Certification Problem

Input (TPDB TRS_Equational/Mixed_AC/BAG_nokinds-noand)

The rewrite relation of the following equational TRS is considered.

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U11(tt)	→	tt	(6)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U121(tt,X)	→	X	(9)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U161(tt,X)	→	X	(16)
U171(tt,A,B)	→	U172(isBag(B),A,B)	(17)
U172(tt,A,B)	→	mult(prod(A),prod(B))	(18)
U181(tt,X)	→	X	(19)
U191(tt,A,B)	→	U192(isBag(B),A,B)	(20)
U192(tt,A,B)	→	plus(sum(A),sum(B))	(21)
U21(tt,V2)	→	U22(isBag(V2))	(22)
U22(tt)	→	tt	(23)
U31(tt)	→	tt	(24)
U41(tt)	→	tt	(25)
U51(tt,V2)	→	U52(isBin(V2))	(26)
U52(tt)	→	tt	(27)
U61(tt,V2)	→	U62(isBin(V2))	(28)
U62(tt)	→	tt	(29)
U71(tt)	→	tt	(30)
U81(tt)	→	tt	(31)
U91(tt)	→	z	(32)
isBag(empty)	→	tt	(33)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBin(z)	→	tt	(36)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
mult(z,X)	→	U91(isBin(X))	(43)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
prod(empty)	→	1(z)	(50)
prod(singl(X))	→	U161(isBin(X),X)	(51)
prod(union(A,B))	→	U171(isBag(A),A,B)	(52)
sum(empty)	→	0(z)	(53)
sum(singl(X))	→	U181(isBin(X),X)	(54)
sum(union(A,B))	→	U191(isBag(A),A,B)	(55)

Associative symbols: mult, plus, union

Commutative symbols: mult, plus, union

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 AC Dependency Pair Transformation

The following set of (strict) dependency pairs is constructed for the TRS.

U101^#(tt,X,Y)	→	U102^#(isBin(Y),X,Y)	(71)
U101^#(tt,X,Y)	→	isBin^#(Y)	(72)
U102^#(tt,X,Y)	→	0^#(mult(X,Y))	(73)
U102^#(tt,X,Y)	→	mult^#(X,Y)	(74)
U111^#(tt,X,Y)	→	U112^#(isBin(Y),X,Y)	(75)
U111^#(tt,X,Y)	→	isBin^#(Y)	(76)
U112^#(tt,X,Y)	→	plus^#(0(mult(X,Y)),Y)	(77)
U112^#(tt,X,Y)	→	0^#(mult(X,Y))	(78)
U112^#(tt,X,Y)	→	mult^#(X,Y)	(79)
U131^#(tt,X,Y)	→	U132^#(isBin(Y),X,Y)	(80)
U131^#(tt,X,Y)	→	isBin^#(Y)	(81)
U132^#(tt,X,Y)	→	0^#(plus(X,Y))	(82)
U132^#(tt,X,Y)	→	plus^#(X,Y)	(83)
U141^#(tt,X,Y)	→	U142^#(isBin(Y),X,Y)	(84)
U141^#(tt,X,Y)	→	isBin^#(Y)	(85)
U142^#(tt,X,Y)	→	plus^#(X,Y)	(86)
U151^#(tt,X,Y)	→	U152^#(isBin(Y),X,Y)	(87)
U151^#(tt,X,Y)	→	isBin^#(Y)	(88)
U152^#(tt,X,Y)	→	0^#(plus(plus(X,Y),1(z)))	(89)
U152^#(tt,X,Y)	→	plus^#(plus(X,Y),1(z))	(90)
U152^#(tt,X,Y)	→	plus^#(X,Y)	(91)
U171^#(tt,A,B)	→	U172^#(isBag(B),A,B)	(92)
U171^#(tt,A,B)	→	isBag^#(B)	(93)
U172^#(tt,A,B)	→	mult^#(prod(A),prod(B))	(94)
U172^#(tt,A,B)	→	prod^#(A)	(95)
U172^#(tt,A,B)	→	prod^#(B)	(96)
U191^#(tt,A,B)	→	U192^#(isBag(B),A,B)	(97)
U191^#(tt,A,B)	→	isBag^#(B)	(98)
U192^#(tt,A,B)	→	plus^#(sum(A),sum(B))	(99)
U192^#(tt,A,B)	→	sum^#(A)	(100)
U192^#(tt,A,B)	→	sum^#(B)	(101)
U21^#(tt,V2)	→	U22^#(isBag(V2))	(102)
U21^#(tt,V2)	→	isBag^#(V2)	(103)
U51^#(tt,V2)	→	U52^#(isBin(V2))	(104)
U51^#(tt,V2)	→	isBin^#(V2)	(105)
U61^#(tt,V2)	→	U62^#(isBin(V2))	(106)
U61^#(tt,V2)	→	isBin^#(V2)	(107)
isBag^#(singl(V1))	→	U11^#(isBin(V1))	(108)
isBag^#(singl(V1))	→	isBin^#(V1)	(109)
isBag^#(union(V1,V2))	→	U21^#(isBag(V1),V2)	(110)
isBag^#(union(V1,V2))	→	isBag^#(V1)	(111)
isBin^#(0(V1))	→	U31^#(isBin(V1))	(112)
isBin^#(0(V1))	→	isBin^#(V1)	(113)
isBin^#(1(V1))	→	U41^#(isBin(V1))	(114)
isBin^#(1(V1))	→	isBin^#(V1)	(115)
isBin^#(mult(V1,V2))	→	U51^#(isBin(V1),V2)	(116)
isBin^#(mult(V1,V2))	→	isBin^#(V1)	(117)
isBin^#(plus(V1,V2))	→	U61^#(isBin(V1),V2)	(118)
isBin^#(plus(V1,V2))	→	isBin^#(V1)	(119)
isBin^#(prod(V1))	→	U71^#(isBag(V1))	(120)
isBin^#(prod(V1))	→	isBag^#(V1)	(121)
isBin^#(sum(V1))	→	U81^#(isBag(V1))	(122)
isBin^#(sum(V1))	→	isBag^#(V1)	(123)
mult^#(z,X)	→	U91^#(isBin(X))	(124)
mult^#(z,X)	→	isBin^#(X)	(125)
mult^#(0(X),Y)	→	U101^#(isBin(X),X,Y)	(126)
mult^#(0(X),Y)	→	isBin^#(X)	(127)
mult^#(1(X),Y)	→	U111^#(isBin(X),X,Y)	(128)
mult^#(1(X),Y)	→	isBin^#(X)	(129)
plus^#(z,X)	→	U121^#(isBin(X),X)	(130)
plus^#(z,X)	→	isBin^#(X)	(131)
plus^#(0(X),0(Y))	→	U131^#(isBin(X),X,Y)	(132)
plus^#(0(X),0(Y))	→	isBin^#(X)	(133)
plus^#(0(X),1(Y))	→	U141^#(isBin(X),X,Y)	(134)
plus^#(0(X),1(Y))	→	isBin^#(X)	(135)
plus^#(1(X),1(Y))	→	U151^#(isBin(X),X,Y)	(136)
plus^#(1(X),1(Y))	→	isBin^#(X)	(137)
prod^#(singl(X))	→	U161^#(isBin(X),X)	(138)
prod^#(singl(X))	→	isBin^#(X)	(139)
prod^#(union(A,B))	→	U171^#(isBag(A),A,B)	(140)
prod^#(union(A,B))	→	isBag^#(A)	(141)
sum^#(empty)	→	0^#(z)	(142)
sum^#(singl(X))	→	U181^#(isBin(X),X)	(143)
sum^#(singl(X))	→	isBin^#(X)	(144)
sum^#(union(A,B))	→	U191^#(isBag(A),A,B)	(145)
sum^#(union(A,B))	→	isBag^#(A)	(146)
mult^#(mult(z,X),ext)	→	mult^#(U91(isBin(X)),ext)	(147)
mult^#(mult(z,X),ext)	→	U91^#(isBin(X))	(148)
mult^#(mult(z,X),ext)	→	isBin^#(X)	(149)
mult^#(mult(0(X),Y),ext)	→	mult^#(U101(isBin(X),X,Y),ext)	(150)
mult^#(mult(0(X),Y),ext)	→	U101^#(isBin(X),X,Y)	(151)
mult^#(mult(0(X),Y),ext)	→	isBin^#(X)	(152)
mult^#(mult(1(X),Y),ext)	→	mult^#(U111(isBin(X),X,Y),ext)	(153)
mult^#(mult(1(X),Y),ext)	→	U111^#(isBin(X),X,Y)	(154)
mult^#(mult(1(X),Y),ext)	→	isBin^#(X)	(155)
plus^#(plus(z,X),ext)	→	plus^#(U121(isBin(X),X),ext)	(156)
plus^#(plus(z,X),ext)	→	U121^#(isBin(X),X)	(157)
plus^#(plus(z,X),ext)	→	isBin^#(X)	(158)
plus^#(plus(0(X),0(Y)),ext)	→	plus^#(U131(isBin(X),X,Y),ext)	(159)
plus^#(plus(0(X),0(Y)),ext)	→	U131^#(isBin(X),X,Y)	(160)
plus^#(plus(0(X),0(Y)),ext)	→	isBin^#(X)	(161)
plus^#(plus(0(X),1(Y)),ext)	→	plus^#(U141(isBin(X),X,Y),ext)	(162)
plus^#(plus(0(X),1(Y)),ext)	→	U141^#(isBin(X),X,Y)	(163)
plus^#(plus(0(X),1(Y)),ext)	→	isBin^#(X)	(164)
plus^#(plus(1(X),1(Y)),ext)	→	plus^#(U151(isBin(X),X,Y),ext)	(165)
plus^#(plus(1(X),1(Y)),ext)	→	U151^#(isBin(X),X,Y)	(166)
plus^#(plus(1(X),1(Y)),ext)	→	isBin^#(X)	(167)
union^#(union(X,empty),ext)	→	union^#(X,ext)	(168)
union^#(union(empty,X),ext)	→	union^#(X,ext)	(169)

The extended rules of the TRS

mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)

give rise to even more dependency pairs (by sharping the root symbols of each rule). Finiteness for all DPs in combination with the equational DPs is proven as follows.

1.1 Dependency Graph Processor

The dependency pairs are split into 6 components.

The 1^st component contains the pair

U172^#(tt,A,B)	→	prod^#(A)	(95)
prod^#(union(A,B))	→	U171^#(isBag(A),A,B)	(140)
U171^#(tt,A,B)	→	U172^#(isBag(B),A,B)	(92)
U172^#(tt,A,B)	→	prod^#(B)	(96)

1.1.1 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[U22(x₁)]	=	1 · x₁
[tt]	=	0
[U81(x₁)]	=	2 · x₁
[isBag(x₁)]	=	2 · x₁
[empty]	=	0
[U41(x₁)]	=	2 · x₁
[isBin(x₁)]	=	2 · x₁
[mult(x₁, x₂)]	=	2 + 3 · x₁ + 3 · x₂
[U51(x₁, x₂)]	=	1 + 2 · x₁ + 3 · x₂
[U61(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[U62(x₁)]	=	1 · x₁
[z]	=	0
[1(x₁)]	=	3 · x₁
[U52(x₁)]	=	1 · x₁
[prod(x₁)]	=	2 · x₁
[U71(x₁)]	=	1 · x₁
[U21(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[union(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[U11(x₁)]	=	2 · x₁
[0(x₁)]	=	2 + 3 · x₁
[U31(x₁)]	=	2 · x₁
[plus(x₁, x₂)]	=	2 + 3 · x₁ + 3 · x₂
[singl(x₁)]	=	3 · x₁
[sum(x₁)]	=	2 · x₁
[U171^#(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 3 · x₃
[U172^#(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[prod^#(x₁)]	=	1 · x₁

together with the usable rules

U22(tt)	→	tt	(23)
U81(tt)	→	tt	(31)
isBag(empty)	→	tt	(33)
U41(tt)	→	tt	(25)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
U61(tt,V2)	→	U62(isBin(V2))	(28)
isBin(z)	→	tt	(36)
isBin(1(V1))	→	U41(isBin(V1))	(38)
U51(tt,V2)	→	U52(isBin(V2))	(26)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
U21(tt,V2)	→	U22(isBag(V2))	(22)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
U11(tt)	→	tt	(6)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
U52(tt)	→	tt	(27)
U71(tt)	→	tt	(30)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
U31(tt)	→	tt	(24)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
U62(tt)	→	tt	(29)

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

prod^#(union(A,B))

→

U171^#(isBag(A),A,B)

(140)

and the rules

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U121(tt,X)	→	X	(9)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U161(tt,X)	→	X	(16)
U171(tt,A,B)	→	U172(isBag(B),A,B)	(17)
U172(tt,A,B)	→	mult(prod(A),prod(B))	(18)
U181(tt,X)	→	X	(19)
U191(tt,A,B)	→	U192(isBag(B),A,B)	(20)
U192(tt,A,B)	→	plus(sum(A),sum(B))	(21)
U51(tt,V2)	→	U52(isBin(V2))	(26)
U91(tt)	→	z	(32)
isBag(empty)	→	tt	(33)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBin(z)	→	tt	(36)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
mult(z,X)	→	U91(isBin(X))	(43)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
prod(empty)	→	1(z)	(50)
prod(singl(X))	→	U161(isBin(X),X)	(51)
prod(union(A,B))	→	U171(isBag(A),A,B)	(52)
sum(empty)	→	0(z)	(53)
sum(singl(X))	→	U181(isBin(X),X)	(54)
sum(union(A,B))	→	U191(isBag(A),A,B)	(55)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair

U102^#(tt,X,Y)	→	mult^#(X,Y)	(74)
mult^#(0(X),Y)	→	U101^#(isBin(X),X,Y)	(126)
U101^#(tt,X,Y)	→	U102^#(isBin(Y),X,Y)	(71)
mult^#(1(X),Y)	→	U111^#(isBin(X),X,Y)	(128)
U111^#(tt,X,Y)	→	U112^#(isBin(Y),X,Y)	(75)
U112^#(tt,X,Y)	→	mult^#(X,Y)	(79)
mult^#(mult(z,X),ext)	→	mult^#(U91(isBin(X)),ext)	(147)
mult^#(mult(0(X),Y),ext)	→	mult^#(U101(isBin(X),X,Y),ext)	(150)
mult^#(mult(0(X),Y),ext)	→	U101^#(isBin(X),X,Y)	(151)
mult^#(mult(1(X),Y),ext)	→	mult^#(U111(isBin(X),X,Y),ext)	(153)
mult^#(mult(1(X),Y),ext)	→	U111^#(isBin(X),X,Y)	(154)
mult^#(mult(x,y),z')	→	mult^#(y,z')	(68)
mult^#(x,y)	→	mult^#(y,x)	(62)
mult^#(mult(x,y),z')	→	mult^#(x,mult(y,z'))	(65)

1.1.2 AC Reduction Pair Processor with Usable Rules

Using the non-linear polynomial interpretation over the naturals

[mult^#(x₁, x₂)]	=	1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[mult(x₁, x₂)]	=	1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[z]	=	0
[U91(x₁)]	=	0
[isBin(x₁)]	=	1
[U112^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ + 1 · x₂ · x₃
[tt]	=	1
[1(x₁)]	=	1 + 1 · x₁
[U111^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₁ · x₂ + 1 · x₁ · x₂ · x₃
[U111(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ + 1 · x₂ · x₃ + 1 · x₁ · x₃
[U102^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₂
[U101^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₂
[0(x₁)]	=	1 · x₁
[U101(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ + 1 · x₂ · x₃
[plus(x₁, x₂)]	=	1 · x₂ + 1 · x₁
[U131(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂
[U141(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₁ · x₂
[U151(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂
[U121(x₁, x₂)]	=	1 · x₁ · x₂
[U11(x₁)]	=	1
[U81(x₁)]	=	1
[U132(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂
[U142(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂
[U152(x₁, x₂, x₃)]	=	1 + 1 · x₁ · x₃ + 1 · x₁ · x₂
[U21(x₁, x₂)]	=	1 · x₁ · x₂
[U22(x₁)]	=	1 · x₁
[isBag(x₁)]	=	1 · x₁
[U41(x₁)]	=	1
[U31(x₁)]	=	1
[U102(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ + 1 · x₂ · x₃
[U71(x₁)]	=	1
[empty]	=	1
[union(x₁, x₂)]	=	1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[singl(x₁)]	=	1
[U51(x₁, x₂)]	=	1
[U52(x₁)]	=	1 · x₁ · x₁
[U61(x₁, x₂)]	=	1 · x₁
[sum(x₁)]	=	0
[prod(x₁)]	=	0
[U112(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₃ + 1 · x₁ · x₂
[U62(x₁)]	=	1 · x₁ · x₁

together with the usable rules

plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U11(tt)	→	tt	(6)
U81(tt)	→	tt	(31)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U21(tt,V2)	→	U22(isBag(V2))	(22)
U41(tt)	→	tt	(25)
U31(tt)	→	tt	(24)
U91(tt)	→	z	(32)
U22(tt)	→	tt	(23)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(z,X)	→	U91(isBin(X))	(43)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
U71(tt)	→	tt	(30)
isBag(empty)	→	tt	(33)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U51(tt,V2)	→	U52(isBin(V2))	(26)
isBin(z)	→	tt	(36)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U121(tt,X)	→	X	(9)
U52(tt)	→	tt	(27)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U61(tt,V2)	→	U62(isBin(V2))	(28)
U62(tt)	→	tt	(29)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)
mult(x,y)	→	mult(y,x)	(56)
mult(mult(x,y),z')	→	mult(x,mult(y,z'))	(59)

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

mult^#(mult(1(X),Y),ext)	→	U111^#(isBin(X),X,Y)	(154)
mult^#(mult(1(X),Y),ext)	→	mult^#(U111(isBin(X),X,Y),ext)	(153)
mult^#(1(X),Y)	→	U111^#(isBin(X),X,Y)	(128)

could be deleted.

1.1.2.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mult^#(0(X),Y)	→	U101^#(isBin(X),X,Y)	(126)
U101^#(tt,X,Y)	→	U102^#(isBin(Y),X,Y)	(71)
U102^#(tt,X,Y)	→	mult^#(X,Y)	(74)
mult^#(mult(z,X),ext)	→	mult^#(U91(isBin(X)),ext)	(147)
mult^#(mult(0(X),Y),ext)	→	mult^#(U101(isBin(X),X,Y),ext)	(150)
mult^#(mult(0(X),Y),ext)	→	U101^#(isBin(X),X,Y)	(151)
mult^#(mult(x,y),z')	→	mult^#(y,z')	(68)
mult^#(x,y)	→	mult^#(y,x)	(62)
mult^#(mult(x,y),z')	→	mult^#(x,mult(y,z'))	(65)

1.1.2.1.1 AC Reduction Pair Processor with Usable Rules

Using the non-linear polynomial interpretation over the naturals

[mult^#(x₁, x₂)]	=	1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[mult(x₁, x₂)]	=	1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[z]	=	0
[U91(x₁)]	=	0
[isBin(x₁)]	=	1
[U101^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₂ · x₃ + 1 · x₁ · x₂
[tt]	=	1
[U102^#(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₁ · x₂ + 1 · x₁ · x₂ · x₃
[0(x₁)]	=	1 + 1 · x₁
[U101(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂ + 1 · x₂ · x₃
[plus(x₁, x₂)]	=	1 · x₂ + 1 · x₁
[U131(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂
[1(x₁)]	=	1 + 1 · x₁
[U141(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂
[U151(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂ + 1 · x₁
[U121(x₁, x₂)]	=	1 · x₁ · x₂
[U41(x₁)]	=	1 · x₁
[U132(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂
[U152(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂ + 1 · x₁
[U21(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₁
[U22(x₁)]	=	1 + 1 · x₁
[isBag(x₁)]	=	1 + 1 · x₁
[U142(x₁, x₂, x₃)]	=	1 + 1 · x₁ · x₃ + 1 · x₁ · x₂
[U71(x₁)]	=	1
[U81(x₁)]	=	1
[U111(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂ + 1 · x₂ · x₃ + 1 · x₁ · x₃
[U112(x₁, x₂, x₃)]	=	1 · x₃ + 1 · x₁ + 1 · x₁ · x₃ + 1 · x₁ · x₂ + 1 · x₁ · x₂ · x₃
[empty]	=	0
[union(x₁, x₂)]	=	1 + 1 · x₂ + 1 · x₁ + 1 · x₁ · x₂
[singl(x₁)]	=	1
[U11(x₁)]	=	1
[U102(x₁, x₂, x₃)]	=	1 + 1 · x₃ + 1 · x₂ + 1 · x₂ · x₃
[U62(x₁)]	=	1
[U61(x₁, x₂)]	=	1 · x₁
[U51(x₁, x₂)]	=	1
[U31(x₁)]	=	1
[sum(x₁)]	=	0
[prod(x₁)]	=	0
[U52(x₁)]	=	1

together with the usable rules

plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U41(tt)	→	tt	(25)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U21(tt,V2)	→	U22(isBag(V2))	(22)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U71(tt)	→	tt	(30)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U81(tt)	→	tt	(31)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
isBag(empty)	→	tt	(33)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
U22(tt)	→	tt	(23)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U121(tt,X)	→	X	(9)
U62(tt)	→	tt	(29)
U11(tt)	→	tt	(6)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U91(tt)	→	z	(32)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(z,X)	→	U91(isBin(X))	(43)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
U61(tt,V2)	→	U62(isBin(V2))	(28)
isBin(z)	→	tt	(36)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U31(tt)	→	tt	(24)
U52(tt)	→	tt	(27)
U51(tt,V2)	→	U52(isBin(V2))	(26)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
0(z)	→	z	(3)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)
mult(x,y)	→	mult(y,x)	(56)
mult(mult(x,y),z')	→	mult(x,mult(y,z'))	(59)

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

mult^#(mult(0(X),Y),ext)	→	U101^#(isBin(X),X,Y)	(151)
mult^#(0(X),Y)	→	U101^#(isBin(X),X,Y)	(126)

could be deleted.

1.1.2.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

mult^#(mult(z,X),ext)	→	mult^#(U91(isBin(X)),ext)	(147)
mult^#(mult(0(X),Y),ext)	→	mult^#(U101(isBin(X),X,Y),ext)	(150)
mult^#(mult(x,y),z')	→	mult^#(y,z')	(68)
mult^#(x,y)	→	mult^#(y,x)	(62)
mult^#(mult(x,y),z')	→	mult^#(x,mult(y,z'))	(65)

1.1.2.1.1.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[mult^#(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[mult(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[z]	=	0
[U91(x₁)]	=	0
[isBin(x₁)]	=	0
[0(x₁)]	=	0
[U101(x₁, x₂, x₃)]	=	0
[U52(x₁)]	=	3
[tt]	=	0
[U11(x₁)]	=	3
[U102(x₁, x₂, x₃)]	=	0
[U152(x₁, x₂, x₃)]	=	0
[plus(x₁, x₂)]	=	2 + 1 · x₁ + 1 · x₂
[1(x₁)]	=	2
[U141(x₁, x₂, x₃)]	=	3
[U142(x₁, x₂, x₃)]	=	3
[U22(x₁)]	=	3
[U31(x₁)]	=	3
[U151(x₁, x₂, x₃)]	=	0
[U111(x₁, x₂, x₃)]	=	3 + 1 · x₃
[U51(x₁, x₂)]	=	3 + 3 · x₂
[U41(x₁)]	=	3
[U61(x₁, x₂)]	=	3 + 3 · x₂
[sum(x₁)]	=	0
[U81(x₁)]	=	3
[isBag(x₁)]	=	0
[prod(x₁)]	=	0
[U71(x₁)]	=	3
[U62(x₁)]	=	3
[U131(x₁, x₂, x₃)]	=	2
[U121(x₁, x₂)]	=	1 · x₂
[U112(x₁, x₂, x₃)]	=	2 + 1 · x₃
[U132(x₁, x₂, x₃)]	=	2
[U21(x₁, x₂)]	=	3 + 3 · x₂
[empty]	=	0
[union(x₁, x₂)]	=	0
[singl(x₁)]	=	0

together with the usable rules

U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(z,X)	→	U91(isBin(X))	(43)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
U91(tt)	→	z	(32)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U121(tt,X)	→	X	(9)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
mult(x,y)	→	mult(y,x)	(56)
mult(mult(x,y),z')	→	mult(x,mult(y,z'))	(59)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)

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

mult^#(mult(x,y),z')	→	mult^#(y,z')	(68)
mult^#(mult(z,X),ext)	→	mult^#(U91(isBin(X)),ext)	(147)
mult^#(mult(0(X),Y),ext)	→	mult^#(U101(isBin(X),X,Y),ext)	(150)

could be deleted.

1.1.2.1.1.1.1.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

The 3^rd component contains the pair

U192^#(tt,A,B)	→	sum^#(A)	(100)
sum^#(union(A,B))	→	U191^#(isBag(A),A,B)	(145)
U191^#(tt,A,B)	→	U192^#(isBag(B),A,B)	(97)
U192^#(tt,A,B)	→	sum^#(B)	(101)

1.1.3 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[U22(x₁)]	=	1 · x₁
[tt]	=	2
[U81(x₁)]	=	2 + 1 · x₁
[isBag(x₁)]	=	2 · x₁
[empty]	=	2
[U41(x₁)]	=	1 · x₁
[isBin(x₁)]	=	1 + 1 · x₁
[mult(x₁, x₂)]	=	1 + 2 · x₁ + 3 · x₂
[U51(x₁, x₂)]	=	1 · x₁ + 3 · x₂
[U61(x₁, x₂)]	=	1 + 1 · x₁ + 2 · x₂
[U62(x₁)]	=	2 · x₁
[z]	=	1
[1(x₁)]	=	1 + 2 · x₁
[U52(x₁)]	=	2 · x₁
[prod(x₁)]	=	3 + 3 · x₁
[U71(x₁)]	=	2 + 1 · x₁
[U21(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[union(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[U11(x₁)]	=	2 + 2 · x₁
[0(x₁)]	=	3 + 3 · x₁
[U31(x₁)]	=	2 · x₁
[plus(x₁, x₂)]	=	3 + 2 · x₁ + 3 · x₂
[singl(x₁)]	=	2 + 2 · x₁
[sum(x₁)]	=	3 + 3 · x₁
[U192^#(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 1 · x₃
[sum^#(x₁)]	=	1 · x₁
[U191^#(x₁, x₂, x₃)]	=	1 · x₁ + 1 · x₂ + 3 · x₃

together with the usable rules

U22(tt)	→	tt	(23)
U81(tt)	→	tt	(31)
isBag(empty)	→	tt	(33)
U41(tt)	→	tt	(25)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
U61(tt,V2)	→	U62(isBin(V2))	(28)
isBin(z)	→	tt	(36)
isBin(1(V1))	→	U41(isBin(V1))	(38)
U51(tt,V2)	→	U52(isBin(V2))	(26)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
U21(tt,V2)	→	U22(isBag(V2))	(22)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
U11(tt)	→	tt	(6)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
U52(tt)	→	tt	(27)
U71(tt)	→	tt	(30)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
U31(tt)	→	tt	(24)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
U62(tt)	→	tt	(29)

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

U192^#(tt,A,B)	→	sum^#(B)	(101)
U191^#(tt,A,B)	→	U192^#(isBag(B),A,B)	(97)
sum^#(union(A,B))	→	U191^#(isBag(A),A,B)	(145)
U192^#(tt,A,B)	→	sum^#(A)	(100)

and the rules

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U11(tt)	→	tt	(6)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U121(tt,X)	→	X	(9)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U161(tt,X)	→	X	(16)
U171(tt,A,B)	→	U172(isBag(B),A,B)	(17)
U172(tt,A,B)	→	mult(prod(A),prod(B))	(18)
U181(tt,X)	→	X	(19)
U191(tt,A,B)	→	U192(isBag(B),A,B)	(20)
U192(tt,A,B)	→	plus(sum(A),sum(B))	(21)
U21(tt,V2)	→	U22(isBag(V2))	(22)
U31(tt)	→	tt	(24)
U52(tt)	→	tt	(27)
U61(tt,V2)	→	U62(isBin(V2))	(28)
U62(tt)	→	tt	(29)
U71(tt)	→	tt	(30)
U81(tt)	→	tt	(31)
U91(tt)	→	z	(32)
isBag(empty)	→	tt	(33)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBin(z)	→	tt	(36)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
mult(z,X)	→	U91(isBin(X))	(43)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
prod(empty)	→	1(z)	(50)
prod(singl(X))	→	U161(isBin(X),X)	(51)
prod(union(A,B))	→	U171(isBag(A),A,B)	(52)
sum(empty)	→	0(z)	(53)
sum(singl(X))	→	U181(isBin(X),X)	(54)
sum(union(A,B))	→	U191(isBag(A),A,B)	(55)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)

could be deleted.

1.1.3.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

The 4^th component contains the pair

U132^#(tt,X,Y)	→	plus^#(X,Y)	(83)
plus^#(0(X),0(Y))	→	U131^#(isBin(X),X,Y)	(132)
U131^#(tt,X,Y)	→	U132^#(isBin(Y),X,Y)	(80)
plus^#(0(X),1(Y))	→	U141^#(isBin(X),X,Y)	(134)
U141^#(tt,X,Y)	→	U142^#(isBin(Y),X,Y)	(84)
U142^#(tt,X,Y)	→	plus^#(X,Y)	(86)
plus^#(1(X),1(Y))	→	U151^#(isBin(X),X,Y)	(136)
U151^#(tt,X,Y)	→	U152^#(isBin(Y),X,Y)	(87)
U152^#(tt,X,Y)	→	plus^#(plus(X,Y),1(z))	(90)
plus^#(plus(z,X),ext)	→	plus^#(U121(isBin(X),X),ext)	(156)
plus^#(plus(0(X),0(Y)),ext)	→	plus^#(U131(isBin(X),X,Y),ext)	(159)
plus^#(plus(0(X),0(Y)),ext)	→	U131^#(isBin(X),X,Y)	(160)
plus^#(plus(0(X),1(Y)),ext)	→	plus^#(U141(isBin(X),X,Y),ext)	(162)
plus^#(plus(0(X),1(Y)),ext)	→	U141^#(isBin(X),X,Y)	(163)
plus^#(plus(1(X),1(Y)),ext)	→	plus^#(U151(isBin(X),X,Y),ext)	(165)
plus^#(plus(1(X),1(Y)),ext)	→	U151^#(isBin(X),X,Y)	(166)
plus^#(plus(x,y),z')	→	plus^#(y,z')	(69)
plus^#(x,y)	→	plus^#(y,x)	(63)
plus^#(plus(x,y),z')	→	plus^#(x,plus(y,z'))	(66)
U152^#(tt,X,Y)	→	plus^#(X,Y)	(91)

1.1.4 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[U152^#(x₁, x₂, x₃)]	=	2 + 2 · x₂ + 2 · x₃
[tt]	=	0
[plus^#(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[U151^#(x₁, x₂, x₃)]	=	3 + 2 · x₂ + 2 · x₃
[isBin(x₁)]	=	0
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[0(x₁)]	=	1 · x₁
[U131^#(x₁, x₂, x₃)]	=	2 · x₂ + 2 · x₃
[1(x₁)]	=	1 + 1 · x₁
[U141(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[z]	=	0
[U141^#(x₁, x₂, x₃)]	=	2 + 2 · x₂ + 2 · x₃
[U142^#(x₁, x₂, x₃)]	=	2 + 2 · x₂ + 2 · x₃
[U132^#(x₁, x₂, x₃)]	=	2 · x₂ + 2 · x₃
[U121(x₁, x₂)]	=	1 · x₂
[U131(x₁, x₂, x₃)]	=	1 · x₂ + 1 · x₃
[U151(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[mult(x₁, x₂)]	=	0
[U51(x₁, x₂)]	=	3 + 3 · x₂
[U31(x₁)]	=	3
[U61(x₁, x₂)]	=	3 + 3 · x₂
[sum(x₁)]	=	0
[U81(x₁)]	=	3
[isBag(x₁)]	=	0
[U41(x₁)]	=	3
[prod(x₁)]	=	0
[U71(x₁)]	=	3
[U132(x₁, x₂, x₃)]	=	1 · x₂ + 1 · x₃
[empty]	=	0
[union(x₁, x₂)]	=	0
[U21(x₁, x₂)]	=	3 + 3 · x₂
[singl(x₁)]	=	0
[U11(x₁)]	=	3
[U22(x₁)]	=	3
[U52(x₁)]	=	3
[U152(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U142(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U62(x₁)]	=	3

together with the usable rules

0(z)	→	z	(3)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U121(tt,X)	→	X	(9)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)

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

U152^#(tt,X,Y)	→	plus^#(X,Y)	(91)
U151^#(tt,X,Y)	→	U152^#(isBin(Y),X,Y)	(87)
plus^#(plus(1(X),1(Y)),ext)	→	U151^#(isBin(X),X,Y)	(166)
plus^#(1(X),1(Y))	→	U151^#(isBin(X),X,Y)	(136)
plus^#(plus(1(X),1(Y)),ext)	→	plus^#(U151(isBin(X),X,Y),ext)	(165)
U142^#(tt,X,Y)	→	plus^#(X,Y)	(86)

could be deleted.

1.1.4.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

U131^#(tt,X,Y)	→	U132^#(isBin(Y),X,Y)	(80)
U132^#(tt,X,Y)	→	plus^#(X,Y)	(83)
plus^#(0(X),0(Y))	→	U131^#(isBin(X),X,Y)	(132)
plus^#(plus(z,X),ext)	→	plus^#(U121(isBin(X),X),ext)	(156)
plus^#(plus(0(X),0(Y)),ext)	→	plus^#(U131(isBin(X),X,Y),ext)	(159)
plus^#(plus(0(X),0(Y)),ext)	→	U131^#(isBin(X),X,Y)	(160)
plus^#(plus(0(X),1(Y)),ext)	→	plus^#(U141(isBin(X),X,Y),ext)	(162)
plus^#(plus(x,y),z')	→	plus^#(y,z')	(69)
plus^#(x,y)	→	plus^#(y,x)	(63)
plus^#(plus(x,y),z')	→	plus^#(x,plus(y,z'))	(66)

1.1.4.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[U131^#(x₁, x₂, x₃)]	=	1 · x₂ + 1 · x₃
[tt]	=	0
[U132^#(x₁, x₂, x₃)]	=	1 · x₂ + 1 · x₃
[isBin(x₁)]	=	0
[plus^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[0(x₁)]	=	1 + 1 · x₁
[z]	=	0
[U121(x₁, x₂)]	=	1 · x₂
[1(x₁)]	=	1 + 1 · x₁
[U141(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U131(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U31(x₁)]	=	3
[U41(x₁)]	=	3
[U132(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U142(x₁, x₂, x₃)]	=	1 + 1 · x₂ + 1 · x₃
[U21(x₁, x₂)]	=	3 + 3 · x₂
[U22(x₁)]	=	3
[isBag(x₁)]	=	0
[mult(x₁, x₂)]	=	0
[U51(x₁, x₂)]	=	3 + 3 · x₂
[U61(x₁, x₂)]	=	3 + 3 · x₂
[sum(x₁)]	=	0
[U81(x₁)]	=	3
[prod(x₁)]	=	0
[U71(x₁)]	=	3
[U52(x₁)]	=	3
[U62(x₁)]	=	3
[empty]	=	0
[union(x₁, x₂)]	=	0
[singl(x₁)]	=	0
[U11(x₁)]	=	3
[U151(x₁, x₂, x₃)]	=	2 + 1 · x₂ + 1 · x₃
[U152(x₁, x₂, x₃)]	=	2 + 1 · x₂ + 1 · x₃

together with the usable rules

0(z)	→	z	(3)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U121(tt,X)	→	X	(9)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)

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

plus^#(plus(0(X),0(Y)),ext)	→	U131^#(isBin(X),X,Y)	(160)
plus^#(0(X),0(Y))	→	U131^#(isBin(X),X,Y)	(132)
plus^#(plus(0(X),1(Y)),ext)	→	plus^#(U141(isBin(X),X,Y),ext)	(162)
plus^#(plus(0(X),0(Y)),ext)	→	plus^#(U131(isBin(X),X,Y),ext)	(159)

could be deleted.

1.1.4.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

plus^#(plus(x,y),z')	→	plus^#(y,z')	(69)
plus^#(plus(z,X),ext)	→	plus^#(U121(isBin(X),X),ext)	(156)
plus^#(x,y)	→	plus^#(y,x)	(63)
plus^#(plus(x,y),z')	→	plus^#(x,plus(y,z'))	(66)

1.1.4.1.1.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[plus^#(x₁, x₂)]	=	2 · x₁ + 2 · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[z]	=	1
[U121(x₁, x₂)]	=	1 · x₂
[isBin(x₁)]	=	0
[U131(x₁, x₂, x₃)]	=	1
[tt]	=	0
[U132(x₁, x₂, x₃)]	=	1
[U11(x₁)]	=	3
[U141(x₁, x₂, x₃)]	=	1
[U142(x₁, x₂, x₃)]	=	1
[U71(x₁)]	=	3
[isBag(x₁)]	=	0
[empty]	=	0
[union(x₁, x₂)]	=	0
[U21(x₁, x₂)]	=	3 + 3 · x₂
[singl(x₁)]	=	0
[0(x₁)]	=	1
[mult(x₁, x₂)]	=	0
[U51(x₁, x₂)]	=	3 + 3 · x₂
[U31(x₁)]	=	3
[U61(x₁, x₂)]	=	3 + 3 · x₂
[sum(x₁)]	=	0
[U81(x₁)]	=	3
[1(x₁)]	=	1
[U41(x₁)]	=	3
[prod(x₁)]	=	0
[U151(x₁, x₂, x₃)]	=	2
[U152(x₁, x₂, x₃)]	=	2
[U62(x₁)]	=	3
[U52(x₁)]	=	3
[U22(x₁)]	=	3

together with the usable rules

U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
0(z)	→	z	(3)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U121(tt,X)	→	X	(9)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)

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

plus^#(plus(z,X),ext)

→

plus^#(U121(isBin(X),X),ext)

(156)

could be deleted.

1.1.4.1.1.1.1.1 AC Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[plus^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	3 + 1 · x₁ + 1 · x₂
[U11(x₁)]	=	3
[tt]	=	0
[0(x₁)]	=	0
[z]	=	0
[U51(x₁, x₂)]	=	3 + 3 · x₂
[U52(x₁)]	=	3
[isBin(x₁)]	=	0
[U131(x₁, x₂, x₃)]	=	2
[1(x₁)]	=	0
[U151(x₁, x₂, x₃)]	=	2
[U141(x₁, x₂, x₃)]	=	3
[U121(x₁, x₂)]	=	1 · x₂
[U132(x₁, x₂, x₃)]	=	0
[U142(x₁, x₂, x₃)]	=	2
[U41(x₁)]	=	3
[mult(x₁, x₂)]	=	0
[U31(x₁)]	=	3
[U61(x₁, x₂)]	=	3 + 3 · x₂
[sum(x₁)]	=	0
[U81(x₁)]	=	3
[isBag(x₁)]	=	0
[prod(x₁)]	=	0
[U71(x₁)]	=	3
[U22(x₁)]	=	3
[empty]	=	0
[union(x₁, x₂)]	=	0
[U21(x₁, x₂)]	=	3 + 3 · x₂
[singl(x₁)]	=	0
[U152(x₁, x₂, x₃)]	=	0
[U62(x₁)]	=	3

together with the usable rules

0(z)	→	z	(3)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U121(tt,X)	→	X	(9)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
plus(x,y)	→	plus(y,x)	(57)

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

plus^#(plus(x,y),z')

→

plus^#(y,z')

(69)

could be deleted.

1.1.4.1.1.1.1.1.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

The 5^th component contains the pair

U21^#(tt,V2)	→	isBag^#(V2)	(103)
isBag^#(singl(V1))	→	isBin^#(V1)	(109)
isBin^#(0(V1))	→	isBin^#(V1)	(113)
isBin^#(1(V1))	→	isBin^#(V1)	(115)
isBin^#(mult(V1,V2))	→	U51^#(isBin(V1),V2)	(116)
U51^#(tt,V2)	→	isBin^#(V2)	(105)
isBin^#(mult(V1,V2))	→	isBin^#(V1)	(117)
isBin^#(plus(V1,V2))	→	U61^#(isBin(V1),V2)	(118)
U61^#(tt,V2)	→	isBin^#(V2)	(107)
isBin^#(plus(V1,V2))	→	isBin^#(V1)	(119)
isBin^#(prod(V1))	→	isBag^#(V1)	(121)
isBag^#(union(V1,V2))	→	U21^#(isBag(V1),V2)	(110)
isBag^#(union(V1,V2))	→	isBag^#(V1)	(111)
isBin^#(sum(V1))	→	isBag^#(V1)	(123)

1.1.5 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[U22(x₁)]	=	1 · x₁
[tt]	=	0
[U81(x₁)]	=	1 · x₁
[isBag(x₁)]	=	2 · x₁
[empty]	=	0
[U41(x₁)]	=	1 · x₁
[isBin(x₁)]	=	2 · x₁
[mult(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[U51(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[U61(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[U62(x₁)]	=	1 · x₁
[z]	=	0
[1(x₁)]	=	3 + 1 · x₁
[U52(x₁)]	=	1 · x₁
[prod(x₁)]	=	2 · x₁
[U71(x₁)]	=	1 · x₁
[U21(x₁, x₂)]	=	1 · x₁ + 3 · x₂
[union(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[U11(x₁)]	=	2 · x₁
[0(x₁)]	=	3 + 3 · x₁
[U31(x₁)]	=	2 · x₁
[plus(x₁, x₂)]	=	3 + 3 · x₁ + 3 · x₂
[singl(x₁)]	=	3 · x₁
[sum(x₁)]	=	2 · x₁
[U51^#(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[isBin^#(x₁)]	=	2 · x₁
[U61^#(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[U21^#(x₁, x₂)]	=	2 · x₁ + 3 · x₂
[isBag^#(x₁)]	=	3 · x₁

together with the usable rules

U22(tt)	→	tt	(23)
U81(tt)	→	tt	(31)
isBag(empty)	→	tt	(33)
U41(tt)	→	tt	(25)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
U61(tt,V2)	→	U62(isBin(V2))	(28)
isBin(z)	→	tt	(36)
isBin(1(V1))	→	U41(isBin(V1))	(38)
U51(tt,V2)	→	U52(isBin(V2))	(26)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
U21(tt,V2)	→	U22(isBag(V2))	(22)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
U11(tt)	→	tt	(6)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
U52(tt)	→	tt	(27)
U71(tt)	→	tt	(30)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
U31(tt)	→	tt	(24)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
U62(tt)	→	tt	(29)

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

isBin^#(plus(V1,V2))	→	isBin^#(V1)	(119)
isBin^#(0(V1))	→	isBin^#(V1)	(113)
isBag^#(singl(V1))	→	isBin^#(V1)	(109)
isBin^#(prod(V1))	→	isBag^#(V1)	(121)
isBin^#(1(V1))	→	isBin^#(V1)	(115)
isBin^#(sum(V1))	→	isBag^#(V1)	(123)
isBag^#(union(V1,V2))	→	U21^#(isBag(V1),V2)	(110)
isBin^#(plus(V1,V2))	→	U61^#(isBin(V1),V2)	(118)
isBin^#(mult(V1,V2))	→	isBin^#(V1)	(117)
isBin^#(mult(V1,V2))	→	U51^#(isBin(V1),V2)	(116)
isBag^#(union(V1,V2))	→	isBag^#(V1)	(111)

and the rules

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U121(tt,X)	→	X	(9)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U161(tt,X)	→	X	(16)
U171(tt,A,B)	→	U172(isBag(B),A,B)	(17)
U172(tt,A,B)	→	mult(prod(A),prod(B))	(18)
U181(tt,X)	→	X	(19)
U191(tt,A,B)	→	U192(isBag(B),A,B)	(20)
U192(tt,A,B)	→	plus(sum(A),sum(B))	(21)
U91(tt)	→	z	(32)
isBag(empty)	→	tt	(33)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBin(z)	→	tt	(36)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
mult(z,X)	→	U91(isBin(X))	(43)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
prod(empty)	→	1(z)	(50)
prod(singl(X))	→	U161(isBin(X),X)	(51)
prod(union(A,B))	→	U171(isBag(A),A,B)	(52)
sum(empty)	→	0(z)	(53)
sum(singl(X))	→	U181(isBin(X),X)	(54)
sum(union(A,B))	→	U191(isBag(A),A,B)	(55)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)

could be deleted.

1.1.5.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 6^th component contains the pair

union^#(union(empty,X),ext)	→	union^#(X,ext)	(169)
union^#(union(X,empty),ext)	→	union^#(X,ext)	(168)
union^#(union(x,y),z')	→	union^#(y,z')	(70)
union^#(x,y)	→	union^#(y,x)	(64)
union^#(union(x,y),z')	→	union^#(x,union(y,z'))	(67)

1.1.6 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[union^#(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[union(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[empty]	=	0

together with the usable rules

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)
union(union(x,y),z')	→	union(x,union(y,z'))	(61)
union(x,y)	→	union(y,x)	(58)

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

union^#(union(empty,X),ext)	→	union^#(X,ext)	(169)
union^#(union(X,empty),ext)	→	union^#(X,ext)	(168)

and the rules

union(X,empty)	→	X	(1)
union(empty,X)	→	X	(2)
0(z)	→	z	(3)
U101(tt,X,Y)	→	U102(isBin(Y),X,Y)	(4)
U102(tt,X,Y)	→	0(mult(X,Y))	(5)
U11(tt)	→	tt	(6)
U111(tt,X,Y)	→	U112(isBin(Y),X,Y)	(7)
U112(tt,X,Y)	→	plus(0(mult(X,Y)),Y)	(8)
U121(tt,X)	→	X	(9)
U131(tt,X,Y)	→	U132(isBin(Y),X,Y)	(10)
U132(tt,X,Y)	→	0(plus(X,Y))	(11)
U141(tt,X,Y)	→	U142(isBin(Y),X,Y)	(12)
U142(tt,X,Y)	→	1(plus(X,Y))	(13)
U151(tt,X,Y)	→	U152(isBin(Y),X,Y)	(14)
U152(tt,X,Y)	→	0(plus(plus(X,Y),1(z)))	(15)
U161(tt,X)	→	X	(16)
U171(tt,A,B)	→	U172(isBag(B),A,B)	(17)
U172(tt,A,B)	→	mult(prod(A),prod(B))	(18)
U181(tt,X)	→	X	(19)
U191(tt,A,B)	→	U192(isBag(B),A,B)	(20)
U192(tt,A,B)	→	plus(sum(A),sum(B))	(21)
U21(tt,V2)	→	U22(isBag(V2))	(22)
U22(tt)	→	tt	(23)
U31(tt)	→	tt	(24)
U41(tt)	→	tt	(25)
U51(tt,V2)	→	U52(isBin(V2))	(26)
U52(tt)	→	tt	(27)
U61(tt,V2)	→	U62(isBin(V2))	(28)
U62(tt)	→	tt	(29)
U71(tt)	→	tt	(30)
U81(tt)	→	tt	(31)
U91(tt)	→	z	(32)
isBag(empty)	→	tt	(33)
isBag(singl(V1))	→	U11(isBin(V1))	(34)
isBag(union(V1,V2))	→	U21(isBag(V1),V2)	(35)
isBin(z)	→	tt	(36)
isBin(0(V1))	→	U31(isBin(V1))	(37)
isBin(1(V1))	→	U41(isBin(V1))	(38)
isBin(mult(V1,V2))	→	U51(isBin(V1),V2)	(39)
isBin(plus(V1,V2))	→	U61(isBin(V1),V2)	(40)
isBin(prod(V1))	→	U71(isBag(V1))	(41)
isBin(sum(V1))	→	U81(isBag(V1))	(42)
mult(z,X)	→	U91(isBin(X))	(43)
mult(0(X),Y)	→	U101(isBin(X),X,Y)	(44)
mult(1(X),Y)	→	U111(isBin(X),X,Y)	(45)
plus(z,X)	→	U121(isBin(X),X)	(46)
plus(0(X),0(Y))	→	U131(isBin(X),X,Y)	(47)
plus(0(X),1(Y))	→	U141(isBin(X),X,Y)	(48)
plus(1(X),1(Y))	→	U151(isBin(X),X,Y)	(49)
prod(empty)	→	1(z)	(50)
prod(singl(X))	→	U161(isBin(X),X)	(51)
prod(union(A,B))	→	U171(isBag(A),A,B)	(52)
sum(empty)	→	0(z)	(53)
sum(singl(X))	→	U181(isBin(X),X)	(54)
sum(union(A,B))	→	U191(isBag(A),A,B)	(55)
mult(mult(z,X),ext)	→	mult(U91(isBin(X)),ext)	(170)
mult(mult(0(X),Y),ext)	→	mult(U101(isBin(X),X,Y),ext)	(171)
mult(mult(1(X),Y),ext)	→	mult(U111(isBin(X),X,Y),ext)	(172)
plus(plus(z,X),ext)	→	plus(U121(isBin(X),X),ext)	(173)
plus(plus(0(X),0(Y)),ext)	→	plus(U131(isBin(X),X,Y),ext)	(174)
plus(plus(0(X),1(Y)),ext)	→	plus(U141(isBin(X),X,Y),ext)	(175)
plus(plus(1(X),1(Y)),ext)	→	plus(U151(isBin(X),X,Y),ext)	(176)
union(union(X,empty),ext)	→	union(X,ext)	(177)
union(union(empty,X),ext)	→	union(X,ext)	(178)

could be deleted.

1.1.6.1 AC Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[union^#(x₁, x₂)]	=	3 · x₁ + 3 · x₂
[union(x₁, x₂)]	=	1 + 1 · x₁ + 1 · x₂
[mult(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[plus(x₁, x₂)]	=	1 · x₁ + 1 · x₂

together with the usable rules

mult(x,y)	→	mult(y,x)	(56)
plus(x,y)	→	plus(y,x)	(57)
union(x,y)	→	union(y,x)	(58)
mult(mult(x,y),z')	→	mult(x,mult(y,z'))	(59)
plus(plus(x,y),z')	→	plus(x,plus(y,z'))	(60)
union(union(x,y),z')	→	union(x,union(y,z'))	(61)

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

union^#(union(x,y),z')

→

union^#(y,z')

(70)

and no rules could be deleted.

1.1.6.1.1 AC Dependency Pair Problem is trivial

There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.