Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/LengthOfFiniteLists_nokinds_noand_GM)

The rewrite relation of the following TRS is considered.

a__zeros	→	cons(0,zeros)	(1)
a__U11(tt)	→	tt	(2)
a__U21(tt)	→	tt	(3)
a__U31(tt)	→	tt	(4)
a__U41(tt,V2)	→	a__U42(a__isNatIList(V2))	(5)
a__U42(tt)	→	tt	(6)
a__U51(tt,V2)	→	a__U52(a__isNatList(V2))	(7)
a__U52(tt)	→	tt	(8)
a__U61(tt,L,N)	→	a__U62(a__isNat(N),L)	(9)
a__U62(tt,L)	→	s(a__length(mark(L)))	(10)
a__isNat(0)	→	tt	(11)
a__isNat(length(V1))	→	a__U11(a__isNatList(V1))	(12)
a__isNat(s(V1))	→	a__U21(a__isNat(V1))	(13)
a__isNatIList(V)	→	a__U31(a__isNatList(V))	(14)
a__isNatIList(zeros)	→	tt	(15)
a__isNatIList(cons(V1,V2))	→	a__U41(a__isNat(V1),V2)	(16)
a__isNatList(nil)	→	tt	(17)
a__isNatList(cons(V1,V2))	→	a__U51(a__isNat(V1),V2)	(18)
a__length(nil)	→	0	(19)
a__length(cons(N,L))	→	a__U61(a__isNatList(L),L,N)	(20)
mark(zeros)	→	a__zeros	(21)
mark(U11(X))	→	a__U11(mark(X))	(22)
mark(U21(X))	→	a__U21(mark(X))	(23)
mark(U31(X))	→	a__U31(mark(X))	(24)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(25)
mark(U42(X))	→	a__U42(mark(X))	(26)
mark(isNatIList(X))	→	a__isNatIList(X)	(27)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(28)
mark(U52(X))	→	a__U52(mark(X))	(29)
mark(isNatList(X))	→	a__isNatList(X)	(30)
mark(U61(X1,X2,X3))	→	a__U61(mark(X1),X2,X3)	(31)
mark(U62(X1,X2))	→	a__U62(mark(X1),X2)	(32)
mark(isNat(X))	→	a__isNat(X)	(33)
mark(length(X))	→	a__length(mark(X))	(34)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(35)
mark(0)	→	0	(36)
mark(tt)	→	tt	(37)
mark(s(X))	→	s(mark(X))	(38)
mark(nil)	→	nil	(39)
a__zeros	→	zeros	(40)
a__U11(X)	→	U11(X)	(41)
a__U21(X)	→	U21(X)	(42)
a__U31(X)	→	U31(X)	(43)
a__U41(X1,X2)	→	U41(X1,X2)	(44)
a__U42(X)	→	U42(X)	(45)
a__isNatIList(X)	→	isNatIList(X)	(46)
a__U51(X1,X2)	→	U51(X1,X2)	(47)
a__U52(X)	→	U52(X)	(48)
a__isNatList(X)	→	isNatList(X)	(49)
a__U61(X1,X2,X3)	→	U61(X1,X2,X3)	(50)
a__U62(X1,X2)	→	U62(X1,X2)	(51)
a__isNat(X)	→	isNat(X)	(52)
a__length(X)	→	length(X)	(53)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a__length(x₁)]

1	1	1
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[tt]

0	0	0
0	0	0
0	0	0

[a__isNat(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U41(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

[0]

0	0	0
0	0	0
0	0	0

[cons(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	1	1
1	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U62(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[nil]

0	0	0
0	0	0
0	0	0

[U42(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U62(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[s(x₁)]

1	0	0
1	0	0
1	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[mark(x₁)]

1	0	0
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[isNatIList(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[U11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U51(x₁, x₂)]

1	1	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	1
0	0	1

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[isNat(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U42(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__zeros]

0	0	0
0	0	0
0	0	0

[a__U31(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[length(x₁)]

1	1	1
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U41(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

[isNatList(x₁)]

1	0	0
0	1	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U61(x₁, x₂, x₃)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[U61(x₁, x₂, x₃)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[a__isNatList(x₁)]

1	0	0
0	1	0
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U31(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[a__U51(x₁, x₂)]

1	1	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	1
0	0	1

· x₂ +

0	0	0
0	0	0
0	0	0

[a__isNatIList(x₁)]

1	1	0
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[a__U52(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U52(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[zeros]

0	0	0
0	0	0
0	0	0

all of the following rules can be deleted.

a__U31(tt)	→	tt	(4)
a__isNatIList(zeros)	→	tt	(15)

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a__length(x₁)]

1	1	1
1	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[tt]

0	0	0
0	0	0
0	0	0

[a__isNat(x₁)]

1	0	0
1	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U41(x₁, x₂)]

1	0	0
1	0	1
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[0]

0	0	0
1	0	0
0	0	0

[cons(x₁, x₂)]

1	0	0
0	0	0
0	0	1

· x₁ +

1	1	1
1	1	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U62(x₁, x₂)]

1	0	0
0	0	1
0	0	0

· x₁ +

1	1	1
1	1	1
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[nil]

0	0	0
1	0	0
0	0	0

[U42(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U62(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	1
1	1	1
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[s(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[mark(x₁)]

1	0	0
0	1	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[isNatIList(x₁)]

1	1	1
1	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U11(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U51(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U11(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[isNat(x₁)]

1	0	0
1	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U42(x₁)]

1	0	0
0	0	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a__zeros]

0	0	0
0	0	0
0	0	0

[a__U31(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[length(x₁)]

1	1	1
1	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U41(x₁, x₂)]

1	0	0
1	0	1
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[isNatList(x₁)]

1	1	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U61(x₁, x₂, x₃)]

1	0	0
0	0	1
0	0	0

· x₁ +

1	1	1
1	1	1
0	0	0

· x₂ +

1	0	1
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[U61(x₁, x₂, x₃)]

1	0	0
0	0	1
0	0	0

· x₁ +

1	1	1
1	1	1
0	0	0

· x₂ +

1	0	1
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[a__isNatList(x₁)]

1	1	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U31(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U51(x₁, x₂)]

1	0	0
0	0	1
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__isNatIList(x₁)]

1	1	1
1	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U52(x₁)]

1	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U52(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[zeros]

0	0	0
0	0	0
0	0	0

all of the following rules can be deleted.

a__isNatList(nil)	→	tt	(17)
a__length(nil)	→	0	(19)

1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a__length(x₁)]

1	0	1
0	0	1
0	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[tt]

1	0	0
0	0	0
0	0	0

[a__isNat(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[a__U41(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
1	0	0
1	0	0

[0]

0	0	0
0	0	0
0	0	0

[cons(x₁, x₂)]

1	1	1
0	1	0
0	0	1

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U62(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
0	1	1
0	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[nil]

0	0	0
1	0	0
1	0	0

[U42(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U62(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	0	0
0	1	1
0	1	1

· x₂ +

0	0	0
0	0	0
0	0	0

[s(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[mark(x₁)]

1	0	1
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[isNatIList(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
1	0	0
1	0	0

[U11(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U51(x₁, x₂)]

1	0	1
0	0	1
0	1	0

· x₁ +

1	0	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U11(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[isNat(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[a__U42(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a__zeros]

0	0	0
0	0	0
0	0	0

[a__U31(x₁)]

1	0	1
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[length(x₁)]

1	0	1
0	0	1
0	1	0

· x₁ +

1	0	0
0	0	0
0	0	0

[U41(x₁, x₂)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
1	0	0
1	0	0

[isNatList(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[a__U61(x₁, x₂, x₃)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	1	1
1	1	1
1	1	1

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[U61(x₁, x₂, x₃)]

1	0	0
0	1	0
0	0	1

· x₁ +

1	0	0
1	1	1
1	1	1

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

[a__isNatList(x₁)]

1	0	1
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

[U21(x₁)]

1	0	0
0	1	1
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U31(x₁)]

1	0	1
0	0	1
0	1	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U51(x₁, x₂)]

1	0	1
0	0	1
0	1	0

· x₁ +

1	0	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__isNatIList(x₁)]

1	1	1
0	0	0
0	0	0

· x₁ +

1	0	0
1	0	0
1	0	0

[a__U52(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U21(x₁)]

1	0	0
0	1	1
0	1	1

· x₁ +

0	0	0
0	0	0
0	0	0

[U52(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[zeros]

0	0	0
0	0	0
0	0	0

all of the following rules can be deleted.

a__isNat(length(V1))	→	a__U11(a__isNatList(V1))	(12)
mark(U41(X1,X2))	→	a__U41(mark(X1),X2)	(25)
mark(nil)	→	nil	(39)
a__isNatIList(X)	→	isNatIList(X)	(46)

1.1.1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals

[a__length(x₁)]

1	1	0
1	0	1
1	0	0

· x₁ +

1	0	0
0	0	0
1	0	0

[tt]

0	0	0
0	0	0
0	0	0

[a__isNat(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U41(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
1	0	0

· x₂ +

1	0	0
1	0	0
0	0	0

[0]

0	0	0
0	0	0
0	0	0

[cons(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
1	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
1	0	0

[a__U62(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
1	1	0
0	0	0

· x₂ +

1	0	0
1	0	0
0	0	0

[U42(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U62(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
1	1	0
0	0	0

· x₂ +

1	0	0
1	0	0
0	0	0

[s(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[mark(x₁)]

1	0	0
0	1	0
0	0	1

· x₁ +

0	0	0
0	0	0
0	0	0

[isNatIList(x₁)]

1	0	1
0	1	0
0	1	1

· x₁ +

1	0	0
1	0	0
0	0	0

[U11(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U51(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__U11(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[isNat(x₁)]

1	0	0
1	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U42(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__zeros]

0	0	0
0	0	0
1	0	0

[a__U31(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[length(x₁)]

1	1	0
1	0	1
1	0	0

· x₁ +

1	0	0
0	0	0
1	0	0

[U41(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂ +

1	0	0
1	0	0
0	0	0

[isNatList(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U61(x₁, x₂, x₃)]

1	0	0
0	0	0
1	1	0

· x₁ +

1	1	0
1	1	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

1	0	0
1	0	0
1	0	0

[U61(x₁, x₂, x₃)]

1	0	0
0	0	0
1	1	0

· x₁ +

1	1	0
1	1	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

1	0	0
1	0	0
1	0	0

[a__isNatList(x₁)]

1	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U31(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U51(x₁, x₂)]

1	0	0
0	0	0
0	0	0

· x₁ +

1	1	0
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

[a__isNatIList(x₁)]

1	0	0
0	1	0
0	1	1

· x₁ +

1	0	0
1	0	0
0	0	0

[a__U52(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[a__U21(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[U52(x₁)]

1	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

[zeros]

0	0	0
0	0	0
1	0	0

all of the following rules can be deleted.

a__isNatIList(V)

→

a__U31(a__isNatList(V))

(14)

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a__U41^#(tt,V2)	→	a__isNatIList^#(V2)	(54)
a__U41^#(tt,V2)	→	a__U42^#(a__isNatIList(V2))	(55)
a__U51^#(tt,V2)	→	a__isNatList^#(V2)	(56)
a__U51^#(tt,V2)	→	a__U52^#(a__isNatList(V2))	(57)
a__U61^#(tt,L,N)	→	a__isNat^#(N)	(58)
a__U61^#(tt,L,N)	→	a__U62^#(a__isNat(N),L)	(59)
a__U62^#(tt,L)	→	mark^#(L)	(60)
a__U62^#(tt,L)	→	a__length^#(mark(L))	(61)
a__isNat^#(s(V1))	→	a__isNat^#(V1)	(62)
a__isNat^#(s(V1))	→	a__U21^#(a__isNat(V1))	(63)
a__isNatIList^#(cons(V1,V2))	→	a__isNat^#(V1)	(64)
a__isNatIList^#(cons(V1,V2))	→	a__U41^#(a__isNat(V1),V2)	(65)
a__isNatList^#(cons(V1,V2))	→	a__isNat^#(V1)	(66)
a__isNatList^#(cons(V1,V2))	→	a__U51^#(a__isNat(V1),V2)	(67)
a__length^#(cons(N,L))	→	a__isNatList^#(L)	(68)
a__length^#(cons(N,L))	→	a__U61^#(a__isNatList(L),L,N)	(69)
mark^#(zeros)	→	a__zeros^#	(70)
mark^#(U11(X))	→	mark^#(X)	(71)
mark^#(U11(X))	→	a__U11^#(mark(X))	(72)
mark^#(U21(X))	→	mark^#(X)	(73)
mark^#(U21(X))	→	a__U21^#(mark(X))	(74)
mark^#(U31(X))	→	mark^#(X)	(75)
mark^#(U31(X))	→	a__U31^#(mark(X))	(76)
mark^#(U42(X))	→	mark^#(X)	(77)
mark^#(U42(X))	→	a__U42^#(mark(X))	(78)
mark^#(isNatIList(X))	→	a__isNatIList^#(X)	(79)
mark^#(U51(X1,X2))	→	mark^#(X1)	(80)
mark^#(U51(X1,X2))	→	a__U51^#(mark(X1),X2)	(81)
mark^#(U52(X))	→	mark^#(X)	(82)
mark^#(U52(X))	→	a__U52^#(mark(X))	(83)
mark^#(isNatList(X))	→	a__isNatList^#(X)	(84)
mark^#(U61(X1,X2,X3))	→	mark^#(X1)	(85)
mark^#(U61(X1,X2,X3))	→	a__U61^#(mark(X1),X2,X3)	(86)
mark^#(U62(X1,X2))	→	mark^#(X1)	(87)
mark^#(U62(X1,X2))	→	a__U62^#(mark(X1),X2)	(88)
mark^#(isNat(X))	→	a__isNat^#(X)	(89)
mark^#(length(X))	→	mark^#(X)	(90)
mark^#(length(X))	→	a__length^#(mark(X))	(91)
mark^#(cons(X1,X2))	→	mark^#(X1)	(92)
mark^#(s(X))	→	mark^#(X)	(93)

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 4 components.

The 1^st component contains the pair

a__length^#(cons(N,L))	→	a__U61^#(a__isNatList(L),L,N)	(69)
a__U61^#(tt,L,N)	→	a__U62^#(a__isNat(N),L)	(59)
a__U62^#(tt,L)	→	mark^#(L)	(60)
mark^#(U11(X))	→	mark^#(X)	(71)
mark^#(U21(X))	→	mark^#(X)	(73)
mark^#(U31(X))	→	mark^#(X)	(75)
mark^#(U42(X))	→	mark^#(X)	(77)
mark^#(U51(X1,X2))	→	mark^#(X1)	(80)
mark^#(U52(X))	→	mark^#(X)	(82)
mark^#(U61(X1,X2,X3))	→	mark^#(X1)	(85)
mark^#(U61(X1,X2,X3))	→	a__U61^#(mark(X1),X2,X3)	(86)
mark^#(U62(X1,X2))	→	mark^#(X1)	(87)
mark^#(U62(X1,X2))	→	a__U62^#(mark(X1),X2)	(88)
a__U62^#(tt,L)	→	a__length^#(mark(L))	(61)
mark^#(length(X))	→	mark^#(X)	(90)
mark^#(length(X))	→	a__length^#(mark(X))	(91)
mark^#(cons(X1,X2))	→	mark^#(X1)	(92)
mark^#(s(X))	→	mark^#(X)	(93)

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the rationals with delta = 1/64

[a__length(x₁)]	=	0 · x₁ + 0
[tt]	=	1
[a__isNat(x₁)]	=	0 · x₁ + 1
[a__U41(x₁, x₂)]	=	1 · x₁ + 0 · x₂ + 0
[0]	=	1
[cons(x₁, x₂)]	=	1 · x₁ + 0 · x₂ + 0
[a__U62(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[U42(x₁)]	=	1 · x₁ + 0
[s(x₁)]	=	0 · x₁ + 0
[U62(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[mark(x₁)]	=	0 · x₁ + 1
[isNatIList(x₁)]	=	0 · x₁ + 0
[U11(x₁)]	=	0 · x₁ + 0
[U51(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[a__U11(x₁)]	=	1/2 · x₁ + 1/2
[isNat(x₁)]	=	0 · x₁ + 1
[a__U42(x₁)]	=	1 · x₁ + 0
[a__zeros]	=	1
[a__U31(x₁)]	=	1/2 · x₁ + 1/2
[length(x₁)]	=	0 · x₁ + 0
[U41(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[isNatList(x₁)]	=	0 · x₁ + 0
[a__U61(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 0 · x₃ + 0
[U61(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 0 · x₃ + 0
[a__isNatList(x₁)]	=	0 · x₁ + 0
[a__U62^#(x₁, x₂)]	=	1/2 · x₁ + 0 · x₂ + 1/2
[U21(x₁)]	=	1 · x₁ + 0
[U31(x₁)]	=	1/2 · x₁ + 1/2
[a__U51(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[a__isNatIList(x₁)]	=	0 · x₁ + 1
[a__U52(x₁)]	=	1 · x₁ + 0
[a__U21(x₁)]	=	1 · x₁ + 0
[a__length^#(x₁)]	=	0 · x₁ + 1/2
[mark^#(x₁)]	=	0 · x₁ + 1
[U52(x₁)]	=	1 · x₁ + 0
[a__U61^#(x₁, x₂, x₃)]	=	1/2 · x₁ + 0 · x₂ + 0 · x₃ + 1/2
[zeros]	=	0

together with the usable rules

a__zeros	→	cons(0,zeros)	(1)
a__U11(tt)	→	tt	(2)
a__U21(tt)	→	tt	(3)
a__U41(tt,V2)	→	a__U42(a__isNatIList(V2))	(5)
a__U42(tt)	→	tt	(6)
a__U51(tt,V2)	→	a__U52(a__isNatList(V2))	(7)
a__U52(tt)	→	tt	(8)
a__U61(tt,L,N)	→	a__U62(a__isNat(N),L)	(9)
a__U62(tt,L)	→	s(a__length(mark(L)))	(10)
a__isNat(0)	→	tt	(11)
a__isNat(s(V1))	→	a__U21(a__isNat(V1))	(13)
a__isNatIList(cons(V1,V2))	→	a__U41(a__isNat(V1),V2)	(16)
a__isNatList(cons(V1,V2))	→	a__U51(a__isNat(V1),V2)	(18)
a__length(cons(N,L))	→	a__U61(a__isNatList(L),L,N)	(20)
mark(zeros)	→	a__zeros	(21)
mark(U11(X))	→	a__U11(mark(X))	(22)
mark(U21(X))	→	a__U21(mark(X))	(23)
mark(U31(X))	→	a__U31(mark(X))	(24)
mark(U42(X))	→	a__U42(mark(X))	(26)
mark(isNatIList(X))	→	a__isNatIList(X)	(27)
mark(U51(X1,X2))	→	a__U51(mark(X1),X2)	(28)
mark(U52(X))	→	a__U52(mark(X))	(29)
mark(isNatList(X))	→	a__isNatList(X)	(30)
mark(U61(X1,X2,X3))	→	a__U61(mark(X1),X2,X3)	(31)
mark(U62(X1,X2))	→	a__U62(mark(X1),X2)	(32)
mark(isNat(X))	→	a__isNat(X)	(33)
mark(length(X))	→	a__length(mark(X))	(34)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(35)
mark(0)	→	0	(36)
mark(tt)	→	tt	(37)
mark(s(X))	→	s(mark(X))	(38)
a__zeros	→	zeros	(40)
a__U11(X)	→	U11(X)	(41)
a__U21(X)	→	U21(X)	(42)
a__U31(X)	→	U31(X)	(43)
a__U41(X1,X2)	→	U41(X1,X2)	(44)
a__U42(X)	→	U42(X)	(45)
a__U51(X1,X2)	→	U51(X1,X2)	(47)
a__U52(X)	→	U52(X)	(48)
a__isNatList(X)	→	isNatList(X)	(49)
a__U61(X1,X2,X3)	→	U61(X1,X2,X3)	(50)
a__U62(X1,X2)	→	U62(X1,X2)	(51)
a__isNat(X)	→	isNat(X)	(52)
a__length(X)	→	length(X)	(53)

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

a__U62^#(tt,L)	→	a__length^#(mark(L))	(61)
mark^#(length(X))	→	a__length^#(mark(X))	(91)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

a__U61^#(tt,L,N)	→	a__U62^#(a__isNat(N),L)	(59)
a__U62^#(tt,L)	→	mark^#(L)	(60)
mark^#(U11(X))	→	mark^#(X)	(71)
mark^#(U21(X))	→	mark^#(X)	(73)
mark^#(U31(X))	→	mark^#(X)	(75)
mark^#(U42(X))	→	mark^#(X)	(77)
mark^#(U51(X1,X2))	→	mark^#(X1)	(80)
mark^#(U52(X))	→	mark^#(X)	(82)
mark^#(U61(X1,X2,X3))	→	mark^#(X1)	(85)
mark^#(U61(X1,X2,X3))	→	a__U61^#(mark(X1),X2,X3)	(86)
mark^#(U62(X1,X2))	→	mark^#(X1)	(87)
mark^#(U62(X1,X2))	→	a__U62^#(mark(X1),X2)	(88)
mark^#(length(X))	→	mark^#(X)	(90)
mark^#(cons(X1,X2))	→	mark^#(X1)	(92)
mark^#(s(X))	→	mark^#(X)	(93)

1.1.1.1.1.1.1.1.1 Subterm Criterion Processor

We use the projection

π(mark^#)	=	1
π(a__U62^#)	=	2
π(a__U61^#)	=	2

and remove the pairs:

mark^#(U11(X))	→	mark^#(X)	(71)
mark^#(U21(X))	→	mark^#(X)	(73)
mark^#(U31(X))	→	mark^#(X)	(75)
mark^#(U42(X))	→	mark^#(X)	(77)
mark^#(U51(X1,X2))	→	mark^#(X1)	(80)
mark^#(U52(X))	→	mark^#(X)	(82)
mark^#(U61(X1,X2,X3))	→	mark^#(X1)	(85)
mark^#(U61(X1,X2,X3))	→	a__U61^#(mark(X1),X2,X3)	(86)
mark^#(U62(X1,X2))	→	mark^#(X1)	(87)
mark^#(U62(X1,X2))	→	a__U62^#(mark(X1),X2)	(88)
mark^#(length(X))	→	mark^#(X)	(90)
mark^#(cons(X1,X2))	→	mark^#(X1)	(92)
mark^#(s(X))	→	mark^#(X)	(93)

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

a__U51^#(tt,V2) → a__isNatList^#(V2) (56)

a__isNatList^#(cons(V1,V2)) → a__U51^#(a__isNat(V1),V2) (67)

1.1.1.1.1.1.2 Size-Change Termination

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

a__U51^#(tt,V2) → a__isNatList^#(V2) (56)

2 ≥ 1

a__isNatList^#(cons(V1,V2)) → a__U51^#(a__isNat(V1),V2) (67)

1 > 2

As there is no critical graph in the transitive closure, there are no infinite chains.
The 3^rd component contains the pair

a__isNatIList^#(cons(V1,V2)) → a__U41^#(a__isNat(V1),V2) (65)

a__U41^#(tt,V2) → a__isNatIList^#(V2) (54)

1.1.1.1.1.1.3 Size-Change Termination

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

a__isNatIList^#(cons(V1,V2)) → a__U41^#(a__isNat(V1),V2) (65)

1 > 2

a__U41^#(tt,V2) → a__isNatIList^#(V2) (54)

2 ≥ 1

As there is no critical graph in the transitive closure, there are no infinite chains.
The 4^th component contains the pair
a__isNat^#(s(V1)) → a__isNat^#(V1) (62)

1.1.1.1.1.1.4 Size-Change Termination

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

a__isNat^#(s(V1)) → a__isNat^#(V1) (62)

1 > 1

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

Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/LengthOfFiniteLists_nokinds_noand_GM)

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

1.1 Rule Removal

1.1.1 Rule Removal

1.1.1.1 Rule Removal

1.1.1.1.1 Dependency Pair Transformation

1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1 Subterm Criterion Processor

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.2 Size-Change Termination

1.1.1.1.1.1.3 Size-Change Termination

1.1.1.1.1.1.4 Size-Change Termination