Certification Problem

Input (TPDB TRS_Innermost/Transformed_CSR_innermost_04/ExSec11_1_Luc02a_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 12 components.

• The 1^st component contains the pair

  top#(ok(X))	→	top#(active(X))	(141)
  top#(mark(X))	→	top#(proper(X))	(139)

  1.1.1 Usable Rules Processor

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

  proper(terms(X))	→	terms(proper(X))	(36)
  proper(cons(X1,X2))	→	cons(proper(X1),proper(X2))	(37)
  proper(recip(X))	→	recip(proper(X))	(38)
  proper(sqr(X))	→	sqr(proper(X))	(39)
  proper(s(X))	→	s(proper(X))	(40)
  proper(0)	→	ok(0)	(41)
  proper(add(X1,X2))	→	add(proper(X1),proper(X2))	(42)
  proper(dbl(X))	→	dbl(proper(X))	(43)
  proper(first(X1,X2))	→	first(proper(X1),proper(X2))	(44)
  proper(nil)	→	ok(nil)	(45)
  proper(half(X))	→	half(proper(X))	(46)
  half(mark(X))	→	mark(half(X))	(35)
  half(ok(X))	→	ok(half(X))	(55)
  first(mark(X1),X2)	→	mark(first(X1,X2))	(33)
  first(X1,mark(X2))	→	mark(first(X1,X2))	(34)
  first(ok(X1),ok(X2))	→	ok(first(X1,X2))	(54)
  dbl(mark(X))	→	mark(dbl(X))	(32)
  dbl(ok(X))	→	ok(dbl(X))	(53)
  add(mark(X1),X2)	→	mark(add(X1,X2))	(30)
  add(X1,mark(X2))	→	mark(add(X1,X2))	(31)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(52)
  s(mark(X))	→	mark(s(X))	(29)
  s(ok(X))	→	ok(s(X))	(51)
  sqr(mark(X))	→	mark(sqr(X))	(28)
  sqr(ok(X))	→	ok(sqr(X))	(50)
  recip(mark(X))	→	mark(recip(X))	(27)
  recip(ok(X))	→	ok(recip(X))	(49)
  cons(mark(X1),X2)	→	mark(cons(X1,X2))	(26)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(48)
  terms(mark(X))	→	mark(terms(X))	(25)
  terms(ok(X))	→	ok(terms(X))	(47)
  active(terms(N))	→	mark(cons(recip(sqr(N)),terms(s(N))))	(1)
  active(sqr(0))	→	mark(0)	(2)
  active(sqr(s(X)))	→	mark(s(add(sqr(X),dbl(X))))	(3)
  active(dbl(0))	→	mark(0)	(4)
  active(dbl(s(X)))	→	mark(s(s(dbl(X))))	(5)
  active(add(0,X))	→	mark(X)	(6)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(7)
  active(first(0,X))	→	mark(nil)	(8)
  active(first(s(X),cons(Y,Z)))	→	mark(cons(Y,first(X,Z)))	(9)
  active(half(0))	→	mark(0)	(10)
  active(half(s(0)))	→	mark(0)	(11)
  active(half(s(s(X))))	→	mark(s(half(X)))	(12)
  active(half(dbl(X)))	→	mark(X)	(13)
  active(terms(X))	→	terms(active(X))	(14)
  active(cons(X1,X2))	→	cons(active(X1),X2)	(15)
  active(recip(X))	→	recip(active(X))	(16)
  active(sqr(X))	→	sqr(active(X))	(17)
  active(s(X))	→	s(active(X))	(18)
  active(add(X1,X2))	→	add(active(X1),X2)	(19)
  active(add(X1,X2))	→	add(X1,active(X2))	(20)
  active(dbl(X))	→	dbl(active(X))	(21)
  active(first(X1,X2))	→	first(active(X1),X2)	(22)
  active(first(X1,X2))	→	first(X1,active(X2))	(23)
  active(half(X))	→	half(active(X))	(24)

  1.1.1.1 Innermost Lhss Removal Processor

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

  active(terms(x0))

  active(cons(x0,x1))

  active(recip(x0))

  active(sqr(x0))

  active(s(x0))

  active(add(x0,x1))

  active(dbl(x0))

  active(first(x0,x1))

  active(half(x0))

  terms(mark(x0))

  cons(mark(x0),x1)

  recip(mark(x0))

  sqr(mark(x0))

  s(mark(x0))

  add(mark(x0),x1)

  add(x0,mark(x1))

  dbl(mark(x0))

  first(mark(x0),x1)

  first(x0,mark(x1))

  half(mark(x0))

  proper(terms(x0))

  proper(cons(x0,x1))

  proper(recip(x0))

  proper(sqr(x0))

  proper(s(x0))

  proper(0)

  proper(add(x0,x1))

  proper(dbl(x0))

  proper(first(x0,x1))

  proper(nil)

  proper(half(x0))

  terms(ok(x0))

  cons(ok(x0),ok(x1))

  recip(ok(x0))

  sqr(ok(x0))

  s(ok(x0))

  add(ok(x0),ok(x1))

  dbl(ok(x0))

  first(ok(x0),ok(x1))

  half(ok(x0))

  1.1.1.1.1 Reduction Pair Processor

  Using the

  prec(top#)	=	0		stat(top#)	=	lex
  prec(mark)	=	1		stat(mark)	=	lex
  prec(terms)	=	7		stat(terms)	=	lex
  prec(sqr)	=	6		stat(sqr)	=	lex
  prec(s)	=	3		stat(s)	=	lex
  prec(0)	=	8		stat(0)	=	lex
  prec(add)	=	4		stat(add)	=	lex
  prec(dbl)	=	5		stat(dbl)	=	lex
  prec(first)	=	3		stat(first)	=	lex
  prec(nil)	=	2		stat(nil)	=	lex
  prec(half)	=	3		stat(half)	=	lex

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

  the pair

  top#(mark(X))

  →

  top#(proper(X))

  (139)

  could be deleted.

  1.1.1.1.1.1 Usable Rules Processor

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

  active(terms(N))	→	mark(cons(recip(sqr(N)),terms(s(N))))	(1)
  active(sqr(0))	→	mark(0)	(2)
  active(sqr(s(X)))	→	mark(s(add(sqr(X),dbl(X))))	(3)
  active(dbl(0))	→	mark(0)	(4)
  active(dbl(s(X)))	→	mark(s(s(dbl(X))))	(5)
  active(add(0,X))	→	mark(X)	(6)
  active(add(s(X),Y))	→	mark(s(add(X,Y)))	(7)
  active(first(0,X))	→	mark(nil)	(8)
  active(first(s(X),cons(Y,Z)))	→	mark(cons(Y,first(X,Z)))	(9)
  active(half(0))	→	mark(0)	(10)
  active(half(s(0)))	→	mark(0)	(11)
  active(half(s(s(X))))	→	mark(s(half(X)))	(12)
  active(half(dbl(X)))	→	mark(X)	(13)
  active(terms(X))	→	terms(active(X))	(14)
  active(cons(X1,X2))	→	cons(active(X1),X2)	(15)
  active(recip(X))	→	recip(active(X))	(16)
  active(sqr(X))	→	sqr(active(X))	(17)
  active(s(X))	→	s(active(X))	(18)
  active(add(X1,X2))	→	add(active(X1),X2)	(19)
  active(add(X1,X2))	→	add(X1,active(X2))	(20)
  active(dbl(X))	→	dbl(active(X))	(21)
  active(first(X1,X2))	→	first(active(X1),X2)	(22)
  active(first(X1,X2))	→	first(X1,active(X2))	(23)
  active(half(X))	→	half(active(X))	(24)
  half(mark(X))	→	mark(half(X))	(35)
  half(ok(X))	→	ok(half(X))	(55)
  first(X1,mark(X2))	→	mark(first(X1,X2))	(34)
  first(ok(X1),ok(X2))	→	ok(first(X1,X2))	(54)
  first(mark(X1),X2)	→	mark(first(X1,X2))	(33)
  dbl(mark(X))	→	mark(dbl(X))	(32)
  dbl(ok(X))	→	ok(dbl(X))	(53)
  add(X1,mark(X2))	→	mark(add(X1,X2))	(31)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(52)
  add(mark(X1),X2)	→	mark(add(X1,X2))	(30)
  s(mark(X))	→	mark(s(X))	(29)
  s(ok(X))	→	ok(s(X))	(51)
  sqr(mark(X))	→	mark(sqr(X))	(28)
  sqr(ok(X))	→	ok(sqr(X))	(50)
  recip(mark(X))	→	mark(recip(X))	(27)
  recip(ok(X))	→	ok(recip(X))	(49)
  cons(mark(X1),X2)	→	mark(cons(X1,X2))	(26)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(48)
  terms(mark(X))	→	mark(terms(X))	(25)
  terms(ok(X))	→	ok(terms(X))	(47)

  1.1.1.1.1.1.1 Innermost Lhss Removal Processor

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

  active(terms(x0))

  active(cons(x0,x1))

  active(recip(x0))

  active(sqr(x0))

  active(s(x0))

  active(add(x0,x1))

  active(dbl(x0))

  active(first(x0,x1))

  active(half(x0))

  terms(mark(x0))

  cons(mark(x0),x1)

  recip(mark(x0))

  sqr(mark(x0))

  s(mark(x0))

  add(mark(x0),x1)

  add(x0,mark(x1))

  dbl(mark(x0))

  first(mark(x0),x1)

  first(x0,mark(x1))

  half(mark(x0))

  terms(ok(x0))

  cons(ok(x0),ok(x1))

  recip(ok(x0))

  sqr(ok(x0))

  s(ok(x0))

  add(ok(x0),ok(x1))

  dbl(ok(x0))

  first(ok(x0),ok(x1))

  half(ok(x0))

  1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor

  Using the linear polynomial interpretation over the naturals

  [active(x1)]	=	2 · x1
  [terms(x1)]	=	2 · x1
  [mark(x1)]	=	1 · x1
  [cons(x1, x2)]	=	1 · x1 + 1 · x2
  [recip(x1)]	=	1 · x1
  [sqr(x1)]	=	2 · x1
  [s(x1)]	=	1 · x1
  [0]	=	0
  [add(x1, x2)]	=	1 · x1 + 2 · x2
  [dbl(x1)]	=	1 · x1
  [first(x1, x2)]	=	2 · x1 + 1 · x2
  [nil]	=	0
  [half(x1)]	=	2 · x1
  [ok(x1)]	=	1 + 2 · x1
  [top#(x1)]	=	1 · x1

  the pair

  top#(ok(X))

  →

  top#(active(X))

  (141)

  and the rules

  proper(terms(X))	→	terms(proper(X))	(36)
  proper(cons(X1,X2))	→	cons(proper(X1),proper(X2))	(37)
  proper(recip(X))	→	recip(proper(X))	(38)
  proper(sqr(X))	→	sqr(proper(X))	(39)
  proper(s(X))	→	s(proper(X))	(40)
  proper(0)	→	ok(0)	(41)
  proper(add(X1,X2))	→	add(proper(X1),proper(X2))	(42)
  proper(dbl(X))	→	dbl(proper(X))	(43)
  proper(first(X1,X2))	→	first(proper(X1),proper(X2))	(44)
  proper(nil)	→	ok(nil)	(45)
  proper(half(X))	→	half(proper(X))	(46)
  terms(ok(X))	→	ok(terms(X))	(47)
  cons(ok(X1),ok(X2))	→	ok(cons(X1,X2))	(48)
  sqr(ok(X))	→	ok(sqr(X))	(50)
  add(ok(X1),ok(X2))	→	ok(add(X1,X2))	(52)
  first(ok(X1),ok(X2))	→	ok(first(X1,X2))	(54)
  half(ok(X))	→	ok(half(X))	(55)
  top(mark(X))	→	top(proper(X))	(56)
  top(ok(X))	→	top(active(X))	(57)

  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#(cons(X1,X2))	→	active#(X1)	(79)
  active#(terms(X))	→	active#(X)	(77)
  active#(recip(X))	→	active#(X)	(81)
  active#(sqr(X))	→	active#(X)	(83)
  active#(s(X))	→	active#(X)	(85)
  active#(add(X1,X2))	→	active#(X1)	(87)
  active#(add(X1,X2))	→	active#(X2)	(89)
  active#(dbl(X))	→	active#(X)	(91)
  active#(first(X1,X2))	→	active#(X1)	(93)
  active#(first(X1,X2))	→	active#(X2)	(95)
  active#(half(X))	→	active#(X)	(97)

  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.

  terms(mark(x0))

  cons(mark(x0),x1)

  recip(mark(x0))

  sqr(mark(x0))

  s(mark(x0))

  add(mark(x0),x1)

  add(x0,mark(x1))

  dbl(mark(x0))

  first(mark(x0),x1)

  first(x0,mark(x1))

  half(mark(x0))

  terms(ok(x0))

  cons(ok(x0),ok(x1))

  recip(ok(x0))

  sqr(ok(x0))

  s(ok(x0))

  add(ok(x0),ok(x1))

  dbl(ok(x0))

  first(ok(x0),ok(x1))

  half(ok(x0))

  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.

  active#(cons(X1,X2))	→	active#(X1)	(79)
  1	>	1
  active#(terms(X))	→	active#(X)	(77)
  1	>	1
  active#(recip(X))	→	active#(X)	(81)
  1	>	1
  active#(sqr(X))	→	active#(X)	(83)
  1	>	1
  active#(s(X))	→	active#(X)	(85)
  1	>	1
  active#(add(X1,X2))	→	active#(X1)	(87)
  1	>	1
  active#(add(X1,X2))	→	active#(X2)	(89)
  1	>	1
  active#(dbl(X))	→	active#(X)	(91)
  1	>	1
  active#(first(X1,X2))	→	active#(X1)	(93)
  1	>	1
  active#(first(X1,X2))	→	active#(X2)	(95)
  1	>	1
  active#(half(X))	→	active#(X)	(97)
  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)	(112)
  proper#(terms(X))	→	proper#(X)	(110)
  proper#(cons(X1,X2))	→	proper#(X2)	(113)
  proper#(recip(X))	→	proper#(X)	(115)
  proper#(sqr(X))	→	proper#(X)	(117)
  proper#(s(X))	→	proper#(X)	(119)
  proper#(add(X1,X2))	→	proper#(X1)	(121)
  proper#(add(X1,X2))	→	proper#(X2)	(122)
  proper#(dbl(X))	→	proper#(X)	(124)
  proper#(first(X1,X2))	→	proper#(X1)	(126)
  proper#(first(X1,X2))	→	proper#(X2)	(127)
  proper#(half(X))	→	proper#(X)	(129)

  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.

  terms(mark(x0))

  cons(mark(x0),x1)

  recip(mark(x0))

  sqr(mark(x0))

  s(mark(x0))

  add(mark(x0),x1)

  add(x0,mark(x1))

  dbl(mark(x0))

  first(mark(x0),x1)

  first(x0,mark(x1))

  half(mark(x0))

  terms(ok(x0))

  cons(ok(x0),ok(x1))

  recip(ok(x0))

  sqr(ok(x0))

  s(ok(x0))

  add(ok(x0),ok(x1))

  dbl(ok(x0))

  first(ok(x0),ok(x1))

  half(ok(x0))

  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.

  proper#(cons(X1,X2))	→	proper#(X1)	(112)
  1	>	1
  proper#(terms(X))	→	proper#(X)	(110)
  1	>	1
  proper#(cons(X1,X2))	→	proper#(X2)	(113)
  1	>	1
  proper#(recip(X))	→	proper#(X)	(115)
  1	>	1
  proper#(sqr(X))	→	proper#(X)	(117)
  1	>	1
  proper#(s(X))	→	proper#(X)	(119)
  1	>	1
  proper#(add(X1,X2))	→	proper#(X1)	(121)
  1	>	1
  proper#(add(X1,X2))	→	proper#(X2)	(122)
  1	>	1
  proper#(dbl(X))	→	proper#(X)	(124)
  1	>	1
  proper#(first(X1,X2))	→	proper#(X1)	(126)
  1	>	1
  proper#(first(X1,X2))	→	proper#(X2)	(127)
  1	>	1
  proper#(half(X))	→	proper#(X)	(129)
  1	>	1

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

• The 4^th component contains the pair

  terms#(ok(X))	→	terms#(X)	(130)
  terms#(mark(X))	→	terms#(X)	(98)

  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.

  There are no lhss.

  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.

  terms#(ok(X))	→	terms#(X)	(130)
  1	>	1
  terms#(mark(X))	→	terms#(X)	(98)
  1	>	1

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

• The 5^th component contains the pair

  cons#(ok(X1),ok(X2))	→	cons#(X1,X2)	(131)
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(99)

  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.

  There are no lhss.

  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.

  cons#(ok(X1),ok(X2))	→	cons#(X1,X2)	(131)
  1	>	1
  2	>	2
  cons#(mark(X1),X2)	→	cons#(X1,X2)	(99)
  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

  recip#(ok(X))	→	recip#(X)	(132)
  recip#(mark(X))	→	recip#(X)	(100)

  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.

  There are no lhss.

  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.

  recip#(ok(X))	→	recip#(X)	(132)
  1	>	1
  recip#(mark(X))	→	recip#(X)	(100)
  1	>	1

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

• The 7^th component contains the pair

  sqr#(ok(X))	→	sqr#(X)	(133)
  sqr#(mark(X))	→	sqr#(X)	(101)

  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.

  There are no lhss.

  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.

  sqr#(ok(X))	→	sqr#(X)	(133)
  1	>	1
  sqr#(mark(X))	→	sqr#(X)	(101)
  1	>	1

  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)	(134)
  s#(mark(X))	→	s#(X)	(102)

  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.

  There are no lhss.

  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.

  s#(ok(X))	→	s#(X)	(134)
  1	>	1
  s#(mark(X))	→	s#(X)	(102)
  1	>	1

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

• The 9^th component contains the pair

  add#(X1,mark(X2))	→	add#(X1,X2)	(104)
  add#(mark(X1),X2)	→	add#(X1,X2)	(103)
  add#(ok(X1),ok(X2))	→	add#(X1,X2)	(135)

  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.

  There are no lhss.

  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.

  add#(X1,mark(X2))	→	add#(X1,X2)	(104)
  1	≥	1
  2	>	2
  add#(mark(X1),X2)	→	add#(X1,X2)	(103)
  1	>	1
  2	≥	2
  add#(ok(X1),ok(X2))	→	add#(X1,X2)	(135)
  1	>	1
  2	>	2

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

• The 10^th component contains the pair

  dbl#(ok(X))	→	dbl#(X)	(136)
  dbl#(mark(X))	→	dbl#(X)	(105)

  1.1.10 Usable Rules Processor

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

  There are no rules.

  1.1.10.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.10.1.1 Size-Change Termination

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

  dbl#(ok(X))	→	dbl#(X)	(136)
  1	>	1
  dbl#(mark(X))	→	dbl#(X)	(105)
  1	>	1

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

• The 11^th component contains the pair

  first#(X1,mark(X2))	→	first#(X1,X2)	(107)
  first#(mark(X1),X2)	→	first#(X1,X2)	(106)
  first#(ok(X1),ok(X2))	→	first#(X1,X2)	(137)

  1.1.11 Usable Rules Processor

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

  There are no rules.

  1.1.11.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.11.1.1 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)	(107)
  1	≥	1
  2	>	2
  first#(mark(X1),X2)	→	first#(X1,X2)	(106)
  1	>	1
  2	≥	2
  first#(ok(X1),ok(X2))	→	first#(X1,X2)	(137)
  1	>	1
  2	>	2

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

• The 12^th component contains the pair

  half#(ok(X))	→	half#(X)	(138)
  half#(mark(X))	→	half#(X)	(108)

  1.1.12 Usable Rules Processor

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

  There are no rules.

  1.1.12.1 Innermost Lhss Removal Processor

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

  There are no lhss.

  1.1.12.1.1 Size-Change Termination

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

  half#(ok(X))	→	half#(X)	(138)
  1	>	1
  half#(mark(X))	→	half#(X)	(108)
  1	>	1

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