Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/raML/longestCommonSubsequence.raml)

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

#abs(#0)	→	#0	(1)
#abs(#neg(@x))	→	#pos(@x)	(2)
#abs(#pos(@x))	→	#pos(@x)	(3)
#abs(#s(@x))	→	#pos(#s(@x))	(4)
#equal(@x,@y)	→	#eq(@x,@y)	(5)
#greater(@x,@y)	→	#ckgt(#compare(@x,@y))	(6)
+(@x,@y)	→	#add(@x,@y)	(7)
firstline(@l)	→	firstline#1(@l)	(8)
firstline#1(::(@x,@xs))	→	::(#abs(#0),firstline(@xs))	(9)
firstline#1(nil)	→	nil	(10)
lcs(@l1,@l2)	→	lcs#1(lcstable(@l1,@l2))	(11)
lcs#1(@m)	→	lcs#2(@m)	(12)
lcs#2(::(@l1,@_@2))	→	lcs#3(@l1)	(13)
lcs#2(nil)	→	#abs(#0)	(14)
lcs#3(::(@len,@_@1))	→	@len	(15)
lcs#3(nil)	→	#abs(#0)	(16)
lcstable(@l1,@l2)	→	lcstable#1(@l1,@l2)	(17)
lcstable#1(::(@x,@xs),@l2)	→	lcstable#2(lcstable(@xs,@l2),@l2,@x)	(18)
lcstable#1(nil,@l2)	→	::(firstline(@l2),nil)	(19)
lcstable#2(@m,@l2,@x)	→	lcstable#3(@m,@l2,@x)	(20)
lcstable#3(::(@l,@ls),@l2,@x)	→	::(newline(@x,@l,@l2),::(@l,@ls))	(21)
lcstable#3(nil,@l2,@x)	→	nil	(22)
max(@a,@b)	→	max#1(#greater(@a,@b),@a,@b)	(23)
max#1(#false,@a,@b)	→	@b	(24)
max#1(#true,@a,@b)	→	@a	(25)
newline(@y,@lastline,@l)	→	newline#1(@l,@lastline,@y)	(26)
newline#1(::(@x,@xs),@lastline,@y)	→	newline#2(@lastline,@x,@xs,@y)	(27)
newline#1(nil,@lastline,@y)	→	nil	(28)
newline#2(::(@belowVal,@lastline'),@x,@xs,@y)	→	newline#3(newline(@y,@lastline',@xs),@belowVal,@lastline',@x,@y)	(29)
newline#2(nil,@x,@xs,@y)	→	nil	(30)
newline#3(@nl,@belowVal,@lastline',@x,@y)	→	newline#4(right(@nl),@belowVal,@lastline',@nl,@x,@y)	(31)
newline#4(@rightVal,@belowVal,@lastline',@nl,@x,@y)	→	newline#5(right(@lastline'),@belowVal,@nl,@rightVal,@x,@y)	(32)
newline#5(@diagVal,@belowVal,@nl,@rightVal,@x,@y)	→	newline#6(newline#7(#equal(@x,@y),@belowVal,@diagVal,@rightVal),@nl)	(33)
newline#6(@elem,@nl)	→	::(@elem,@nl)	(34)
newline#7(#false,@belowVal,@diagVal,@rightVal)	→	max(@belowVal,@rightVal)	(35)
newline#7(#true,@belowVal,@diagVal,@rightVal)	→	+(@diagVal,#pos(#s(#0)))	(36)
right(@l)	→	right#1(@l)	(37)
right#1(::(@x,@xs))	→	@x	(38)
right#1(nil)	→	#abs(#0)	(39)

and S is the following TRS.

#add(#0,@y)	→	@y	(40)
#add(#neg(#s(#0)),@y)	→	#pred(@y)	(41)
#add(#neg(#s(#s(@x))),@y)	→	#pred(#add(#pos(#s(@x)),@y))	(42)
#add(#pos(#s(#0)),@y)	→	#succ(@y)	(43)
#add(#pos(#s(#s(@x))),@y)	→	#succ(#add(#pos(#s(@x)),@y))	(44)
#and(#false,#false)	→	#false	(45)
#and(#false,#true)	→	#false	(46)
#and(#true,#false)	→	#false	(47)
#and(#true,#true)	→	#true	(48)
#ckgt(#EQ)	→	#false	(49)
#ckgt(#GT)	→	#true	(50)
#ckgt(#LT)	→	#false	(51)
#compare(#0,#0)	→	#EQ	(52)
#compare(#0,#neg(@y))	→	#GT	(53)
#compare(#0,#pos(@y))	→	#LT	(54)
#compare(#0,#s(@y))	→	#LT	(55)
#compare(#neg(@x),#0)	→	#LT	(56)
#compare(#neg(@x),#neg(@y))	→	#compare(@y,@x)	(57)
#compare(#neg(@x),#pos(@y))	→	#LT	(58)
#compare(#pos(@x),#0)	→	#GT	(59)
#compare(#pos(@x),#neg(@y))	→	#GT	(60)
#compare(#pos(@x),#pos(@y))	→	#compare(@x,@y)	(61)
#compare(#s(@x),#0)	→	#GT	(62)
#compare(#s(@x),#s(@y))	→	#compare(@x,@y)	(63)
#eq(#0,#0)	→	#true	(64)
#eq(#0,#neg(@y))	→	#false	(65)
#eq(#0,#pos(@y))	→	#false	(66)
#eq(#0,#s(@y))	→	#false	(67)
#eq(#neg(@x),#0)	→	#false	(68)
#eq(#neg(@x),#neg(@y))	→	#eq(@x,@y)	(69)
#eq(#neg(@x),#pos(@y))	→	#false	(70)
#eq(#pos(@x),#0)	→	#false	(71)
#eq(#pos(@x),#neg(@y))	→	#false	(72)
#eq(#pos(@x),#pos(@y))	→	#eq(@x,@y)	(73)
#eq(#s(@x),#0)	→	#false	(74)
#eq(#s(@x),#s(@y))	→	#eq(@x,@y)	(75)
#eq(::(@x_1,@x_2),::(@y_1,@y_2))	→	#and(#eq(@x_1,@y_1),#eq(@x_2,@y_2))	(76)
#eq(::(@x_1,@x_2),nil)	→	#false	(77)
#eq(nil,::(@y_1,@y_2))	→	#false	(78)
#eq(nil,nil)	→	#true	(79)
#pred(#0)	→	#neg(#s(#0))	(80)
#pred(#neg(#s(@x)))	→	#neg(#s(#s(@x)))	(81)
#pred(#pos(#s(#0)))	→	#0	(82)
#pred(#pos(#s(#s(@x))))	→	#pos(#s(@x))	(83)
#succ(#0)	→	#pos(#s(#0))	(84)
#succ(#neg(#s(#0)))	→	#0	(85)
#succ(#neg(#s(#s(@x))))	→	#neg(#s(@x))	(86)
#succ(#pos(#s(@x)))	→	#pos(#s(#s(@x)))	(87)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n²).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

#abs^#(#0)

→

c48

(88)

originates from

#abs(#0)

→

(1)

#abs^#(#neg(z0))

→

c49

(90)

originates from

#abs(#neg(z0))

→

#pos(z0)

(89)

#abs^#(#pos(z0))

→

c50

(92)

originates from

#abs(#pos(z0))

→

#pos(z0)

(91)

#abs^#(#s(z0))

→

c51

(94)

originates from

#abs(#s(z0))

→

#pos(#s(z0))

(93)

#equal^#(z0,z1)

→

c52(#eq^#(z0,z1))

(96)

originates from

#equal(z0,z1)

→

#eq(z0,z1)

(95)

#greater^#(z0,z1)

→

c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))

(98)

originates from

#greater(z0,z1)

→

#ckgt(#compare(z0,z1))

(97)

+^#(z0,z1)

→

c54(#add^#(z0,z1))

(100)

originates from

+(z0,z1)

→

#add(z0,z1)

(99)

firstline^#(z0)

→

c55(firstline#1^#(z0))

(102)

originates from

firstline(z0)

→

firstline#1(z0)

(101)

firstline#1^#(::(z0,z1))

→

c56(#abs^#(#0),firstline^#(z1))

(104)

originates from

firstline#1(::(z0,z1))

→

::(#abs(#0),firstline(z1))

(103)

firstline#1^#(nil)

→

c57

(105)

originates from

firstline#1(nil)

→

nil

(10)

lcs^#(z0,z1)

→

c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))

(107)

originates from

lcs(z0,z1)

→

lcs#1(lcstable(z0,z1))

(106)

lcs#1^#(z0)

→

c59(lcs#2^#(z0))

(109)

originates from

lcs#1(z0)

→

lcs#2(z0)

(108)

lcs#2^#(::(z0,z1))

→

c60(lcs#3^#(z0))

(111)

originates from

lcs#2(::(z0,z1))

→

lcs#3(z0)

(110)

lcs#2^#(nil)

→

c61(#abs^#(#0))

(112)

originates from

lcs#2(nil)

→

#abs(#0)

(14)

lcs#3^#(::(z0,z1))

→

c62

(114)

originates from

lcs#3(::(z0,z1))

→

(113)

lcs#3^#(nil)

→

c63(#abs^#(#0))

(115)

originates from

lcs#3(nil)

→

#abs(#0)

(16)

lcstable^#(z0,z1)

→

c64(lcstable#1^#(z0,z1))

(117)

originates from

lcstable(z0,z1)

→

lcstable#1(z0,z1)

(116)

lcstable#1^#(::(z0,z1),z2)

→

c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))

(119)

originates from

lcstable#1(::(z0,z1),z2)

→

lcstable#2(lcstable(z1,z2),z2,z0)

(118)

lcstable#1^#(nil,z0)

→

c66(firstline^#(z0))

(121)

originates from

lcstable#1(nil,z0)

→

::(firstline(z0),nil)

(120)

lcstable#2^#(z0,z1,z2)

→

c67(lcstable#3^#(z0,z1,z2))

(123)

originates from

lcstable#2(z0,z1,z2)

→

lcstable#3(z0,z1,z2)

(122)

lcstable#3^#(::(z0,z1),z2,z3)

→

c68(newline^#(z3,z0,z2))

(125)

originates from

lcstable#3(::(z0,z1),z2,z3)

→

::(newline(z3,z0,z2),::(z0,z1))

(124)

lcstable#3^#(nil,z0,z1)

→

c69

(127)

originates from

lcstable#3(nil,z0,z1)

→

nil

(126)

max^#(z0,z1)

→

c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))

(129)

originates from

max(z0,z1)

→

max#1(#greater(z0,z1),z0,z1)

(128)

max#1^#(#false,z0,z1)

→

c71

(131)

originates from

max#1(#false,z0,z1)

→

(130)

max#1^#(#true,z0,z1)

→

c72

(133)

originates from

max#1(#true,z0,z1)

→

(132)

newline^#(z0,z1,z2)

→

c73(newline#1^#(z2,z1,z0))

(135)

originates from

newline(z0,z1,z2)

→

newline#1(z2,z1,z0)

(134)

newline#1^#(::(z0,z1),z2,z3)

→

c74(newline#2^#(z2,z0,z1,z3))

(137)

originates from

newline#1(::(z0,z1),z2,z3)

→

newline#2(z2,z0,z1,z3)

(136)

newline#1^#(nil,z0,z1)

→

c75

(139)

originates from

newline#1(nil,z0,z1)

→

nil

(138)

newline#2^#(::(z0,z1),z2,z3,z4)

→

c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))

(141)

originates from

newline#2(::(z0,z1),z2,z3,z4)

→

newline#3(newline(z4,z1,z3),z0,z1,z2,z4)

(140)

newline#2^#(nil,z0,z1,z2)

→

c77

(143)

originates from

newline#2(nil,z0,z1,z2)

→

nil

(142)

newline#3^#(z0,z1,z2,z3,z4)

→

c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))

(145)

originates from

newline#3(z0,z1,z2,z3,z4)

→

newline#4(right(z0),z1,z2,z0,z3,z4)

(144)

newline#4^#(z0,z1,z2,z3,z4,z5)

→

c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))

(147)

originates from

newline#4(z0,z1,z2,z3,z4,z5)

→

newline#5(right(z2),z1,z3,z0,z4,z5)

(146)

newline#5^#(z0,z1,z2,z3,z4,z5)

→

c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))

(149)

originates from

newline#5(z0,z1,z2,z3,z4,z5)

→

newline#6(newline#7(#equal(z4,z5),z1,z0,z3),z2)

(148)

newline#6^#(z0,z1)

→

c81

(151)

originates from

newline#6(z0,z1)

→

::(z0,z1)

(150)

newline#7^#(#false,z0,z1,z2)

→

c82(max^#(z0,z2))

(153)

originates from

newline#7(#false,z0,z1,z2)

→

max(z0,z2)

(152)

newline#7^#(#true,z0,z1,z2)

→

c83(+^#(z1,#pos(#s(#0))))

(155)

originates from

newline#7(#true,z0,z1,z2)

→

+(z1,#pos(#s(#0)))

(154)

right^#(z0)

→

c84(right#1^#(z0))

(157)

originates from

right(z0)

→

right#1(z0)

(156)

right#1^#(::(z0,z1))

→

c85

(159)

originates from

right#1(::(z0,z1))

→

(158)

right#1^#(nil)

→

c86(#abs^#(#0))

(160)

originates from

right#1(nil)

→

#abs(#0)

(39)

#add^#(#0,z0)

→

(162)

originates from

#add(#0,z0)

→

(161)

#add^#(#neg(#s(#0)),z0)

→

c1(#pred^#(z0))

(164)

originates from

#add(#neg(#s(#0)),z0)

→

#pred(z0)

(163)

#add^#(#neg(#s(#s(z0))),z1)

→

c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))

(166)

originates from

#add(#neg(#s(#s(z0))),z1)

→

#pred(#add(#pos(#s(z0)),z1))

(165)

#add^#(#pos(#s(#0)),z0)

→

c3(#succ^#(z0))

(168)

originates from

#add(#pos(#s(#0)),z0)

→

#succ(z0)

(167)

#add^#(#pos(#s(#s(z0))),z1)

→

c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))

(170)

originates from

#add(#pos(#s(#s(z0))),z1)

→

#succ(#add(#pos(#s(z0)),z1))

(169)

#and^#(#false,#false)

→

(171)

originates from

#and(#false,#false)

→

#false

(45)

#and^#(#false,#true)

→

(172)

originates from

#and(#false,#true)

→

#false

(46)

#and^#(#true,#false)

→

(173)

originates from

#and(#true,#false)

→

#false

(47)

#and^#(#true,#true)

→

(174)

originates from

#and(#true,#true)

→

#true

(48)

#ckgt^#(#EQ)

→

(175)

originates from

#ckgt(#EQ)

→

#false

(49)

#ckgt^#(#GT)

→

c10

(176)

originates from

#ckgt(#GT)

→

#true

(50)

#ckgt^#(#LT)

→

c11

(177)

originates from

#ckgt(#LT)

→

#false

(51)

#compare^#(#0,#0)

→

c12

(178)

originates from

#compare(#0,#0)

→

#EQ

(52)

#compare^#(#0,#neg(z0))

→

c13

(180)

originates from

#compare(#0,#neg(z0))

→

#GT

(179)

#compare^#(#0,#pos(z0))

→

c14

(182)

originates from

#compare(#0,#pos(z0))

→

#LT

(181)

#compare^#(#0,#s(z0))

→

c15

(184)

originates from

#compare(#0,#s(z0))

→

#LT

(183)

#compare^#(#neg(z0),#0)

→

c16

(186)

originates from

#compare(#neg(z0),#0)

→

#LT

(185)

#compare^#(#neg(z0),#neg(z1))

→

c17(#compare^#(z1,z0))

(188)

originates from

#compare(#neg(z0),#neg(z1))

→

#compare(z1,z0)

(187)

#compare^#(#neg(z0),#pos(z1))

→

c18

(190)

originates from

#compare(#neg(z0),#pos(z1))

→

#LT

(189)

#compare^#(#pos(z0),#0)

→

c19

(192)

originates from

#compare(#pos(z0),#0)

→

#GT

(191)

#compare^#(#pos(z0),#neg(z1))

→

c20

(194)

originates from

#compare(#pos(z0),#neg(z1))

→

#GT

(193)

#compare^#(#pos(z0),#pos(z1))

→

c21(#compare^#(z0,z1))

(196)

originates from

#compare(#pos(z0),#pos(z1))

→

#compare(z0,z1)

(195)

#compare^#(#s(z0),#0)

→

c22

(198)

originates from

#compare(#s(z0),#0)

→

#GT

(197)

#compare^#(#s(z0),#s(z1))

→

c23(#compare^#(z0,z1))

(200)

originates from

#compare(#s(z0),#s(z1))

→

#compare(z0,z1)

(199)

#eq^#(#0,#0)

→

c24

(201)

originates from

#eq(#0,#0)

→

#true

(64)

#eq^#(#0,#neg(z0))

→

c25

(203)

originates from

#eq(#0,#neg(z0))

→

#false

(202)

#eq^#(#0,#pos(z0))

→

c26

(205)

originates from

#eq(#0,#pos(z0))

→

#false

(204)

#eq^#(#0,#s(z0))

→

c27

(207)

originates from

#eq(#0,#s(z0))

→

#false

(206)

#eq^#(#neg(z0),#0)

→

c28

(209)

originates from

#eq(#neg(z0),#0)

→

#false

(208)

#eq^#(#neg(z0),#neg(z1))

→

c29(#eq^#(z0,z1))

(211)

originates from

#eq(#neg(z0),#neg(z1))

→

#eq(z0,z1)

(210)

#eq^#(#neg(z0),#pos(z1))

→

c30

(213)

originates from

#eq(#neg(z0),#pos(z1))

→

#false

(212)

#eq^#(#pos(z0),#0)

→

c31

(215)

originates from

#eq(#pos(z0),#0)

→

#false

(214)

#eq^#(#pos(z0),#neg(z1))

→

c32

(217)

originates from

#eq(#pos(z0),#neg(z1))

→

#false

(216)

#eq^#(#pos(z0),#pos(z1))

→

c33(#eq^#(z0,z1))

(219)

originates from

#eq(#pos(z0),#pos(z1))

→

#eq(z0,z1)

(218)

#eq^#(#s(z0),#0)

→

c34

(221)

originates from

#eq(#s(z0),#0)

→

#false

(220)

#eq^#(#s(z0),#s(z1))

→

c35(#eq^#(z0,z1))

(223)

originates from

#eq(#s(z0),#s(z1))

→

#eq(z0,z1)

(222)

#eq^#(::(z0,z1),::(z2,z3))

→

c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))

(225)

originates from

#eq(::(z0,z1),::(z2,z3))

→

#and(#eq(z0,z2),#eq(z1,z3))

(224)

#eq^#(::(z0,z1),nil)

→

c37

(227)

originates from

#eq(::(z0,z1),nil)

→

#false

(226)

#eq^#(nil,::(z0,z1))

→

c38

(229)

originates from

#eq(nil,::(z0,z1))

→

#false

(228)

#eq^#(nil,nil)

→

c39

(230)

originates from

#eq(nil,nil)

→

#true

(79)

#pred^#(#0)

→

c40

(231)

originates from

#pred(#0)

→

#neg(#s(#0))

(80)

#pred^#(#neg(#s(z0)))

→

c41

(233)

originates from

#pred(#neg(#s(z0)))

→

#neg(#s(#s(z0)))

(232)

#pred^#(#pos(#s(#0)))

→

c42

(234)

originates from

#pred(#pos(#s(#0)))

→

(82)

#pred^#(#pos(#s(#s(z0))))

→

c43

(236)

originates from

#pred(#pos(#s(#s(z0))))

→

#pos(#s(z0))

(235)

#succ^#(#0)

→

c44

(237)

originates from

#succ(#0)

→

#pos(#s(#0))

(84)

#succ^#(#neg(#s(#0)))

→

c45

(238)

originates from

#succ(#neg(#s(#0)))

→

(85)

#succ^#(#neg(#s(#s(z0))))

→

c46

(240)

originates from

#succ(#neg(#s(#s(z0))))

→

#neg(#s(z0))

(239)

#succ^#(#pos(#s(z0)))

→

c47

(242)

originates from

#succ(#pos(#s(z0)))

→

#pos(#s(#s(z0)))

(241)

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

#add^#(#0,z0)

#add^#(#neg(#s(#0)),z0)

#add^#(#neg(#s(#s(z0))),z1)

#add^#(#pos(#s(#0)),z0)

#add^#(#pos(#s(#s(z0))),z1)

#and^#(#false,#false)

#and^#(#false,#true)

#and^#(#true,#false)

#and^#(#true,#true)

#ckgt^#(#EQ)

#ckgt^#(#GT)

#ckgt^#(#LT)

#compare^#(#0,#0)

#compare^#(#0,#neg(z0))

#compare^#(#0,#pos(z0))

#compare^#(#0,#s(z0))

#compare^#(#neg(z0),#0)

#compare^#(#neg(z0),#neg(z1))

#compare^#(#neg(z0),#pos(z1))

#compare^#(#pos(z0),#0)

#compare^#(#pos(z0),#neg(z1))

#compare^#(#pos(z0),#pos(z1))

#compare^#(#s(z0),#0)

#compare^#(#s(z0),#s(z1))

#eq^#(#0,#0)

#eq^#(#0,#neg(z0))

#eq^#(#0,#pos(z0))

#eq^#(#0,#s(z0))

#eq^#(#neg(z0),#0)

#eq^#(#neg(z0),#neg(z1))

#eq^#(#neg(z0),#pos(z1))

#eq^#(#pos(z0),#0)

#eq^#(#pos(z0),#neg(z1))

#eq^#(#pos(z0),#pos(z1))

#eq^#(#s(z0),#0)

#eq^#(#s(z0),#s(z1))

#eq^#(::(z0,z1),::(z2,z3))

#eq^#(::(z0,z1),nil)

#eq^#(nil,::(z0,z1))

#eq^#(nil,nil)

#pred^#(#0)

#pred^#(#neg(#s(z0)))

#pred^#(#pos(#s(#0)))

#pred^#(#pos(#s(#s(z0))))

#succ^#(#0)

#succ^#(#neg(#s(#0)))

#succ^#(#neg(#s(#s(z0))))

#succ^#(#pos(#s(z0)))

#abs^#(#0)

#abs^#(#neg(z0))

#abs^#(#pos(z0))

#abs^#(#s(z0))

#equal^#(z0,z1)

#greater^#(z0,z1)

+^#(z0,z1)

firstline^#(z0)

firstline#1^#(::(z0,z1))

firstline#1^#(nil)

lcs^#(z0,z1)

lcs#1^#(z0)

lcs#2^#(::(z0,z1))

lcs#2^#(nil)

lcs#3^#(::(z0,z1))

lcs#3^#(nil)

lcstable^#(z0,z1)

lcstable#1^#(::(z0,z1),z2)

lcstable#1^#(nil,z0)

lcstable#2^#(z0,z1,z2)

lcstable#3^#(::(z0,z1),z2,z3)

lcstable#3^#(nil,z0,z1)

max^#(z0,z1)

max#1^#(#false,z0,z1)

max#1^#(#true,z0,z1)

newline^#(z0,z1,z2)

newline#1^#(::(z0,z1),z2,z3)

newline#1^#(nil,z0,z1)

newline#2^#(::(z0,z1),z2,z3,z4)

newline#2^#(nil,z0,z1,z2)

newline#3^#(z0,z1,z2,z3,z4)

newline#4^#(z0,z1,z2,z3,z4,z5)

newline#5^#(z0,z1,z2,z3,z4,z5)

newline#6^#(z0,z1)

newline#7^#(#false,z0,z1,z2)

newline#7^#(#true,z0,z1,z2)

right^#(z0)

right#1^#(::(z0,z1))

right#1^#(nil)

1.1 Usable Rules

We remove the following rules since they are not usable.

#abs(#neg(z0))	→	#pos(z0)	(89)
#abs(#pos(z0))	→	#pos(z0)	(91)
#abs(#s(z0))	→	#pos(#s(z0))	(93)
lcs(z0,z1)	→	lcs#1(lcstable(z0,z1))	(106)
lcs#1(z0)	→	lcs#2(z0)	(108)
lcs#2(::(z0,z1))	→	lcs#3(z0)	(110)
lcs#2(nil)	→	#abs(#0)	(14)
lcs#3(::(z0,z1))	→	z0	(113)
lcs#3(nil)	→	#abs(#0)	(16)

1.1.1 Rule Shifting

The rules

lcstable#2^#(z0,z1,z2)

→

c67(lcstable#3^#(z0,z1,z2))

(123)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#compare(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[right(x₁)]	=	1 + 1 · x₁
[right#1(x₁)]	=	1 + 1 · x₁
[#abs(x₁)]	=	1 + 1 · x₁
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[newline#6(x₁, x₂)]	=	1 + 1 · x₂
[newline#7(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[#greater(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 1 · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁
[#add^#(x₁, x₂)]	=	1 · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	1 · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#2^#(x₁, x₂, x₃)]	=	1
[lcstable#3^#(x₁, x₂, x₃)]	=	0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	1
[#GT]	=	1
[#LT]	=	1
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1 Rule Shifting

The rules

lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#compare(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[right(x₁)]	=	1 + 1 · x₁
[right#1(x₁)]	=	1 + 1 · x₁
[#abs(x₁)]	=	1 + 1 · x₁
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[newline#6(x₁, x₂)]	=	1 + 1 · x₂
[newline#7(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[#greater(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 1 · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁
[#add^#(x₁, x₂)]	=	1 · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	1 · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₂
[lcs#1^#(x₁)]	=	1
[lcs#2^#(x₁)]	=	1
[lcs#3^#(x₁)]	=	1
[lcstable^#(x₁, x₂)]	=	1 · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	1 · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	0
[lcstable#3^#(x₁, x₂, x₃)]	=	0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	1
[#GT]	=	1
[#LT]	=	1
[::(x₁, x₂)]	=	0
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1 Rule Shifting

The rules

#abs^#(#pos(z0))	→	c50	(92)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	3
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[#compare(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline(x₁, x₂, x₃)]	=	3 · x₃ + 0
[newline#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline#2(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[newline#3(x₁,...,x₅)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[newline#4(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[right(x₁)]	=	0
[right#1(x₁)]	=	3
[#abs(x₁)]	=	3
[newline#5(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[newline#6(x₁, x₂)]	=	3 + 3 · x₂
[newline#7(x₁,...,x₄)]	=	3 · x₄ + 0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[max#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	3
[+(x₁, x₂)]	=	3 + 3 · x₁
[#pred(x₁)]	=	3
[firstline(x₁)]	=	3 + 3 · x₁
[firstline#1(x₁)]	=	3
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 2 · x₁
[lcs#1^#(x₁)]	=	2
[lcs#2^#(x₁)]	=	2
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₁ + 0
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	2
[lcstable#3^#(x₁, x₂, x₃)]	=	1
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	1
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	3
[#GT]	=	3
[#LT]	=	3
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1 Rule Shifting

The rules

lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₂
[#succ(x₁)]	=	3
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[#compare(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline(x₁, x₂, x₃)]	=	3 · x₃ + 0
[newline#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline#2(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[newline#3(x₁,...,x₅)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[newline#4(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[right(x₁)]	=	2 + 2 · x₁
[right#1(x₁)]	=	2 + 2 · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[newline#6(x₁, x₂)]	=	3 + 3 · x₂
[newline#7(x₁,...,x₄)]	=	2 + 2 · x₁ + 2 · x₂ + 2 · x₃ + 2 · x₄
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂
[max#1(x₁, x₂, x₃)]	=	1 + 2 · x₂ + 2 · x₃
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	3
[+(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₂
[#pred(x₁)]	=	3 + 1 · x₁
[firstline(x₁)]	=	3 + 3 · x₁
[firstline#1(x₁)]	=	3
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	3 + 3 · x₁
[lcs#1^#(x₁)]	=	1
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 + 2 · x₁
[lcstable#1^#(x₁, x₂)]	=	2 + 2 · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	2 + 1 · x₃
[lcstable#3^#(x₁, x₂, x₃)]	=	1 + 1 · x₃
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	1
[#true]	=	1
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[#0]	=	0
[#neg(x₁)]	=	1 · x₁ + 0
[#EQ]	=	3
[#GT]	=	3
[#LT]	=	3
[::(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[nil]	=	2

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1 Rule Shifting

The rules

#abs^#(#neg(z0))	→	c49	(90)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	3
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[#compare(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline(x₁, x₂, x₃)]	=	2
[newline#1(x₁, x₂, x₃)]	=	2
[newline#2(x₁,...,x₄)]	=	2
[newline#3(x₁,...,x₅)]	=	0
[newline#4(x₁,...,x₆)]	=	0
[right(x₁)]	=	0
[right#1(x₁)]	=	3
[#abs(x₁)]	=	3
[newline#5(x₁,...,x₆)]	=	0
[newline#6(x₁, x₂)]	=	0
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[max#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	3
[+(x₁, x₂)]	=	3 + 3 · x₁
[#pred(x₁)]	=	3
[firstline(x₁)]	=	3 + 3 · x₁
[firstline#1(x₁)]	=	3
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	0
[lcstable#1^#(x₁, x₂)]	=	0
[lcstable#2^#(x₁, x₂, x₃)]	=	0
[lcstable#3^#(x₁, x₂, x₃)]	=	0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	1
[#EQ]	=	3
[#GT]	=	3
[#LT]	=	3
[::(x₁, x₂)]	=	0
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#compare(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[right(x₁)]	=	1 + 1 · x₁
[right#1(x₁)]	=	1 + 1 · x₁
[#abs(x₁)]	=	1 + 1 · x₁
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[newline#6(x₁, x₂)]	=	1 + 1 · x₂
[newline#7(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[#greater(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#ckgt(x₁)]	=	0
[+(x₁, x₂)]	=	1 + 1 · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 1 · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁
[#add^#(x₁, x₂)]	=	1 · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	1 · x₂ + 0
[firstline^#(x₁)]	=	1 · x₁ + 0
[firstline#1^#(x₁)]	=	1 · x₁ + 0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[lcs#1^#(x₁)]	=	1
[lcs#2^#(x₁)]	=	1
[lcs#3^#(x₁)]	=	1
[lcstable^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[lcstable#2^#(x₁, x₂, x₃)]	=	1
[lcstable#3^#(x₁, x₂, x₃)]	=	1
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1
[newline#1^#(x₁, x₂, x₃)]	=	1
[newline#2^#(x₁,...,x₄)]	=	1
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	1
[#GT]	=	1
[#LT]	=	1
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	1

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	3
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[#compare(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline#2(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[newline#3(x₁,...,x₅)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[newline#4(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[right(x₁)]	=	0
[right#1(x₁)]	=	3
[#abs(x₁)]	=	3
[newline#5(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[newline#6(x₁, x₂)]	=	3 + 3 · x₂
[newline#7(x₁,...,x₄)]	=	3 · x₂ + 0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[max#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	3
[+(x₁, x₂)]	=	3 + 3 · x₁
[#pred(x₁)]	=	3
[firstline(x₁)]	=	3 + 3 · x₁
[firstline#1(x₁)]	=	3
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	2 + 1 · x₁
[firstline#1^#(x₁)]	=	1 · x₁ + 0
[lcs^#(x₁, x₂)]	=	3 + 2 · x₁ + 1 · x₂
[lcs#1^#(x₁)]	=	2
[lcs#2^#(x₁)]	=	2
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 + 2 · x₁ + 1 · x₂
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₂
[lcstable#2^#(x₁, x₂, x₃)]	=	0
[lcstable#3^#(x₁, x₂, x₃)]	=	0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	3
[#GT]	=	3
[#LT]	=	3
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	1

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

#abs^#(#s(z0))

→

c51

(94)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	3
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[#compare(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[lcstable#2(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[lcstable#3(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[newline#2(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[newline#3(x₁,...,x₅)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅
[newline#4(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[right(x₁)]	=	2 · x₁ + 0
[right#1(x₁)]	=	2 · x₁ + 0
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆
[newline#6(x₁, x₂)]	=	3 + 3 · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[max#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	3
[+(x₁, x₂)]	=	3 + 3 · x₁
[#pred(x₁)]	=	3
[firstline(x₁)]	=	3 + 3 · x₁
[firstline#1(x₁)]	=	3
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	3 · x₁ + 0
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	0
[lcstable#1^#(x₁, x₂)]	=	0
[lcstable#2^#(x₁, x₂, x₃)]	=	0
[lcstable#3^#(x₁, x₂, x₃)]	=	0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	0
[newline#1^#(x₁, x₂, x₃)]	=	0
[newline#2^#(x₁,...,x₄)]	=	0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	1
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	3
[#GT]	=	3
[#LT]	=	3
[::(x₁, x₂)]	=	1 · x₁ + 0
[nil]	=	2

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline#1^#(::(z0,z1),z2,z3)

→

c74(newline#2^#(z2,z0,z1,z3))

(137)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 2 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 2 · x₂ · x₂ + 1 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 · x₃ + 0
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	1
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 2 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁ + 2 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	2
[max#1^#(x₁, x₂, x₃)]	=	2
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	2
[newline#5^#(x₁,...,x₆)]	=	2
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₁ · x₁
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	1
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

There are 108 ruless (increase limit for explicit display).

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline#2^#(::(z0,z1),z2,z3,z4)

→

c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))

(141)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 1 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 1 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 1 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₁ · x₂
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₁ · x₂
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	2
[lcs#2^#(x₁)]	=	2
[lcs#3^#(x₁)]	=	2
[lcstable^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	2
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline^#(z0,z1,z2)

→

c73(newline#1^#(z2,z1,z0))

(135)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#1^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	2 + 2 · x₂
[lcstable#3^#(x₁, x₂, x₃)]	=	2 + 2 · x₂
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 + 1 · x₃
[newline#1^#(x₁, x₂, x₃)]	=	1 + 1 · x₁
[newline#2^#(x₁,...,x₄)]	=	2 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	0
[newline#4^#(x₁,...,x₆)]	=	0
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	2
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 2 · x₂ + 2 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	2
[newline#5^#(x₁,...,x₆)]	=	2
[newline#6^#(x₁, x₂)]	=	1
[newline#7^#(x₁,...,x₄)]	=	1
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	2
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

newline#5^#(z0,z1,z2,z3,z4,z5)

→

c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))

(149)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 2 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	1 · x₁ + 0 + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0 + 1 · x₁ · x₁
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0 + 1 · x₁ · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	1
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	1
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 · x₁ + 0 + 1 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 1 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₂ + 0 + 1 · x₁ · x₂
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	1
[right#1^#(x₁)]	=	1
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

#equal^#(z0,z1)

→

c52(#eq^#(z0,z1))

(96)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	1
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 1 · x₁ + 2 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂
[lcstable#1^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	1
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	1
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

#abs^#(#0)

→

c48

(88)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	1 + 2 · x₁ + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	1 · x₁ · x₁ + 0
[firstline#1^#(x₁)]	=	1 · x₁ · x₁ + 0
[lcs^#(x₁, x₂)]	=	2 + 1 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	1
[lcs#2^#(x₁)]	=	1
[lcs#3^#(x₁)]	=	1
[lcstable^#(x₁, x₂)]	=	1 · x₂ · x₂ + 0 + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#1^#(x₁, x₂)]	=	1 · x₂ · x₂ + 0 + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	1
[right#1^#(x₁)]	=	1
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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

#greater^#(z0,z1)

→

c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))

(98)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	1
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	2 · x₁ + 0
[firstline#1^#(x₁)]	=	2 · x₁ + 0
[lcs^#(x₁, x₂)]	=	1 + 2 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcstable#1^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	1
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	1
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	1
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	1
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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

max^#(z0,z1)

→

c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))

(129)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 2 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 · x₁ + 0 + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#1^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0
[max^#(x₁, x₂)]	=	1
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	1
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	1
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	1
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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

right^#(z0)

→

c84(right#1^#(z0))

(157)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	0
[right#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	1 + 2 · x₁ + 2 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	1 · x₁ · x₂ + 0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂ + 2 · x₂ · x₂ + 2 · x₁ · x₂ + 1 · x₁ · x₁
[lcs#1^#(x₁)]	=	2
[lcs#2^#(x₁)]	=	2
[lcs#3^#(x₁)]	=	1
[lcstable^#(x₁, x₂)]	=	2 · x₂ + 0 + 2 · x₁ · x₂
[lcstable#1^#(x₁, x₂)]	=	2 · x₁ · x₂ + 0
[lcstable#2^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[lcstable#3^#(x₁, x₂, x₃)]	=	2 · x₂ + 0
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	2 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	2 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	2 + 2 · x₃
[newline#3^#(x₁,...,x₅)]	=	2
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	0
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	0
[right^#(x₁)]	=	1
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	2
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	2 + 1 · x₂
[nil]	=	0

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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

+^#(z0,z1)

→

c54(#add^#(z0,z1))

(100)

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

[c]	=	0
[c1(x₁)]	=	1 · x₁ + 0
[c2(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c3(x₁)]	=	1 · x₁ + 0
[c4(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9]	=	0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13]	=	0
[c14]	=	0
[c15]	=	0
[c16]	=	0
[c17(x₁)]	=	1 · x₁ + 0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21(x₁)]	=	1 · x₁ + 0
[c22]	=	0
[c23(x₁)]	=	1 · x₁ + 0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29(x₁)]	=	1 · x₁ + 0
[c30]	=	0
[c31]	=	0
[c32]	=	0
[c33(x₁)]	=	1 · x₁ + 0
[c34]	=	0
[c35(x₁)]	=	1 · x₁ + 0
[c36(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c37]	=	0
[c38]	=	0
[c39]	=	0
[c40]	=	0
[c41]	=	0
[c42]	=	0
[c43]	=	0
[c44]	=	0
[c45]	=	0
[c46]	=	0
[c47]	=	0
[c48]	=	0
[c49]	=	0
[c50]	=	0
[c51]	=	0
[c52(x₁)]	=	1 · x₁ + 0
[c53(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c57]	=	0
[c58(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c59(x₁)]	=	1 · x₁ + 0
[c60(x₁)]	=	1 · x₁ + 0
[c61(x₁)]	=	1 · x₁ + 0
[c62]	=	0
[c63(x₁)]	=	1 · x₁ + 0
[c64(x₁)]	=	1 · x₁ + 0
[c65(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c66(x₁)]	=	1 · x₁ + 0
[c67(x₁)]	=	1 · x₁ + 0
[c68(x₁)]	=	1 · x₁ + 0
[c69]	=	0
[c70(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c71]	=	0
[c72]	=	0
[c73(x₁)]	=	1 · x₁ + 0
[c74(x₁)]	=	1 · x₁ + 0
[c75]	=	0
[c76(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c77]	=	0
[c78(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c79(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c80(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c81]	=	0
[c82(x₁)]	=	1 · x₁ + 0
[c83(x₁)]	=	1 · x₁ + 0
[c84(x₁)]	=	1 · x₁ + 0
[c85]	=	0
[c86(x₁)]	=	1 · x₁ + 0
[#add(x₁, x₂)]	=	0
[#succ(x₁)]	=	1
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	1
[#compare(x₁, x₂)]	=	0
[lcstable(x₁, x₂)]	=	0
[lcstable#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[lcstable#3(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline(x₁, x₂, x₃)]	=	0
[newline#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#2(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[newline#3(x₁,...,x₅)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₁ · x₅ + 1 · x₁ · x₁ + 1 · x₂ · x₁ + 1 · x₃ · x₁ + 1 · x₄ · x₁ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₂ · x₄ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₃ · x₃
[newline#4(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[right(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[right#1(x₁)]	=	2 · x₁ + 0 + 2 · x₁ · x₁
[#abs(x₁)]	=	0
[newline#5(x₁,...,x₆)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₆ · x₆ + 1 · x₅ · x₆ + 1 · x₄ · x₆ + 1 · x₃ · x₆ + 1 · x₂ · x₆ + 1 · x₅ · x₅ + 1 · x₄ · x₅ + 1 · x₃ · x₅ + 1 · x₂ · x₅ + 1 · x₂ · x₂ + 1 · x₃ · x₂ + 1 · x₄ · x₂ + 1 · x₄ · x₄ + 1 · x₃ · x₄ + 1 · x₃ · x₃
[newline#6(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₂ · x₂
[newline#7(x₁,...,x₄)]	=	0
[#equal(x₁, x₂)]	=	0
[max(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[max#1(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₃ · x₃ + 1 · x₂ · x₃ + 1 · x₂ · x₂
[#greater(x₁, x₂)]	=	0
[#ckgt(x₁)]	=	1
[+(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₂ · x₂ + 1 · x₁ · x₂ + 1 · x₁ · x₁
[#pred(x₁)]	=	1
[firstline(x₁)]	=	2 + 1 · x₁ + 1 · x₁ · x₁
[firstline#1(x₁)]	=	1 + 1 · x₁ + 1 · x₁ · x₁
[#add^#(x₁, x₂)]	=	0
[#and^#(x₁, x₂)]	=	0
[#ckgt^#(x₁)]	=	0
[#compare^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#pred^#(x₁)]	=	0
[#succ^#(x₁)]	=	0
[#abs^#(x₁)]	=	2 · x₁ + 0 + 1 · x₁ · x₁
[#equal^#(x₁, x₂)]	=	0
[#greater^#(x₁, x₂)]	=	0
[+^#(x₁, x₂)]	=	1
[firstline^#(x₁)]	=	0
[firstline#1^#(x₁)]	=	0
[lcs^#(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂ + 1 · x₂ · x₂ + 2 · x₁ · x₂ + 2 · x₁ · x₁
[lcs#1^#(x₁)]	=	0
[lcs#2^#(x₁)]	=	0
[lcs#3^#(x₁)]	=	0
[lcstable^#(x₁, x₂)]	=	2 · x₂ + 0 + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#1^#(x₁, x₂)]	=	2 · x₂ + 0 + 1 · x₁ · x₂ + 1 · x₁ · x₁
[lcstable#2^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 2 · x₃
[lcstable#3^#(x₁, x₂, x₃)]	=	1 · x₂ + 0 + 1 · x₃
[max^#(x₁, x₂)]	=	0
[max#1^#(x₁, x₂, x₃)]	=	0
[newline^#(x₁, x₂, x₃)]	=	1 · x₃ + 0
[newline#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[newline#2^#(x₁,...,x₄)]	=	1 + 1 · x₃
[newline#3^#(x₁,...,x₅)]	=	1
[newline#4^#(x₁,...,x₆)]	=	1
[newline#5^#(x₁,...,x₆)]	=	1
[newline#6^#(x₁, x₂)]	=	0
[newline#7^#(x₁,...,x₄)]	=	1
[right^#(x₁)]	=	0
[right#1^#(x₁)]	=	0
[#false]	=	0
[#true]	=	0
[#pos(x₁)]	=	0
[#s(x₁)]	=	0
[#0]	=	0
[#neg(x₁)]	=	0
[#EQ]	=	0
[#GT]	=	0
[#LT]	=	0
[::(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[nil]	=	2

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

#add^#(#0,z0)	→	c	(162)
#add^#(#neg(#s(#0)),z0)	→	c1(#pred^#(z0))	(164)
#add^#(#neg(#s(#s(z0))),z1)	→	c2(#pred^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(166)
#add^#(#pos(#s(#0)),z0)	→	c3(#succ^#(z0))	(168)
#add^#(#pos(#s(#s(z0))),z1)	→	c4(#succ^#(#add(#pos(#s(z0)),z1)),#add^#(#pos(#s(z0)),z1))	(170)
#and^#(#false,#false)	→	c5	(171)
#and^#(#false,#true)	→	c6	(172)
#and^#(#true,#false)	→	c7	(173)
#and^#(#true,#true)	→	c8	(174)
#ckgt^#(#EQ)	→	c9	(175)
#ckgt^#(#GT)	→	c10	(176)
#ckgt^#(#LT)	→	c11	(177)
#compare^#(#0,#0)	→	c12	(178)
#compare^#(#0,#neg(z0))	→	c13	(180)
#compare^#(#0,#pos(z0))	→	c14	(182)
#compare^#(#0,#s(z0))	→	c15	(184)
#compare^#(#neg(z0),#0)	→	c16	(186)
#compare^#(#neg(z0),#neg(z1))	→	c17(#compare^#(z1,z0))	(188)
#compare^#(#neg(z0),#pos(z1))	→	c18	(190)
#compare^#(#pos(z0),#0)	→	c19	(192)
#compare^#(#pos(z0),#neg(z1))	→	c20	(194)
#compare^#(#pos(z0),#pos(z1))	→	c21(#compare^#(z0,z1))	(196)
#compare^#(#s(z0),#0)	→	c22	(198)
#compare^#(#s(z0),#s(z1))	→	c23(#compare^#(z0,z1))	(200)
#eq^#(#0,#0)	→	c24	(201)
#eq^#(#0,#neg(z0))	→	c25	(203)
#eq^#(#0,#pos(z0))	→	c26	(205)
#eq^#(#0,#s(z0))	→	c27	(207)
#eq^#(#neg(z0),#0)	→	c28	(209)
#eq^#(#neg(z0),#neg(z1))	→	c29(#eq^#(z0,z1))	(211)
#eq^#(#neg(z0),#pos(z1))	→	c30	(213)
#eq^#(#pos(z0),#0)	→	c31	(215)
#eq^#(#pos(z0),#neg(z1))	→	c32	(217)
#eq^#(#pos(z0),#pos(z1))	→	c33(#eq^#(z0,z1))	(219)
#eq^#(#s(z0),#0)	→	c34	(221)
#eq^#(#s(z0),#s(z1))	→	c35(#eq^#(z0,z1))	(223)
#eq^#(::(z0,z1),::(z2,z3))	→	c36(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(225)
#eq^#(::(z0,z1),nil)	→	c37	(227)
#eq^#(nil,::(z0,z1))	→	c38	(229)
#eq^#(nil,nil)	→	c39	(230)
#pred^#(#0)	→	c40	(231)
#pred^#(#neg(#s(z0)))	→	c41	(233)
#pred^#(#pos(#s(#0)))	→	c42	(234)
#pred^#(#pos(#s(#s(z0))))	→	c43	(236)
#succ^#(#0)	→	c44	(237)
#succ^#(#neg(#s(#0)))	→	c45	(238)
#succ^#(#neg(#s(#s(z0))))	→	c46	(240)
#succ^#(#pos(#s(z0)))	→	c47	(242)
#abs^#(#0)	→	c48	(88)
#abs^#(#neg(z0))	→	c49	(90)
#abs^#(#pos(z0))	→	c50	(92)
#abs^#(#s(z0))	→	c51	(94)
#equal^#(z0,z1)	→	c52(#eq^#(z0,z1))	(96)
#greater^#(z0,z1)	→	c53(#ckgt^#(#compare(z0,z1)),#compare^#(z0,z1))	(98)
+^#(z0,z1)	→	c54(#add^#(z0,z1))	(100)
firstline^#(z0)	→	c55(firstline#1^#(z0))	(102)
firstline#1^#(::(z0,z1))	→	c56(#abs^#(#0),firstline^#(z1))	(104)
firstline#1^#(nil)	→	c57	(105)
lcs^#(z0,z1)	→	c58(lcs#1^#(lcstable(z0,z1)),lcstable^#(z0,z1))	(107)
lcs#1^#(z0)	→	c59(lcs#2^#(z0))	(109)
lcs#2^#(::(z0,z1))	→	c60(lcs#3^#(z0))	(111)
lcs#2^#(nil)	→	c61(#abs^#(#0))	(112)
lcs#3^#(::(z0,z1))	→	c62	(114)
lcs#3^#(nil)	→	c63(#abs^#(#0))	(115)
lcstable^#(z0,z1)	→	c64(lcstable#1^#(z0,z1))	(117)
lcstable#1^#(::(z0,z1),z2)	→	c65(lcstable#2^#(lcstable(z1,z2),z2,z0),lcstable^#(z1,z2))	(119)
lcstable#1^#(nil,z0)	→	c66(firstline^#(z0))	(121)
lcstable#2^#(z0,z1,z2)	→	c67(lcstable#3^#(z0,z1,z2))	(123)
lcstable#3^#(::(z0,z1),z2,z3)	→	c68(newline^#(z3,z0,z2))	(125)
lcstable#3^#(nil,z0,z1)	→	c69	(127)
max^#(z0,z1)	→	c70(max#1^#(#greater(z0,z1),z0,z1),#greater^#(z0,z1))	(129)
max#1^#(#false,z0,z1)	→	c71	(131)
max#1^#(#true,z0,z1)	→	c72	(133)
newline^#(z0,z1,z2)	→	c73(newline#1^#(z2,z1,z0))	(135)
newline#1^#(::(z0,z1),z2,z3)	→	c74(newline#2^#(z2,z0,z1,z3))	(137)
newline#1^#(nil,z0,z1)	→	c75	(139)
newline#2^#(::(z0,z1),z2,z3,z4)	→	c76(newline#3^#(newline(z4,z1,z3),z0,z1,z2,z4),newline^#(z4,z1,z3))	(141)
newline#2^#(nil,z0,z1,z2)	→	c77	(143)
newline#3^#(z0,z1,z2,z3,z4)	→	c78(newline#4^#(right(z0),z1,z2,z0,z3,z4),right^#(z0))	(145)
newline#4^#(z0,z1,z2,z3,z4,z5)	→	c79(newline#5^#(right(z2),z1,z3,z0,z4,z5),right^#(z2))	(147)
newline#5^#(z0,z1,z2,z3,z4,z5)	→	c80(newline#6^#(newline#7(#equal(z4,z5),z1,z0,z3),z2),newline#7^#(#equal(z4,z5),z1,z0,z3),#equal^#(z4,z5))	(149)
newline#6^#(z0,z1)	→	c81	(151)
newline#7^#(#false,z0,z1,z2)	→	c82(max^#(z0,z2))	(153)
newline#7^#(#true,z0,z1,z2)	→	c83(+^#(z1,#pos(#s(#0))))	(155)
right^#(z0)	→	c84(right#1^#(z0))	(157)
right#1^#(::(z0,z1))	→	c85	(159)
right#1^#(nil)	→	c86(#abs^#(#0))	(160)

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).