Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/MYNAT_complete-noand_Z)

The rewrite relation of the following TRS is considered.

U101(tt,M,N)	→	U102(isNatKind(activate(M)),activate(M),activate(N))	(1)
U102(tt,M,N)	→	U103(isNat(activate(N)),activate(M),activate(N))	(2)
U103(tt,M,N)	→	U104(isNatKind(activate(N)),activate(M),activate(N))	(3)
U104(tt,M,N)	→	plus(x(activate(N),activate(M)),activate(N))	(4)
U11(tt,V1,V2)	→	U12(isNatKind(activate(V1)),activate(V1),activate(V2))	(5)
U12(tt,V1,V2)	→	U13(isNatKind(activate(V2)),activate(V1),activate(V2))	(6)
U13(tt,V1,V2)	→	U14(isNatKind(activate(V2)),activate(V1),activate(V2))	(7)
U14(tt,V1,V2)	→	U15(isNat(activate(V1)),activate(V2))	(8)
U15(tt,V2)	→	U16(isNat(activate(V2)))	(9)
U16(tt)	→	tt	(10)
U21(tt,V1)	→	U22(isNatKind(activate(V1)),activate(V1))	(11)
U22(tt,V1)	→	U23(isNat(activate(V1)))	(12)
U23(tt)	→	tt	(13)
U31(tt,V1,V2)	→	U32(isNatKind(activate(V1)),activate(V1),activate(V2))	(14)
U32(tt,V1,V2)	→	U33(isNatKind(activate(V2)),activate(V1),activate(V2))	(15)
U33(tt,V1,V2)	→	U34(isNatKind(activate(V2)),activate(V1),activate(V2))	(16)
U34(tt,V1,V2)	→	U35(isNat(activate(V1)),activate(V2))	(17)
U35(tt,V2)	→	U36(isNat(activate(V2)))	(18)
U36(tt)	→	tt	(19)
U41(tt,V2)	→	U42(isNatKind(activate(V2)))	(20)
U42(tt)	→	tt	(21)
U51(tt)	→	tt	(22)
U61(tt,V2)	→	U62(isNatKind(activate(V2)))	(23)
U62(tt)	→	tt	(24)
U71(tt,N)	→	U72(isNatKind(activate(N)),activate(N))	(25)
U72(tt,N)	→	activate(N)	(26)
U81(tt,M,N)	→	U82(isNatKind(activate(M)),activate(M),activate(N))	(27)
U82(tt,M,N)	→	U83(isNat(activate(N)),activate(M),activate(N))	(28)
U83(tt,M,N)	→	U84(isNatKind(activate(N)),activate(M),activate(N))	(29)
U84(tt,M,N)	→	s(plus(activate(N),activate(M)))	(30)
U91(tt,N)	→	U92(isNatKind(activate(N)))	(31)
U92(tt)	→	0	(32)
isNat(n__0)	→	tt	(33)
isNat(n__plus(V1,V2))	→	U11(isNatKind(activate(V1)),activate(V1),activate(V2))	(34)
isNat(n__s(V1))	→	U21(isNatKind(activate(V1)),activate(V1))	(35)
isNat(n__x(V1,V2))	→	U31(isNatKind(activate(V1)),activate(V1),activate(V2))	(36)
isNatKind(n__0)	→	tt	(37)
isNatKind(n__plus(V1,V2))	→	U41(isNatKind(activate(V1)),activate(V2))	(38)
isNatKind(n__s(V1))	→	U51(isNatKind(activate(V1)))	(39)
isNatKind(n__x(V1,V2))	→	U61(isNatKind(activate(V1)),activate(V2))	(40)
plus(N,0)	→	U71(isNat(N),N)	(41)
plus(N,s(M))	→	U81(isNat(M),M,N)	(42)
x(N,0)	→	U91(isNat(N),N)	(43)
x(N,s(M))	→	U101(isNat(M),M,N)	(44)
0	→	n__0	(45)
plus(X1,X2)	→	n__plus(X1,X2)	(46)
s(X)	→	n__s(X)	(47)
x(X1,X2)	→	n__x(X1,X2)	(48)
activate(n__0)	→	0	(49)
activate(n__plus(X1,X2))	→	plus(X1,X2)	(50)
activate(n__s(X))	→	s(X)	(51)
activate(n__x(X1,X2))	→	x(X1,X2)	(52)
activate(X)	→	X	(53)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

There are 124 ruless (increase limit for explicit display).

1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

There are 113 ruless (increase limit for explicit display).

1.1.1 Reduction Pair Processor

Using the

prec(U102^#)	=	15	stat(U102^#)	=	mul
prec(tt)	=	2	stat(tt)	=	mul
prec(U103^#)	=	15	stat(U103^#)	=	mul
prec(isNat)	=	2	stat(isNat)	=	mul
prec(U104^#)	=	15	stat(U104^#)	=	mul
prec(isNatKind)	=	2	stat(isNatKind)	=	mul
prec(plus^#)	=	8	stat(plus^#)	=	mul
prec(x)	=	15	stat(x)	=	mul
prec(0)	=	0	stat(0)	=	mul
prec(U71^#)	=	3	stat(U71^#)	=	mul
prec(U72^#)	=	1	stat(U72^#)	=	mul
prec(n__plus)	=	8	stat(n__plus)	=	mul
prec(U11^#)	=	8	stat(U11^#)	=	mul
prec(U12^#)	=	8	stat(U12^#)	=	mul
prec(U13^#)	=	8	stat(U13^#)	=	mul
prec(U14^#)	=	8	stat(U14^#)	=	mul
prec(U41^#)	=	4	stat(U41^#)	=	mul
prec(n__x)	=	15	stat(n__x)	=	mul
prec(x^#)	=	15	stat(x^#)	=	mul
prec(U91^#)	=	9	stat(U91^#)	=	lex
prec(n__s)	=	5	stat(n__s)	=	mul
prec(U61^#)	=	10	stat(U61^#)	=	lex
prec(U21^#)	=	2	stat(U21^#)	=	lex
prec(U22^#)	=	2	stat(U22^#)	=	lex
prec(U31^#)	=	13	stat(U31^#)	=	mul
prec(U32^#)	=	13	stat(U32^#)	=	mul
prec(U33^#)	=	12	stat(U33^#)	=	mul
prec(U34^#)	=	11	stat(U34^#)	=	lex
prec(s)	=	5	stat(s)	=	mul
prec(U101^#)	=	15	stat(U101^#)	=	mul
prec(U81^#)	=	8	stat(U81^#)	=	mul
prec(U82^#)	=	8	stat(U82^#)	=	mul
prec(U83^#)	=	8	stat(U83^#)	=	mul
prec(U84^#)	=	8	stat(U84^#)	=	mul
prec(n__0)	=	0	stat(n__0)	=	mul
prec(U102)	=	15	stat(U102)	=	mul
prec(U103)	=	15	stat(U103)	=	mul
prec(U104)	=	15	stat(U104)	=	mul
prec(plus)	=	8	stat(plus)	=	mul
prec(U71)	=	7	stat(U71)	=	mul
prec(U72)	=	6	stat(U72)	=	lex
prec(U101)	=	15	stat(U101)	=	mul
prec(U11)	=	2	stat(U11)	=	mul
prec(U31)	=	2	stat(U31)	=	mul
prec(U61)	=	2	stat(U61)	=	mul
prec(U91)	=	14	stat(U91)	=	mul
prec(U22)	=	2	stat(U22)	=	mul
prec(U81)	=	8	stat(U81)	=	mul
prec(U82)	=	8	stat(U82)	=	mul
prec(U83)	=	8	stat(U83)	=	mul
prec(U84)	=	8	stat(U84)	=	mul
prec(U12)	=	2	stat(U12)	=	mul
prec(U13)	=	2	stat(U13)	=	mul
prec(U14)	=	2	stat(U14)	=	mul

π(U102^#)	=	[1,2,3]
π(tt)	=	[]
π(U103^#)	=	[1,2,3]
π(isNat)	=	[]
π(activate)	=	1
π(U104^#)	=	[1,2,3]
π(isNatKind)	=	[]
π(plus^#)	=	[1,2]
π(x)	=	[1,2]
π(0)	=	[]
π(U71^#)	=	[2]
π(U72^#)	=	[1,2]
π(activate^#)	=	1
π(n__plus)	=	[1,2]
π(isNat^#)	=	1
π(U11^#)	=	[2,3]
π(U12^#)	=	[2,3]
π(U13^#)	=	[2,3]
π(U14^#)	=	[2,3]
π(U15^#)	=	2
π(isNatKind^#)	=	1
π(U41^#)	=	[2]
π(n__x)	=	[1,2]
π(x^#)	=	[1,2]
π(U91^#)	=	[2]
π(n__s)	=	[1]
π(U61^#)	=	[1,2]
π(U21^#)	=	[1,2]
π(U22^#)	=	[1,2]
π(U31^#)	=	[2,3]
π(U32^#)	=	[2,3]
π(U33^#)	=	[1,2,3]
π(U34^#)	=	[1,3,2]
π(U35^#)	=	2
π(s)	=	[1]
π(U101^#)	=	[1,2,3]
π(U81^#)	=	[1,2,3]
π(U82^#)	=	[1,2,3]
π(U83^#)	=	[1,2,3]
π(U84^#)	=	[1,2,3]
π(n__0)	=	[]
π(U102)	=	[1,2,3]
π(U103)	=	[1,2,3]
π(U104)	=	[1,2,3]
π(plus)	=	[1,2]
π(U71)	=	[2]
π(U72)	=	[1,2]
π(U101)	=	[1,2,3]
π(U11)	=	[]
π(U21)	=	1
π(U31)	=	[]
π(U41)	=	1
π(U51)	=	1
π(U61)	=	[]
π(U91)	=	[1,2]
π(U62)	=	1
π(U22)	=	[]
π(U23)	=	1
π(U32)	=	1
π(U33)	=	1
π(U34)	=	1
π(U35)	=	1
π(U36)	=	1
π(U92)	=	1
π(U81)	=	[1,2,3]
π(U82)	=	[1,2,3]
π(U83)	=	[1,2,3]
π(U84)	=	[1,2,3]
π(U42)	=	1
π(U12)	=	[]
π(U13)	=	[]
π(U14)	=	[]
π(U15)	=	1
π(U16)	=	1

the pairs

U104^#(tt,M,N)	→	plus^#(x(activate(N),activate(M)),activate(N))	(66)
plus^#(N,0)	→	U71^#(isNat(N),N)	(166)
U71^#(tt,N)	→	U72^#(isNatKind(activate(N)),activate(N))	(120)
U72^#(tt,N)	→	activate^#(N)	(123)
plus^#(N,0)	→	isNat^#(N)	(167)
U14^#(tt,V1,V2)	→	U15^#(isNat(activate(V1)),activate(V2))	(82)
isNat^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(145)
isNatKind^#(n__plus(V1,V2))	→	U41^#(isNatKind(activate(V1)),activate(V2))	(155)
U41^#(tt,V2)	→	isNatKind^#(activate(V2))	(115)
isNatKind^#(n__plus(V1,V2))	→	isNatKind^#(activate(V1))	(156)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V1)	(157)
x^#(N,0)	→	U91^#(isNat(N),N)	(170)
U91^#(tt,N)	→	isNatKind^#(activate(N))	(141)
isNatKind^#(n__plus(V1,V2))	→	activate^#(V2)	(158)
isNatKind^#(n__s(V1))	→	isNatKind^#(activate(V1))	(160)
isNatKind^#(n__s(V1))	→	activate^#(V1)	(161)
isNatKind^#(n__x(V1,V2))	→	U61^#(isNatKind(activate(V1)),activate(V2))	(162)
U61^#(tt,V2)	→	isNatKind^#(activate(V2))	(118)
isNatKind^#(n__x(V1,V2))	→	isNatKind^#(activate(V1))	(163)
isNatKind^#(n__x(V1,V2))	→	activate^#(V1)	(164)
isNatKind^#(n__x(V1,V2))	→	activate^#(V2)	(165)
U61^#(tt,V2)	→	activate^#(V2)	(119)
U91^#(tt,N)	→	activate^#(N)	(142)
x^#(N,0)	→	isNat^#(N)	(171)
isNat^#(n__plus(V1,V2))	→	activate^#(V1)	(146)
isNat^#(n__plus(V1,V2))	→	activate^#(V2)	(147)
isNat^#(n__s(V1))	→	U21^#(isNatKind(activate(V1)),activate(V1))	(148)
U22^#(tt,V1)	→	isNat^#(activate(V1))	(93)
isNat^#(n__s(V1))	→	isNatKind^#(activate(V1))	(149)
isNat^#(n__s(V1))	→	activate^#(V1)	(150)
isNat^#(n__x(V1,V2))	→	U31^#(isNatKind(activate(V1)),activate(V1),activate(V2))	(151)
U32^#(tt,V1,V2)	→	U33^#(isNatKind(activate(V2)),activate(V1),activate(V2))	(99)
U33^#(tt,V1,V2)	→	U34^#(isNatKind(activate(V2)),activate(V1),activate(V2))	(103)
U34^#(tt,V1,V2)	→	U35^#(isNat(activate(V1)),activate(V2))	(107)
isNat^#(n__x(V1,V2))	→	isNatKind^#(activate(V1))	(152)
isNat^#(n__x(V1,V2))	→	activate^#(V1)	(153)
isNat^#(n__x(V1,V2))	→	activate^#(V2)	(154)
U34^#(tt,V1,V2)	→	isNat^#(activate(V1))	(108)
U34^#(tt,V1,V2)	→	activate^#(V1)	(109)
U34^#(tt,V1,V2)	→	activate^#(V2)	(110)
U33^#(tt,V1,V2)	→	isNatKind^#(activate(V2))	(104)
U33^#(tt,V1,V2)	→	activate^#(V2)	(105)
U33^#(tt,V1,V2)	→	activate^#(V1)	(106)
U32^#(tt,V1,V2)	→	isNatKind^#(activate(V2))	(100)
U32^#(tt,V1,V2)	→	activate^#(V2)	(101)
U32^#(tt,V1,V2)	→	activate^#(V1)	(102)
U31^#(tt,V1,V2)	→	isNatKind^#(activate(V1))	(96)
U31^#(tt,V1,V2)	→	activate^#(V1)	(97)
U31^#(tt,V1,V2)	→	activate^#(V2)	(98)
U22^#(tt,V1)	→	activate^#(V1)	(94)
U21^#(tt,V1)	→	isNatKind^#(activate(V1))	(90)
U21^#(tt,V1)	→	activate^#(V1)	(91)
x^#(N,s(M))	→	U101^#(isNat(M),M,N)	(172)
U102^#(tt,M,N)	→	isNat^#(activate(N))	(59)
U102^#(tt,M,N)	→	activate^#(N)	(60)
U102^#(tt,M,N)	→	activate^#(M)	(61)
U101^#(tt,M,N)	→	isNatKind^#(activate(M))	(55)
U101^#(tt,M,N)	→	activate^#(M)	(56)
U101^#(tt,M,N)	→	activate^#(N)	(57)
x^#(N,s(M))	→	isNat^#(M)	(173)
U41^#(tt,V2)	→	activate^#(V2)	(116)
U14^#(tt,V1,V2)	→	isNat^#(activate(V1))	(83)
U14^#(tt,V1,V2)	→	activate^#(V1)	(84)
U14^#(tt,V1,V2)	→	activate^#(V2)	(85)
U13^#(tt,V1,V2)	→	isNatKind^#(activate(V2))	(79)
U13^#(tt,V1,V2)	→	activate^#(V2)	(80)
U13^#(tt,V1,V2)	→	activate^#(V1)	(81)
U12^#(tt,V1,V2)	→	isNatKind^#(activate(V2))	(75)
U12^#(tt,V1,V2)	→	activate^#(V2)	(76)
U12^#(tt,V1,V2)	→	activate^#(V1)	(77)
U11^#(tt,V1,V2)	→	isNatKind^#(activate(V1))	(71)
U11^#(tt,V1,V2)	→	activate^#(V1)	(72)
U11^#(tt,V1,V2)	→	activate^#(V2)	(73)
plus^#(N,s(M))	→	U81^#(isNat(M),M,N)	(168)
U84^#(tt,M,N)	→	plus^#(activate(N),activate(M))	(137)
plus^#(N,s(M))	→	isNat^#(M)	(169)
U84^#(tt,M,N)	→	activate^#(N)	(138)
U84^#(tt,M,N)	→	activate^#(M)	(139)
U83^#(tt,M,N)	→	isNatKind^#(activate(N))	(133)
U83^#(tt,M,N)	→	activate^#(N)	(134)
U83^#(tt,M,N)	→	activate^#(M)	(135)
U82^#(tt,M,N)	→	isNat^#(activate(N))	(129)
U82^#(tt,M,N)	→	activate^#(N)	(130)
U82^#(tt,M,N)	→	activate^#(M)	(131)
U81^#(tt,M,N)	→	isNatKind^#(activate(M))	(125)
U81^#(tt,M,N)	→	activate^#(M)	(126)
U81^#(tt,M,N)	→	activate^#(N)	(127)
U71^#(tt,N)	→	isNatKind^#(activate(N))	(121)
U71^#(tt,N)	→	activate^#(N)	(122)
U104^#(tt,M,N)	→	x^#(activate(N),activate(M))	(67)
U104^#(tt,M,N)	→	activate^#(N)	(68)
U104^#(tt,M,N)	→	activate^#(M)	(69)
U103^#(tt,M,N)	→	isNatKind^#(activate(N))	(63)
U103^#(tt,M,N)	→	activate^#(N)	(64)
U103^#(tt,M,N)	→	activate^#(M)	(65)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.