Certification Problem

Input (TPDB Runtime_Complexity_Innermost_Rewriting/raML/bfs.raml)

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

#equal(@x,@y)	→	#eq(@x,@y)	(1)
appendreverse(@toreverse,@sofar)	→	appendreverse#1(@toreverse,@sofar)	(2)
appendreverse#1(::(@a,@as),@sofar)	→	appendreverse(@as,::(@a,@sofar))	(3)
appendreverse#1(nil,@sofar)	→	@sofar	(4)
bfs(@queue,@futurequeue,@x)	→	bfs#1(@queue,@futurequeue,@x)	(5)
bfs#1(::(@t,@ts),@futurequeue,@x)	→	bfs#3(@t,@futurequeue,@ts,@x)	(6)
bfs#1(nil,@futurequeue,@x)	→	bfs#2(@futurequeue,@x)	(7)
bfs#2(::(@t,@ts),@x)	→	bfs(reverse(::(@t,@ts)),nil,@x)	(8)
bfs#2(nil,@x)	→	leaf	(9)
bfs#3(leaf,@futurequeue,@ts,@x)	→	bfs(@ts,@futurequeue,@x)	(10)
bfs#3(node(@y,@t1,@t2),@futurequeue,@ts,@x)	→	bfs#4(#equal(@x,@y),@futurequeue,@t1,@t2,@ts,@x,@y)	(11)
bfs#4(#false,@futurequeue,@t1,@t2,@ts,@x,@y)	→	bfs(@ts,::(@t2,::(@t1,@futurequeue)),@x)	(12)
bfs#4(#true,@futurequeue,@t1,@t2,@ts,@x,@y)	→	node(@y,@t1,@t2)	(13)
bfs2(@t,@x)	→	bfs2#1(dobfs(@t,@x),@x)	(14)
bfs2#1(@t',@x)	→	dobfs(@t',@x)	(15)
dfs(@queue,@x)	→	dfs#1(@queue,@x)	(16)
dfs#1(::(@t,@ts),@x)	→	dfs#2(@t,@t,@ts,@x)	(17)
dfs#1(nil,@x)	→	leaf	(18)
dfs#2(leaf,@t,@ts,@x)	→	dfs(@ts,@x)	(19)
dfs#2(node(@a,@t1,@t2),@t,@ts,@x)	→	dfs#3(#equal(@a,@x),@t,@t1,@t2,@ts,@x)	(20)
dfs#3(#false,@t,@t1,@t2,@ts,@x)	→	dfs(::(@t1,::(@t2,@ts)),@x)	(21)
dfs#3(#true,@t,@t1,@t2,@ts,@x)	→	@t	(22)
dobfs(@t,@x)	→	bfs(::(@t,nil),nil,@x)	(23)
dodfs(@t,@x)	→	dfs(::(@t,nil),@x)	(24)
reverse(@xs)	→	appendreverse(@xs,nil)	(25)

and S is the following TRS.

#and(#false,#false)	→	#false	(26)
#and(#false,#true)	→	#false	(27)
#and(#true,#false)	→	#false	(28)
#and(#true,#true)	→	#true	(29)
#eq(#0,#0)	→	#true	(30)
#eq(#0,#neg(@y))	→	#false	(31)
#eq(#0,#pos(@y))	→	#false	(32)
#eq(#0,#s(@y))	→	#false	(33)
#eq(#neg(@x),#0)	→	#false	(34)
#eq(#neg(@x),#neg(@y))	→	#eq(@x,@y)	(35)
#eq(#neg(@x),#pos(@y))	→	#false	(36)
#eq(#pos(@x),#0)	→	#false	(37)
#eq(#pos(@x),#neg(@y))	→	#false	(38)
#eq(#pos(@x),#pos(@y))	→	#eq(@x,@y)	(39)
#eq(#s(@x),#0)	→	#false	(40)
#eq(#s(@x),#s(@y))	→	#eq(@x,@y)	(41)
#eq(::(@x_1,@x_2),::(@y_1,@y_2))	→	#and(#eq(@x_1,@y_1),#eq(@x_2,@y_2))	(42)
#eq(::(@x_1,@x_2),leaf)	→	#false	(43)
#eq(::(@x_1,@x_2),nil)	→	#false	(44)
#eq(::(@x_1,@x_2),node(@y_1,@y_2,@y_3))	→	#false	(45)
#eq(leaf,::(@y_1,@y_2))	→	#false	(46)
#eq(leaf,leaf)	→	#true	(47)
#eq(leaf,nil)	→	#false	(48)
#eq(leaf,node(@y_1,@y_2,@y_3))	→	#false	(49)
#eq(nil,::(@y_1,@y_2))	→	#false	(50)
#eq(nil,leaf)	→	#false	(51)
#eq(nil,nil)	→	#true	(52)
#eq(nil,node(@y_1,@y_2,@y_3))	→	#false	(53)
#eq(node(@x_1,@x_2,@x_3),::(@y_1,@y_2))	→	#false	(54)
#eq(node(@x_1,@x_2,@x_3),leaf)	→	#false	(55)
#eq(node(@x_1,@x_2,@x_3),nil)	→	#false	(56)
#eq(node(@x_1,@x_2,@x_3),node(@y_1,@y_2,@y_3))	→	#and(#eq(@x_1,@y_1),#and(#eq(@x_2,@y_2),#eq(@x_3,@y_3)))	(57)

The evaluation strategy is innermost.

Property / Task

Determine bounds on the runtime complexity.

Answer / Result

An upperbound for the complexity is O(n³).

Proof (by AProVE @ termCOMP 2023)

1 Dependency Tuples

We get the following set of dependency tuples:

#equal^#(z0,z1)

→

c32(#eq^#(z0,z1))

(59)

originates from

#equal(z0,z1)

→

#eq(z0,z1)

(58)

appendreverse^#(z0,z1)

→

c33(appendreverse#1^#(z0,z1))

(61)

originates from

appendreverse(z0,z1)

→

appendreverse#1(z0,z1)

(60)

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

→

c34(appendreverse^#(z1,::(z0,z2)))

(63)

originates from

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

→

appendreverse(z1,::(z0,z2))

(62)

appendreverse#1^#(nil,z0)

→

c35

(65)

originates from

appendreverse#1(nil,z0)

→

(64)

bfs^#(z0,z1,z2)

→

c36(bfs#1^#(z0,z1,z2))

(67)

originates from

bfs(z0,z1,z2)

→

bfs#1(z0,z1,z2)

(66)

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

→

c37(bfs#3^#(z0,z2,z1,z3))

(69)

originates from

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

→

bfs#3(z0,z2,z1,z3)

(68)

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

→

c38(bfs#2^#(z0,z1))

(71)

originates from

bfs#1(nil,z0,z1)

→

bfs#2(z0,z1)

(70)

bfs#2^#(::(z0,z1),z2)

→

c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))

(73)

originates from

bfs#2(::(z0,z1),z2)

→

bfs(reverse(::(z0,z1)),nil,z2)

(72)

bfs#2^#(nil,z0)

→

c40

(75)

originates from

bfs#2(nil,z0)

→

leaf

(74)

bfs#3^#(leaf,z0,z1,z2)

→

c41(bfs^#(z1,z0,z2))

(77)

originates from

bfs#3(leaf,z0,z1,z2)

→

bfs(z1,z0,z2)

(76)

bfs#3^#(node(z0,z1,z2),z3,z4,z5)

→

c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))

(79)

originates from

bfs#3(node(z0,z1,z2),z3,z4,z5)

→

bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)

(78)

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

→

c43(bfs^#(z3,::(z2,::(z1,z0)),z4))

(81)

originates from

bfs#4(#false,z0,z1,z2,z3,z4,z5)

→

bfs(z3,::(z2,::(z1,z0)),z4)

(80)

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

→

c44

(83)

originates from

bfs#4(#true,z0,z1,z2,z3,z4,z5)

→

node(z5,z1,z2)

(82)

bfs2^#(z0,z1)

→

c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))

(85)

originates from

bfs2(z0,z1)

→

bfs2#1(dobfs(z0,z1),z1)

(84)

bfs2#1^#(z0,z1)

→

c46(dobfs^#(z0,z1))

(87)

originates from

bfs2#1(z0,z1)

→

dobfs(z0,z1)

(86)

dfs^#(z0,z1)

→

c47(dfs#1^#(z0,z1))

(89)

originates from

dfs(z0,z1)

→

dfs#1(z0,z1)

(88)

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

→

c48(dfs#2^#(z0,z0,z1,z2))

(91)

originates from

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

→

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

(90)

dfs#1^#(nil,z0)

→

c49

(93)

originates from

dfs#1(nil,z0)

→

leaf

(92)

dfs#2^#(leaf,z0,z1,z2)

→

c50(dfs^#(z1,z2))

(95)

originates from

dfs#2(leaf,z0,z1,z2)

→

dfs(z1,z2)

(94)

dfs#2^#(node(z0,z1,z2),z3,z4,z5)

→

c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))

(97)

originates from

dfs#2(node(z0,z1,z2),z3,z4,z5)

→

dfs#3(#equal(z0,z5),z3,z1,z2,z4,z5)

(96)

dfs#3^#(#false,z0,z1,z2,z3,z4)

→

c52(dfs^#(::(z1,::(z2,z3)),z4))

(99)

originates from

dfs#3(#false,z0,z1,z2,z3,z4)

→

dfs(::(z1,::(z2,z3)),z4)

(98)

dfs#3^#(#true,z0,z1,z2,z3,z4)

→

c53

(101)

originates from

dfs#3(#true,z0,z1,z2,z3,z4)

→

(100)

dobfs^#(z0,z1)

→

c54(bfs^#(::(z0,nil),nil,z1))

(103)

originates from

dobfs(z0,z1)

→

bfs(::(z0,nil),nil,z1)

(102)

dodfs^#(z0,z1)

→

c55(dfs^#(::(z0,nil),z1))

(105)

originates from

dodfs(z0,z1)

→

dfs(::(z0,nil),z1)

(104)

reverse^#(z0)

→

c56(appendreverse^#(z0,nil))

(107)

originates from

reverse(z0)

→

appendreverse(z0,nil)

(106)

#and^#(#false,#false)

→

(108)

originates from

#and(#false,#false)

→

#false

(26)

#and^#(#false,#true)

→

(109)

originates from

#and(#false,#true)

→

#false

(27)

#and^#(#true,#false)

→

(110)

originates from

#and(#true,#false)

→

#false

(28)

#and^#(#true,#true)

→

(111)

originates from

#and(#true,#true)

→

#true

(29)

#eq^#(#0,#0)

→

(112)

originates from

#eq(#0,#0)

→

#true

(30)

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

→

(114)

originates from

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

→

#false

(113)

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

→

(116)

originates from

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

→

#false

(115)

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

→

(118)

originates from

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

→

#false

(117)

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

→

(120)

originates from

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

→

#false

(119)

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

→

c9(#eq^#(z0,z1))

(122)

originates from

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

→

#eq(z0,z1)

(121)

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

→

c10

(124)

originates from

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

→

#false

(123)

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

→

c11

(126)

originates from

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

→

#false

(125)

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

→

c12

(128)

originates from

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

→

#false

(127)

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

→

c13(#eq^#(z0,z1))

(130)

originates from

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

→

#eq(z0,z1)

(129)

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

→

c14

(132)

originates from

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

→

#false

(131)

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

→

c15(#eq^#(z0,z1))

(134)

originates from

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

→

#eq(z0,z1)

(133)

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

→

c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))

(136)

originates from

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

→

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

(135)

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

→

c17

(138)

originates from

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

→

#false

(137)

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

→

c18

(140)

originates from

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

→

#false

(139)

#eq^#(::(z0,z1),node(z2,z3,z4))

→

c19

(142)

originates from

#eq(::(z0,z1),node(z2,z3,z4))

→

#false

(141)

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

→

c20

(144)

originates from

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

→

#false

(143)

#eq^#(leaf,leaf)

→

c21

(145)

originates from

#eq(leaf,leaf)

→

#true

(47)

#eq^#(leaf,nil)

→

c22

(146)

originates from

#eq(leaf,nil)

→

#false

(48)

#eq^#(leaf,node(z0,z1,z2))

→

c23

(148)

originates from

#eq(leaf,node(z0,z1,z2))

→

#false

(147)

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

→

c24

(150)

originates from

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

→

#false

(149)

#eq^#(nil,leaf)

→

c25

(151)

originates from

#eq(nil,leaf)

→

#false

(51)

#eq^#(nil,nil)

→

c26

(152)

originates from

#eq(nil,nil)

→

#true

(52)

#eq^#(nil,node(z0,z1,z2))

→

c27

(154)

originates from

#eq(nil,node(z0,z1,z2))

→

#false

(153)

#eq^#(node(z0,z1,z2),::(z3,z4))

→

c28

(156)

originates from

#eq(node(z0,z1,z2),::(z3,z4))

→

#false

(155)

#eq^#(node(z0,z1,z2),leaf)

→

c29

(158)

originates from

#eq(node(z0,z1,z2),leaf)

→

#false

(157)

#eq^#(node(z0,z1,z2),nil)

→

c30

(160)

originates from

#eq(node(z0,z1,z2),nil)

→

#false

(159)

#eq^#(node(z0,z1,z2),node(z3,z4,z5))

→

c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))

(162)

originates from

#eq(node(z0,z1,z2),node(z3,z4,z5))

→

#and(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5)))

(161)

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

#and^#(#false,#false)

#and^#(#false,#true)

#and^#(#true,#false)

#and^#(#true,#true)

#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),leaf)

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

#eq^#(::(z0,z1),node(z2,z3,z4))

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

#eq^#(leaf,leaf)

#eq^#(leaf,nil)

#eq^#(leaf,node(z0,z1,z2))

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

#eq^#(nil,leaf)

#eq^#(nil,nil)

#eq^#(nil,node(z0,z1,z2))

#eq^#(node(z0,z1,z2),::(z3,z4))

#eq^#(node(z0,z1,z2),leaf)

#eq^#(node(z0,z1,z2),nil)

#eq^#(node(z0,z1,z2),node(z3,z4,z5))

#equal^#(z0,z1)

appendreverse^#(z0,z1)

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

appendreverse#1^#(nil,z0)

bfs^#(z0,z1,z2)

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

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

bfs#2^#(::(z0,z1),z2)

bfs#2^#(nil,z0)

bfs#3^#(leaf,z0,z1,z2)

bfs#3^#(node(z0,z1,z2),z3,z4,z5)

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

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

bfs2^#(z0,z1)

bfs2#1^#(z0,z1)

dfs^#(z0,z1)

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

dfs#1^#(nil,z0)

dfs#2^#(leaf,z0,z1,z2)

dfs#2^#(node(z0,z1,z2),z3,z4,z5)

dfs#3^#(#false,z0,z1,z2,z3,z4)

dfs#3^#(#true,z0,z1,z2,z3,z4)

dobfs^#(z0,z1)

dodfs^#(z0,z1)

reverse^#(z0)

1.1 Usable Rules

We remove the following rules since they are not usable.

bfs2(z0,z1)	→	bfs2#1(dobfs(z0,z1),z1)	(84)
bfs2#1(z0,z1)	→	dobfs(z0,z1)	(86)
dfs(z0,z1)	→	dfs#1(z0,z1)	(88)
dfs#1(::(z0,z1),z2)	→	dfs#2(z0,z0,z1,z2)	(90)
dfs#1(nil,z0)	→	leaf	(92)
dfs#2(leaf,z0,z1,z2)	→	dfs(z1,z2)	(94)
dfs#2(node(z0,z1,z2),z3,z4,z5)	→	dfs#3(#equal(z0,z5),z3,z1,z2,z4,z5)	(96)
dfs#3(#false,z0,z1,z2,z3,z4)	→	dfs(::(z1,::(z2,z3)),z4)	(98)
dfs#3(#true,z0,z1,z2,z3,z4)	→	z0	(100)
dodfs(z0,z1)	→	dfs(::(z0,nil),z1)	(104)

1.1.1 Rule Shifting

The rules

bfs2#1^#(z0,z1)

→

c46(dobfs^#(z0,z1))

(87)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	2 · x₂ + 0
[#and(x₁, x₂)]	=	2
[reverse(x₁)]	=	0
[appendreverse(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[appendreverse#1(x₁, x₂)]	=	3 + 3 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[bfs(x₁, x₂, x₃)]	=	3 + 3 · x₃
[bfs#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[bfs#3(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[bfs#4(x₁,...,x₇)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆ + 3 · x₇
[bfs#2(x₁, x₂)]	=	3 + 3 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	2 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	2 + 1 · x₂
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 · x₁ + 0
[reverse^#(x₁)]	=	0
[#0]	=	0
[#true]	=	0
[#neg(x₁)]	=	1 · x₁ + 0
[#false]	=	0
[#pos(x₁)]	=	1 · x₁ + 0
[#s(x₁)]	=	1 · x₁ + 0
[::(x₁, x₂)]	=	2
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	2 + 1 · x₁

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1 Rule Shifting

The rules

dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs#1^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs#2^#(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₃ + 1 · x₄
[dfs#3^#(x₁,...,x₆)]	=	1 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	0
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1 Rule Shifting

The rules

bfs2^#(z0,z1)

→

c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))

(85)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	1 · x₂ + 0
[appendreverse#1^#(x₁, x₂)]	=	1 · x₂ + 0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 · x₂ + 0
[dfs^#(x₁, x₂)]	=	1 · x₂ + 0
[dfs#1^#(x₁, x₂)]	=	1 · x₂ + 0
[dfs#2^#(x₁,...,x₄)]	=	1 · x₄ + 0
[dfs#3^#(x₁,...,x₆)]	=	1 · x₆ + 0
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 · x₂ + 0
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	1
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	0
[leaf]	=	1
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1 Rule Shifting

The rules

dodfs^#(z0,z1)

→

c55(dfs^#(::(z0,nil),z1))

(105)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	1 · x₂ + 0
[appendreverse#1^#(x₁, x₂)]	=	1 · x₂ + 0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 · x₂ + 0
[dfs#1^#(x₁, x₂)]	=	1 · x₂ + 0
[dfs#2^#(x₁,...,x₄)]	=	1 · x₄ + 0
[dfs#3^#(x₁,...,x₆)]	=	1 · x₆ + 0
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	0
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	0
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1 Rule Shifting

The rules

dfs^#(z0,z1)

→

c47(dfs#1^#(z0,z1))

(89)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs#2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[dfs#3^#(x₁,...,x₆)]	=	1 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	0
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1.1 Rule Shifting

The rules

dfs#2^#(node(z0,z1,z2),z3,z4,z5)

→

c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))

(97)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs#2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[dfs#3^#(x₁,...,x₆)]	=	1 · x₃ + 0 + 1 · x₄ + 1 · x₅ + 1 · x₆
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	1
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

dfs#3^#(#false,z0,z1,z2,z3,z4)

→

c52(dfs^#(::(z1,::(z2,z3)),z4))

(99)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1
[appendreverse(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs#2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[dfs#3^#(x₁,...,x₆)]	=	1 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	0
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c48(dfs#2^#(z0,z0,z1,z2))

(91)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1
[reverse(x₁)]	=	1 + 1 · x₁
[appendreverse(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#equal(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dobfs(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	1 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₂
[dfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs#1^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs#2^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₃ + 1 · x₄
[dfs#3^#(x₁,...,x₆)]	=	1 + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₆
[dobfs^#(x₁, x₂)]	=	0
[dodfs^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	1
[#true]	=	1
[#neg(x₁)]	=	1 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	1 + 1 · x₁
[#s(x₁)]	=	1 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

dobfs^#(z0,z1)

→

c54(bfs^#(::(z0,nil),nil,z1))

(103)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	0
[appendreverse(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[appendreverse#1(x₁, x₂)]	=	3 + 3 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	1 + 3 · x₁ + 3 · x₂
[bfs(x₁, x₂, x₃)]	=	3 + 3 · x₃
[bfs#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[bfs#3(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[bfs#4(x₁,...,x₇)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆ + 3 · x₇
[bfs#2(x₁, x₂)]	=	3 + 3 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	0
[bfs#1^#(x₁, x₂, x₃)]	=	0
[bfs#2^#(x₁, x₂)]	=	0
[bfs#3^#(x₁,...,x₄)]	=	0
[bfs#4^#(x₁,...,x₇)]	=	0
[bfs2^#(x₁, x₂)]	=	3 + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	2 + 1 · x₂
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	1
[dodfs^#(x₁, x₂)]	=	1 · x₂ + 0
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	0
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)

1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs#3^#(leaf,z0,z1,z2)

→

c41(bfs^#(z1,z0,z2))

(77)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	1 · x₁ + 0
[appendreverse(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	1 + 2 · x₁ + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 2 · x₂ + 1 · x₃
[bfs#1(x₁, x₂, x₃)]	=	1 + 2 · x₁ + 2 · x₂ + 1 · x₃
[bfs#3(x₁,...,x₄)]	=	1 + 2 · x₁ + 2 · x₂ + 2 · x₃ + 1 · x₄
[bfs#4(x₁,...,x₇)]	=	1 + 2 · x₂ + 2 · x₃ + 2 · x₄ + 2 · x₅ + 1 · x₆ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 + 2 · x₁ + 1 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#2^#(x₁, x₂)]	=	1 · x₁ + 0
[bfs#3^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4^#(x₁,...,x₇)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs2^#(x₁, x₂)]	=	1 + 3 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 · x₁ + 0
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	1 · x₁ + 0
[dodfs^#(x₁, x₂)]	=	1 + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	1
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs#2^#(nil,z0)

→

c40

(75)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	1
[appendreverse(x₁, x₂)]	=	1 · x₂ + 0
[appendreverse#1(x₁, x₂)]	=	1 · x₂ + 0
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	2 + 3 · x₁ + 3 · x₂
[bfs(x₁, x₂, x₃)]	=	3 + 3 · x₃
[bfs#1(x₁, x₂, x₃)]	=	3 + 3 · x₂ + 3 · x₃
[bfs#3(x₁,...,x₄)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄
[bfs#4(x₁,...,x₇)]	=	3 + 3 · x₂ + 3 · x₃ + 3 · x₄ + 3 · x₅ + 3 · x₆ + 3 · x₇
[bfs#2(x₁, x₂)]	=	3 + 3 · x₂
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0
[bfs#2^#(x₁, x₂)]	=	1
[bfs#3^#(x₁,...,x₄)]	=	1 · x₃ + 0
[bfs#4^#(x₁,...,x₇)]	=	1 · x₅ + 0
[bfs2^#(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	2 + 1 · x₂
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	1
[dodfs^#(x₁, x₂)]	=	1 + 2 · x₁ + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₂ + 0
[leaf]	=	0
[nil]	=	1
[node(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
appendreverse#1(nil,z0)	→	z0	(64)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)

1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs#3^#(node(z0,z1,z2),z3,z4,z5)

→

c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))

(79)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	1 · x₁ + 0
[appendreverse(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	2 · x₁ + 0
[bfs(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 2 · x₂
[bfs#1(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 2 · x₂
[bfs#3(x₁,...,x₄)]	=	2 · x₁ + 0 + 2 · x₂ + 2 · x₃
[bfs#4(x₁,...,x₇)]	=	1 + 2 · x₂ + 2 · x₃ + 2 · x₄ + 2 · x₅ + 1 · x₇
[bfs#2(x₁, x₂)]	=	2 · x₁ + 0
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#2^#(x₁, x₂)]	=	1 · x₁ + 0
[bfs#3^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4^#(x₁,...,x₇)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs2^#(x₁, x₂)]	=	1 + 3 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs^#(x₁, x₂)]	=	2 · x₁ + 0
[dfs#1^#(x₁, x₂)]	=	2 · x₁ + 0
[dfs#2^#(x₁,...,x₄)]	=	2 · x₁ + 0 + 2 · x₃
[dfs#3^#(x₁,...,x₆)]	=	2 · x₃ + 0 + 2 · x₄ + 2 · x₅
[dobfs^#(x₁, x₂)]	=	1 · x₁ + 0
[dodfs^#(x₁, x₂)]	=	1 + 3 · x₁ + 3 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

#equal^#(z0,z1)

→

c32(#eq^#(z0,z1))

(59)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	1 · x₁ + 0
[appendreverse(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	1 + 1 · x₁
[bfs(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#3(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs#2(x₁, x₂)]	=	1 · x₁ + 0
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	1
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#2^#(x₁, x₂)]	=	1 · x₁ + 0
[bfs#3^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4^#(x₁,...,x₇)]	=	1 · x₂ + 0 + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs2^#(x₁, x₂)]	=	1 + 3 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[dfs^#(x₁, x₂)]	=	2 · x₁ + 0
[dfs#1^#(x₁, x₂)]	=	2 · x₁ + 0
[dfs#2^#(x₁,...,x₄)]	=	2 · x₁ + 0 + 2 · x₃
[dfs#3^#(x₁,...,x₆)]	=	2 · x₃ + 0 + 2 · x₄ + 2 · x₅
[dobfs^#(x₁, x₂)]	=	1 · x₁ + 0
[dodfs^#(x₁, x₂)]	=	1 + 3 · x₁ + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	0
[#and(x₁, x₂)]	=	0
[reverse(x₁)]	=	1 · x₁ + 0
[appendreverse(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[#equal(x₁, x₂)]	=	0
[dobfs(x₁, x₂)]	=	2 · x₁ + 0 + 1 · x₂
[bfs(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 2 · x₂
[bfs#1(x₁, x₂, x₃)]	=	2 · x₁ + 0 + 2 · x₂
[bfs#3(x₁,...,x₄)]	=	2 · x₁ + 0 + 2 · x₂ + 2 · x₃
[bfs#4(x₁,...,x₇)]	=	1 + 2 · x₂ + 2 · x₃ + 2 · x₄ + 2 · x₅ + 1 · x₇
[bfs#2(x₁, x₂)]	=	2 · x₁ + 0
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#2^#(x₁, x₂)]	=	1 · x₁ + 0
[bfs#3^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4^#(x₁,...,x₇)]	=	1 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs2^#(x₁, x₂)]	=	1 + 3 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₁
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	1 · x₁ + 0
[dodfs^#(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[reverse^#(x₁)]	=	0
[#0]	=	3
[#true]	=	0
[#neg(x₁)]	=	3 + 1 · x₁
[#false]	=	0
[#pos(x₁)]	=	3 + 1 · x₁
[#s(x₁)]	=	3 + 1 · x₁
[::(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c37(bfs#3^#(z0,z2,z1,z3))

(69)

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

[c]	=	0
[c1]	=	0
[c2]	=	0
[c3]	=	0
[c4]	=	0
[c5]	=	0
[c6]	=	0
[c7]	=	0
[c8]	=	0
[c9(x₁)]	=	1 · x₁ + 0
[c10]	=	0
[c11]	=	0
[c12]	=	0
[c13(x₁)]	=	1 · x₁ + 0
[c14]	=	0
[c15(x₁)]	=	1 · x₁ + 0
[c16(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[c17]	=	0
[c18]	=	0
[c19]	=	0
[c20]	=	0
[c21]	=	0
[c22]	=	0
[c23]	=	0
[c24]	=	0
[c25]	=	0
[c26]	=	0
[c27]	=	0
[c28]	=	0
[c29]	=	0
[c30]	=	0
[c31(x₁,...,x₅)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[c32(x₁)]	=	1 · x₁ + 0
[c33(x₁)]	=	1 · x₁ + 0
[c34(x₁)]	=	1 · x₁ + 0
[c35]	=	0
[c36(x₁)]	=	1 · x₁ + 0
[c37(x₁)]	=	1 · x₁ + 0
[c38(x₁)]	=	1 · x₁ + 0
[c39(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c40]	=	0
[c41(x₁)]	=	1 · x₁ + 0
[c42(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c43(x₁)]	=	1 · x₁ + 0
[c44]	=	0
[c45(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c46(x₁)]	=	1 · x₁ + 0
[c47(x₁)]	=	1 · x₁ + 0
[c48(x₁)]	=	1 · x₁ + 0
[c49]	=	0
[c50(x₁)]	=	1 · x₁ + 0
[c51(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[c52(x₁)]	=	1 · x₁ + 0
[c53]	=	0
[c54(x₁)]	=	1 · x₁ + 0
[c55(x₁)]	=	1 · x₁ + 0
[c56(x₁)]	=	1 · x₁ + 0
[#eq(x₁, x₂)]	=	1
[#and(x₁, x₂)]	=	1 · x₁ + 0
[reverse(x₁)]	=	1 · x₁ + 0
[appendreverse(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[appendreverse#1(x₁, x₂)]	=	1 · x₁ + 0 + 1 · x₂
[#equal(x₁, x₂)]	=	1
[dobfs(x₁, x₂)]	=	1 + 1 · x₁
[bfs(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#3(x₁,...,x₄)]	=	1 + 1 · x₁ + 1 · x₂ + 1 · x₃
[bfs#4(x₁,...,x₇)]	=	3 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅ + 1 · x₇
[bfs#2(x₁, x₂)]	=	1 · x₁ + 0
[#and^#(x₁, x₂)]	=	0
[#eq^#(x₁, x₂)]	=	0
[#equal^#(x₁, x₂)]	=	0
[appendreverse^#(x₁, x₂)]	=	0
[appendreverse#1^#(x₁, x₂)]	=	0
[bfs^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#1^#(x₁, x₂, x₃)]	=	1 · x₁ + 0 + 1 · x₂
[bfs#2^#(x₁, x₂)]	=	1 · x₁ + 0
[bfs#3^#(x₁,...,x₄)]	=	1 · x₁ + 0 + 1 · x₂ + 1 · x₃
[bfs#4^#(x₁,...,x₇)]	=	2 · x₁ + 0 + 1 · x₂ + 1 · x₃ + 1 · x₄ + 1 · x₅
[bfs2^#(x₁, x₂)]	=	3 + 3 · x₁ + 1 · x₂
[bfs2#1^#(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[dfs^#(x₁, x₂)]	=	0
[dfs#1^#(x₁, x₂)]	=	0
[dfs#2^#(x₁,...,x₄)]	=	0
[dfs#3^#(x₁,...,x₆)]	=	0
[dobfs^#(x₁, x₂)]	=	1 + 1 · x₁
[dodfs^#(x₁, x₂)]	=	0
[reverse^#(x₁)]	=	0
[#0]	=	0
[#true]	=	1
[#neg(x₁)]	=	1 · x₁ + 0
[#false]	=	1
[#pos(x₁)]	=	1 · x₁ + 0
[#s(x₁)]	=	1 · x₁ + 0
[::(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[leaf]	=	0
[nil]	=	0
[node(x₁, x₂, x₃)]	=	3 + 1 · x₁ + 1 · x₂ + 1 · x₃

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

There are 104 ruless (increase limit for explicit display).

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs^#(z0,z1,z2)

→

c36(bfs#1^#(z0,z1,z2))

(67)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c47(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c32(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

0	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

3	0
0	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

0	0
0	0

· x₄ +

0	0
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
0	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
4	0

[dodfs^#(x₁, x₂)]

4	0
1	0

0	4
0	0

· x₁ +

1	4
0	1

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
0	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

1	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

1	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
0	0

[bfs#2^#(x₁, x₂)]

0	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

4	0
4	0

0	4
0	0

· x₁ +

1	4
0	0

· x₂

[bfs2#1^#(x₁, x₂)]

1	0
0	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

3	0
0	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
1	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[appendreverse(x₁, x₂)]

0	0
0	0

1	0
0	1

· x₁ +

1	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

0	0
1	0

1	0
0	1

· x₁ +

0	1
0	1

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

0	0
0	0

1	0
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

1	0
0	1

· x₁ +

1	0
0	1

· x₂

[::(x₁, x₂)]

1	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

bfs#2^#(::(z0,z1),z2)

→

c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))

(73)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
4	0

1	2
0	0

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c47(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c32(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
1	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

1	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

3	0
0	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

0	0
0	0

· x₄ +

0	0
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
0	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
0	0

[dodfs^#(x₁, x₂)]

4	0
4	0

1	3
0	1

· x₁ +

1	1
0	1

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
0	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

1	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
0	0

[bfs#2^#(x₁, x₂)]

1	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

4	0
4	0

0	4
0	0

· x₁ +

1	1
0	0

· x₂

[bfs2#1^#(x₁, x₂)]

1	0
1	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

3	0
0	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
0	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
1	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[appendreverse(x₁, x₂)]

1	0
0	0

3	0
0	1

· x₁ +

2	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

4	2
0	0

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

2	0
0	0

4	0
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

3	0
0	1

· x₁ +

2	0
0	1

· x₂

[::(x₁, x₂)]

1	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
1	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Rule Shifting

The rules

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

→

c38(bfs#2^#(z0,z1))

(71)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
4	0

1	0
0	1

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c47(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c32(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

0	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

3	0
0	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

0	0
0	0

· x₄ +

0	0
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
0	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
0	0

[dodfs^#(x₁, x₂)]

4	0
1	0

0	4
0	0

· x₁ +

1	4
0	1

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
4	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
4	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

1	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
2	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

1	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

1	0
2	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
0	0

[bfs#2^#(x₁, x₂)]

0	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

4	0
4	0

0	4
0	0

· x₁ +

1	4
0	0

· x₂

[bfs2#1^#(x₁, x₂)]

1	0
0	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

1	0
0	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

3	0
0	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
1	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
1	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[appendreverse(x₁, x₂)]

0	0
0	0

4	0
0	1

· x₁ +

1	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

0	0
0	0

4	4
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

4	0
0	1

· x₁ +

1	0
0	1

· x₂

[::(x₁, x₂)]

1	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
1	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
1	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

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

reverse^#(z0)

→

c56(appendreverse^#(z0,nil))

(107)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

1	0
1	0

0	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
2	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
4	0

1	0
0	1

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c47(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c32(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

0	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

4	0
0	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

0	0
0	0

· x₄ +

0	0
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
1	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
0	0

[dodfs^#(x₁, x₂)]

3	0
1	0

1	3
0	0

· x₁ +

1	1
0	1

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
1	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
1	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	2
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	1
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
0	0

[bfs#2^#(x₁, x₂)]

0	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

4	0
4	0

0	4
0	1

· x₁ +

1	4
0	0

· x₂

[bfs2#1^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

4	0
0	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
0	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[appendreverse(x₁, x₂)]

0	0
0	0

2	0
0	1

· x₁ +

1	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

2	0
0	0

0	0
0	1

· x₁ +

2	4
0	0

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

0	0
0	0

2	1
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

2	0
0	1

· x₁ +

1	0
0	1

· x₂

[::(x₁, x₂)]

2	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

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

appendreverse#1^#(nil,z0)

→

c35

(65)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

1	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

1	0
0	0

0	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
4	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c47(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c32(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

0	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

4	0
0	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂ +

0	0
0	0

· x₃ +

0	0
0	0

· x₄ +

0	0
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
0	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

0	0
4	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

1	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
0	0

[dodfs^#(x₁, x₂)]

2	0
1	0

0	1
0	1

· x₁ +

1	3
0	0

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
4	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
4	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
2	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

0	0
4	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

0	0
2	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
0	0

[bfs#2^#(x₁, x₂)]

0	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

1	0
4	0

0	4
0	0

· x₁ +

1	4
0	1

· x₂

[bfs2#1^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

4	0
0	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
0	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

4	2
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[appendreverse(x₁, x₂)]

0	0
0	0

4	0
0	1

· x₁ +

4	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

0	0
0	0

1	0
0	1

· x₁ +

0	0
0	0

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

0	0
0	0

4	0
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

4	0
0	1

· x₁ +

4	0
0	1

· x₂

[::(x₁, x₂)]

2	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
2	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

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

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

→

c34(appendreverse^#(z1,::(z0,z2)))

(63)

are strictly oriented by the following linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

0	0
0	0

· x₂

[reverse^#(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#and^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0
0	0

0	4
0	0

· x₁ +

0	0
0	0

· x₂ +

0	4
0	0

· x₃ +

1	0
0	0

· x₄

[c21]

0	0
0	0

[c5]

0	0
0	0

[c16(x₁, x₂, x₃)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃

[c17]

0	0
0	0

[c6]

0	0
0	0

[c51(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c20]

0	0
0	0

[c34(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c45(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c52(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c47(x₁)]

0	0
1	0

1	0
0	0

· x₁

[c32(x₁)]

0	0
1	0

1	0
0	1

· x₁

[c33(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c18]

0	0
0	0

[c46(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c7]

0	0
0	0

[#equal^#(x₁, x₂)]

0	0
1	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[c40]

0	0
0	0

[c23]

0	0
0	0

[c]

0	0
0	0

[c44]

2	0
1	0

[c8]

0	0
0	0

[c14]

0	0
0	0

[dfs#3^#(x₁,...,x₆)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	4
0	0

· x₃ +

0	4
0	0

· x₄ +

0	4
0	0

· x₅ +

1	0
0	0

· x₆

[c56(x₁)]

0	0
0	0

1	0
0	0

· x₁

[bfs#3^#(x₁,...,x₄)]

0	0
4	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	0
0	0

· x₄

[#eq^#(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

0	0
0	0

· x₂

[c37(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c35]

0	0
0	0

[dodfs^#(x₁, x₂)]

3	0
4	0

0	4
0	0

· x₁ +

1	1
0	0

· x₂

[c28]

0	0
0	0

[dfs#1^#(x₁, x₂)]

0	0
4	0

0	4
0	0

· x₁ +

1	0
0	0

· x₂

[c49]

0	0
4	0

[c3]

0	0
0	0

[bfs^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c25]

0	0
0	0

[c38(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c1]

0	0
0	0

[c12]

0	0
0	0

[c42(x₁, x₂)]

0	0
0	0

1	4
0	0

· x₁ +

1	2
0	0

· x₂

[c30]

0	0
0	0

[dobfs^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

0	0
0	0

· x₂

[c2]

0	0
0	0

[c9(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c13(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c55(x₁)]

0	0
0	0

1	0
0	0

· x₁

[dfs^#(x₁, x₂)]

0	0
2	0

0	4
0	0

· x₁ +

1	0
0	0

· x₂

[c24]

0	0
0	0

[bfs#1^#(x₁, x₂, x₃)]

0	0
0	0

0	2
0	0

· x₁ +

1	2
0	0

· x₂ +

0	0
0	0

· x₃

[c26]

0	0
0	0

[c39(x₁, x₂)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂

[c11]

0	0
0	0

[c54(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c19]

0	0
0	0

[c53]

0	0
4	0

[bfs#2^#(x₁, x₂)]

0	0
0	0

1	2
0	0

· x₁ +

0	0
0	0

· x₂

[bfs2^#(x₁, x₂)]

4	0
3	0

0	4
0	0

· x₁ +

1	4
0	0

· x₂

[bfs2#1^#(x₁, x₂)]

0	0
0	0

0	2
0	0

· x₁ +

1	0
0	0

· x₂

[c31(x₁,...,x₅)]

0	0
0	0

1	0
0	0

· x₁ +

1	0
0	0

· x₂ +

1	0
0	0

· x₃ +

1	0
0	0

· x₄ +

1	0
0	0

· x₅

[c10]

0	0
0	0

[c27]

0	0
0	0

[c41(x₁)]

0	0
2	0

1	0
0	0

· x₁

[c48(x₁)]

0	0
2	0

1	0
0	0

· x₁

[c43(x₁)]

0	0
1	0

1	0
0	0

· x₁

[c50(x₁)]

0	0
0	0

1	0
0	0

· x₁

[c4]

0	0
0	0

[bfs#4^#(x₁,...,x₇)]

2	0
1	0

0	0
0	0

· x₁ +

1	2
0	0

· x₂ +

0	2
0	0

· x₃ +

0	2
0	0

· x₄ +

0	2
0	0

· x₅ +

0	0
0	0

· x₆ +

0	0
0	0

· x₇

[c29]

0	0
0	0

[c36(x₁)]

0	0
0	0

1	0
0	1

· x₁

[c22]

0	0
0	0

[c15(x₁)]

0	0
0	0

1	0
0	0

· x₁

[#0]

0	0
0	0

[#eq(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[leaf]

0	0
0	0

[#equal(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0
4	0

0	0
0	0

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	1

· x₄ +

0	0
0	1

· x₅ +

0	0
0	0

· x₆ +

0	0
0	1

· x₇

[appendreverse(x₁, x₂)]

0	0
0	0

2	0
0	1

· x₁ +

2	0
0	1

· x₂

[#neg(x₁)]

0	0
0	0

0	0
0	0

· x₁

[dobfs(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

1	0
0	0

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	0

· x₃

[#and(x₁, x₂)]

0	0
0	0

0	0
0	0

· x₁ +

0	0
0	0

· x₂

[nil]

0	0
0	0

[#false]

0	0
0	0

[reverse(x₁)]

0	0
0	0

3	1
0	1

· x₁

[appendreverse#1(x₁, x₂)]

0	0
0	0

2	0
0	1

· x₁ +

2	0
0	1

· x₂

[::(x₁, x₂)]

1	0
0	0

0	0
0	1

· x₁ +

1	0
0	1

· x₂

[bfs#2(x₁, x₂)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	0

· x₂

[#pos(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#s(x₁)]

0	0
0	0

0	0
0	0

· x₁

[#true]

0	0
0	0

[node(x₁, x₂, x₃)]

0	0
4	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃

[bfs#3(x₁,...,x₄)]

0	0
0	0

0	0
0	1

· x₁ +

0	0
0	1

· x₂ +

0	0
0	1

· x₃ +

0	0
0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

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

appendreverse^#(z0,z1)

→

c33(appendreverse#1^#(z0,z1))

(61)

are strictly oriented by the following linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the rationals with delta = 1

[appendreverse^#(x₁, x₂)]

1	0	0
0	0	0
0	0	0

0	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[reverse^#(x₁)]

1	0	0
0	0	0
0	0	0

0	0	1
0	0	0
0	0	0

· x₁

[#and^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[dfs#2^#(x₁,...,x₄)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

· x₃ +

1	0	0
0	0	0
0	0	0

· x₄

[c21]

0	0	0
0	0	0
0	0	0

[c5]

0	0	0
0	0	0
0	0	0

[c16(x₁, x₂, x₃)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃

[c17]

0	0	0
1	0	0
0	0	0

[c6]

0	0	0
0	0	0
0	0	0

[c51(x₁, x₂)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c20]

0	0	0
0	0	0
1	0	0

[c34(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c45(x₁, x₂)]

0	0	0
0	0	0
3	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c52(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c47(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	2
0	0	1

· x₁

[c32(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c33(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	2
0	0	1

· x₁

[c18]

0	0	0
1	0	0
0	0	0

[c46(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	2
0	0	1

· x₁

[c7]

0	0	0
0	0	0
0	0	0

[#equal^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[c40]

0	0	0
0	0	0
0	0	0

[c23]

0	0	0
0	0	0
0	0	0

[c]

0	0	0
0	0	0
0	0	0

[c44]

3	0	0
0	0	0
0	0	0

[c8]

0	0	0
0	0	0
0	0	0

[c14]

0	0	0
0	0	0
0	0	0

[dfs#3^#(x₁,...,x₆)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	1	2
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

· x₄ +

0	0	0
0	0	0
0	0	0

· x₅ +

1	0	0
0	0	0
0	0	0

· x₆

[c56(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[bfs#3^#(x₁,...,x₄)]

0	0	0
0	0	0
0	0	0

0	1	0
0	0	0
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	1	0
0	0	0
0	0	0

· x₃ +

0	0	0
0	0	0
0	0	0

· x₄

[#eq^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	1
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	1

· x₂

[appendreverse#1^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[c37(x₁)]

0	0	0
2	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[c35]

0	0	0
0	0	0
0	0	0

[dodfs^#(x₁, x₂)]

1	0	0
1	0	0
3	0	0

0	1	1
0	0	1
0	0	0

· x₁ +

1	1	1
0	0	1
0	0	0

· x₂

[c28]

0	0	0
0	0	0
1	0	0

[dfs#1^#(x₁, x₂)]

0	0	0
1	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c49]

0	0	0
1	0	0
0	0	0

[c3]

0	0	0
0	0	0
0	0	0

[bfs^#(x₁, x₂, x₃)]

0	0	0
0	0	0
0	0	0

0	1	0
0	0	0
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

· x₃

[c25]

0	0	0
0	0	0
0	0	0

[c38(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[c1]

0	0	0
0	0	0
0	0	0

[c12]

0	0	0
0	0	0
0	0	0

[c42(x₁, x₂)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c30]

0	0	0
0	0	0
0	0	0

[dobfs^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	1	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[c2]

0	0	0
0	0	0
0	0	0

[c9(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c13(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c55(x₁)]

0	0	0
0	0	0
3	0	0

1	0	0
0	0	0
0	0	1

· x₁

[dfs^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c24]

0	0	0
0	0	0
1	0	0

[bfs#1^#(x₁, x₂, x₃)]

0	0	0
3	0	0
0	0	0

0	1	0
0	0	1
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	0	0
0	0	0
0	0	0

· x₃

[c26]

0	0	0
0	0	0
0	0	0

[c39(x₁, x₂)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c11]

0	0	0
0	0	0
0	0	0

[c54(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c19]

0	0	0
1	0	0
0	0	0

[c53]

0	0	0
0	0	0
0	0	0

[bfs#2^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

1	1	1
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[bfs2^#(x₁, x₂)]

1	0	0
1	0	0
3	0	0

1	3	1
0	0	1
0	0	0

· x₁ +

1	3	1
0	0	1
0	0	0

· x₂

[bfs2#1^#(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	2	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂

[c31(x₁,...,x₅)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁ +

1	0	0
0	0	0
0	0	0

· x₂ +

1	0	0
0	0	0
0	0	0

· x₃ +

1	0	0
0	0	0
0	0	0

· x₄ +

1	0	0
0	0	0
0	0	0

· x₅

[c10]

0	0	0
0	0	0
0	0	0

[c27]

0	0	0
0	0	0
0	0	0

[c41(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[c48(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[c43(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[c50(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	1

· x₁

[c4]

0	0	0
0	0	0
0	0	0

[bfs#4^#(x₁,...,x₇)]

3	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

1	1	1
0	0	0
0	0	0

· x₂ +

0	1	0
0	0	0
0	0	0

· x₃ +

0	1	0
0	0	0
0	0	0

· x₄ +

0	1	0
0	0	0
0	0	0

· x₅ +

0	0	0
0	0	0
0	0	0

· x₆ +

0	0	0
0	0	0
0	0	0

· x₇

[c29]

0	0	0
0	0	0
0	0	0

[c36(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	2
0	0	1

· x₁

[c22]

0	0	0
0	0	0
0	0	0

[c15(x₁)]

0	0	0
0	0	0
0	0	0

1	0	0
0	0	0
0	0	0

· x₁

[#0]

0	0	0
0	0	0
0	0	0

[#eq(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[bfs(x₁, x₂, x₃)]

0	0	0
0	0	0
0	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂ +

0	0	0
0	1	0
0	0	0

· x₃

[leaf]

0	0	0
0	0	0
0	0	0

[#equal(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[bfs#4(x₁,...,x₇)]

0	0	0
3	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂ +

0	0	0
0	1	2
0	0	0

· x₃ +

0	0	0
0	1	0
0	0	0

· x₄ +

0	0	0
0	1	0
0	0	0

· x₅ +

0	0	0
0	1	0
0	0	0

· x₆ +

0	0	0
0	0	0
0	0	0

· x₇

[appendreverse(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	1
0	1	0
0	0	2

· x₁ +

1	0	0
0	1	0
0	0	2

· x₂

[#neg(x₁)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁

[dobfs(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	2

· x₂

[bfs#1(x₁, x₂, x₃)]

0	0	0
0	0	0
0	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂ +

0	0	0
0	1	0
0	0	0

· x₃

[#and(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	0	0
0	0	0

· x₂

[nil]

0	0	0
0	0	0
0	0	0

[#false]

0	0	0
0	0	0
0	0	0

[reverse(x₁)]

0	0	0
0	0	0
0	0	0

0	0	1
0	1	0
0	0	2

· x₁

[appendreverse#1(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	1
0	1	0
0	0	2

· x₁ +

1	0	0
0	1	0
0	0	2

· x₂

[::(x₁, x₂)]

1	0	0
0	0	0
1	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	1

· x₂

[bfs#2(x₁, x₂)]

0	0	0
0	0	0
0	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂

[#pos(x₁)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁

[#s(x₁)]

0	0	0
0	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁

[#true]

0	0	0
0	0	0
0	0	0

[node(x₁, x₂, x₃)]

0	0	0
3	0	0
0	0	0

0	0	0
0	0	0
0	0	0

· x₁ +

0	0	0
0	1	2
0	0	0

· x₂ +

0	0	0
0	1	0
0	0	0

· x₃

[bfs#3(x₁,...,x₄)]

0	0	0
0	0	0
0	0	0

0	0	0
0	1	0
0	0	0

· x₁ +

0	0	0
0	1	0
0	0	0

· x₂ +

0	0	0
0	1	0
0	0	0

· x₃ +

0	0	0
0	1	0
0	0	0

· x₄

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

#and^#(#false,#false)	→	c	(108)
#and^#(#false,#true)	→	c1	(109)
#and^#(#true,#false)	→	c2	(110)
#and^#(#true,#true)	→	c3	(111)
#eq^#(#0,#0)	→	c4	(112)
#eq^#(#0,#neg(z0))	→	c5	(114)
#eq^#(#0,#pos(z0))	→	c6	(116)
#eq^#(#0,#s(z0))	→	c7	(118)
#eq^#(#neg(z0),#0)	→	c8	(120)
#eq^#(#neg(z0),#neg(z1))	→	c9(#eq^#(z0,z1))	(122)
#eq^#(#neg(z0),#pos(z1))	→	c10	(124)
#eq^#(#pos(z0),#0)	→	c11	(126)
#eq^#(#pos(z0),#neg(z1))	→	c12	(128)
#eq^#(#pos(z0),#pos(z1))	→	c13(#eq^#(z0,z1))	(130)
#eq^#(#s(z0),#0)	→	c14	(132)
#eq^#(#s(z0),#s(z1))	→	c15(#eq^#(z0,z1))	(134)
#eq^#(::(z0,z1),::(z2,z3))	→	c16(#and^#(#eq(z0,z2),#eq(z1,z3)),#eq^#(z0,z2),#eq^#(z1,z3))	(136)
#eq^#(::(z0,z1),leaf)	→	c17	(138)
#eq^#(::(z0,z1),nil)	→	c18	(140)
#eq^#(::(z0,z1),node(z2,z3,z4))	→	c19	(142)
#eq^#(leaf,::(z0,z1))	→	c20	(144)
#eq^#(leaf,leaf)	→	c21	(145)
#eq^#(leaf,nil)	→	c22	(146)
#eq^#(leaf,node(z0,z1,z2))	→	c23	(148)
#eq^#(nil,::(z0,z1))	→	c24	(150)
#eq^#(nil,leaf)	→	c25	(151)
#eq^#(nil,nil)	→	c26	(152)
#eq^#(nil,node(z0,z1,z2))	→	c27	(154)
#eq^#(node(z0,z1,z2),::(z3,z4))	→	c28	(156)
#eq^#(node(z0,z1,z2),leaf)	→	c29	(158)
#eq^#(node(z0,z1,z2),nil)	→	c30	(160)
#eq^#(node(z0,z1,z2),node(z3,z4,z5))	→	c31(#and^#(#eq(z0,z3),#and(#eq(z1,z4),#eq(z2,z5))),#eq^#(z0,z3),#and^#(#eq(z1,z4),#eq(z2,z5)),#eq^#(z1,z4),#eq^#(z2,z5))	(162)
#equal^#(z0,z1)	→	c32(#eq^#(z0,z1))	(59)
appendreverse^#(z0,z1)	→	c33(appendreverse#1^#(z0,z1))	(61)
appendreverse#1^#(::(z0,z1),z2)	→	c34(appendreverse^#(z1,::(z0,z2)))	(63)
appendreverse#1^#(nil,z0)	→	c35	(65)
bfs^#(z0,z1,z2)	→	c36(bfs#1^#(z0,z1,z2))	(67)
bfs#1^#(::(z0,z1),z2,z3)	→	c37(bfs#3^#(z0,z2,z1,z3))	(69)
bfs#1^#(nil,z0,z1)	→	c38(bfs#2^#(z0,z1))	(71)
bfs#2^#(::(z0,z1),z2)	→	c39(bfs^#(reverse(::(z0,z1)),nil,z2),reverse^#(::(z0,z1)))	(73)
bfs#2^#(nil,z0)	→	c40	(75)
bfs#3^#(leaf,z0,z1,z2)	→	c41(bfs^#(z1,z0,z2))	(77)
bfs#3^#(node(z0,z1,z2),z3,z4,z5)	→	c42(bfs#4^#(#equal(z5,z0),z3,z1,z2,z4,z5,z0),#equal^#(z5,z0))	(79)
bfs#4^#(#false,z0,z1,z2,z3,z4,z5)	→	c43(bfs^#(z3,::(z2,::(z1,z0)),z4))	(81)
bfs#4^#(#true,z0,z1,z2,z3,z4,z5)	→	c44	(83)
bfs2^#(z0,z1)	→	c45(bfs2#1^#(dobfs(z0,z1),z1),dobfs^#(z0,z1))	(85)
bfs2#1^#(z0,z1)	→	c46(dobfs^#(z0,z1))	(87)
dfs^#(z0,z1)	→	c47(dfs#1^#(z0,z1))	(89)
dfs#1^#(::(z0,z1),z2)	→	c48(dfs#2^#(z0,z0,z1,z2))	(91)
dfs#1^#(nil,z0)	→	c49	(93)
dfs#2^#(leaf,z0,z1,z2)	→	c50(dfs^#(z1,z2))	(95)
dfs#2^#(node(z0,z1,z2),z3,z4,z5)	→	c51(dfs#3^#(#equal(z0,z5),z3,z1,z2,z4,z5),#equal^#(z0,z5))	(97)
dfs#3^#(#false,z0,z1,z2,z3,z4)	→	c52(dfs^#(::(z1,::(z2,z3)),z4))	(99)
dfs#3^#(#true,z0,z1,z2,z3,z4)	→	c53	(101)
dobfs^#(z0,z1)	→	c54(bfs^#(::(z0,nil),nil,z1))	(103)
dodfs^#(z0,z1)	→	c55(dfs^#(::(z0,nil),z1))	(105)
reverse^#(z0)	→	c56(appendreverse^#(z0,nil))	(107)
reverse(z0)	→	appendreverse(z0,nil)	(106)
bfs#4(#true,z0,z1,z2,z3,z4,z5)	→	node(z5,z1,z2)	(82)
appendreverse#1(nil,z0)	→	z0	(64)
dobfs(z0,z1)	→	bfs(::(z0,nil),nil,z1)	(102)
bfs#1(::(z0,z1),z2,z3)	→	bfs#3(z0,z2,z1,z3)	(68)
bfs#2(::(z0,z1),z2)	→	bfs(reverse(::(z0,z1)),nil,z2)	(72)
appendreverse#1(::(z0,z1),z2)	→	appendreverse(z1,::(z0,z2))	(62)
bfs#4(#false,z0,z1,z2,z3,z4,z5)	→	bfs(z3,::(z2,::(z1,z0)),z4)	(80)
appendreverse(z0,z1)	→	appendreverse#1(z0,z1)	(60)
bfs(z0,z1,z2)	→	bfs#1(z0,z1,z2)	(66)
bfs#3(leaf,z0,z1,z2)	→	bfs(z1,z0,z2)	(76)
bfs#3(node(z0,z1,z2),z3,z4,z5)	→	bfs#4(#equal(z5,z0),z3,z1,z2,z4,z5,z0)	(78)
bfs#2(nil,z0)	→	leaf	(74)
bfs#1(nil,z0,z1)	→	bfs#2(z0,z1)	(70)

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).