Certification Problem

Input (TPDB TRS_Standard/Secret_05_TRS/cime5)

The rewrite relation of the following TRS is considered.

intersect'ii'in(cons(X,X0),cons(X,X1))	→	intersect'ii'out	(1)
intersect'ii'in(Xs,cons(X0,Ys))	→	u'1'1(intersect'ii'in(Xs,Ys))	(2)
u'1'1(intersect'ii'out)	→	intersect'ii'out	(3)
intersect'ii'in(cons(X0,Xs),Ys)	→	u'2'1(intersect'ii'in(Xs,Ys))	(4)
u'2'1(intersect'ii'out)	→	intersect'ii'out	(5)
reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF)	→	u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))	(6)
u'3'1(reduce'ii'out)	→	reduce'ii'out	(7)
reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF)	→	u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))	(8)
u'4'1(reduce'ii'out)	→	reduce'ii'out	(9)
reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF)	→	u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))	(10)
u'5'1(reduce'ii'out)	→	reduce'ii'out	(11)
reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)	(12)
u'6'1(reduce'ii'out,F2,Fs,Gs,NF)	→	u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))	(13)
u'6'2(reduce'ii'out)	→	reduce'ii'out	(14)
reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))	(15)
u'7'1(reduce'ii'out)	→	reduce'ii'out	(16)
reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF)	→	u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))	(17)
u'8'1(reduce'ii'out)	→	reduce'ii'out	(18)
reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF)	→	u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))	(19)
u'9'1(reduce'ii'out)	→	reduce'ii'out	(20)
reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))	(21)
u'10'1(reduce'ii'out)	→	reduce'ii'out	(22)
reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)	→	u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))	(23)
u'11'1(reduce'ii'out)	→	reduce'ii'out	(24)
reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)	(25)
u'12'1(reduce'ii'out,Fs,G2,Gs,NF)	→	u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))	(26)
u'12'2(reduce'ii'out)	→	reduce'ii'out	(27)
reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))	(28)
u'13'1(reduce'ii'out)	→	reduce'ii'out	(29)
reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))	(30)
u'14'1(reduce'ii'out)	→	reduce'ii'out	(31)
reduce'ii'in(sequent(nil,nil),sequent(F1,F2))	→	u'15'1(intersect'ii'in(F1,F2))	(32)
u'15'1(intersect'ii'out)	→	reduce'ii'out	(33)
tautology'i'in(F)	→	u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))	(34)
u'16'1(reduce'ii'out)	→	tautology'i'out	(35)

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.

intersect'ii'in^#(Xs,cons(X0,Ys))	→	u'1'1^#(intersect'ii'in(Xs,Ys))	(36)
intersect'ii'in^#(Xs,cons(X0,Ys))	→	intersect'ii'in^#(Xs,Ys)	(37)
intersect'ii'in^#(cons(X0,Xs),Ys)	→	u'2'1^#(intersect'ii'in(Xs,Ys))	(38)
intersect'ii'in^#(cons(X0,Xs),Ys)	→	intersect'ii'in^#(Xs,Ys)	(39)
reduce'ii'in^#(sequent(cons(if(A,B),Fs),Gs),NF)	→	u'3'1^#(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))	(40)
reduce'ii'in^#(sequent(cons(if(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)	(41)
reduce'ii'in^#(sequent(cons(iff(A,B),Fs),Gs),NF)	→	u'4'1^#(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))	(42)
reduce'ii'in^#(sequent(cons(iff(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)	(43)
reduce'ii'in^#(sequent(cons(x'2a(F1,F2),Fs),Gs),NF)	→	u'5'1^#(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))	(44)
reduce'ii'in^#(sequent(cons(x'2a(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,cons(F2,Fs)),Gs),NF)	(45)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	u'6'1^#(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)	(46)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,Fs),Gs),NF)	(47)
u'6'1^#(reduce'ii'out,F2,Fs,Gs,NF)	→	u'6'2^#(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))	(48)
u'6'1^#(reduce'ii'out,F2,Fs,Gs,NF)	→	reduce'ii'in^#(sequent(cons(F2,Fs),Gs),NF)	(49)
reduce'ii'in^#(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	u'7'1^#(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))	(50)
reduce'ii'in^#(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(Fs,cons(F1,Gs)),NF)	(51)
reduce'ii'in^#(sequent(Fs,cons(if(A,B),Gs)),NF)	→	u'8'1^#(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))	(52)
reduce'ii'in^#(sequent(Fs,cons(if(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)	(53)
reduce'ii'in^#(sequent(Fs,cons(iff(A,B),Gs)),NF)	→	u'9'1^#(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))	(54)
reduce'ii'in^#(sequent(Fs,cons(iff(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)	(55)
reduce'ii'in^#(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	u'10'1^#(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))	(56)
reduce'ii'in^#(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	reduce'ii'in^#(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))	(57)
reduce'ii'in^#(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)	→	u'11'1^#(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))	(58)
reduce'ii'in^#(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,cons(G2,Gs))),NF)	(59)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	u'12'1^#(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)	(60)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,Gs)),NF)	(61)
u'12'1^#(reduce'ii'out,Fs,G2,Gs,NF)	→	u'12'2^#(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))	(62)
u'12'1^#(reduce'ii'out,Fs,G2,Gs,NF)	→	reduce'ii'in^#(sequent(Fs,cons(G2,Gs)),NF)	(63)
reduce'ii'in^#(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	u'13'1^#(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))	(64)
reduce'ii'in^#(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	reduce'ii'in^#(sequent(cons(G1,Fs),Gs),NF)	(65)
reduce'ii'in^#(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	u'14'1^#(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))	(66)
reduce'ii'in^#(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	reduce'ii'in^#(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))	(67)
reduce'ii'in^#(sequent(nil,nil),sequent(F1,F2))	→	u'15'1^#(intersect'ii'in(F1,F2))	(68)
reduce'ii'in^#(sequent(nil,nil),sequent(F1,F2))	→	intersect'ii'in^#(F1,F2)	(69)
tautology'i'in^#(F)	→	u'16'1^#(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))	(70)
tautology'i'in^#(F)	→	reduce'ii'in^#(sequent(nil,cons(F,nil)),sequent(nil,nil))	(71)

1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	u'6'1^#(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)	(46)
u'6'1^#(reduce'ii'out,F2,Fs,Gs,NF)	→	reduce'ii'in^#(sequent(cons(F2,Fs),Gs),NF)	(49)
reduce'ii'in^#(sequent(cons(if(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)	(41)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,Fs),Gs),NF)	(47)
reduce'ii'in^#(sequent(cons(iff(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)	(43)
reduce'ii'in^#(sequent(cons(x'2a(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,cons(F2,Fs)),Gs),NF)	(45)
reduce'ii'in^#(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(Fs,cons(F1,Gs)),NF)	(51)
reduce'ii'in^#(sequent(Fs,cons(if(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)	(53)
reduce'ii'in^#(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	reduce'ii'in^#(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))	(57)
reduce'ii'in^#(sequent(Fs,cons(iff(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)	(55)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	u'12'1^#(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)	(60)
u'12'1^#(reduce'ii'out,Fs,G2,Gs,NF)	→	reduce'ii'in^#(sequent(Fs,cons(G2,Gs)),NF)	(63)
reduce'ii'in^#(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,cons(G2,Gs))),NF)	(59)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,Gs)),NF)	(61)
reduce'ii'in^#(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	reduce'ii'in^#(sequent(cons(G1,Fs),Gs),NF)	(65)
reduce'ii'in^#(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	reduce'ii'in^#(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))	(67)

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[u'12'1^#(x₁,...,x₅)]	=	2 · x₂ + 2 · x₃ + 2 · x₄ + 2 · x₅
[u'6'1^#(x₁,...,x₅)]	=	2 · x₂ + 2 · x₃ + 2 · x₄ + 2 · x₅
[reduce'ii'in(x₁, x₂)]	=	-2 + 2 · x₂
[sequent(x₁, x₂)]	=	x₁ + x₂
[cons(x₁, x₂)]	=	x₁ + x₂
[if(x₁, x₂)]	=	x₁ + x₂
[u'3'1(x₁)]	=	2
[x'2b(x₁, x₂)]	=	x₁ + x₂
[x'2d(x₁)]	=	x₁
[iff(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂
[u'4'1(x₁)]	=	-2
[x'2a(x₁, x₂)]	=	1 + x₁ + x₂
[u'5'1(x₁)]	=	-2 + 2 · x₁
[u'6'1(x₁,...,x₅)]	=	-2 + x₃
[u'7'1(x₁)]	=	-2 + 2 · x₁
[u'8'1(x₁)]	=	-2
[u'9'1(x₁)]	=	1
[p(x₁)]	=	0
[u'10'1(x₁)]	=	2
[u'11'1(x₁)]	=	2
[u'12'1(x₁,...,x₅)]	=	2 + x₃
[u'13'1(x₁)]	=	-2
[nil]	=	0
[u'14'1(x₁)]	=	-2
[reduce'ii'out]	=	0
[u'15'1(x₁)]	=	-2
[intersect'ii'in(x₁, x₂)]	=	0
[u'12'2(x₁)]	=	-2
[u'6'2(x₁)]	=	2 + 2 · x₁
[intersect'ii'out]	=	2
[u'1'1(x₁)]	=	-2
[u'2'1(x₁)]	=	-2
[reduce'ii'in^#(x₁, x₂)]	=	2 · x₁ + 2 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

reduce'ii'in^#(sequent(cons(iff(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)	(43)
reduce'ii'in^#(sequent(cons(x'2a(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,cons(F2,Fs)),Gs),NF)	(45)
reduce'ii'in^#(sequent(Fs,cons(iff(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)	(55)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	u'12'1^#(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)	(60)
reduce'ii'in^#(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,Gs)),NF)	(61)

could be deleted.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

The 1^st component contains the pair

u'6'1^#(reduce'ii'out,F2,Fs,Gs,NF)	→	reduce'ii'in^#(sequent(cons(F2,Fs),Gs),NF)	(49)
reduce'ii'in^#(sequent(cons(if(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)	(41)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	u'6'1^#(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)	(46)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,Fs),Gs),NF)	(47)
reduce'ii'in^#(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(Fs,cons(F1,Gs)),NF)	(51)
reduce'ii'in^#(sequent(Fs,cons(if(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)	(53)
reduce'ii'in^#(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	reduce'ii'in^#(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))	(57)
reduce'ii'in^#(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(G1,cons(G2,Gs))),NF)	(59)
reduce'ii'in^#(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	reduce'ii'in^#(sequent(cons(G1,Fs),Gs),NF)	(65)
reduce'ii'in^#(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	reduce'ii'in^#(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))	(67)

1.1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[u'6'1^#(x₁,...,x₅)]	=	1 + 2 · x₂ + 2 · x₃ + 2 · x₄
[reduce'ii'in(x₁, x₂)]	=	-2
[sequent(x₁, x₂)]	=	-2 + 2 · x₁ + 2 · x₂
[cons(x₁, x₂)]	=	2 + x₁ + x₂
[if(x₁, x₂)]	=	2 + x₁ + 2 · x₂
[u'3'1(x₁)]	=	-2
[x'2b(x₁, x₂)]	=	2 + 2 · x₁ + x₂
[x'2d(x₁)]	=	x₁
[iff(x₁, x₂)]	=	0
[u'4'1(x₁)]	=	-2
[x'2a(x₁, x₂)]	=	-2
[u'5'1(x₁)]	=	2
[u'6'1(x₁,...,x₅)]	=	-2 + 2 · x₂ + x₄ + 2 · x₅
[u'7'1(x₁)]	=	-2
[u'8'1(x₁)]	=	-2
[u'9'1(x₁)]	=	-2
[p(x₁)]	=	2
[u'10'1(x₁)]	=	-2
[u'11'1(x₁)]	=	2
[u'12'1(x₁,...,x₅)]	=	1 + x₃
[u'13'1(x₁)]	=	-2
[reduce'ii'out]	=	0
[u'14'1(x₁)]	=	1
[nil]	=	1
[u'15'1(x₁)]	=	-2
[intersect'ii'in(x₁, x₂)]	=	0
[u'12'2(x₁)]	=	2
[u'6'2(x₁)]	=	2
[intersect'ii'out]	=	2
[u'1'1(x₁)]	=	-2
[u'2'1(x₁)]	=	-2
[reduce'ii'in^#(x₁, x₂)]	=	-2 + x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

u'6'1^#(reduce'ii'out,F2,Fs,Gs,NF)	→	reduce'ii'in^#(sequent(cons(F2,Fs),Gs),NF)	(49)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	u'6'1^#(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)	(46)
reduce'ii'in^#(sequent(cons(x'2b(F1,F2),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(F1,Fs),Gs),NF)	(47)
reduce'ii'in^#(sequent(cons(p(V),Fs),Gs),sequent(Left,Right))	→	reduce'ii'in^#(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))	(57)
reduce'ii'in^#(sequent(nil,cons(p(V),Gs)),sequent(Left,Right))	→	reduce'ii'in^#(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))	(67)

could be deleted.

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[sequent(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[cons(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[if(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[x'2b(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[x'2d(x₁)]	=	1 · x₁
[reduce'ii'in^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[reduce'ii'in^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[sequent(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[cons(x₁, x₂)]	=	2 + 2 · x₁ + 1 · x₂
[if(x₁, x₂)]	=	2 + 2 · x₁ + 2 · x₂
[x'2b(x₁, x₂)]	=	1 + 1 · x₁ + 2 · x₂
[x'2d(x₁)]	=	1 + 2 · x₁

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pairs

reduce'ii'in^#(sequent(cons(if(A,B),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)	(41)
reduce'ii'in^#(sequent(cons(x'2d(F1),Fs),Gs),NF)	→	reduce'ii'in^#(sequent(Fs,cons(F1,Gs)),NF)	(51)
reduce'ii'in^#(sequent(Fs,cons(if(A,B),Gs)),NF)	→	reduce'ii'in^#(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)	(53)
reduce'ii'in^#(sequent(Fs,cons(x'2d(G1),Gs)),NF)	→	reduce'ii'in^#(sequent(cons(G1,Fs),Gs),NF)	(65)

and no rules could be deleted.

1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over the naturals

[reduce'ii'in^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[sequent(x₁, x₂)]	=	1 · x₁ + 2 · x₂
[cons(x₁, x₂)]	=	2 · x₁ + 1 · x₂
[x'2b(x₁, x₂)]	=	2 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the pair

reduce'ii'in^#(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF)

→

reduce'ii'in^#(sequent(Fs,cons(G1,cons(G2,Gs))),NF)

(59)

and no rules could be deleted.

1.1.1.1.1.1.1.1.1 P is empty

There are no pairs anymore.

The 2^nd component contains the pair

intersect'ii'in^#(cons(X0,Xs),Ys) → intersect'ii'in^#(Xs,Ys) (39)

intersect'ii'in^#(Xs,cons(X0,Ys)) → intersect'ii'in^#(Xs,Ys) (37)

1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals

[cons(x₁, x₂)] = 1 · x₁ + 1 · x₂

[intersect'ii'in^#(x₁, x₂)] = 1 · x₁ + 1 · x₂

having no usable rules (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.
1.1.2.1 Size-Change Termination

Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.

intersect'ii'in^#(cons(X0,Xs),Ys) → intersect'ii'in^#(Xs,Ys) (39)

1 > 1

2 ≥ 2

intersect'ii'in^#(Xs,cons(X0,Ys)) → intersect'ii'in^#(Xs,Ys) (37)

1 ≥ 1

2 > 2

As there is no critical graph in the transitive closure, there are no infinite chains.

[cons(x₁, x₂)]	=	1 · x₁ + 1 · x₂
[intersect'ii'in^#(x₁, x₂)]	=	1 · x₁ + 1 · x₂

Certification Problem

Input (TPDB TRS_Standard/Secret_05_TRS/cime5)

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1 Dependency Graph Processor

1.1.1.1.1 Reduction Pair Processor with Usable Rules

1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1.1.1.1.1.1.1 P is empty

1.1.2 Monotonic Reduction Pair Processor with Usable Rules

1.1.2.1 Size-Change Termination