Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup01)

The rewrite relation of the following TRS is considered.

a12(a12(a12(a12(x1))))	→	x1	(1)
a13(a13(a13(a13(x1))))	→	x1	(2)
a14(a14(a14(a14(x1))))	→	x1	(3)
a15(a15(a15(a15(x1))))	→	x1	(4)
a16(a16(a16(a16(x1))))	→	x1	(5)
a23(a23(a23(a23(x1))))	→	x1	(6)
a24(a24(a24(a24(x1))))	→	x1	(7)
a25(a25(a25(a25(x1))))	→	x1	(8)
a26(a26(a26(a26(x1))))	→	x1	(9)
a34(a34(a34(a34(x1))))	→	x1	(10)
a35(a35(a35(a35(x1))))	→	x1	(11)
a36(a36(a36(a36(x1))))	→	x1	(12)
a45(a45(a45(a45(x1))))	→	x1	(13)
a46(a46(a46(a46(x1))))	→	x1	(14)
a56(a56(a56(a56(x1))))	→	x1	(15)
a13(a13(x1))	→	a12(a12(a23(a23(a12(a12(x1))))))	(16)
a14(a14(x1))	→	a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))	(17)
a15(a15(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))	(18)
a16(a16(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))	(19)
a24(a24(x1))	→	a23(a23(a34(a34(a23(a23(x1))))))	(20)
a25(a25(x1))	→	a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))	(21)
a26(a26(x1))	→	a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))	(22)
a35(a35(x1))	→	a34(a34(a45(a45(a34(a34(x1))))))	(23)
a36(a36(x1))	→	a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))	(24)
a46(a46(x1))	→	a45(a45(a56(a56(a45(a45(x1))))))	(25)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1))))))))))))	→	x1	(27)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)
a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a23(a23(a45(a45(x1))))	→	a45(a45(a23(a23(x1))))	(33)
a23(a23(a56(a56(x1))))	→	a56(a56(a23(a23(x1))))	(34)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)

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

[a12(x₁)]	=	1 · x₁ + 1
[a13(x₁)]	=	1 · x₁ + 4
[a14(x₁)]	=	1 · x₁ + 6
[a15(x₁)]	=	1 · x₁ + 8
[a16(x₁)]	=	1 · x₁ + 10
[a23(x₁)]	=	1 · x₁ + 1
[a24(x₁)]	=	1 · x₁ + 4
[a25(x₁)]	=	1 · x₁ + 6
[a26(x₁)]	=	1 · x₁ + 8
[a34(x₁)]	=	1 · x₁ + 1
[a35(x₁)]	=	1 · x₁ + 4
[a36(x₁)]	=	1 · x₁ + 6
[a45(x₁)]	=	1 · x₁ + 1
[a46(x₁)]	=	1 · x₁ + 4
[a56(x₁)]	=	1 · x₁ + 1

all of the following rules can be deleted.

a12(a12(a12(a12(x1))))	→	x1	(1)
a13(a13(a13(a13(x1))))	→	x1	(2)
a14(a14(a14(a14(x1))))	→	x1	(3)
a15(a15(a15(a15(x1))))	→	x1	(4)
a16(a16(a16(a16(x1))))	→	x1	(5)
a23(a23(a23(a23(x1))))	→	x1	(6)
a24(a24(a24(a24(x1))))	→	x1	(7)
a25(a25(a25(a25(x1))))	→	x1	(8)
a26(a26(a26(a26(x1))))	→	x1	(9)
a34(a34(a34(a34(x1))))	→	x1	(10)
a35(a35(a35(a35(x1))))	→	x1	(11)
a36(a36(a36(a36(x1))))	→	x1	(12)
a45(a45(a45(a45(x1))))	→	x1	(13)
a46(a46(a46(a46(x1))))	→	x1	(14)
a56(a56(a56(a56(x1))))	→	x1	(15)
a13(a13(x1))	→	a12(a12(a23(a23(a12(a12(x1))))))	(16)
a14(a14(x1))	→	a12(a12(a23(a23(a34(a34(a23(a23(a12(a12(x1))))))))))	(17)
a15(a15(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))	(18)
a16(a16(x1))	→	a12(a12(a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(a12(a12(x1))))))))))))))))))	(19)
a24(a24(x1))	→	a23(a23(a34(a34(a23(a23(x1))))))	(20)
a25(a25(x1))	→	a23(a23(a34(a34(a45(a45(a34(a34(a23(a23(x1))))))))))	(21)
a26(a26(x1))	→	a23(a23(a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(a23(a23(x1))))))))))))))	(22)
a35(a35(x1))	→	a34(a34(a45(a45(a34(a34(x1))))))	(23)
a36(a36(x1))	→	a34(a34(a45(a45(a56(a56(a45(a45(a34(a34(x1))))))))))	(24)
a46(a46(x1))	→	a45(a45(a56(a56(a45(a45(x1))))))	(25)
a12(a12(a23(a23(a12(a12(a23(a23(a12(a12(a23(a23(x1))))))))))))	→	x1	(26)
a23(a23(a34(a34(a23(a23(a34(a34(a23(a23(a34(a34(x1))))))))))))	→	x1	(27)
a34(a34(a45(a45(a34(a34(a45(a45(a34(a34(a45(a45(x1))))))))))))	→	x1	(28)
a45(a45(a56(a56(a45(a45(a56(a56(a45(a45(a56(a56(x1))))))))))))	→	x1	(29)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a12^#(a12(a34(a34(x1))))	→	a34^#(a34(a12(a12(x1))))	(36)
a12^#(a12(a34(a34(x1))))	→	a34^#(a12(a12(x1)))	(37)
a12^#(a12(a34(a34(x1))))	→	a12^#(a12(x1))	(38)
a12^#(a12(a34(a34(x1))))	→	a12^#(x1)	(39)
a12^#(a12(a45(a45(x1))))	→	a12^#(a12(x1))	(40)
a12^#(a12(a45(a45(x1))))	→	a12^#(x1)	(41)
a12^#(a12(a56(a56(x1))))	→	a12^#(a12(x1))	(42)
a12^#(a12(a56(a56(x1))))	→	a12^#(x1)	(43)
a23^#(a23(a45(a45(x1))))	→	a23^#(a23(x1))	(44)
a23^#(a23(a45(a45(x1))))	→	a23^#(x1)	(45)
a23^#(a23(a56(a56(x1))))	→	a23^#(a23(x1))	(46)
a23^#(a23(a56(a56(x1))))	→	a23^#(x1)	(47)
a34^#(a34(a56(a56(x1))))	→	a34^#(a34(x1))	(48)
a34^#(a34(a56(a56(x1))))	→	a34^#(x1)	(49)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

a12^#(a12(a34(a34(x1))))	→	a12^#(x1)	(39)
a12^#(a12(a34(a34(x1))))	→	a12^#(a12(x1))	(38)
a12^#(a12(a45(a45(x1))))	→	a12^#(a12(x1))	(40)
a12^#(a12(a45(a45(x1))))	→	a12^#(x1)	(41)
a12^#(a12(a56(a56(x1))))	→	a12^#(a12(x1))	(42)
a12^#(a12(a56(a56(x1))))	→	a12^#(x1)	(43)

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[a12(x₁)]	=	1 · x₁
[a34(x₁)]	=	1 · x₁
[a45(x₁)]	=	1 · x₁
[a56(x₁)]	=	1 · x₁
[a12^#(x₁)]	=	1 · x₁

together with the usable rules

a12(a12(a34(a34(x1))))	→	a34(a34(a12(a12(x1))))	(30)
a12(a12(a45(a45(x1))))	→	a45(a45(a12(a12(x1))))	(31)
a12(a12(a56(a56(x1))))	→	a56(a56(a12(a12(x1))))	(32)
a34(a34(a56(a56(x1))))	→	a56(a56(a34(a34(x1))))	(35)

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

1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a12(x₁)]	=	1 · x₁
[a34(x₁)]	=	1 + 1 · x₁
[a45(x₁)]	=	1 + 3 · x₁
[a56(x₁)]	=	1 · x₁
[a12^#(x₁)]	=	2 · x₁

the pairs

a12^#(a12(a34(a34(x1))))	→	a12^#(x1)	(39)
a12^#(a12(a34(a34(x1))))	→	a12^#(a12(x1))	(38)
a12^#(a12(a45(a45(x1))))	→	a12^#(a12(x1))	(40)
a12^#(a12(a45(a45(x1))))	→	a12^#(x1)	(41)

and no rules could be deleted.

1.1.1.1.1.1 Monotonic Reduction Pair Processor

Using the linear polynomial interpretation over the naturals

[a12(x₁)]	=	1 · x₁
[a34(x₁)]	=	3 · x₁
[a45(x₁)]	=	2 · x₁
[a56(x₁)]	=	1 + 2 · x₁
[a12^#(x₁)]	=	2 · x₁

the pairs

a12^#(a12(a56(a56(x1))))	→	a12^#(a12(x1))	(42)
a12^#(a12(a56(a56(x1))))	→	a12^#(x1)	(43)

and the rule

a34(a34(a56(a56(x1))))

→

a56(a56(a34(a34(x1))))

(35)

could be deleted.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

a23^#(a23(a45(a45(x1)))) → a23^#(x1) (45)

a23^#(a23(a45(a45(x1)))) → a23^#(a23(x1)) (44)

a23^#(a23(a56(a56(x1)))) → a23^#(a23(x1)) (46)

a23^#(a23(a56(a56(x1)))) → a23^#(x1) (47)

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

[a23(x₁)] = 1 · x₁

[a45(x₁)] = 1 · x₁

[a56(x₁)] = 1 · x₁

[a23^#(x₁)] = 1 · x₁

together with the usable rules

a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1)))) (33)

a23(a23(a56(a56(x1)))) → a56(a56(a23(a23(x1)))) (34)

(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.2.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.2.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[a23(x₁)] = 3 + 3 · x₁

[a45(x₁)] = 2 + 2 · x₁

[a56(x₁)] = 3 + 3 · x₁

[a23^#(x₁)] = 2 · x₁

the pairs

a23^#(a23(a45(a45(x1)))) → a23^#(x1) (45)

a23^#(a23(a45(a45(x1)))) → a23^#(a23(x1)) (44)

a23^#(a23(a56(a56(x1)))) → a23^#(a23(x1)) (46)

a23^#(a23(a56(a56(x1)))) → a23^#(x1) (47)

and the rule
a23(a23(a45(a45(x1)))) → a45(a45(a23(a23(x1)))) (33)
could be deleted.
1.1.1.2.1.1.1 P is empty
There are no pairs anymore.
The 3^rd component contains the pair

a34^#(a34(a56(a56(x1)))) → a34^#(x1) (49)

a34^#(a34(a56(a56(x1)))) → a34^#(a34(x1)) (48)

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

[a34(x₁)] = 1 · x₁

[a56(x₁)] = 1 · x₁

[a34^#(x₁)] = 1 · x₁

together with the usable rule
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1)))) (35)
(w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.1.3.1 Switch to Innermost Termination
The TRS does not have overlaps with the pairs and is locally confluent:
20
Hence, it suffices to show innermost termination in the following.
1.1.1.3.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[a34(x₁)] = 1 + 2 · x₁

[a56(x₁)] = 1 · x₁

[a34^#(x₁)] = 2 · x₁

the pair
a34^#(a34(a56(a56(x1)))) → a34^#(x1) (49)
and no rules could be deleted.
1.1.1.3.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals

[a34(x₁)] = 3 · x₁

[a56(x₁)] = 3 + 3 · x₁

[a34^#(x₁)] = 2 · x₁

the pair
a34^#(a34(a56(a56(x1)))) → a34^#(a34(x1)) (48)
and the rule
a34(a34(a56(a56(x1)))) → a56(a56(a34(a34(x1)))) (35)
could be deleted.
1.1.1.3.1.1.1.1 P is empty
There are no pairs anymore.

[a23(x₁)]	=	3 + 3 · x₁
[a45(x₁)]	=	2 + 2 · x₁
[a56(x₁)]	=	3 + 3 · x₁
[a23^#(x₁)]	=	2 · x₁

Certification Problem

Input (TPDB SRS_Standard/Trafo_06/dup01)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Rule Removal

1.1 Dependency Pair Transformation

1.1.1 Dependency Graph Processor

1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1 Monotonic Reduction Pair Processor

1.1.1.1.1.1.1 P is empty

1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.2.1 Switch to Innermost Termination

1.1.1.2.1.1 Monotonic Reduction Pair Processor

1.1.1.2.1.1.1 P is empty

1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.3.1 Switch to Innermost Termination

1.1.1.3.1.1 Monotonic Reduction Pair Processor

1.1.1.3.1.1.1 Monotonic Reduction Pair Processor

1.1.1.3.1.1.1.1 P is empty