Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex1_GL02a_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))	(78)
  top#(mark(X))	→	top#(proper(X))	(76)

  1.1.1 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [proper(x1)]	=	1 · x1
  [eq(x1, x2)]	=	2 · x1 + 1 · x2
  [0]	=	0
  [ok(x1)]	=	2 · x1
  [true]	=	0
  [s(x1)]	=	1 · x1
  [false]	=	0
  [inf(x1)]	=	2 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [take(x1, x2)]	=	1 · x1 + 2 · x2
  [nil]	=	0
  [length(x1)]	=	2 · x1
  [mark(x1)]	=	1 · x1
  [active(x1)]	=	2 · x1
  [top#(x1)]	=	1 · x1

  together with the usable rules

  proper(eq(X1,X2))	→	eq(proper(X1),proper(X2))	(17)
  proper(0)	→	ok(0)	(18)
  proper(true)	→	ok(true)	(19)
  proper(s(X))	→	s(proper(X))	(20)
  proper(false)	→	ok(false)	(21)
  proper(inf(X))	→	inf(proper(X))	(22)
  proper(cons(X1,X2))	→	cons(proper(X1),proper(X2))	(23)
  proper(take(X1,X2))	→	take(proper(X1),proper(X2))	(24)
  proper(nil)	→	ok(nil)	(25)
  proper(length(X))	→	length(proper(X))	(26)
  length(mark(X))	→	mark(length(X))	(16)
  length(ok(X))	→	ok(length(X))	(32)
  take(mark(X1),X2)	→	mark(take(X1,X2))	(14)
  take(X1,mark(X2))	→	mark(take(X1,X2))	(15)
  take(ok(X1),ok(X2))	→	ok(take(X1,X2))	(31)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(30)
  inf(mark(X))	→	mark(inf(X))	(13)
  inf(ok(X))	→	ok(inf(X))	(29)
  s(ok(X))	→	ok(s(X))	(28)
  eq(ok(X1),ok(X2))	→	ok(eq(X1,X2))	(27)
  active(eq(0,0))	→	mark(true)	(1)
  active(eq(s(X),s(Y)))	→	mark(eq(X,Y))	(2)
  active(eq(X,Y))	→	mark(false)	(3)
  active(inf(X))	→	mark(cons(X,inf(s(X))))	(4)
  active(take(0,X))	→	mark(nil)	(5)
  active(take(s(X),cons(Y,L)))	→	mark(cons(Y,take(X,L)))	(6)
  active(length(nil))	→	mark(0)	(7)
  active(length(cons(X,L)))	→	mark(s(length(L)))	(8)
  active(inf(X))	→	inf(active(X))	(9)
  active(take(X1,X2))	→	take(active(X1),X2)	(10)
  active(take(X1,X2))	→	take(X1,active(X2))	(11)
  active(length(X))	→	length(active(X))	(12)

  (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 matrix interpretations of dimension 2 with strict dimension 1 over the integers

  [top#(x1)]	=	0 0 + 0 1 0 0 · x1
  [ok(x1)]	=	0 0 + 1 0 0 1 · x1
  [active(x1)]	=	0 0 + 0 0 0 1 · x1
  [mark(x1)]	=	0 1 + 0 0 0 1 · x1
  [proper(x1)]	=	0 0 + 1 0 0 1 · x1
  [eq(x1, x2)]	=	0 1 + 0 0 0 0 · x1 + 0 0 1 0 · x2
  [0]	=	0 0
  [true]	=	0 0
  [s(x1)]	=	1 0 + 1 0 0 0 · x1
  [false]	=	0 0
  [inf(x1)]	=	0 1 + 0 0 0 1 · x1
  [cons(x1, x2)]	=	0 0 + 0 0 0 0 · x1 + 0 0 0 0 · x2
  [take(x1, x2)]	=	0 1 + 0 0 0 1 · x1 + 0 0 0 1 · x2
  [nil]	=	0 0
  [length(x1)]	=	0 1 + 0 0 0 1 · x1

  the pair

  top#(mark(X))

  →

  top#(proper(X))

  (76)

  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
  [eq(x1, x2)]	=	1 · x1 + 2 · x2
  [0]	=	0
  [mark(x1)]	=	1 · x1
  [true]	=	0
  [s(x1)]	=	1 · x1
  [false]	=	0
  [inf(x1)]	=	2 · x1
  [cons(x1, x2)]	=	2 · x1 + 1 · x2
  [take(x1, x2)]	=	1 · x1 + 2 · x2
  [nil]	=	0
  [length(x1)]	=	1 · x1
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	1 · x1

  together with the usable rules

  active(eq(0,0))	→	mark(true)	(1)
  active(eq(s(X),s(Y)))	→	mark(eq(X,Y))	(2)
  active(eq(X,Y))	→	mark(false)	(3)
  active(inf(X))	→	mark(cons(X,inf(s(X))))	(4)
  active(take(0,X))	→	mark(nil)	(5)
  active(take(s(X),cons(Y,L)))	→	mark(cons(Y,take(X,L)))	(6)
  active(length(nil))	→	mark(0)	(7)
  active(length(cons(X,L)))	→	mark(s(length(L)))	(8)
  active(inf(X))	→	inf(active(X))	(9)
  active(take(X1,X2))	→	take(active(X1),X2)	(10)
  active(take(X1,X2))	→	take(X1,active(X2))	(11)
  active(length(X))	→	length(active(X))	(12)
  length(mark(X))	→	mark(length(X))	(16)
  length(ok(X))	→	ok(length(X))	(32)
  take(mark(X1),X2)	→	mark(take(X1,X2))	(14)
  take(X1,mark(X2))	→	mark(take(X1,X2))	(15)
  take(ok(X1),ok(X2))	→	ok(take(X1,X2))	(31)
  inf(mark(X))	→	mark(inf(X))	(13)
  inf(ok(X))	→	ok(inf(X))	(29)
  s(ok(X))	→	ok(s(X))	(28)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(30)
  eq(ok(X1),ok(X2))	→	ok(eq(X1,X2))	(27)

  (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
  [eq(x1, x2)]	=	2 · x1 + 1 · x2
  [0]	=	1
  [mark(x1)]	=	1 · x1
  [true]	=	1
  [s(x1)]	=	1 · x1
  [false]	=	0
  [inf(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [take(x1, x2)]	=	1 · x1 + 2 · x2
  [nil]	=	2
  [length(x1)]	=	2 · x1
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	2 · x1

  the rules

  active(eq(0,0))	→	mark(true)	(1)
  active(length(nil))	→	mark(0)	(7)

  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
  [eq(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [false]	=	0
  [inf(x1)]	=	2 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [take(x1, x2)]	=	2 · x1 + 1 · x2
  [0]	=	1
  [nil]	=	2
  [length(x1)]	=	2 · x1
  [ok(x1)]	=	2 · x1
  [top#(x1)]	=	1 · x1

  together with the usable rules

  active(eq(s(X),s(Y)))	→	mark(eq(X,Y))	(2)
  active(eq(X,Y))	→	mark(false)	(3)
  active(inf(X))	→	mark(cons(X,inf(s(X))))	(4)
  active(take(0,X))	→	mark(nil)	(5)
  active(take(s(X),cons(Y,L)))	→	mark(cons(Y,take(X,L)))	(6)
  active(length(cons(X,L)))	→	mark(s(length(L)))	(8)
  active(inf(X))	→	inf(active(X))	(9)
  active(take(X1,X2))	→	take(active(X1),X2)	(10)
  active(take(X1,X2))	→	take(X1,active(X2))	(11)
  active(length(X))	→	length(active(X))	(12)
  length(mark(X))	→	mark(length(X))	(16)
  length(ok(X))	→	ok(length(X))	(32)
  take(mark(X1),X2)	→	mark(take(X1,X2))	(14)
  take(X1,mark(X2))	→	mark(take(X1,X2))	(15)
  take(ok(X1),ok(X2))	→	ok(take(X1,X2))	(31)
  inf(mark(X))	→	mark(inf(X))	(13)
  inf(ok(X))	→	ok(inf(X))	(29)
  s(ok(X))	→	ok(s(X))	(28)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(30)
  eq(ok(X1),ok(X2))	→	ok(eq(X1,X2))	(27)

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

  active(take(0,X))

  →

  mark(nil)

  (5)

  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)]	=	2 · x1
  [eq(x1, x2)]	=	1 + 1 · x1 + 2 · x2
  [s(x1)]	=	1 · x1
  [mark(x1)]	=	1 · x1
  [false]	=	1
  [inf(x1)]	=	2 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [take(x1, x2)]	=	1 + 2 · x1 + 2 · x2
  [length(x1)]	=	2 + 2 · x1
  [ok(x1)]	=	2 + 2 · x1
  [top#(x1)]	=	1 · x1

  the pair

  top#(ok(X))

  →

  top#(active(X))

  (78)

  and the rules

  active(eq(s(X),s(Y)))	→	mark(eq(X,Y))	(2)
  active(eq(X,Y))	→	mark(false)	(3)
  active(take(s(X),cons(Y,L)))	→	mark(cons(Y,take(X,L)))	(6)
  active(length(cons(X,L)))	→	mark(s(length(L)))	(8)
  active(take(X1,X2))	→	take(active(X1),X2)	(10)
  active(take(X1,X2))	→	take(X1,active(X2))	(11)
  active(length(X))	→	length(active(X))	(12)
  take(ok(X1),ok(X2))	→	ok(take(X1,X2))	(31)
  inf(ok(X))	→	ok(inf(X))	(29)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(30)
  eq(ok(X1),ok(X2))	→	ok(eq(X1,X2))	(27)

  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#(take(X1,X2))	→	active#(X1)	(46)
  active#(inf(X))	→	active#(X)	(44)
  active#(take(X1,X2))	→	active#(X2)	(48)
  active#(length(X))	→	active#(X)	(50)

  1.1.2 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [take(x1, x2)]	=	1 · x1 + 1 · x2
  [inf(x1)]	=	1 · x1
  [length(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#(take(X1,X2))	→	active#(X1)	(46)
  1	>	1
  active#(inf(X))	→	active#(X)	(44)
  1	>	1
  active#(take(X1,X2))	→	active#(X2)	(48)
  1	>	1
  active#(length(X))	→	active#(X)	(50)
  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#(eq(X1,X2))	→	proper#(X2)	(57)
  proper#(eq(X1,X2))	→	proper#(X1)	(56)
  proper#(s(X))	→	proper#(X)	(59)
  proper#(inf(X))	→	proper#(X)	(61)
  proper#(cons(X1,X2))	→	proper#(X1)	(63)
  proper#(cons(X1,X2))	→	proper#(X2)	(64)
  proper#(take(X1,X2))	→	proper#(X1)	(66)
  proper#(take(X1,X2))	→	proper#(X2)	(67)
  proper#(length(X))	→	proper#(X)	(69)

  1.1.3 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [eq(x1, x2)]	=	1 · x1 + 1 · x2
  [s(x1)]	=	1 · x1
  [inf(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [take(x1, x2)]	=	1 · x1 + 1 · x2
  [length(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#(eq(X1,X2))	→	proper#(X2)	(57)
  1	>	1
  proper#(eq(X1,X2))	→	proper#(X1)	(56)
  1	>	1
  proper#(s(X))	→	proper#(X)	(59)
  1	>	1
  proper#(inf(X))	→	proper#(X)	(61)
  1	>	1
  proper#(cons(X1,X2))	→	proper#(X1)	(63)
  1	>	1
  proper#(cons(X1,X2))	→	proper#(X2)	(64)
  1	>	1
  proper#(take(X1,X2))	→	proper#(X1)	(66)
  1	>	1
  proper#(take(X1,X2))	→	proper#(X2)	(67)
  1	>	1
  proper#(length(X))	→	proper#(X)	(69)
  1	>	1

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

• The 4^th component contains the pair

  inf#(ok(X))	→	inf#(X)	(72)
  inf#(mark(X))	→	inf#(X)	(51)

  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
  [inf#(x1)]	=	1 · x1

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

  1.1.4.1 Size-Change Termination

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

  inf#(ok(X))	→	inf#(X)	(72)
  1	>	1
  inf#(mark(X))	→	inf#(X)	(51)
  1	>	1

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

• The 5^th component contains the pair

  take#(X1,mark(X2))	→	take#(X1,X2)	(53)
  take#(mark(X1),X2)	→	take#(X1,X2)	(52)
  take#(ok(X1),ok(X2))	→	take#(X1,X2)	(74)

  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
  [take#(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.

  take#(X1,mark(X2))	→	take#(X1,X2)	(53)
  1	≥	1
  2	>	2
  take#(mark(X1),X2)	→	take#(X1,X2)	(52)
  1	>	1
  2	≥	2
  take#(ok(X1),ok(X2))	→	take#(X1,X2)	(74)
  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

  length#(ok(X))	→	length#(X)	(75)
  length#(mark(X))	→	length#(X)	(54)

  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
  [length#(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.

  length#(ok(X))	→	length#(X)	(75)
  1	>	1
  length#(mark(X))	→	length#(X)	(54)
  1	>	1

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

• The 7^th component contains the pair

  eq#(ok(X1),ok(X2))

  →

  eq#(X1,X2)

  (70)

  1.1.7 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(x1)]	=	1 · x1
  [eq#(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.

  eq#(ok(X1),ok(X2))	→	eq#(X1,X2)	(70)
  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

  s#(ok(X))

  →

  s#(X)

  (71)

  1.1.8 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.

  1.1.8.1 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)	(71)
  1	>	1

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

• The 9^th component contains the pair

  cons#(ok(X1),ok(X2))

  →

  cons#(X1,X2)

  (73)

  1.1.9 Monotonic Reduction Pair Processor with Usable Rules

  Using the linear polynomial interpretation over the naturals

  [ok(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.

  1.1.9.1 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)	(73)
  1	>	1
  2	>	2

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