Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/Frederiksen_Others/gexgcd)

The relative rewrite relation R/S is considered where R is the following TRS

m2(S(0),b,res,True)	→	False	(1)
m2(S(S(x)),b,res,True)	→	True	(2)
m2(0,b,res,True)	→	False	(3)
m3(S(0),b,res,t)	→	False	(4)
m3(S(S(x)),b,res,t)	→	True	(5)
m3(0,b,res,t)	→	False	(6)
l8(res,y,res',True,mtmp,t)	→	res	(7)
l5(x,y,res,tmp,mtmp,True)	→	0	(8)
help1(S(0))	→	False	(9)
help1(S(S(x)))	→	True	(10)
e4(a,b,res,False)	→	False	(11)
e4(a,b,res,True)	→	True	(12)
e2(a,b,res,False)	→	False	(13)
l15(x,y,res,tmp,False,t)	→	l16(x,y,gcd(y,0),tmp,False,t)	(14)
l15(x,y,res,tmp,True,t)	→	l16(x,y,gcd(y,S(0)),tmp,True,t)	(15)
l13(x,y,res,tmp,False,t)	→	l16(x,y,gcd(0,y),tmp,False,t)	(16)
l13(x,y,res,tmp,True,t)	→	l16(x,y,gcd(S(0),y),tmp,True,t)	(17)
m4(S(x'),S(x),res,t)	→	m5(S(x'),S(x),monus(x',x),t)	(18)
m2(a,b,res,False)	→	m4(a,b,res,False)	(19)
l8(x,y,res,False,mtmp,t)	→	l10(x,y,res,False,mtmp,t)	(20)
l5(x,y,res,tmp,mtmp,False)	→	l7(x,y,res,tmp,mtmp,False)	(21)
l2(x,y,res,tmp,mtmp,False)	→	l3(x,y,res,tmp,mtmp,False)	(22)
l2(x,y,res,tmp,mtmp,True)	→	res	(23)
l11(x,y,res,tmp,mtmp,False)	→	l14(x,y,res,tmp,mtmp,False)	(24)
l11(x,y,res,tmp,mtmp,True)	→	l12(x,y,res,tmp,mtmp,True)	(25)
help1(0)	→	False	(26)
e2(a,b,res,True)	→	e3(a,b,res,True)	(27)
bool2Nat(False)	→	0	(28)
bool2Nat(True)	→	S(0)	(29)
m1(a,x,res,t)	→	m2(a,x,res,False)	(30)
l9(res,y,res',tmp,mtmp,t)	→	res	(31)
l6(x,y,res,tmp,mtmp,t)	→	0	(32)
l4(x',x,res,tmp,mtmp,t)	→	l5(x',x,res,tmp,mtmp,False)	(33)
l1(x,y,res,tmp,mtmp,t)	→	l2(x,y,res,tmp,mtmp,False)	(34)
e7(a,b,res,t)	→	False	(35)
e6(a,b,res,t)	→	False	(36)
e5(a,b,res,t)	→	True	(37)
monus(a,b)	→	m1(a,b,False,False)	(38)
m5(a,b,res,t)	→	res	(39)
l7(x,y,res,tmp,mtmp,t)	→	l8(x,y,res,equal0(x,y),mtmp,t)	(40)
l3(x,y,res,tmp,mtmp,t)	→	l4(x,y,0,tmp,mtmp,t)	(41)
l16(x,y,res,tmp,mtmp,t)	→	res	(42)
l14(x,y,res,tmp,mtmp,t)	→	l15(x,y,res,tmp,monus(x,y),t)	(43)
l12(x,y,res,tmp,mtmp,t)	→	l13(x,y,res,tmp,monus(x,y),t)	(44)
l10(x,y,res,tmp,mtmp,t)	→	l11(x,y,res,tmp,mtmp,<(x,y))	(45)
gcd(x,y)	→	l1(x,y,0,False,False,False)	(46)
equal0(a,b)	→	e1(a,b,False,False)	(47)
e8(a,b,res,t)	→	res	(48)
e3(a,b,res,t)	→	e4(a,b,res,<(b,a))	(49)
e1(a,b,res,t)	→	e2(a,b,res,<(a,b))	(50)

and S is the following TRS.

<(S(x),S(y))	→	<(x,y)	(51)
<(0,S(y))	→	True	(52)
<(x,0)	→	False	(53)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

m2^#(S(0),z0,z1,True)

→

(55)

originates from

m2(S(0),z0,z1,True)

→

False

(54)

m2^#(S(S(z0)),z1,z2,True)

→

(57)

originates from

m2(S(S(z0)),z1,z2,True)

→

True

(56)

m2^#(0,z0,z1,True)

→

(59)

originates from

m2(0,z0,z1,True)

→

False

(58)

m2^#(z0,z1,z2,False)

→

c6(m4^#(z0,z1,z2,False))

(61)

originates from

m2(z0,z1,z2,False)

→

m4(z0,z1,z2,False)

(60)

m3^#(S(0),z0,z1,z2)

→

(63)

originates from

m3(S(0),z0,z1,z2)

→

False

(62)

m3^#(S(S(z0)),z1,z2,z3)

→

(65)

originates from

m3(S(S(z0)),z1,z2,z3)

→

True

(64)

m3^#(0,z0,z1,z2)

→

(67)

originates from

m3(0,z0,z1,z2)

→

False

(66)

l8^#(z0,z1,z2,True,z3,z4)

→

c10

(69)

originates from

l8(z0,z1,z2,True,z3,z4)

→

(68)

l8^#(z0,z1,z2,False,z3,z4)

→

c11(l10^#(z0,z1,z2,False,z3,z4))

(71)

originates from

l8(z0,z1,z2,False,z3,z4)

→

l10(z0,z1,z2,False,z3,z4)

(70)

l5^#(z0,z1,z2,z3,z4,True)

→

c12

(73)

originates from

l5(z0,z1,z2,z3,z4,True)

→

(72)

l5^#(z0,z1,z2,z3,z4,False)

→

c13(l7^#(z0,z1,z2,z3,z4,False))

(75)

originates from

l5(z0,z1,z2,z3,z4,False)

→

l7(z0,z1,z2,z3,z4,False)

(74)

help1^#(S(0))

→

c14

(76)

originates from

help1(S(0))

→

False

(9)

help1^#(S(S(z0)))

→

c15

(78)

originates from

help1(S(S(z0)))

→

True

(77)

help1^#(0)

→

c16

(79)

originates from

help1(0)

→

False

(26)

e4^#(z0,z1,z2,False)

→

c17

(81)

originates from

e4(z0,z1,z2,False)

→

False

(80)

e4^#(z0,z1,z2,True)

→

c18

(83)

originates from

e4(z0,z1,z2,True)

→

True

(82)

e2^#(z0,z1,z2,False)

→

c19

(85)

originates from

e2(z0,z1,z2,False)

→

False

(84)

e2^#(z0,z1,z2,True)

→

c20(e3^#(z0,z1,z2,True))

(87)

originates from

e2(z0,z1,z2,True)

→

e3(z0,z1,z2,True)

(86)

l15^#(z0,z1,z2,z3,False,z4)

→

c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))

(89)

originates from

l15(z0,z1,z2,z3,False,z4)

→

l16(z0,z1,gcd(z1,0),z3,False,z4)

(88)

l15^#(z0,z1,z2,z3,True,z4)

→

c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))

(91)

originates from

l15(z0,z1,z2,z3,True,z4)

→

l16(z0,z1,gcd(z1,S(0)),z3,True,z4)

(90)

l13^#(z0,z1,z2,z3,False,z4)

→

c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))

(93)

originates from

l13(z0,z1,z2,z3,False,z4)

→

l16(z0,z1,gcd(0,z1),z3,False,z4)

(92)

l13^#(z0,z1,z2,z3,True,z4)

→

c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))

(95)

originates from

l13(z0,z1,z2,z3,True,z4)

→

l16(z0,z1,gcd(S(0),z1),z3,True,z4)

(94)

m4^#(S(z0),S(z1),z2,z3)

→

c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))

(97)

originates from

m4(S(z0),S(z1),z2,z3)

→

m5(S(z0),S(z1),monus(z0,z1),z3)

(96)

l2^#(z0,z1,z2,z3,z4,False)

→

c26(l3^#(z0,z1,z2,z3,z4,False))

(99)

originates from

l2(z0,z1,z2,z3,z4,False)

→

l3(z0,z1,z2,z3,z4,False)

(98)

l2^#(z0,z1,z2,z3,z4,True)

→

c27

(101)

originates from

l2(z0,z1,z2,z3,z4,True)

→

(100)

l11^#(z0,z1,z2,z3,z4,False)

→

c28(l14^#(z0,z1,z2,z3,z4,False))

(103)

originates from

l11(z0,z1,z2,z3,z4,False)

→

l14(z0,z1,z2,z3,z4,False)

(102)

l11^#(z0,z1,z2,z3,z4,True)

→

c29(l12^#(z0,z1,z2,z3,z4,True))

(105)

originates from

l11(z0,z1,z2,z3,z4,True)

→

l12(z0,z1,z2,z3,z4,True)

(104)

bool2Nat^#(False)

→

c30

(106)

originates from

bool2Nat(False)

→

(28)

bool2Nat^#(True)

→

c31

(107)

originates from

bool2Nat(True)

→

S(0)

(29)

m1^#(z0,z1,z2,z3)

→

c32(m2^#(z0,z1,z2,False))

(109)

originates from

m1(z0,z1,z2,z3)

→

m2(z0,z1,z2,False)

(108)

l9^#(z0,z1,z2,z3,z4,z5)

→

c33

(111)

originates from

l9(z0,z1,z2,z3,z4,z5)

→

(110)

l6^#(z0,z1,z2,z3,z4,z5)

→

c34

(113)

originates from

l6(z0,z1,z2,z3,z4,z5)

→

(112)

l4^#(z0,z1,z2,z3,z4,z5)

→

c35(l5^#(z0,z1,z2,z3,z4,False))

(115)

originates from

l4(z0,z1,z2,z3,z4,z5)

→

l5(z0,z1,z2,z3,z4,False)

(114)

l1^#(z0,z1,z2,z3,z4,z5)

→

c36(l2^#(z0,z1,z2,z3,z4,False))

(117)

originates from

l1(z0,z1,z2,z3,z4,z5)

→

l2(z0,z1,z2,z3,z4,False)

(116)

e7^#(z0,z1,z2,z3)

→

c37

(119)

originates from

e7(z0,z1,z2,z3)

→

False

(118)

e6^#(z0,z1,z2,z3)

→

c38

(121)

originates from

e6(z0,z1,z2,z3)

→

False

(120)

e5^#(z0,z1,z2,z3)

→

c39

(123)

originates from

e5(z0,z1,z2,z3)

→

True

(122)

monus^#(z0,z1)

→

c40(m1^#(z0,z1,False,False))

(125)

originates from

monus(z0,z1)

→

m1(z0,z1,False,False)

(124)

m5^#(z0,z1,z2,z3)

→

c41

(127)

originates from

m5(z0,z1,z2,z3)

→

(126)

l7^#(z0,z1,z2,z3,z4,z5)

→

c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))

(129)

originates from

l7(z0,z1,z2,z3,z4,z5)

→

l8(z0,z1,z2,equal0(z0,z1),z4,z5)

(128)

l3^#(z0,z1,z2,z3,z4,z5)

→

c43(l4^#(z0,z1,0,z3,z4,z5))

(131)

originates from

l3(z0,z1,z2,z3,z4,z5)

→

l4(z0,z1,0,z3,z4,z5)

(130)

l16^#(z0,z1,z2,z3,z4,z5)

→

c44

(133)

originates from

l16(z0,z1,z2,z3,z4,z5)

→

(132)

l14^#(z0,z1,z2,z3,z4,z5)

→

c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))

(135)

originates from

l14(z0,z1,z2,z3,z4,z5)

→

l15(z0,z1,z2,z3,monus(z0,z1),z5)

(134)

l12^#(z0,z1,z2,z3,z4,z5)

→

c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))

(137)

originates from

l12(z0,z1,z2,z3,z4,z5)

→

l13(z0,z1,z2,z3,monus(z0,z1),z5)

(136)

l10^#(z0,z1,z2,z3,z4,z5)

→

c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))

(139)

originates from

l10(z0,z1,z2,z3,z4,z5)

→

l11(z0,z1,z2,z3,z4,<(z0,z1))

(138)

gcd^#(z0,z1)

→

c48(l1^#(z0,z1,0,False,False,False))

(141)

originates from

gcd(z0,z1)

→

l1(z0,z1,0,False,False,False)

(140)

equal0^#(z0,z1)

→

c49(e1^#(z0,z1,False,False))

(143)

originates from

equal0(z0,z1)

→

e1(z0,z1,False,False)

(142)

e8^#(z0,z1,z2,z3)

→

c50

(145)

originates from

e8(z0,z1,z2,z3)

→

(144)

e3^#(z0,z1,z2,z3)

→

c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))

(147)

originates from

e3(z0,z1,z2,z3)

→

e4(z0,z1,z2,<(z1,z0))

(146)

e1^#(z0,z1,z2,z3)

→

c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))

(149)

originates from

e1(z0,z1,z2,z3)

→

e2(z0,z1,z2,<(z0,z1))

(148)

<^#(S(z0),S(z1))

→

c(<^#(z0,z1))

(151)

originates from

<(S(z0),S(z1))

→

<(z0,z1)

(150)

<^#(0,S(z0))

→

(153)

originates from

<(0,S(z0))

→

True

(152)

<^#(z0,0)

→

(155)

originates from

<(z0,0)

→

False

(154)

Moreover, we add the following terms to the innermost strategy.

<^#(S(z0),S(z1))

<^#(0,S(z0))

<^#(z0,0)

m2^#(S(0),z0,z1,True)

m2^#(S(S(z0)),z1,z2,True)

m2^#(0,z0,z1,True)

m2^#(z0,z1,z2,False)

m3^#(S(0),z0,z1,z2)

m3^#(S(S(z0)),z1,z2,z3)

m3^#(0,z0,z1,z2)

l8^#(z0,z1,z2,True,z3,z4)

l8^#(z0,z1,z2,False,z3,z4)

l5^#(z0,z1,z2,z3,z4,True)

l5^#(z0,z1,z2,z3,z4,False)

help1^#(S(0))

help1^#(S(S(z0)))

help1^#(0)

e4^#(z0,z1,z2,False)

e4^#(z0,z1,z2,True)

e2^#(z0,z1,z2,False)

e2^#(z0,z1,z2,True)

l15^#(z0,z1,z2,z3,False,z4)

l15^#(z0,z1,z2,z3,True,z4)

l13^#(z0,z1,z2,z3,False,z4)

l13^#(z0,z1,z2,z3,True,z4)

m4^#(S(z0),S(z1),z2,z3)

l2^#(z0,z1,z2,z3,z4,False)

l2^#(z0,z1,z2,z3,z4,True)

l11^#(z0,z1,z2,z3,z4,False)

l11^#(z0,z1,z2,z3,z4,True)

bool2Nat^#(False)

bool2Nat^#(True)

m1^#(z0,z1,z2,z3)

l9^#(z0,z1,z2,z3,z4,z5)

l6^#(z0,z1,z2,z3,z4,z5)

l4^#(z0,z1,z2,z3,z4,z5)

l1^#(z0,z1,z2,z3,z4,z5)

e7^#(z0,z1,z2,z3)

e6^#(z0,z1,z2,z3)

e5^#(z0,z1,z2,z3)

monus^#(z0,z1)

m5^#(z0,z1,z2,z3)

l7^#(z0,z1,z2,z3,z4,z5)

l3^#(z0,z1,z2,z3,z4,z5)

l16^#(z0,z1,z2,z3,z4,z5)

l14^#(z0,z1,z2,z3,z4,z5)

l12^#(z0,z1,z2,z3,z4,z5)

l10^#(z0,z1,z2,z3,z4,z5)

gcd^#(z0,z1)

equal0^#(z0,z1)

e8^#(z0,z1,z2,z3)

e3^#(z0,z1,z2,z3)

e1^#(z0,z1,z2,z3)

1.1 Usable Rules

We remove the following rules since they are not usable.

m2(S(0),z0,z1,True)	→	False	(54)
m2(S(S(z0)),z1,z2,True)	→	True	(56)
m2(0,z0,z1,True)	→	False	(58)
m3(S(0),z0,z1,z2)	→	False	(62)
m3(S(S(z0)),z1,z2,z3)	→	True	(64)
m3(0,z0,z1,z2)	→	False	(66)
l5(z0,z1,z2,z3,z4,True)	→	0	(72)
help1(S(0))	→	False	(9)
help1(S(S(z0)))	→	True	(77)
help1(0)	→	False	(26)
l2(z0,z1,z2,z3,z4,True)	→	z2	(100)
bool2Nat(False)	→	0	(28)
bool2Nat(True)	→	S(0)	(29)
l9(z0,z1,z2,z3,z4,z5)	→	z0	(110)
l6(z0,z1,z2,z3,z4,z5)	→	0	(112)
e7(z0,z1,z2,z3)	→	False	(118)
e6(z0,z1,z2,z3)	→	False	(120)
e5(z0,z1,z2,z3)	→	True	(122)
e8(z0,z1,z2,z3)	→	z2	(144)

1.1.1 Rule Shifting

The rules

e6^#(z0,z1,z2,z3)	→	c38	(121)
e8^#(z0,z1,z2,z3)	→	c50	(145)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 · x₁ + 0 + 2 · x₂
[l1(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 3 · x₃
[l2(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 3 · x₃
[l3(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 2 · x₃
[l4(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 3 · x₃
[l5(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 2 · x₃
[l7(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 2 · x₃
[l8(x₁,...,x₆)]	=	1 · x₁ + 0 + 2 · x₂ + 1 · x₃
[equal0(x₁, x₂)]	=	2
[e1(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e2(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e4(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[l10(x₁,...,x₆)]	=	2 · x₂ + 0 + 1 · x₃
[l11(x₁,...,x₆)]	=	2 · x₂ + 0 + 3 · x₆
[l14(x₁,...,x₆)]	=	2 + 1 · x₂ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₂ + 3 · x₅
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[m2(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[m4(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[m5(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₄
[l16(x₁,...,x₆)]	=	1 · x₃ + 0
[l12(x₁,...,x₆)]	=	1 + 2 · x₂
[l13(x₁,...,x₆)]	=	1 + 2 · x₂ + 1 · x₅
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	2 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	3 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	0
[l13^#(x₁,...,x₆)]	=	0
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	2 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	3 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	3 · x₃ + 0
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	1
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	0
[l7^#(x₁,...,x₆)]	=	2 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	2 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	0
[l12^#(x₁,...,x₆)]	=	0
[l10^#(x₁,...,x₆)]	=	2 · x₃ + 0
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	1
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	2
[0]	=	0
[True]	=	3
[False]	=	3

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)

1.1.1.1 Rule Shifting

The rules

m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m2(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m4(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m3^#(x₁,...,x₄)]	=	1 + 1 · x₁
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	1 + 1 · x₁
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[m4^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	1 + 1 · x₁
[m1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l9^#(x₁,...,x₆)]	=	1
[l6^#(x₁,...,x₆)]	=	1
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0 + 1 · x₅ + 1 · x₆
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	0
[0]	=	0
[True]	=	0
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1 Rule Shifting

The rules

l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₃
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1 Rule Shifting

The rules

e7^#(z0,z1,z2,z3)	→	c37	(119)
e5^#(z0,z1,z2,z3)	→	c39	(123)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	1
[m1(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m2(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m4(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[m4^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	1
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	1
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0 + 1 · x₅ + 1 · x₆
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	0
[0]	=	0
[True]	=	0
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)

1.1.1.1.1.1.1 Rule Shifting

The rules

m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	1
[m1(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m2(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m4(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄
[m4^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

l15^#(z0,z1,z2,z3,False,z4)

→

c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))

(89)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₄ + 0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	0
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l13^#(z0,z1,z2,z3,False,z4)

→

c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))

(93)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₂ + 0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₄ + 0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	0
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

gcd^#(z0,z1)

→

c48(l1^#(z0,z1,0,False,False,False))

(141)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l15^#(z0,z1,z2,z3,True,z4)

→

c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))

(91)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m2(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m4(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄
[m4^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l2^#(z0,z1,z2,z3,z4,False)

→

c26(l3^#(z0,z1,z2,z3,z4,False))

(99)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l13^#(z0,z1,z2,z3,True,z4)

→

c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))

(95)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m2(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m4(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l16^#(z0,z1,z2,z3,z4,z5)

→

c44

(133)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 · x₃ + 0
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	1
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	0
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

m5^#(z0,z1,z2,z3)

→

c41

(127)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m2(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m4(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	1 · x₁ + 0
[m5^#(x₁,...,x₄)]	=	1 + 1 · x₃ + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0
[l14^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l10^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	1 + 1 · x₁
[0]	=	0
[True]	=	1
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l1^#(z0,z1,z2,z3,z4,z5)

→

c36(l2^#(z0,z1,z2,z3,z4,False))

(117)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1
[e2^#(x₁,...,x₄)]	=	1
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	1
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1
[e1^#(x₁,...,x₄)]	=	1
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l5^#(z0,z1,z2,z3,z4,False)

→

c13(l7^#(z0,z1,z2,z3,z4,False))

(75)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l3^#(z0,z1,z2,z3,z4,z5)

→

c43(l4^#(z0,z1,0,z3,z4,z5))

(131)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l4^#(z0,z1,z2,z3,z4,z5)

→

c35(l5^#(z0,z1,z2,z3,z4,False))

(115)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l7^#(z0,z1,z2,z3,z4,z5)

→

c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))

(129)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

equal0^#(z0,z1)

→

c49(e1^#(z0,z1,False,False))

(143)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	1
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l11^#(z0,z1,z2,z3,z4,False)

→

c28(l14^#(z0,z1,z2,z3,z4,False))

(103)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₃
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l12^#(z0,z1,z2,z3,z4,z5)

→

c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))

(137)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₃
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

m4^#(S(z0),S(z1),z2,z3)

→

c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))

(97)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m2(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m4(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[m5(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[l5^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[l15^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[l2^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l1^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	1 · x₁ + 0
[m5^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[l7^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l16^#(x₁,...,x₆)]	=	1 · x₄ + 0
[l14^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l10^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[gcd^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[e1^#(x₁,...,x₄)]	=	1 · x₃ + 0 + 1 · x₄
[S(x₁)]	=	1 + 1 · x₁
[0]	=	0
[True]	=	1
[False]	=	0

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

e1^#(z0,z1,z2,z3)

→

c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))

(149)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 + 1 · x₁
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	1
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	1
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l8^#(z0,z1,z2,False,z3,z4)

→

c11(l10^#(z0,z1,z2,False,z3,z4))

(71)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 · x₁ + 0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

l10^#(z0,z1,z2,z3,z4,z5)

→

c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))

(139)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₃
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

e3^#(z0,z1,z2,z3)

→

c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))

(147)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 · x₁ + 0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	1
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	1
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	1
[e1^#(x₁,...,x₄)]	=	1
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

e2^#(z0,z1,z2,True)

→

c20(e3^#(z0,z1,z2,True))

(87)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[l1(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l2(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l3(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l7(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l8(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₅ + 1 · x₆
[equal0(x₁, x₂)]	=	1 · x₁ + 0
[e1(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[<(x₁, x₂)]	=	1 · x₁ + 0
[e3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[e4(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l10(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l11(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[l14(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l15(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l16(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l12(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[l13(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l5^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	1
[l15^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₅
[m4^#(x₁,...,x₄)]	=	0
[l2^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 · x₃ + 0
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	0
[m5^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l12^#(x₁,...,x₆)]	=	1 · x₃ + 0
[l10^#(x₁,...,x₆)]	=	1 · x₃ + 0
[gcd^#(x₁, x₂)]	=	1
[equal0^#(x₁, x₂)]	=	1
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	1
[S(x₁)]	=	0
[0]	=	0
[True]	=	1
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

m1^#(z0,z1,z2,z3)

→

c32(m2^#(z0,z1,z2,False))

(109)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	3
[l1(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l2(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l3(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l4(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l5(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l7(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l8(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₅ + 3 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e2(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e4(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[l10(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l11(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[l14(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l15(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₃ + 0
[l16(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l12(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l13(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₁ + 0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₃
[l5^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 2 · x₃
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	2 · x₂ + 0 + 2 · x₃ + 3 · x₅
[l13^#(x₁,...,x₆)]	=	2 · x₂ + 0 + 2 · x₃ + 3 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₁ + 0
[l2^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 3 · x₃
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 + 1 · x₁
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 3 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 3 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	1 + 1 · x₁
[m5^#(x₁,...,x₄)]	=	0
[l7^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 2 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 2 · x₁ + 2 · x₂ + 3 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₃
[l12^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₃
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₃
[gcd^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	1 + 1 · x₁
[0]	=	0
[True]	=	1
[False]	=	2

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

monus^#(z0,z1)

→

c40(m1^#(z0,z1,False,False))

(125)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	3
[l1(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l2(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l3(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l4(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l5(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l7(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l8(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₅ + 3 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e2(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e4(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[l10(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l11(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[l14(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l15(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₃ + 0
[l16(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l12(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l13(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 · x₁ + 0
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l5^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 2 · x₃ + 1 · x₆
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 2 · x₃ + 3 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 2 · x₃ + 3 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₁ + 0
[l2^#(x₁,...,x₆)]	=	1 · x₁ + 0 + 1 · x₂ + 3 · x₃ + 1 · x₆
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 · x₁ + 0
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 3 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 3 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	1 + 1 · x₁
[m5^#(x₁,...,x₄)]	=	0
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[l16^#(x₁,...,x₆)]	=	2 · x₅ + 0
[l14^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l12^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[gcd^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	2 + 1 · x₁
[0]	=	0
[True]	=	3
[False]	=	1

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

m2^#(z0,z1,z2,False)

→

c6(m4^#(z0,z1,z2,False))

(61)

are strictly oriented by the following linear polynomial interpretation over the naturals

[c(x₁)]	=	1 · x₁ + 0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6(x₁)]	=	1 · x₁ + 0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11(x₁)]	=	1 · x₁ + 0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20(x₁)]	=	1 · x₁ + 0
[c21(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c22(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c23(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c24(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c25(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c26(x₁)]	=	1 · x₁ + 0
[c27]	=	0
[c28(x₁)]	=	1 · x₁ + 0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32(x₁)]	=	1 · x₁ + 0
[c33]	=	0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁)]	=	1 · x₁ + 0
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40(x₁)]	=	1 · x₁ + 0
[c41]	=	0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c47(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c48(x₁)]	=	1 · x₁ + 0
[c49(x₁)]	=	1 · x₁ + 0
[c50]	=	0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[gcd(x₁, x₂)]	=	3
[l1(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l2(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l3(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l4(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l5(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l7(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l8(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₅ + 3 · x₆
[equal0(x₁, x₂)]	=	0
[e1(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e2(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[<(x₁, x₂)]	=	0
[e3(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄
[e4(x₁,...,x₄)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃
[l10(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l11(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[l14(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l15(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₆
[monus(x₁, x₂)]	=	0
[m1(x₁,...,x₄)]	=	0
[m2(x₁,...,x₄)]	=	0
[m4(x₁,...,x₄)]	=	0
[m5(x₁,...,x₄)]	=	1 · x₃ + 0
[l16(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l12(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[l13(x₁,...,x₆)]	=	3 + 3 · x₁ + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₆
[<^#(x₁, x₂)]	=	0
[m2^#(x₁,...,x₄)]	=	1 + 1 · x₁
[m3^#(x₁,...,x₄)]	=	0
[l8^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l5^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 3 · x₃
[help1^#(x₁)]	=	0
[e4^#(x₁,...,x₄)]	=	0
[e2^#(x₁,...,x₄)]	=	0
[l15^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 2 · x₃ + 2 · x₅
[l13^#(x₁,...,x₆)]	=	1 · x₂ + 0 + 2 · x₃ + 2 · x₅
[m4^#(x₁,...,x₄)]	=	1 · x₁ + 0
[l2^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l11^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[bool2Nat^#(x₁)]	=	0
[m1^#(x₁,...,x₄)]	=	1 + 1 · x₁
[l9^#(x₁,...,x₆)]	=	0
[l6^#(x₁,...,x₆)]	=	0
[l4^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 3 · x₃
[l1^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 3 · x₃
[e7^#(x₁,...,x₄)]	=	0
[e6^#(x₁,...,x₄)]	=	0
[e5^#(x₁,...,x₄)]	=	0
[monus^#(x₁, x₂)]	=	1 + 1 · x₁
[m5^#(x₁,...,x₄)]	=	0
[l7^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l3^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l16^#(x₁,...,x₆)]	=	0
[l14^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l12^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[l10^#(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₃
[gcd^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[equal0^#(x₁, x₂)]	=	0
[e8^#(x₁,...,x₄)]	=	0
[e3^#(x₁,...,x₄)]	=	0
[e1^#(x₁,...,x₄)]	=	0
[S(x₁)]	=	2 + 1 · x₁
[0]	=	0
[True]	=	2
[False]	=	2

which has the intended complexity. Here, only the following usable rules have been considered:

<^#(S(z0),S(z1))	→	c(<^#(z0,z1))	(151)
<^#(0,S(z0))	→	c1	(153)
<^#(z0,0)	→	c2	(155)
m2^#(S(0),z0,z1,True)	→	c3	(55)
m2^#(S(S(z0)),z1,z2,True)	→	c4	(57)
m2^#(0,z0,z1,True)	→	c5	(59)
m2^#(z0,z1,z2,False)	→	c6(m4^#(z0,z1,z2,False))	(61)
m3^#(S(0),z0,z1,z2)	→	c7	(63)
m3^#(S(S(z0)),z1,z2,z3)	→	c8	(65)
m3^#(0,z0,z1,z2)	→	c9	(67)
l8^#(z0,z1,z2,True,z3,z4)	→	c10	(69)
l8^#(z0,z1,z2,False,z3,z4)	→	c11(l10^#(z0,z1,z2,False,z3,z4))	(71)
l5^#(z0,z1,z2,z3,z4,True)	→	c12	(73)
l5^#(z0,z1,z2,z3,z4,False)	→	c13(l7^#(z0,z1,z2,z3,z4,False))	(75)
help1^#(S(0))	→	c14	(76)
help1^#(S(S(z0)))	→	c15	(78)
help1^#(0)	→	c16	(79)
e4^#(z0,z1,z2,False)	→	c17	(81)
e4^#(z0,z1,z2,True)	→	c18	(83)
e2^#(z0,z1,z2,False)	→	c19	(85)
e2^#(z0,z1,z2,True)	→	c20(e3^#(z0,z1,z2,True))	(87)
l15^#(z0,z1,z2,z3,False,z4)	→	c21(l16^#(z0,z1,gcd(z1,0),z3,False,z4),gcd^#(z1,0))	(89)
l15^#(z0,z1,z2,z3,True,z4)	→	c22(l16^#(z0,z1,gcd(z1,S(0)),z3,True,z4),gcd^#(z1,S(0)))	(91)
l13^#(z0,z1,z2,z3,False,z4)	→	c23(l16^#(z0,z1,gcd(0,z1),z3,False,z4),gcd^#(0,z1))	(93)
l13^#(z0,z1,z2,z3,True,z4)	→	c24(l16^#(z0,z1,gcd(S(0),z1),z3,True,z4),gcd^#(S(0),z1))	(95)
m4^#(S(z0),S(z1),z2,z3)	→	c25(m5^#(S(z0),S(z1),monus(z0,z1),z3),monus^#(z0,z1))	(97)
l2^#(z0,z1,z2,z3,z4,False)	→	c26(l3^#(z0,z1,z2,z3,z4,False))	(99)
l2^#(z0,z1,z2,z3,z4,True)	→	c27	(101)
l11^#(z0,z1,z2,z3,z4,False)	→	c28(l14^#(z0,z1,z2,z3,z4,False))	(103)
l11^#(z0,z1,z2,z3,z4,True)	→	c29(l12^#(z0,z1,z2,z3,z4,True))	(105)
bool2Nat^#(False)	→	c30	(106)
bool2Nat^#(True)	→	c31	(107)
m1^#(z0,z1,z2,z3)	→	c32(m2^#(z0,z1,z2,False))	(109)
l9^#(z0,z1,z2,z3,z4,z5)	→	c33	(111)
l6^#(z0,z1,z2,z3,z4,z5)	→	c34	(113)
l4^#(z0,z1,z2,z3,z4,z5)	→	c35(l5^#(z0,z1,z2,z3,z4,False))	(115)
l1^#(z0,z1,z2,z3,z4,z5)	→	c36(l2^#(z0,z1,z2,z3,z4,False))	(117)
e7^#(z0,z1,z2,z3)	→	c37	(119)
e6^#(z0,z1,z2,z3)	→	c38	(121)
e5^#(z0,z1,z2,z3)	→	c39	(123)
monus^#(z0,z1)	→	c40(m1^#(z0,z1,False,False))	(125)
m5^#(z0,z1,z2,z3)	→	c41	(127)
l7^#(z0,z1,z2,z3,z4,z5)	→	c42(l8^#(z0,z1,z2,equal0(z0,z1),z4,z5),equal0^#(z0,z1))	(129)
l3^#(z0,z1,z2,z3,z4,z5)	→	c43(l4^#(z0,z1,0,z3,z4,z5))	(131)
l16^#(z0,z1,z2,z3,z4,z5)	→	c44	(133)
l14^#(z0,z1,z2,z3,z4,z5)	→	c45(l15^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(135)
l12^#(z0,z1,z2,z3,z4,z5)	→	c46(l13^#(z0,z1,z2,z3,monus(z0,z1),z5),monus^#(z0,z1))	(137)
l10^#(z0,z1,z2,z3,z4,z5)	→	c47(l11^#(z0,z1,z2,z3,z4,<(z0,z1)),<^#(z0,z1))	(139)
gcd^#(z0,z1)	→	c48(l1^#(z0,z1,0,False,False,False))	(141)
equal0^#(z0,z1)	→	c49(e1^#(z0,z1,False,False))	(143)
e8^#(z0,z1,z2,z3)	→	c50	(145)
e3^#(z0,z1,z2,z3)	→	c51(e4^#(z0,z1,z2,<(z1,z0)),<^#(z1,z0))	(147)
e1^#(z0,z1,z2,z3)	→	c52(e2^#(z0,z1,z2,<(z0,z1)),<^#(z0,z1))	(149)
m1(z0,z1,z2,z3)	→	m2(z0,z1,z2,False)	(108)
m4(S(z0),S(z1),z2,z3)	→	m5(S(z0),S(z1),monus(z0,z1),z3)	(96)
m5(z0,z1,z2,z3)	→	z2	(126)
m2(z0,z1,z2,False)	→	m4(z0,z1,z2,False)	(60)
monus(z0,z1)	→	m1(z0,z1,False,False)	(124)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 R is empty

There are no rules in the TRS R. Hence, R/S has complexity O(1).