Certification Problem

Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/Ex2_Luc02a_iGM)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 9 components.

• The 1^st component contains the pair

  mark#(cons(X1,X2))	→	active#(cons(mark(X1),X2))	(70)
  active#(terms(N))	→	mark#(cons(recip(sqr(N)),terms(s(N))))	(42)
  mark#(cons(X1,X2))	→	mark#(X1)	(72)
  mark#(terms(X))	→	active#(terms(mark(X)))	(67)
  active#(sqr(s(X)))	→	mark#(s(add(sqr(X),dbl(X))))	(49)
  mark#(s(X))	→	active#(s(mark(X)))	(79)
  active#(dbl(s(X)))	→	mark#(s(s(dbl(X))))	(55)
  mark#(s(X))	→	mark#(X)	(81)
  mark#(terms(X))	→	mark#(X)	(69)
  mark#(recip(X))	→	active#(recip(mark(X)))	(73)
  active#(add(0,X))	→	mark#(X)	(59)
  mark#(recip(X))	→	mark#(X)	(75)
  mark#(sqr(X))	→	active#(sqr(mark(X)))	(76)
  active#(add(s(X),Y))	→	mark#(s(add(X,Y)))	(60)
  active#(first(s(X),cons(Y,Z)))	→	mark#(cons(Y,first(X,Z)))	(64)
  mark#(sqr(X))	→	mark#(X)	(78)
  mark#(add(X1,X2))	→	active#(add(mark(X1),mark(X2)))	(83)
  mark#(add(X1,X2))	→	mark#(X1)	(85)
  mark#(add(X1,X2))	→	mark#(X2)	(86)
  mark#(dbl(X))	→	active#(dbl(mark(X)))	(87)
  mark#(dbl(X))	→	mark#(X)	(89)
  mark#(first(X1,X2))	→	active#(first(mark(X1),mark(X2)))	(90)
  mark#(first(X1,X2))	→	mark#(X1)	(92)
  mark#(first(X1,X2))	→	mark#(X2)	(93)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the

  prec(mark#)	=	1		stat(mark#)	=	lex
  prec(cons)	=	1		stat(cons)	=	lex
  prec(mark)	=	2		stat(mark)	=	lex
  prec(terms)	=	1		stat(terms)	=	lex
  prec(recip)	=	0		stat(recip)	=	lex
  prec(sqr)	=	1		stat(sqr)	=	lex
  prec(s)	=	1		stat(s)	=	lex
  prec(add)	=	1		stat(add)	=	lex
  prec(dbl)	=	1		stat(dbl)	=	lex
  prec(0)	=	2		stat(0)	=	lex
  prec(first)	=	1		stat(first)	=	lex
  prec(active)	=	0		stat(active)	=	lex
  prec(nil)	=	1		stat(nil)	=	lex

  π(mark#)	=	[]
  π(cons)	=	[]
  π(active#)	=	1
  π(mark)	=	[]
  π(terms)	=	[]
  π(recip)	=	[]
  π(sqr)	=	[]
  π(s)	=	[]
  π(add)	=	[]
  π(dbl)	=	[]
  π(0)	=	[]
  π(first)	=	[]
  π(active)	=	[]
  π(nil)	=	[]

  together with the usable rules

  cons(X1,mark(X2))	→	cons(X1,X2)	(23)
  cons(mark(X1),X2)	→	cons(X1,X2)	(22)
  cons(active(X1),X2)	→	cons(X1,X2)	(24)
  cons(X1,active(X2))	→	cons(X1,X2)	(25)
  terms(active(X))	→	terms(X)	(21)
  terms(mark(X))	→	terms(X)	(20)
  s(active(X))	→	s(X)	(31)
  s(mark(X))	→	s(X)	(30)
  recip(active(X))	→	recip(X)	(27)
  recip(mark(X))	→	recip(X)	(26)
  sqr(active(X))	→	sqr(X)	(29)
  sqr(mark(X))	→	sqr(X)	(28)
  add(X1,mark(X2))	→	add(X1,X2)	(33)
  add(mark(X1),X2)	→	add(X1,X2)	(32)
  add(active(X1),X2)	→	add(X1,X2)	(34)
  add(X1,active(X2))	→	add(X1,X2)	(35)
  dbl(active(X))	→	dbl(X)	(37)
  dbl(mark(X))	→	dbl(X)	(36)
  first(X1,mark(X2))	→	first(X1,X2)	(39)
  first(mark(X1),X2)	→	first(X1,X2)	(38)
  first(active(X1),X2)	→	first(X1,X2)	(40)
  first(X1,active(X2))	→	first(X1,X2)	(41)

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

  mark#(recip(X))

  →

  active#(recip(mark(X)))

  (73)

  could be deleted.

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [active#(x1)]	=	-2 + 2 · x1
  [add(x1, x2)]	=	2
  [cons(x1, x2)]	=	2
  [dbl(x1)]	=	2
  [first(x1, x2)]	=	2
  [s(x1)]	=	0
  [sqr(x1)]	=	2
  [terms(x1)]	=	2
  [mark(x1)]	=	2
  [active(x1)]	=	-2 + x1
  [recip(x1)]	=	-2
  [0]	=	0
  [nil]	=	0
  [mark#(x1)]	=	2

  together with the usable rules

  cons(X1,mark(X2))	→	cons(X1,X2)	(23)
  cons(mark(X1),X2)	→	cons(X1,X2)	(22)
  cons(active(X1),X2)	→	cons(X1,X2)	(24)
  cons(X1,active(X2))	→	cons(X1,X2)	(25)
  terms(active(X))	→	terms(X)	(21)
  terms(mark(X))	→	terms(X)	(20)
  s(active(X))	→	s(X)	(31)
  s(mark(X))	→	s(X)	(30)
  sqr(active(X))	→	sqr(X)	(29)
  sqr(mark(X))	→	sqr(X)	(28)
  add(X1,mark(X2))	→	add(X1,X2)	(33)
  add(mark(X1),X2)	→	add(X1,X2)	(32)
  add(active(X1),X2)	→	add(X1,X2)	(34)
  add(X1,active(X2))	→	add(X1,X2)	(35)
  dbl(active(X))	→	dbl(X)	(37)
  dbl(mark(X))	→	dbl(X)	(36)
  first(X1,mark(X2))	→	first(X1,X2)	(39)
  first(mark(X1),X2)	→	first(X1,X2)	(38)
  first(active(X1),X2)	→	first(X1,X2)	(40)
  first(X1,active(X2))	→	first(X1,X2)	(41)

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

  mark#(s(X))

  →

  active#(s(mark(X)))

  (79)

  could be deleted.

  1.1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [active#(x1)]	=	-2 + 2 · x1
  [add(x1, x2)]	=	2
  [cons(x1, x2)]	=	-2
  [dbl(x1)]	=	2
  [first(x1, x2)]	=	2
  [sqr(x1)]	=	2
  [terms(x1)]	=	2
  [mark(x1)]	=	-2
  [active(x1)]	=	2
  [recip(x1)]	=	-2
  [s(x1)]	=	-2 + 2 · x1
  [0]	=	0
  [nil]	=	0
  [mark#(x1)]	=	2

  together with the usable rules

  cons(X1,mark(X2))	→	cons(X1,X2)	(23)
  cons(mark(X1),X2)	→	cons(X1,X2)	(22)
  cons(active(X1),X2)	→	cons(X1,X2)	(24)
  cons(X1,active(X2))	→	cons(X1,X2)	(25)
  terms(active(X))	→	terms(X)	(21)
  terms(mark(X))	→	terms(X)	(20)
  sqr(active(X))	→	sqr(X)	(29)
  sqr(mark(X))	→	sqr(X)	(28)
  add(X1,mark(X2))	→	add(X1,X2)	(33)
  add(mark(X1),X2)	→	add(X1,X2)	(32)
  add(active(X1),X2)	→	add(X1,X2)	(34)
  add(X1,active(X2))	→	add(X1,X2)	(35)
  dbl(active(X))	→	dbl(X)	(37)
  dbl(mark(X))	→	dbl(X)	(36)
  first(X1,mark(X2))	→	first(X1,X2)	(39)
  first(mark(X1),X2)	→	first(X1,X2)	(38)
  first(active(X1),X2)	→	first(X1,X2)	(40)
  first(X1,active(X2))	→	first(X1,X2)	(41)

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

  mark#(cons(X1,X2))

  →

  active#(cons(mark(X1),X2))

  (70)

  could be deleted.

  1.1.1.1.1.1 Reduction Pair Processor

  Using the

  prec(terms)	=	6		stat(terms)	=	lex
  prec(mark#)	=	1		stat(mark#)	=	mul
  prec(recip)	=	1		stat(recip)	=	mul
  prec(sqr)	=	5		stat(sqr)	=	lex
  prec(s)	=	2		stat(s)	=	lex
  prec(add)	=	3		stat(add)	=	mul
  prec(dbl)	=	4		stat(dbl)	=	lex
  prec(0)	=	1		stat(0)	=	mul
  prec(first)	=	7		stat(first)	=	lex
  prec(nil)	=	0		stat(nil)	=	mul

  π(active#)	=	1
  π(terms)	=	[1]
  π(mark#)	=	[1]
  π(cons)	=	1
  π(recip)	=	[1]
  π(sqr)	=	[1]
  π(s)	=	[1]
  π(mark)	=	1
  π(add)	=	[1,2]
  π(dbl)	=	[1]
  π(0)	=	[]
  π(first)	=	[1,2]
  π(active)	=	1
  π(nil)	=	[]

  the pairs

  active#(terms(N))	→	mark#(cons(recip(sqr(N)),terms(s(N))))	(42)
  mark#(terms(X))	→	active#(terms(mark(X)))	(67)
  active#(sqr(s(X)))	→	mark#(s(add(sqr(X),dbl(X))))	(49)
  active#(dbl(s(X)))	→	mark#(s(s(dbl(X))))	(55)
  mark#(s(X))	→	mark#(X)	(81)
  mark#(terms(X))	→	mark#(X)	(69)
  active#(add(0,X))	→	mark#(X)	(59)
  mark#(recip(X))	→	mark#(X)	(75)
  mark#(sqr(X))	→	active#(sqr(mark(X)))	(76)
  active#(add(s(X),Y))	→	mark#(s(add(X,Y)))	(60)
  active#(first(s(X),cons(Y,Z)))	→	mark#(cons(Y,first(X,Z)))	(64)
  mark#(sqr(X))	→	mark#(X)	(78)
  mark#(add(X1,X2))	→	active#(add(mark(X1),mark(X2)))	(83)
  mark#(add(X1,X2))	→	mark#(X1)	(85)
  mark#(add(X1,X2))	→	mark#(X2)	(86)
  mark#(dbl(X))	→	active#(dbl(mark(X)))	(87)
  mark#(dbl(X))	→	mark#(X)	(89)
  mark#(first(X1,X2))	→	active#(first(mark(X1),mark(X2)))	(90)
  mark#(first(X1,X2))	→	mark#(X1)	(92)
  mark#(first(X1,X2))	→	mark#(X2)	(93)

  could be deleted.

  1.1.1.1.1.1.1 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.1.1.1.1.1.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  cons(mark(x0),x1)

  cons(x0,mark(x1))

  cons(active(x0),x1)

  cons(x0,active(x1))

  1.1.1.1.1.1.1.1.1 Size-Change Termination

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

  mark#(cons(X1,X2))	→	mark#(X1)	(72)
  1	>	1

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

• The 2^nd component contains the pair

  terms#(active(X))	→	terms#(X)	(96)
  terms#(mark(X))	→	terms#(X)	(95)

  1.1.2 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.2.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.2.1.1 Size-Change Termination

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

  terms#(active(X))	→	terms#(X)	(96)
  1	>	1
  terms#(mark(X))	→	terms#(X)	(95)
  1	>	1

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

• The 3^rd component contains the pair

  cons#(X1,mark(X2))	→	cons#(X1,X2)	(98)
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(97)
  cons#(active(X1),X2)	→	cons#(X1,X2)	(99)
  cons#(X1,active(X2))	→	cons#(X1,X2)	(100)

  1.1.3 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.3.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.3.1.1 Size-Change Termination

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

  cons#(X1,mark(X2))	→	cons#(X1,X2)	(98)
  1	≥	1
  2	>	2
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(97)
  1	>	1
  2	≥	2
  cons#(active(X1),X2)	→	cons#(X1,X2)	(99)
  1	>	1
  2	≥	2
  cons#(X1,active(X2))	→	cons#(X1,X2)	(100)
  1	≥	1
  2	>	2

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

• The 4^th component contains the pair

  recip#(active(X))	→	recip#(X)	(102)
  recip#(mark(X))	→	recip#(X)	(101)

  1.1.4 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.4.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.4.1.1 Size-Change Termination

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

  recip#(active(X))	→	recip#(X)	(102)
  1	>	1
  recip#(mark(X))	→	recip#(X)	(101)
  1	>	1

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

• The 5^th component contains the pair

  sqr#(active(X))	→	sqr#(X)	(104)
  sqr#(mark(X))	→	sqr#(X)	(103)

  1.1.5 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.5.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.5.1.1 Size-Change Termination

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

  sqr#(active(X))	→	sqr#(X)	(104)
  1	>	1
  sqr#(mark(X))	→	sqr#(X)	(103)
  1	>	1

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

• The 6^th component contains the pair

  s#(active(X))	→	s#(X)	(106)
  s#(mark(X))	→	s#(X)	(105)

  1.1.6 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.6.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.6.1.1 Size-Change Termination

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

  s#(active(X))	→	s#(X)	(106)
  1	>	1
  s#(mark(X))	→	s#(X)	(105)
  1	>	1

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

• The 7^th component contains the pair

  add#(X1,mark(X2))	→	add#(X1,X2)	(108)
  add#(mark(X1),X2)	→	add#(X1,X2)	(107)
  add#(active(X1),X2)	→	add#(X1,X2)	(109)
  add#(X1,active(X2))	→	add#(X1,X2)	(110)

  1.1.7 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.7.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.7.1.1 Size-Change Termination

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

  add#(X1,mark(X2))	→	add#(X1,X2)	(108)
  1	≥	1
  2	>	2
  add#(mark(X1),X2)	→	add#(X1,X2)	(107)
  1	>	1
  2	≥	2
  add#(active(X1),X2)	→	add#(X1,X2)	(109)
  1	>	1
  2	≥	2
  add#(X1,active(X2))	→	add#(X1,X2)	(110)
  1	≥	1
  2	>	2

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

• The 8^th component contains the pair

  dbl#(active(X))	→	dbl#(X)	(112)
  dbl#(mark(X))	→	dbl#(X)	(111)

  1.1.8 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.8.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.8.1.1 Size-Change Termination

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

  dbl#(active(X))	→	dbl#(X)	(112)
  1	>	1
  dbl#(mark(X))	→	dbl#(X)	(111)
  1	>	1

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

• The 9^th component contains the pair

  first#(X1,mark(X2))	→	first#(X1,X2)	(114)
  first#(mark(X1),X2)	→	first#(X1,X2)	(113)
  first#(active(X1),X2)	→	first#(X1,X2)	(115)
  first#(X1,active(X2))	→	first#(X1,X2)	(116)

  1.1.9 Usable Rules Processor

  We restrict the rewrite rules to the following usable rules of the DP problem.

  There are no rules.

  1.1.9.1 Innermost Lhss Removal Processor

  We restrict the innermost strategy to the following left hand sides.

  active(terms(x0))

  active(sqr(0))

  active(sqr(s(x0)))

  active(dbl(0))

  active(dbl(s(x0)))

  active(add(0,x0))

  active(add(s(x0),x1))

  active(first(0,x0))

  active(first(s(x0),cons(x1,x2)))

  mark(terms(x0))

  mark(cons(x0,x1))

  mark(recip(x0))

  mark(sqr(x0))

  mark(s(x0))

  mark(0)

  mark(add(x0,x1))

  mark(dbl(x0))

  mark(first(x0,x1))

  mark(nil)

  1.1.9.1.1 Size-Change Termination

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

  first#(X1,mark(X2))	→	first#(X1,X2)	(114)
  1	≥	1
  2	>	2
  first#(mark(X1),X2)	→	first#(X1,X2)	(113)
  1	>	1
  2	≥	2
  first#(active(X1),X2)	→	first#(X1,X2)	(115)
  1	>	1
  2	≥	2
  first#(X1,active(X2))	→	first#(X1,X2)	(116)
  1	≥	1
  2	>	2

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