Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_complete_FR)

The rewrite relation of the following TRS is considered.

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

U11^#(tt,V1,V2)	→	activate^#(V2)	(30)
U11^#(tt,V1,V2)	→	activate^#(V1)	(31)
U11^#(tt,V1,V2)	→	isNat^#(activate(V1))	(32)
U11^#(tt,V1,V2)	→	U12^#(isNat(activate(V1)),activate(V2))	(33)
U12^#(tt,V2)	→	activate^#(V2)	(34)
U12^#(tt,V2)	→	isNat^#(activate(V2))	(35)
U12^#(tt,V2)	→	U13^#(isNat(activate(V2)))	(36)
U21^#(tt,V1)	→	activate^#(V1)	(37)
U21^#(tt,V1)	→	isNat^#(activate(V1))	(38)
U21^#(tt,V1)	→	U22^#(isNat(activate(V1)))	(39)
U31^#(tt,N)	→	activate^#(N)	(40)
U41^#(tt,M,N)	→	activate^#(M)	(41)
U41^#(tt,M,N)	→	activate^#(N)	(42)
U41^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(43)
U41^#(tt,M,N)	→	s^#(plus(activate(N),activate(M)))	(44)
and^#(tt,X)	→	activate^#(X)	(45)
isNat^#(n__plus(V1,V2))	→	activate^#(V2)	(46)
isNat^#(n__plus(V1,V2))	→	activate^#(V1)	(47)
isNat^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(48)
isNat^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(49)
isNat^#(n__plus(V1,V2))	→	U11^#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(50)
isNat^#(n__s(V1))	→	activate^#(V1)	(51)
isNat^#(n__s(V1))	→	isNatKind^#(activate(V1))	(52)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(53)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V2)	(54)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V1)	(55)
isNatKind^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(56)
isNatKind^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(57)
isNatKind^#(n__s(V1))	→	activate^#(V1)	(58)
isNatKind^#(n__s(V1))	→	isNatKind^#(activate(V1))	(59)
plus^#(N,0)	→	isNat^#(N)	(60)
plus^#(N,0)	→	and^#(isNat(N),n__isNatKind(N))	(61)
plus^#(N,0)	→	U31^#(and(isNat(N),n__isNatKind(N)),N)	(62)
plus^#(N,s(M))	→	isNat^#(M)	(63)
plus^#(N,s(M))	→	and^#(isNat(M),n__isNatKind(M))	(64)
plus^#(N,s(M))	→	and^#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))	(65)
plus^#(N,s(M))	→	U41^#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(66)
activate^#(n__0)	→	0^#	(67)
activate^#(n__plus(X1,X2))	→	activate^#(X2)	(68)
activate^#(n__plus(X1,X2))	→	activate^#(X1)	(69)
activate^#(n__plus(X1,X2))	→	plus^#(activate(X1),activate(X2))	(70)
activate^#(n__isNatKind(X))	→	isNatKind^#(X)	(71)
activate^#(n__s(X))	→	activate^#(X)	(72)
activate^#(n__s(X))	→	s^#(activate(X))	(73)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(74)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(75)
activate^#(n__isNat(X))	→	isNat^#(X)	(76)

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

isNatKind^#(n__s(V1))	→	isNatKind^#(activate(V1))	(59)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V2)	(54)
activate^#(n__plus(X1,X2))	→	activate^#(X2)	(68)
activate^#(n__plus(X1,X2))	→	activate^#(X1)	(69)
activate^#(n__plus(X1,X2))	→	plus^#(activate(X1),activate(X2))	(70)
plus^#(N,0)	→	isNat^#(N)	(60)
isNat^#(n__plus(V1,V2))	→	activate^#(V2)	(46)
activate^#(n__isNatKind(X))	→	isNatKind^#(X)	(71)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V1)	(55)
activate^#(n__s(X))	→	activate^#(X)	(72)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(74)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(75)
and^#(tt,X)	→	activate^#(X)	(45)
activate^#(n__isNat(X))	→	isNat^#(X)	(76)
isNat^#(n__plus(V1,V2))	→	activate^#(V1)	(47)
isNat^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(48)
isNatKind^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(56)
isNatKind^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(57)
isNatKind^#(n__s(V1))	→	activate^#(V1)	(58)
isNat^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(49)
isNat^#(n__plus(V1,V2))	→	U11^#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(50)
U11^#(tt,V1,V2)	→	activate^#(V2)	(30)
U11^#(tt,V1,V2)	→	activate^#(V1)	(31)
U11^#(tt,V1,V2)	→	isNat^#(activate(V1))	(32)
isNat^#(n__s(V1))	→	activate^#(V1)	(51)
isNat^#(n__s(V1))	→	isNatKind^#(activate(V1))	(52)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(53)
U21^#(tt,V1)	→	activate^#(V1)	(37)
U21^#(tt,V1)	→	isNat^#(activate(V1))	(38)
U11^#(tt,V1,V2)	→	U12^#(isNat(activate(V1)),activate(V2))	(33)
U12^#(tt,V2)	→	activate^#(V2)	(34)
U12^#(tt,V2)	→	isNat^#(activate(V2))	(35)
plus^#(N,0)	→	and^#(isNat(N),n__isNatKind(N))	(61)
plus^#(N,0)	→	U31^#(and(isNat(N),n__isNatKind(N)),N)	(62)
U31^#(tt,N)	→	activate^#(N)	(40)
plus^#(N,s(M))	→	isNat^#(M)	(63)
plus^#(N,s(M))	→	and^#(isNat(M),n__isNatKind(M))	(64)
plus^#(N,s(M))	→	and^#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))	(65)
plus^#(N,s(M))	→	U41^#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(66)
U41^#(tt,M,N)	→	activate^#(M)	(41)
U41^#(tt,M,N)	→	activate^#(N)	(42)
U41^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(43)

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[0]	=	0
[U12(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[isNatKind(x₁)]	=	0 · x₁ + -∞
[U31(x₁, x₂)]	=	0 · x₁ + 7 · x₂ + 3
[U11(x₁, x₂, x₃)]	=	0 · x₁ + 4 · x₂ + 0 · x₃ + 0
[isNat(x₁)]	=	0 · x₁ + 0
[n__isNatKind(x₁)]	=	0 · x₁ + -∞
[U11^#(x₁, x₂, x₃)]	=	4 · x₁ + 4 · x₂ + 6 · x₃ + -∞
[U21^#(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + 0
[n__isNat(x₁)]	=	0 · x₁ + 0
[and^#(x₁, x₂)]	=	0 · x₁ + 2 · x₂ + 0
[n__s(x₁)]	=	0 · x₁ + 0
[activate^#(x₁)]	=	0 · x₁ + 0
[U31^#(x₁, x₂)]	=	2 · x₁ + 1 · x₂ + 0
[U13(x₁)]	=	0 · x₁ + 0
[isNatKind^#(x₁)]	=	0 · x₁ + 0
[plus(x₁, x₂)]	=	7 · x₁ + 6 · x₂ + -∞
[tt]	=	0
[U22(x₁)]	=	-∞ · x₁ + 0
[n__and(x₁, x₂)]	=	2 · x₁ + 2 · x₂ + -∞
[plus^#(x₁, x₂)]	=	4 · x₁ + 4 · x₂ + 0
[n__plus(x₁, x₂)]	=	7 · x₁ + 6 · x₂ + -∞
[and(x₁, x₂)]	=	2 · x₁ + 2 · x₂ + -∞
[isNat^#(x₁)]	=	0 · x₁ + 0
[U12^#(x₁, x₂)]	=	4 · x₁ + 6 · x₂ + 0
[s(x₁)]	=	0 · x₁ + 0
[U41(x₁, x₂, x₃)]	=	-∞ · x₁ + 6 · x₂ + 7 · x₃ + 4
[n__0]	=	0
[U21(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + -∞
[U41^#(x₁, x₂, x₃)]	=	0 · x₁ + 4 · x₂ + 4 · x₃ + 0
[activate(x₁)]	=	0 · x₁ + -∞

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

plus^#(N,0)	→	isNat^#(N)	(60)
U11^#(tt,V1,V2)	→	activate^#(V2)	(30)
U11^#(tt,V1,V2)	→	activate^#(V1)	(31)
U11^#(tt,V1,V2)	→	isNat^#(activate(V1))	(32)
U12^#(tt,V2)	→	activate^#(V2)	(34)
U12^#(tt,V2)	→	isNat^#(activate(V2))	(35)
plus^#(N,0)	→	and^#(isNat(N),n__isNatKind(N))	(61)
U31^#(tt,N)	→	activate^#(N)	(40)
plus^#(N,s(M))	→	isNat^#(M)	(63)
plus^#(N,s(M))	→	and^#(isNat(M),n__isNatKind(M))	(64)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

isNatKind^#(n__s(V1))	→	isNatKind^#(activate(V1))	(59)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V2)	(54)
activate^#(n__plus(X1,X2))	→	activate^#(X2)	(68)
activate^#(n__plus(X1,X2))	→	activate^#(X1)	(69)
activate^#(n__plus(X1,X2))	→	plus^#(activate(X1),activate(X2))	(70)
plus^#(N,s(M))	→	and^#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))	(65)
and^#(tt,X)	→	activate^#(X)	(45)
activate^#(n__isNatKind(X))	→	isNatKind^#(X)	(71)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V1)	(55)
activate^#(n__s(X))	→	activate^#(X)	(72)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(74)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(75)
activate^#(n__isNat(X))	→	isNat^#(X)	(76)
isNat^#(n__plus(V1,V2))	→	activate^#(V2)	(46)
isNat^#(n__plus(V1,V2))	→	activate^#(V1)	(47)
isNat^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(48)
isNatKind^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(56)
isNatKind^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(57)
isNatKind^#(n__s(V1))	→	activate^#(V1)	(58)
isNat^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(49)
isNat^#(n__s(V1))	→	activate^#(V1)	(51)
isNat^#(n__s(V1))	→	isNatKind^#(activate(V1))	(52)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(53)
U21^#(tt,V1)	→	activate^#(V1)	(37)
U21^#(tt,V1)	→	isNat^#(activate(V1))	(38)
plus^#(N,s(M))	→	U41^#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(66)
U41^#(tt,M,N)	→	activate^#(M)	(41)
U41^#(tt,M,N)	→	activate^#(N)	(42)
U41^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(43)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[0]	=	0
[U12(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + 0
[isNatKind(x₁)]	=	0 · x₁ + 0
[U31(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + 0
[U11(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 0 · x₃ + 0
[isNat(x₁)]	=	0 · x₁ + 0
[n__isNatKind(x₁)]	=	0 · x₁ + 0
[U21^#(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[n__isNat(x₁)]	=	0 · x₁ + 0
[and^#(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[n__s(x₁)]	=	0 · x₁ + 0
[activate^#(x₁)]	=	0 · x₁ + 0
[U13(x₁)]	=	-∞ · x₁ + 0
[isNatKind^#(x₁)]	=	0 · x₁ + 0
[plus(x₁, x₂)]	=	4 · x₁ + 0 · x₂ + 4
[tt]	=	0
[U22(x₁)]	=	-∞ · x₁ + 0
[n__and(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[plus^#(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[n__plus(x₁, x₂)]	=	4 · x₁ + 0 · x₂ + 4
[and(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[isNat^#(x₁)]	=	0 · x₁ + -∞
[s(x₁)]	=	0 · x₁ + 0
[U41(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 4 · x₃ + 4
[n__0]	=	0
[U21(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 0 · x₃ + 0
[activate(x₁)]	=	0 · x₁ + 0

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

activate^#(n__plus(X1,X2))	→	activate^#(X1)	(69)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V1)	(55)
isNat^#(n__plus(V1,V2))	→	activate^#(V1)	(47)
isNat^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(48)
isNatKind^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(56)

could be deleted.

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[0]	=	0
[U12(x₁, x₂)]	=	0 · x₁ + 4 · x₂ + 0
[isNatKind(x₁)]	=	2 · x₁ + 0
[U31(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[U11(x₁, x₂, x₃)]	=	-∞ · x₁ + 6 · x₂ + 4 · x₃ + 3
[isNat(x₁)]	=	3 · x₁ + 0
[n__isNatKind(x₁)]	=	2 · x₁ + 0
[U21^#(x₁, x₂)]	=	0 · x₁ + 3 · x₂ + 0
[n__isNat(x₁)]	=	3 · x₁ + 0
[and^#(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[n__s(x₁)]	=	0 · x₁ + -∞
[activate^#(x₁)]	=	0 · x₁ + -∞
[U13(x₁)]	=	0 · x₁ + 0
[isNatKind^#(x₁)]	=	0 · x₁ + 0
[plus(x₁, x₂)]	=	3 · x₁ + 4 · x₂ + 0
[tt]	=	0
[U22(x₁)]	=	0 · x₁ + 0
[n__and(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[plus^#(x₁, x₂)]	=	3 · x₁ + 3 · x₂ + 0
[n__plus(x₁, x₂)]	=	3 · x₁ + 4 · x₂ + 0
[and(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[isNat^#(x₁)]	=	3 · x₁ + 0
[s(x₁)]	=	0 · x₁ + -∞
[U41(x₁, x₂, x₃)]	=	-∞ · x₁ + 4 · x₂ + 3 · x₃ + 0
[n__0]	=	0
[U21(x₁, x₂)]	=	0 · x₁ + 3 · x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	0 · x₁ + 3 · x₂ + 3 · x₃ + 0
[activate(x₁)]	=	0 · x₁ + -∞

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

isNatKind^#(n__plus(V1,V2))	→	activate^#(V2)	(54)
activate^#(n__plus(X1,X2))	→	activate^#(X2)	(68)
isNat^#(n__plus(V1,V2))	→	activate^#(V2)	(46)
isNat^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(49)
isNat^#(n__s(V1))	→	activate^#(V1)	(51)
U21^#(tt,V1)	→	activate^#(V1)	(37)
U41^#(tt,M,N)	→	activate^#(M)	(41)
U41^#(tt,M,N)	→	activate^#(N)	(42)

could be deleted.

1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[0]	=	1
[U12(x₁, x₂)]	=	-∞ · x₁ + 3 · x₂ + -∞
[isNatKind(x₁)]	=	3 · x₁ + -∞
[U31(x₁, x₂)]	=	-∞ · x₁ + 4 · x₂ + -∞
[U11(x₁, x₂, x₃)]	=	-∞ · x₁ + 0 · x₂ + 4 · x₃ + -∞
[isNat(x₁)]	=	3 · x₁ + -∞
[n__isNatKind(x₁)]	=	3 · x₁ + -∞
[U21^#(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + 0
[n__isNat(x₁)]	=	3 · x₁ + -∞
[and^#(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + -∞
[n__s(x₁)]	=	0 · x₁ + 0
[activate^#(x₁)]	=	0 · x₁ + -∞
[U13(x₁)]	=	0 · x₁ + -∞
[isNatKind^#(x₁)]	=	0 · x₁ + -∞
[plus(x₁, x₂)]	=	4 · x₁ + 4 · x₂ + 0
[tt]	=	0
[U22(x₁)]	=	-∞ · x₁ + 0
[n__and(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + -∞
[plus^#(x₁, x₂)]	=	4 · x₁ + 4 · x₂ + 0
[n__plus(x₁, x₂)]	=	4 · x₁ + 4 · x₂ + 0
[and(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + -∞
[isNat^#(x₁)]	=	0 · x₁ + -∞
[s(x₁)]	=	0 · x₁ + 0
[U41(x₁, x₂, x₃)]	=	-∞ · x₁ + 4 · x₂ + 4 · x₃ + 0
[n__0]	=	1
[U21(x₁, x₂)]	=	0 · x₁ + 2 · x₂ + -∞
[U41^#(x₁, x₂, x₃)]	=	-∞ · x₁ + 4 · x₂ + 4 · x₃ + 4
[activate(x₁)]	=	0 · x₁ + -∞

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

activate^#(n__isNatKind(X))	→	isNatKind^#(X)	(71)
activate^#(n__and(X1,X2))	→	activate^#(X1)	(74)
activate^#(n__and(X1,X2))	→	and^#(activate(X1),X2)	(75)
activate^#(n__isNat(X))	→	isNat^#(X)	(76)
isNatKind^#(n__plus(V1,V2))	→	and^#(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(57)

could be deleted.

1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

The 1^st component contains the pair

U21^#(tt,V1)	→	isNat^#(activate(V1))	(38)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(53)

1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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

[0]	=	0
[U12(x₁, x₂)]	=	0 · x₁ + 1/2 · x₂ + 1
[isNatKind(x₁)]	=	0 · x₁ + 0
[U31(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 1/2
[U11(x₁, x₂, x₃)]	=	0 · x₁ + 1/2 · x₂ + 1/2 · x₃ + 1
[isNat(x₁)]	=	2 · x₁ + 0
[n__isNatKind(x₁)]	=	0 · x₁ + 0
[U21^#(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 1/2
[n__isNat(x₁)]	=	2 · x₁ + 0
[n__s(x₁)]	=	1 · x₁ + 1/2
[U13(x₁)]	=	0 · x₁ + 1
[plus(x₁, x₂)]	=	2 · x₁ + 1 · x₂ + 1
[tt]	=	0
[U22(x₁)]	=	0 · x₁ + 1
[n__and(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[n__plus(x₁, x₂)]	=	2 · x₁ + 1 · x₂ + 1
[and(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[isNat^#(x₁)]	=	1 · x₁ + 0
[s(x₁)]	=	1 · x₁ + 1/2
[U41(x₁, x₂, x₃)]	=	0 · x₁ + 1 · x₂ + 2 · x₃ + 3/2
[n__0]	=	0
[U21(x₁, x₂)]	=	0 · x₁ + 3/2 · x₂ + 1
[activate(x₁)]	=	1 · x₁ + 0

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

U21^#(tt,V1)

→

isNat^#(activate(V1))

(38)

could be deleted.

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

isNatKind^#(n__s(V1))

→

isNatKind^#(activate(V1))

(59)

1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[0]	=	0
[U12(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[isNatKind(x₁)]	=	2 · x₁ + 0
[U31(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 1
[U11(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 0 · x₃ + 0
[isNat(x₁)]	=	0 · x₁ + 0
[n__isNatKind(x₁)]	=	2 · x₁ + 0
[n__isNat(x₁)]	=	0 · x₁ + 0
[n__s(x₁)]	=	1 · x₁ + 2
[U13(x₁)]	=	0 · x₁ + 0
[isNatKind^#(x₁)]	=	4 · x₁ + 4
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 1
[tt]	=	0
[U22(x₁)]	=	0 · x₁ + 0
[n__and(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 2
[n__plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 1
[and(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 2
[s(x₁)]	=	1 · x₁ + 2
[U41(x₁, x₂, x₃)]	=	0 · x₁ + 1 · x₂ + 1 · x₃ + 3
[n__0]	=	0
[U21(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[activate(x₁)]	=	1 · x₁ + 0

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

isNatKind^#(n__s(V1))

→

isNatKind^#(activate(V1))

(59)

could be deleted.

1.1.1.1.1.1.1.1.2.1 P is empty

There are no pairs anymore.

The 3^rd component contains the pair

activate^#(n__plus(X1,X2))	→	plus^#(activate(X1),activate(X2))	(70)
plus^#(N,s(M))	→	and^#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))	(65)
and^#(tt,X)	→	activate^#(X)	(45)
activate^#(n__s(X))	→	activate^#(X)	(72)
plus^#(N,s(M))	→	U41^#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(66)
U41^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(43)

1.1.1.1.1.1.1.1.3 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the arctic semiring over the integers

[0]	=	0
[U12(x₁, x₂)]	=	0 · x₁ + -∞ · x₂ + 0
[isNatKind(x₁)]	=	6 · x₁ + 0
[U31(x₁, x₂)]	=	2 · x₁ + 12 · x₂ + 3
[U11(x₁, x₂, x₃)]	=	2 · x₁ + 4 · x₂ + 6 · x₃ + 0
[isNat(x₁)]	=	-∞ · x₁ + 0
[n__isNatKind(x₁)]	=	-∞ · x₁ + 0
[n__isNat(x₁)]	=	-∞ · x₁ + 0
[and^#(x₁, x₂)]	=	-∞ · x₁ + 1 · x₂ + 0
[n__s(x₁)]	=	0 · x₁ + -∞
[activate^#(x₁)]	=	1 · x₁ + 0
[U13(x₁)]	=	2 · x₁ + 4
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 0
[tt]	=	2
[U22(x₁)]	=	2 · x₁ + 1
[n__and(x₁, x₂)]	=	0 · x₁ + 0 · x₂ + 0
[plus^#(x₁, x₂)]	=	-∞ · x₁ + -∞ · x₂ + 2
[n__plus(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 1
[and(x₁, x₂)]	=	-∞ · x₁ + 0 · x₂ + -∞
[s(x₁)]	=	2 · x₁ + 0
[U41(x₁, x₂, x₃)]	=	0 · x₁ + 0 · x₂ + 4 · x₃ + 0
[n__0]	=	1
[U21(x₁, x₂)]	=	0 · x₁ + 6 · x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	-∞ · x₁ + -∞ · x₂ + -∞ · x₃ + 2
[activate(x₁)]	=	0 · x₁ + 0

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

plus^#(N,s(M))

→

and^#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N)))

(65)

could be deleted.

1.1.1.1.1.1.1.1.3.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair
activate^#(n__s(X)) → activate^#(X) (72)

1.1.1.1.1.1.1.1.3.1.1 Size-Change Termination

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

activate^#(n__s(X)) → activate^#(X) (72)

1 > 1

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

The 2^nd component contains the pair

plus^#(N,s(M))	→	U41^#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(66)
U41^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(43)

1.1.1.1.1.1.1.1.3.1.2 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[0]	=	0
[U12(x₁, x₂)]	=	1 · x₁ + 0 · x₂ + 0
[isNatKind(x₁)]	=	0 · x₁ + 0
[U31(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[U11(x₁, x₂, x₃)]	=	0 · x₁ + 1 · x₂ + 0 · x₃ + 0
[isNat(x₁)]	=	1 · x₁ + 0
[n__isNatKind(x₁)]	=	0 · x₁ + 0
[n__isNat(x₁)]	=	1 · x₁ + 0
[n__s(x₁)]	=	1 · x₁ + 1
[U13(x₁)]	=	0 · x₁ + 0
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 0
[tt]	=	0
[U22(x₁)]	=	0 · x₁ + 0
[n__and(x₁, x₂)]	=	0 · x₁ + 2 · x₂ + 0
[plus^#(x₁, x₂)]	=	4 · x₁ + 1 · x₂ + 1
[n__plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂ + 0
[and(x₁, x₂)]	=	0 · x₁ + 2 · x₂ + 0
[s(x₁)]	=	1 · x₁ + 1
[U41(x₁, x₂, x₃)]	=	0 · x₁ + 1 · x₂ + 1 · x₃ + 1
[n__0]	=	0
[U21(x₁, x₂)]	=	0 · x₁ + 1 · x₂ + 0
[U41^#(x₁, x₂, x₃)]	=	0 · x₁ + 1 · x₂ + 4 · x₃ + 2
[activate(x₁)]	=	1 · x₁ + 0

together with the usable rules

U11(tt,V1,V2)	→	U12(isNat(activate(V1)),activate(V2))	(1)
U12(tt,V2)	→	U13(isNat(activate(V2)))	(2)
U13(tt)	→	tt	(3)
U21(tt,V1)	→	U22(isNat(activate(V1)))	(4)
U22(tt)	→	tt	(5)
U31(tt,N)	→	activate(N)	(6)
U41(tt,M,N)	→	s(plus(activate(N),activate(M)))	(7)
and(tt,X)	→	activate(X)	(8)
isNat(n__0)	→	tt	(9)
isNat(n__plus(V1,V2))	→	U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2))	(10)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(11)
isNatKind(n__0)	→	tt	(12)
isNatKind(n__plus(V1,V2))	→	and(isNatKind(activate(V1)),n__isNatKind(activate(V2)))	(13)
isNatKind(n__s(V1))	→	isNatKind(activate(V1))	(14)
plus(N,0)	→	U31(and(isNat(N),n__isNatKind(N)),N)	(15)
plus(N,s(M))	→	U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N)	(16)
0	→	n__0	(17)
plus(X1,X2)	→	n__plus(X1,X2)	(18)
isNatKind(X)	→	n__isNatKind(X)	(19)
s(X)	→	n__s(X)	(20)
and(X1,X2)	→	n__and(X1,X2)	(21)
isNat(X)	→	n__isNat(X)	(22)
activate(n__0)	→	0	(23)
activate(n__plus(X1,X2))	→	plus(activate(X1),activate(X2))	(24)
activate(n__isNatKind(X))	→	isNatKind(X)	(25)
activate(n__s(X))	→	s(activate(X))	(26)
activate(n__and(X1,X2))	→	and(activate(X1),X2)	(27)
activate(n__isNat(X))	→	isNat(X)	(28)
activate(X)	→	X	(29)

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

U41^#(tt,M,N)

→

plus^#(activate(N),activate(M))

(43)

could be deleted.

1.1.1.1.1.1.1.1.3.1.2.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/PEANO_complete_FR)

Property / Task

Answer / Result

Proof (by ttt2 @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

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 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.2 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.2.1 P is empty

1.1.1.1.1.1.1.1.3 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.3.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.3.1.1 Size-Change Termination

1.1.1.1.1.1.1.1.3.1.2 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.3.1.2.1 Dependency Graph Processor