 2

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

Certification Problem

Input (TPDB TRS_Standard/CiME_04/log2)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

1.1 Rule Removal

1.1.1 Dependency Pair Transformation

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Size-Change Termination

1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.2.1 Size-Change Termination

1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.3.1 Monotonic Reduction Pair Processor

1.1.1.1.3.1.1 P is empty

1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.4.1 Monotonic Reduction Pair Processor

1.1.1.1.4.1.1 P is empty

1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.5.1 Size-Change Termination