Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex2_Luc03b_C)

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

  top#(ok(X))	→	top#(active(X))	(91)
  top#(mark(X))	→	top#(proper(X))	(89)

  1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [proper(x1)]	=	1 · x1
  [0]	=	0
  [ok(x1)]	=	2 · x1
  [s(x1)]	=	1 · x1
  [nil]	=	0
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	2 · x1 + 2 · x2
  [len(x1)]	=	2 · x1
  [mark(x1)]	=	1 · x1
  [active(x1)]	=	2 · x1
  [top#(x1)]	=	1 · x1

  together with the usable rules

  proper(0)	→	ok(0)	(22)
  proper(s(X))	→	s(proper(X))	(23)
  proper(nil)	→	ok(nil)	(24)
  proper(cons(X1,X2))	→	cons(proper(X1),proper(X2))	(25)
  proper(fst(X1,X2))	→	fst(proper(X1),proper(X2))	(26)
  proper(from(X))	→	from(proper(X))	(27)
  proper(add(X1,X2))	→	add(proper(X1),proper(X2))	(28)
  proper(len(X))	→	len(proper(X))	(29)
  len(mark(X))	→	mark(len(X))	(21)
  len(ok(X))	→	ok(len(X))	(35)
  add(mark(X1),X2)	→	mark(add(X1,X2))	(19)
  add(X1,mark(X2))	→	mark(add(X1,X2))	(20)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(34)
  from(mark(X))	→	mark(from(X))	(18)
  from(ok(X))	→	ok(from(X))	(33)
  fst(mark(X1),X2)	→	mark(fst(X1,X2))	(16)
  fst(X1,mark(X2))	→	mark(fst(X1,X2))	(17)
  fst(ok(X1),ok(X2))	→	ok(fst(X1,X2))	(32)
  cons(mark(X1),X2)	→	mark(cons(X1,X2))	(15)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(31)
  s(ok(X))	→	ok(s(X))	(30)
  active(fst(0,Z))	→	mark(nil)	(1)
  active(fst(s(X),cons(Y,Z)))	→	mark(cons(Y,fst(X,Z)))	(2)
  active(from(X))	→	mark(cons(X,from(s(X))))	(3)
  active(add(0,X))	→	mark(X)	(4)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(5)
  active(len(nil))	→	mark(0)	(6)
  active(len(cons(X,Z)))	→	mark(s(len(Z)))	(7)
  active(cons(X1,X2))	→	cons(active(X1),X2)	(8)
  active(fst(X1,X2))	→	fst(active(X1),X2)	(9)
  active(fst(X1,X2))	→	fst(X1,active(X2))	(10)
  active(from(X))	→	from(active(X))	(11)
  active(add(X1,X2))	→	add(active(X1),X2)	(12)
  active(add(X1,X2))	→	add(X1,active(X2))	(13)
  active(len(X))	→	len(active(X))	(14)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1 Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [top#(x1)]	=	2 + x1
  [active(x1)]	=	x1
  [fst(x1, x2)]	=	1 + 2 · x1 + x2
  [0]	=	0
  [mark(x1)]	=	1 + x1
  [nil]	=	0
  [s(x1)]	=	-2
  [cons(x1, x2)]	=	x1
  [from(x1)]	=	1 + 2 · x1
  [add(x1, x2)]	=	1 + x1 + 2 · x2
  [len(x1)]	=	1 + 2 · x1
  [proper(x1)]	=	x1
  [ok(x1)]	=	x1

  the pair

  top#(mark(X))

  →

  top#(proper(X))

  (89)

  could be deleted.

  1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [active(x1)]	=	2 · x1
  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [0]	=	0
  [mark(x1)]	=	1 · x1
  [nil]	=	0
  [s(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	2 · x1 + 1 · x2
  [len(x1)]	=	2 · x1
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	2 · x1

  together with the usable rules

  active(fst(0,Z))	→	mark(nil)	(1)
  active(fst(s(X),cons(Y,Z)))	→	mark(cons(Y,fst(X,Z)))	(2)
  active(from(X))	→	mark(cons(X,from(s(X))))	(3)
  active(add(0,X))	→	mark(X)	(4)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(5)
  active(len(nil))	→	mark(0)	(6)
  active(len(cons(X,Z)))	→	mark(s(len(Z)))	(7)
  active(cons(X1,X2))	→	cons(active(X1),X2)	(8)
  active(fst(X1,X2))	→	fst(active(X1),X2)	(9)
  active(fst(X1,X2))	→	fst(X1,active(X2))	(10)
  active(from(X))	→	from(active(X))	(11)
  active(add(X1,X2))	→	add(active(X1),X2)	(12)
  active(add(X1,X2))	→	add(X1,active(X2))	(13)
  active(len(X))	→	len(active(X))	(14)
  len(mark(X))	→	mark(len(X))	(21)
  len(ok(X))	→	ok(len(X))	(35)
  add(mark(X1),X2)	→	mark(add(X1,X2))	(19)
  add(X1,mark(X2))	→	mark(add(X1,X2))	(20)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(34)
  from(mark(X))	→	mark(from(X))	(18)
  from(ok(X))	→	ok(from(X))	(33)
  fst(mark(X1),X2)	→	mark(fst(X1,X2))	(16)
  fst(X1,mark(X2))	→	mark(fst(X1,X2))	(17)
  fst(ok(X1),ok(X2))	→	ok(fst(X1,X2))	(32)
  cons(mark(X1),X2)	→	mark(cons(X1,X2))	(15)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(31)
  s(ok(X))	→	ok(s(X))	(30)

  (w.r.t. the implicit argument filter of the reduction pair), the rule could be deleted.

  1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [active(x1)]	=	2 · x1
  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [0]	=	2
  [mark(x1)]	=	1 · x1
  [nil]	=	1
  [s(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	2 · x1 + 1 · x2
  [len(x1)]	=	1 · x1
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	2 · x1

  the rules

  active(fst(0,Z))	→	mark(nil)	(1)
  active(add(0,X))	→	mark(X)	(4)

  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

  [active(x1)]	=	2 · x1
  [fst(x1, x2)]	=	2 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [mark(x1)]	=	1 · x1
  [from(x1)]	=	2 · x1
  [add(x1, x2)]	=	2 · x1 + 2 · x2
  [len(x1)]	=	2 · x1
  [nil]	=	1
  [0]	=	2
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	2 · x1

  together with the usable rules

  active(fst(s(X),cons(Y,Z)))	→	mark(cons(Y,fst(X,Z)))	(2)
  active(from(X))	→	mark(cons(X,from(s(X))))	(3)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(5)
  active(len(nil))	→	mark(0)	(6)
  active(len(cons(X,Z)))	→	mark(s(len(Z)))	(7)
  active(cons(X1,X2))	→	cons(active(X1),X2)	(8)
  active(fst(X1,X2))	→	fst(active(X1),X2)	(9)
  active(fst(X1,X2))	→	fst(X1,active(X2))	(10)
  active(from(X))	→	from(active(X))	(11)
  active(add(X1,X2))	→	add(active(X1),X2)	(12)
  active(add(X1,X2))	→	add(X1,active(X2))	(13)
  active(len(X))	→	len(active(X))	(14)
  len(mark(X))	→	mark(len(X))	(21)
  len(ok(X))	→	ok(len(X))	(35)
  add(mark(X1),X2)	→	mark(add(X1,X2))	(19)
  add(X1,mark(X2))	→	mark(add(X1,X2))	(20)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(34)
  from(mark(X))	→	mark(from(X))	(18)
  from(ok(X))	→	ok(from(X))	(33)
  fst(mark(X1),X2)	→	mark(fst(X1,X2))	(16)
  fst(X1,mark(X2))	→	mark(fst(X1,X2))	(17)
  fst(ok(X1),ok(X2))	→	ok(fst(X1,X2))	(32)
  cons(mark(X1),X2)	→	mark(cons(X1,X2))	(15)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(31)
  s(ok(X))	→	ok(s(X))	(30)

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

  active(len(nil))

  →

  mark(0)

  (6)

  could be deleted.

  1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [active(x1)]	=	1 + 2 · x1
  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [mark(x1)]	=	1 · x1
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	1 · x1 + 1 · x2
  [len(x1)]	=	1 · x1
  [ok(x1)]	=	2 + 2 · x1
  [top#(x1)]	=	1 · x1

  the pair

  top#(ok(X))

  →

  top#(active(X))

  (91)

  and the rules

  active(fst(s(X),cons(Y,Z)))	→	mark(cons(Y,fst(X,Z)))	(2)
  active(from(X))	→	mark(cons(X,from(s(X))))	(3)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(5)
  active(len(cons(X,Z)))	→	mark(s(len(Z)))	(7)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(34)
  fst(ok(X1),ok(X2))	→	ok(fst(X1,X2))	(32)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(31)

  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

  active#(fst(X1,X2))	→	active#(X1)	(50)
  active#(cons(X1,X2))	→	active#(X1)	(48)
  active#(fst(X1,X2))	→	active#(X2)	(52)
  active#(from(X))	→	active#(X)	(54)
  active#(add(X1,X2))	→	active#(X1)	(56)
  active#(add(X1,X2))	→	active#(X2)	(58)
  active#(len(X))	→	active#(X)	(60)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	1 · x1 + 1 · x2
  [len(x1)]	=	1 · x1
  [active#(x1)]	=	1 · x1

  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.

  active#(fst(X1,X2))	→	active#(X1)	(50)
  1	>	1
  active#(cons(X1,X2))	→	active#(X1)	(48)
  1	>	1
  active#(fst(X1,X2))	→	active#(X2)	(52)
  1	>	1
  active#(from(X))	→	active#(X)	(54)
  1	>	1
  active#(add(X1,X2))	→	active#(X1)	(56)
  1	>	1
  active#(add(X1,X2))	→	active#(X2)	(58)
  1	>	1
  active#(len(X))	→	active#(X)	(60)
  1	>	1

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

• The 3^rd component contains the pair

  proper#(cons(X1,X2))	→	proper#(X1)	(71)
  proper#(s(X))	→	proper#(X)	(69)
  proper#(cons(X1,X2))	→	proper#(X2)	(72)
  proper#(fst(X1,X2))	→	proper#(X1)	(74)
  proper#(fst(X1,X2))	→	proper#(X2)	(75)
  proper#(from(X))	→	proper#(X)	(77)
  proper#(add(X1,X2))	→	proper#(X1)	(79)
  proper#(add(X1,X2))	→	proper#(X2)	(80)
  proper#(len(X))	→	proper#(X)	(82)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [fst(x1, x2)]	=	1 · x1 + 1 · x2
  [from(x1)]	=	1 · x1
  [add(x1, x2)]	=	1 · x1 + 1 · x2
  [len(x1)]	=	1 · x1
  [proper#(x1)]	=	1 · x1

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

  1.1.3.1 Size-Change Termination

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

  proper#(cons(X1,X2))	→	proper#(X1)	(71)
  1	>	1
  proper#(s(X))	→	proper#(X)	(69)
  1	>	1
  proper#(cons(X1,X2))	→	proper#(X2)	(72)
  1	>	1
  proper#(fst(X1,X2))	→	proper#(X1)	(74)
  1	>	1
  proper#(fst(X1,X2))	→	proper#(X2)	(75)
  1	>	1
  proper#(from(X))	→	proper#(X)	(77)
  1	>	1
  proper#(add(X1,X2))	→	proper#(X1)	(79)
  1	>	1
  proper#(add(X1,X2))	→	proper#(X2)	(80)
  1	>	1
  proper#(len(X))	→	proper#(X)	(82)
  1	>	1

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

• The 4^th component contains the pair

  cons#(ok(X1),ok(X2))	→	cons#(X1,X2)	(84)
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(61)

  1.1.4 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [cons#(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.

  1.1.4.1 Size-Change Termination

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

  cons#(ok(X1),ok(X2))	→	cons#(X1,X2)	(84)
  1	>	1
  2	>	2
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(61)
  1	>	1
  2	≥	2

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

• The 5^th component contains the pair

  fst#(X1,mark(X2))	→	fst#(X1,X2)	(63)
  fst#(mark(X1),X2)	→	fst#(X1,X2)	(62)
  fst#(ok(X1),ok(X2))	→	fst#(X1,X2)	(85)

  1.1.5 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [mark(x1)]	=	1 · x1
  [ok(x1)]	=	1 · x1
  [fst#(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.

  1.1.5.1 Size-Change Termination

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

  fst#(X1,mark(X2))	→	fst#(X1,X2)	(63)
  1	≥	1
  2	>	2
  fst#(mark(X1),X2)	→	fst#(X1,X2)	(62)
  1	>	1
  2	≥	2
  fst#(ok(X1),ok(X2))	→	fst#(X1,X2)	(85)
  1	>	1
  2	>	2

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

• The 6^th component contains the pair

  from#(ok(X))	→	from#(X)	(86)
  from#(mark(X))	→	from#(X)	(64)

  1.1.6 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [from#(x1)]	=	1 · x1

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

  1.1.6.1 Size-Change Termination

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

  from#(ok(X))	→	from#(X)	(86)
  1	>	1
  from#(mark(X))	→	from#(X)	(64)
  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)	(66)
  add#(mark(X1),X2)	→	add#(X1,X2)	(65)
  add#(ok(X1),ok(X2))	→	add#(X1,X2)	(87)

  1.1.7 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [mark(x1)]	=	1 · x1
  [ok(x1)]	=	1 · x1
  [add#(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.

  1.1.7.1 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)	(66)
  1	≥	1
  2	>	2
  add#(mark(X1),X2)	→	add#(X1,X2)	(65)
  1	>	1
  2	≥	2
  add#(ok(X1),ok(X2))	→	add#(X1,X2)	(87)
  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

  len#(ok(X))	→	len#(X)	(88)
  len#(mark(X))	→	len#(X)	(67)

  1.1.8 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [len#(x1)]	=	1 · x1

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

  1.1.8.1 Size-Change Termination

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

  len#(ok(X))	→	len#(X)	(88)
  1	>	1
  len#(mark(X))	→	len#(X)	(67)
  1	>	1

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

• The 9^th component contains the pair

  s#(ok(X))

  →

  s#(X)

  (83)

  1.1.9 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(x1)]	=	1 · x1
  [s#(x1)]	=	1 · x1

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

  1.1.9.1 Size-Change Termination

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

  s#(ok(X))	→	s#(X)	(83)
  1	>	1

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