Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExAppendixB_AEL03_GM)

The rewrite relation of the following TRS is considered.

a__from(X)	→	cons(mark(X),from(s(X)))	(1)
a__2ndspos(0,Z)	→	rnil	(2)
a__2ndspos(s(N),cons(X,Z))	→	a__2ndspos(s(mark(N)),cons2(X,mark(Z)))	(3)
a__2ndspos(s(N),cons2(X,cons(Y,Z)))	→	rcons(posrecip(mark(Y)),a__2ndsneg(mark(N),mark(Z)))	(4)
a__2ndsneg(0,Z)	→	rnil	(5)
a__2ndsneg(s(N),cons(X,Z))	→	a__2ndsneg(s(mark(N)),cons2(X,mark(Z)))	(6)
a__2ndsneg(s(N),cons2(X,cons(Y,Z)))	→	rcons(negrecip(mark(Y)),a__2ndspos(mark(N),mark(Z)))	(7)
a__pi(X)	→	a__2ndspos(mark(X),a__from(0))	(8)
a__plus(0,Y)	→	mark(Y)	(9)
a__plus(s(X),Y)	→	s(a__plus(mark(X),mark(Y)))	(10)
a__times(0,Y)	→	0	(11)
a__times(s(X),Y)	→	a__plus(mark(Y),a__times(mark(X),mark(Y)))	(12)
a__square(X)	→	a__times(mark(X),mark(X))	(13)
mark(from(X))	→	a__from(mark(X))	(14)
mark(2ndspos(X1,X2))	→	a__2ndspos(mark(X1),mark(X2))	(15)
mark(2ndsneg(X1,X2))	→	a__2ndsneg(mark(X1),mark(X2))	(16)
mark(pi(X))	→	a__pi(mark(X))	(17)
mark(plus(X1,X2))	→	a__plus(mark(X1),mark(X2))	(18)
mark(times(X1,X2))	→	a__times(mark(X1),mark(X2))	(19)
mark(square(X))	→	a__square(mark(X))	(20)
mark(0)	→	0	(21)
mark(s(X))	→	s(mark(X))	(22)
mark(posrecip(X))	→	posrecip(mark(X))	(23)
mark(negrecip(X))	→	negrecip(mark(X))	(24)
mark(nil)	→	nil	(25)
mark(cons(X1,X2))	→	cons(mark(X1),X2)	(26)
mark(cons2(X1,X2))	→	cons2(X1,mark(X2))	(27)
mark(rnil)	→	rnil	(28)
mark(rcons(X1,X2))	→	rcons(mark(X1),mark(X2))	(29)
a__from(X)	→	from(X)	(30)
a__2ndspos(X1,X2)	→	2ndspos(X1,X2)	(31)
a__2ndsneg(X1,X2)	→	2ndsneg(X1,X2)	(32)
a__pi(X)	→	pi(X)	(33)
a__plus(X1,X2)	→	plus(X1,X2)	(34)
a__times(X1,X2)	→	times(X1,X2)	(35)
a__square(X)	→	square(X)	(36)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

a__from^#(X)	→	mark^#(X)	(37)
a__2ndspos^#(s(N),cons(X,Z))	→	a__2ndspos^#(s(mark(N)),cons2(X,mark(Z)))	(38)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(N)	(39)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(Z)	(40)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(41)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndsneg^#(mark(N),mark(Z))	(42)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(43)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(44)
a__2ndsneg^#(s(N),cons(X,Z))	→	a__2ndsneg^#(s(mark(N)),cons2(X,mark(Z)))	(45)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(N)	(46)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(Z)	(47)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(48)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndspos^#(mark(N),mark(Z))	(49)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(50)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(51)
a__pi^#(X)	→	a__2ndspos^#(mark(X),a__from(0))	(52)
a__pi^#(X)	→	mark^#(X)	(53)
a__pi^#(X)	→	a__from^#(0)	(54)
a__plus^#(0,Y)	→	mark^#(Y)	(55)
a__plus^#(s(X),Y)	→	a__plus^#(mark(X),mark(Y))	(56)
a__plus^#(s(X),Y)	→	mark^#(X)	(57)
a__plus^#(s(X),Y)	→	mark^#(Y)	(58)
a__times^#(s(X),Y)	→	a__plus^#(mark(Y),a__times(mark(X),mark(Y)))	(59)
a__times^#(s(X),Y)	→	mark^#(Y)	(60)
a__times^#(s(X),Y)	→	a__times^#(mark(X),mark(Y))	(61)
a__times^#(s(X),Y)	→	mark^#(X)	(62)
a__square^#(X)	→	a__times^#(mark(X),mark(X))	(63)
a__square^#(X)	→	mark^#(X)	(64)
mark^#(from(X))	→	a__from^#(mark(X))	(65)
mark^#(from(X))	→	mark^#(X)	(66)
mark^#(2ndspos(X1,X2))	→	a__2ndspos^#(mark(X1),mark(X2))	(67)
mark^#(2ndspos(X1,X2))	→	mark^#(X1)	(68)
mark^#(2ndspos(X1,X2))	→	mark^#(X2)	(69)
mark^#(2ndsneg(X1,X2))	→	a__2ndsneg^#(mark(X1),mark(X2))	(70)
mark^#(2ndsneg(X1,X2))	→	mark^#(X1)	(71)
mark^#(2ndsneg(X1,X2))	→	mark^#(X2)	(72)
mark^#(pi(X))	→	a__pi^#(mark(X))	(73)
mark^#(pi(X))	→	mark^#(X)	(74)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(75)
mark^#(plus(X1,X2))	→	mark^#(X1)	(76)
mark^#(plus(X1,X2))	→	mark^#(X2)	(77)
mark^#(times(X1,X2))	→	a__times^#(mark(X1),mark(X2))	(78)
mark^#(times(X1,X2))	→	mark^#(X1)	(79)
mark^#(times(X1,X2))	→	mark^#(X2)	(80)
mark^#(square(X))	→	a__square^#(mark(X))	(81)
mark^#(square(X))	→	mark^#(X)	(82)
mark^#(s(X))	→	mark^#(X)	(83)
mark^#(posrecip(X))	→	mark^#(X)	(84)
mark^#(negrecip(X))	→	mark^#(X)	(85)
mark^#(cons(X1,X2))	→	mark^#(X1)	(86)
mark^#(cons2(X1,X2))	→	mark^#(X2)	(87)
mark^#(rcons(X1,X2))	→	mark^#(X1)	(88)
mark^#(rcons(X1,X2))	→	mark^#(X2)	(89)

1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[a__from^#(x₁)]

· x₁

[mark^#(x₁)]

· x₁

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[mark(x₁)]

-∞

· x₁

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[a__pi^#(x₁)]

· x₁

[a__from(x₁)]

· x₁

[0]

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[a__square^#(x₁)]

· x₁

[from(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[pi(x₁)]

· x₁

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[times(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

· x₁

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pairs

mark^#(square(X))	→	a__square^#(mark(X))	(81)
mark^#(square(X))	→	mark^#(X)	(82)

could be deleted.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(from(X))	→	a__from^#(mark(X))	(65)
a__from^#(X)	→	mark^#(X)	(37)
mark^#(from(X))	→	mark^#(X)	(66)
mark^#(2ndspos(X1,X2))	→	a__2ndspos^#(mark(X1),mark(X2))	(67)
a__2ndspos^#(s(N),cons(X,Z))	→	a__2ndspos^#(s(mark(N)),cons2(X,mark(Z)))	(38)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(41)
mark^#(2ndspos(X1,X2))	→	mark^#(X1)	(68)
mark^#(2ndspos(X1,X2))	→	mark^#(X2)	(69)
mark^#(2ndsneg(X1,X2))	→	a__2ndsneg^#(mark(X1),mark(X2))	(70)
a__2ndsneg^#(s(N),cons(X,Z))	→	a__2ndsneg^#(s(mark(N)),cons2(X,mark(Z)))	(45)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(48)
mark^#(2ndsneg(X1,X2))	→	mark^#(X1)	(71)
mark^#(2ndsneg(X1,X2))	→	mark^#(X2)	(72)
mark^#(pi(X))	→	a__pi^#(mark(X))	(73)
a__pi^#(X)	→	a__2ndspos^#(mark(X),a__from(0))	(52)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(N)	(39)
mark^#(pi(X))	→	mark^#(X)	(74)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(75)
a__plus^#(0,Y)	→	mark^#(Y)	(55)
mark^#(plus(X1,X2))	→	mark^#(X1)	(76)
mark^#(plus(X1,X2))	→	mark^#(X2)	(77)
mark^#(times(X1,X2))	→	a__times^#(mark(X1),mark(X2))	(78)
a__times^#(s(X),Y)	→	a__plus^#(mark(Y),a__times(mark(X),mark(Y)))	(59)
a__plus^#(s(X),Y)	→	a__plus^#(mark(X),mark(Y))	(56)
a__plus^#(s(X),Y)	→	mark^#(X)	(57)
mark^#(times(X1,X2))	→	mark^#(X1)	(79)
mark^#(times(X1,X2))	→	mark^#(X2)	(80)
mark^#(s(X))	→	mark^#(X)	(83)
mark^#(posrecip(X))	→	mark^#(X)	(84)
mark^#(negrecip(X))	→	mark^#(X)	(85)
mark^#(cons(X1,X2))	→	mark^#(X1)	(86)
mark^#(cons2(X1,X2))	→	mark^#(X2)	(87)
mark^#(rcons(X1,X2))	→	mark^#(X1)	(88)
mark^#(rcons(X1,X2))	→	mark^#(X2)	(89)
a__plus^#(s(X),Y)	→	mark^#(Y)	(58)
a__times^#(s(X),Y)	→	mark^#(Y)	(60)
a__times^#(s(X),Y)	→	a__times^#(mark(X),mark(Y))	(61)
a__times^#(s(X),Y)	→	mark^#(X)	(62)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(Z)	(40)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndsneg^#(mark(N),mark(Z))	(42)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(N)	(46)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(Z)	(47)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndspos^#(mark(N),mark(Z))	(49)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(43)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(44)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(50)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(51)
a__pi^#(X)	→	mark^#(X)	(53)
a__pi^#(X)	→	a__from^#(0)	(54)

1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[from(x₁)]

-∞

· x₁

[a__from^#(x₁)]

· x₁

[mark(x₁)]

-∞

· x₁

[2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi^#(x₁)]

· x₁

[a__from(x₁)]

-∞

· x₁

[0]

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(pi(X))

→

mark^#(X)

(74)

could be deleted.

1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[from(x₁)]

-∞

· x₁

[a__from^#(x₁)]

· x₁

[mark(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi^#(x₁)]

· x₁

[a__from(x₁)]

· x₁

[0]

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

-∞

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(pi(X))

→

a__pi^#(mark(X))

(73)

could be deleted.

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__from^#(X)	→	mark^#(X)	(37)
mark^#(from(X))	→	a__from^#(mark(X))	(65)
mark^#(from(X))	→	mark^#(X)	(66)
mark^#(2ndspos(X1,X2))	→	a__2ndspos^#(mark(X1),mark(X2))	(67)
a__2ndspos^#(s(N),cons(X,Z))	→	a__2ndspos^#(s(mark(N)),cons2(X,mark(Z)))	(38)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(41)
mark^#(2ndspos(X1,X2))	→	mark^#(X1)	(68)
mark^#(2ndspos(X1,X2))	→	mark^#(X2)	(69)
mark^#(2ndsneg(X1,X2))	→	a__2ndsneg^#(mark(X1),mark(X2))	(70)
a__2ndsneg^#(s(N),cons(X,Z))	→	a__2ndsneg^#(s(mark(N)),cons2(X,mark(Z)))	(45)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(48)
mark^#(2ndsneg(X1,X2))	→	mark^#(X1)	(71)
mark^#(2ndsneg(X1,X2))	→	mark^#(X2)	(72)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(75)
a__plus^#(0,Y)	→	mark^#(Y)	(55)
mark^#(plus(X1,X2))	→	mark^#(X1)	(76)
mark^#(plus(X1,X2))	→	mark^#(X2)	(77)
mark^#(times(X1,X2))	→	a__times^#(mark(X1),mark(X2))	(78)
a__times^#(s(X),Y)	→	a__plus^#(mark(Y),a__times(mark(X),mark(Y)))	(59)
a__plus^#(s(X),Y)	→	a__plus^#(mark(X),mark(Y))	(56)
a__plus^#(s(X),Y)	→	mark^#(X)	(57)
mark^#(times(X1,X2))	→	mark^#(X1)	(79)
mark^#(times(X1,X2))	→	mark^#(X2)	(80)
mark^#(s(X))	→	mark^#(X)	(83)
mark^#(posrecip(X))	→	mark^#(X)	(84)
mark^#(negrecip(X))	→	mark^#(X)	(85)
mark^#(cons(X1,X2))	→	mark^#(X1)	(86)
mark^#(cons2(X1,X2))	→	mark^#(X2)	(87)
mark^#(rcons(X1,X2))	→	mark^#(X1)	(88)
mark^#(rcons(X1,X2))	→	mark^#(X2)	(89)
a__plus^#(s(X),Y)	→	mark^#(Y)	(58)
a__times^#(s(X),Y)	→	mark^#(Y)	(60)
a__times^#(s(X),Y)	→	a__times^#(mark(X),mark(Y))	(61)
a__times^#(s(X),Y)	→	mark^#(X)	(62)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndspos^#(mark(N),mark(Z))	(49)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(N)	(39)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(Z)	(40)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndsneg^#(mark(N),mark(Z))	(42)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(N)	(46)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(Z)	(47)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(50)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(51)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(43)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(44)

1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[a__from^#(x₁)]

· x₁

[mark^#(x₁)]

· x₁

[from(x₁)]

· x₁

[mark(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

-∞

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(from(X))

→

mark^#(X)

(66)

could be deleted.

1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[a__from^#(x₁)]

· x₁

[mark^#(x₁)]

· x₁

[from(x₁)]

· x₁

[mark(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

-∞

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

a__from^#(X)

→

mark^#(X)

(37)

could be deleted.

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mark^#(2ndspos(X1,X2))	→	a__2ndspos^#(mark(X1),mark(X2))	(67)
a__2ndspos^#(s(N),cons(X,Z))	→	a__2ndspos^#(s(mark(N)),cons2(X,mark(Z)))	(38)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(41)
mark^#(2ndspos(X1,X2))	→	mark^#(X1)	(68)
mark^#(2ndspos(X1,X2))	→	mark^#(X2)	(69)
mark^#(2ndsneg(X1,X2))	→	a__2ndsneg^#(mark(X1),mark(X2))	(70)
a__2ndsneg^#(s(N),cons(X,Z))	→	a__2ndsneg^#(s(mark(N)),cons2(X,mark(Z)))	(45)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(48)
mark^#(2ndsneg(X1,X2))	→	mark^#(X1)	(71)
mark^#(2ndsneg(X1,X2))	→	mark^#(X2)	(72)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(75)
a__plus^#(0,Y)	→	mark^#(Y)	(55)
mark^#(plus(X1,X2))	→	mark^#(X1)	(76)
mark^#(plus(X1,X2))	→	mark^#(X2)	(77)
mark^#(times(X1,X2))	→	a__times^#(mark(X1),mark(X2))	(78)
a__times^#(s(X),Y)	→	a__plus^#(mark(Y),a__times(mark(X),mark(Y)))	(59)
a__plus^#(s(X),Y)	→	a__plus^#(mark(X),mark(Y))	(56)
a__plus^#(s(X),Y)	→	mark^#(X)	(57)
mark^#(times(X1,X2))	→	mark^#(X1)	(79)
mark^#(times(X1,X2))	→	mark^#(X2)	(80)
mark^#(s(X))	→	mark^#(X)	(83)
mark^#(posrecip(X))	→	mark^#(X)	(84)
mark^#(negrecip(X))	→	mark^#(X)	(85)
mark^#(cons(X1,X2))	→	mark^#(X1)	(86)
mark^#(cons2(X1,X2))	→	mark^#(X2)	(87)
mark^#(rcons(X1,X2))	→	mark^#(X1)	(88)
mark^#(rcons(X1,X2))	→	mark^#(X2)	(89)
a__plus^#(s(X),Y)	→	mark^#(Y)	(58)
a__times^#(s(X),Y)	→	mark^#(Y)	(60)
a__times^#(s(X),Y)	→	a__times^#(mark(X),mark(Y))	(61)
a__times^#(s(X),Y)	→	mark^#(X)	(62)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndspos^#(mark(N),mark(Z))	(49)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(N)	(39)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(Z)	(40)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndsneg^#(mark(N),mark(Z))	(42)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(N)	(46)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(Z)	(47)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(50)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(51)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(43)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(44)

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

-∞

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

-∞

· x₁ +

· x₂

[posrecip(x₁)]

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

a__times^#(s(X),Y)

→

mark^#(X)

(62)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

-∞

· x₁ +

· x₂

[posrecip(x₁)]

· x₁

[negrecip(x₁)]

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pairs

mark^#(2ndspos(X1,X2))	→	mark^#(X2)	(69)
mark^#(2ndsneg(X1,X2))	→	mark^#(X2)	(72)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

-∞

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pairs

mark^#(2ndspos(X1,X2))	→	mark^#(X1)	(68)
mark^#(2ndsneg(X1,X2))	→	mark^#(X1)	(71)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(times(X1,X2))

→

mark^#(X1)

(79)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[mark(x₁)]

-∞

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

a__plus^#(s(X),Y)

→

mark^#(X)

(57)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

-∞

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

· x₁ +

· x₂

[from(x₁)]

-∞

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(rcons(X1,X2))

→

mark^#(X1)

(88)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

-∞

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

-∞

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[nil]

[rnil]

the pair

mark^#(negrecip(X))

→

mark^#(X)

(85)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

-∞

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[negrecip(x₁)]

-∞

· x₁

[nil]

[rnil]

the pair

a__times^#(s(X),Y)

→

mark^#(Y)

(60)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

-∞

· x₁

[cons(x₁, x₂)]

-∞

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[posrecip(x₁)]

· x₁

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

-∞

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

· x₁ +

· x₂

[square(x₁)]

-∞

· x₁

[a__square(x₁)]

· x₁

[negrecip(x₁)]

-∞

· x₁

[nil]

[rnil]

the pair

mark^#(posrecip(X))

→

mark^#(X)

(84)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

[mark^#(x₁)]

· x₁

[2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndspos^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[mark(x₁)]

· x₁

[s(x₁)]

· x₁

[cons(x₁, x₂)]

· x₁ +

· x₂

[cons2(x₁, x₂)]

-∞

· x₁ +

· x₂

[2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[a__2ndsneg^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__plus^#(x₁, x₂)]

· x₁ +

· x₂

[0]

[times(x₁, x₂)]

· x₁ +

· x₂

[a__times^#(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__times(x₁, x₂)]

· x₁ +

· x₂

[rcons(x₁, x₂)]

-∞

· x₁ +

· x₂

[from(x₁)]

· x₁

[a__from(x₁)]

· x₁

[a__2ndspos(x₁, x₂)]

-∞

· x₁ +

· x₂

[a__2ndsneg(x₁, x₂)]

· x₁ +

· x₂

[pi(x₁)]

· x₁

[a__pi(x₁)]

· x₁

[a__plus(x₁, x₂)]

-∞

· x₁ +

· x₂

[square(x₁)]

· x₁

[a__square(x₁)]

· x₁

[posrecip(x₁)]

-∞

· x₁

[negrecip(x₁)]

-∞

· x₁

[nil]

[rnil]

the pairs

a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(41)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Y)	(48)
mark^#(cons(X1,X2))	→	mark^#(X1)	(86)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

Using the

prec(2ndspos)	=	6	stat(2ndspos)	=	mul
prec(a__2ndspos^#)	=	2	stat(a__2ndspos^#)	=	lex
prec(s)	=	1	stat(s)	=	mul
prec(2ndsneg)	=	6	stat(2ndsneg)	=	mul
prec(a__2ndsneg^#)	=	2	stat(a__2ndsneg^#)	=	lex
prec(plus)	=	7	stat(plus)	=	mul
prec(a__plus^#)	=	0	stat(a__plus^#)	=	lex
prec(0)	=	8	stat(0)	=	mul
prec(times)	=	9	stat(times)	=	mul
prec(a__times^#)	=	9	stat(a__times^#)	=	mul
prec(a__times)	=	9	stat(a__times)	=	mul
prec(rcons)	=	3	stat(rcons)	=	mul
prec(from)	=	10	stat(from)	=	lex
prec(a__from)	=	10	stat(a__from)	=	lex
prec(a__2ndspos)	=	6	stat(a__2ndspos)	=	mul
prec(a__2ndsneg)	=	6	stat(a__2ndsneg)	=	mul
prec(pi)	=	10	stat(pi)	=	lex
prec(a__pi)	=	10	stat(a__pi)	=	lex
prec(a__plus)	=	7	stat(a__plus)	=	mul
prec(square)	=	11	stat(square)	=	mul
prec(a__square)	=	11	stat(a__square)	=	mul
prec(posrecip)	=	6	stat(posrecip)	=	mul
prec(negrecip)	=	4	stat(negrecip)	=	mul
prec(nil)	=	12	stat(nil)	=	mul
prec(rnil)	=	5	stat(rnil)	=	mul

π(mark^#)	=	1
π(2ndspos)	=	[1,2]
π(a__2ndspos^#)	=	[2,1]
π(mark)	=	1
π(s)	=	[1]
π(cons)	=	2
π(cons2)	=	2
π(2ndsneg)	=	[1,2]
π(a__2ndsneg^#)	=	[2,1]
π(plus)	=	[1,2]
π(a__plus^#)	=	[2,1]
π(0)	=	[]
π(times)	=	[1,2]
π(a__times^#)	=	[1,2]
π(a__times)	=	[1,2]
π(rcons)	=	[1,2]
π(from)	=	[]
π(a__from)	=	[]
π(a__2ndspos)	=	[1,2]
π(a__2ndsneg)	=	[1,2]
π(pi)	=	[1]
π(a__pi)	=	[1]
π(a__plus)	=	[1,2]
π(square)	=	[1]
π(a__square)	=	[1]
π(posrecip)	=	[]
π(negrecip)	=	[]
π(nil)	=	[]
π(rnil)	=	[]

the pairs

mark^#(2ndspos(X1,X2))	→	a__2ndspos^#(mark(X1),mark(X2))	(67)
mark^#(2ndsneg(X1,X2))	→	a__2ndsneg^#(mark(X1),mark(X2))	(70)
mark^#(plus(X1,X2))	→	a__plus^#(mark(X1),mark(X2))	(75)
a__plus^#(0,Y)	→	mark^#(Y)	(55)
mark^#(plus(X1,X2))	→	mark^#(X1)	(76)
mark^#(plus(X1,X2))	→	mark^#(X2)	(77)
a__times^#(s(X),Y)	→	a__plus^#(mark(Y),a__times(mark(X),mark(Y)))	(59)
a__plus^#(s(X),Y)	→	a__plus^#(mark(X),mark(Y))	(56)
mark^#(times(X1,X2))	→	mark^#(X2)	(80)
mark^#(s(X))	→	mark^#(X)	(83)
mark^#(rcons(X1,X2))	→	mark^#(X2)	(89)
a__plus^#(s(X),Y)	→	mark^#(Y)	(58)
a__times^#(s(X),Y)	→	a__times^#(mark(X),mark(Y))	(61)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndspos^#(mark(N),mark(Z))	(49)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(N)	(39)
a__2ndspos^#(s(N),cons(X,Z))	→	mark^#(Z)	(40)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	a__2ndsneg^#(mark(N),mark(Z))	(42)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(N)	(46)
a__2ndsneg^#(s(N),cons(X,Z))	→	mark^#(Z)	(47)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(50)
a__2ndsneg^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(51)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(N)	(43)
a__2ndspos^#(s(N),cons2(X,cons(Y,Z)))	→	mark^#(Z)	(44)

could be deleted.

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair
mark^#(cons2(X1,X2)) → mark^#(X2) (87)

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

[cons2(x₁, x₂)] = 1 · x₁ + 1 · x₂

[mark^#(x₁)] = 1 · x₁

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

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

mark^#(cons2(X1,X2)) → mark^#(X2) (87)

1 > 1

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

[cons2(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[mark^#(x₁)]	=	1 · x₁

Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/ExAppendixB_AEL03_GM)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Reduction Pair Processor

1.1.1 Dependency Graph Processor

1.1.1.1 Reduction Pair Processor

1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination