Certification Problem

Input (TPDB TRS_Standard/AProVE_07/thiemann36)

The rewrite relation of the following TRS is considered.

le(0,y)	→	true	(1)
le(s(x),0)	→	false	(2)
le(s(x),s(y))	→	le(x,y)	(3)
quot(x,0)	→	quotZeroErro	(4)
quot(x,s(y))	→	quotIter(x,s(y),0,0,0)	(5)
quotIter(x,s(y),z,u,v)	→	if(le(x,z),x,s(y),z,u,v)	(6)
if(true,x,y,z,u,v)	→	v	(7)
if(false,x,y,z,u,v)	→	if2(le(y,s(u)),x,y,s(z),s(u),v)	(8)
if2(false,x,y,z,u,v)	→	quotIter(x,y,z,u,v)	(9)
if2(true,x,y,z,u,v)	→	quotIter(x,y,z,0,s(v))	(10)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

Hence, it suffices to show innermost termination in the following.

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.

le^#(s(x),s(y))	→	le^#(x,y)	(11)
quot^#(x,s(y))	→	quotIter^#(x,s(y),0,0,0)	(12)
quotIter^#(x,s(y),z,u,v)	→	if^#(le(x,z),x,s(y),z,u,v)	(13)
quotIter^#(x,s(y),z,u,v)	→	le^#(x,z)	(14)
if^#(false,x,y,z,u,v)	→	if2^#(le(y,s(u)),x,y,s(z),s(u),v)	(15)
if^#(false,x,y,z,u,v)	→	le^#(y,s(u))	(16)
if2^#(false,x,y,z,u,v)	→	quotIter^#(x,y,z,u,v)	(17)
if2^#(true,x,y,z,u,v)	→	quotIter^#(x,y,z,0,s(v))	(18)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

quotIter^#(x,s(y),z,u,v)	→	if^#(le(x,z),x,s(y),z,u,v)	(13)
if^#(false,x,y,z,u,v)	→	if2^#(le(y,s(u)),x,y,s(z),s(u),v)	(15)
if2^#(false,x,y,z,u,v)	→	quotIter^#(x,y,z,u,v)	(17)
if2^#(true,x,y,z,u,v)	→	quotIter^#(x,y,z,0,s(v))	(18)

1.1.1.1 Usable Rules Processor

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

le(0,y)	→	true	(1)
le(s(x),s(y))	→	le(x,y)	(3)
le(s(x),0)	→	false	(2)

1.1.1.1.1 Innermost Lhss Removal Processor

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

le(0,x0)

le(s(x0),0)

le(s(x0),s(x1))

1.1.1.1.1.1 Reduction Pair Processor

We apply the generic reduction pair processor using the linear polynomial interpretation over the naturals

[c]	=	-2
[if^#(x₁,...,x₆)]	=	-1 · x₁ + 1 · x₂ + -1 · x₄ + -1
[quotIter^#(x₁,...,x₅)]	=	1 · x₁ + -1 · x₃ + -1
[if2^#(x₁,...,x₆)]	=	-1 · x₁ + 1 · x₂ + -1 · x₄ + -1
[s(x₁)]	=	1 · x₁ + 1
[le(x₁, x₂)]	=	0
[false]	=	0
[true]	=	0
[0]	=	0

The pair(s)

if^#(false,x,y,z,u,v)

→

if2^#(le(y,s(u)),x,y,s(z),s(u),v)

(15)

are strictly oriented and the pair(s)

if^#(false,x,y,z,u,v)

→

if2^#(le(y,s(u)),x,y,s(z),s(u),v)

(15)

are bounded w.r.t. the constant c.

The following constraints are generated for the pairs.

(if^#(false,x107,s(x31),x106,x33,x34)≥c⟹if^#(false,s(x107),s(x31),s(x106),x33,x34)≥c)
if^#(false,s(x111),s(x31),0,x33,x34)≥c
quotIter^#(x10,s(x16),x12,x13,x14)≥if^#(le(x10,x12),x10,s(x16),x12,x13,x14)
quotIter^#(x20,s(x26),x22,0,s(x24))≥if^#(le(x20,x22),x20,s(x26),x22,0,s(x24))
(if^#(false,x107,s(x31),x106,x33,x34) > if2^#(le(s(x31),s(x33)),x107,s(x31),s(x106),s(x33),x34)⟹if^#(false,s(x107),s(x31),s(x106),x33,x34) > if2^#(le(s(x31),s(x33)),s(x107),s(x31),s(s(x106)),s(x33),x34))
if^#(false,s(x111),s(x31),0,x33,x34) > if2^#(le(s(x31),s(x33)),s(x111),s(x31),s(0),s(x33),x34)
(if2^#(false,x60,s(x123),s(x62),s(x122),x64)≥quotIter^#(x60,s(x123),s(x62),s(x122),x64)⟹if2^#(false,x60,s(s(x123)),s(x62),s(s(x122)),x64)≥quotIter^#(x60,s(s(x123)),s(x62),s(s(x122)),x64))
if2^#(false,x60,s(s(x127)),s(x62),s(0),x64)≥quotIter^#(x60,s(s(x127)),s(x62),s(0),x64)
if2^#(true,x85,0,s(x87),s(x88),x89)≥quotIter^#(x85,0,s(x87),0,s(x89))
if2^#(true,x85,s(0),s(x87),s(x137),x89)≥quotIter^#(x85,s(0),s(x87),0,s(x89))
(if2^#(true,x85,s(x139),s(x87),s(x138),x89)≥quotIter^#(x85,s(x139),s(x87),0,s(x89))⟹if2^#(true,x85,s(s(x139)),s(x87),s(s(x138)),x89)≥quotIter^#(x85,s(s(x139)),s(x87),0,s(x89)))

The details are shown below:

For the chain we build the initial constraint
(if^#(le(x30,x32),x30,s(x31),x32,x33,x34)→^*if^#(false,x35,x36,x37,x38,x39)⟹if^#(false,x35,x36,x37,x38,x39)≥c)

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x30,x32)→^*false⟹x30→^*x35⟹s(x31)→^*x36⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x35,x36,x37,x38,x39)≥c)
- Applying Rule "Variable in Equation" allows to substitute x35 by x30 which results in
  (le(x30,x32)→^*false⟹s(x31)→^*x36⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,x36,x37,x38,x39)≥c)
- Applying Rule "Variable in Equation" allows to substitute x36 by s(x31) which results in
  (le(x30,x32)→^*false⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,s(x31),x37,x38,x39)≥c)
- Applying Rule "Variable in Equation" allows to substitute x37 by x32 which results in
  (le(x30,x32)→^*false⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,s(x31),x32,x38,x39)≥c)
- Applying Rule "Variable in Equation" allows to substitute x38 by x33 which results in
  (le(x30,x32)→^*false⟹x34→^*x39⟹if^#(false,x30,s(x31),x32,x33,x39)≥c)
- Applying Rule "Variable in Equation" allows to substitute x39 by x34 which results in
  (le(x30,x32)→^*false⟹if^#(false,x30,s(x31),x32,x33,x34)≥c)
- Applying Rule "Induction" on le(x30,x32)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if^#(false,0,s(x31),x105,x33,x34)≥c)
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x107,x106)→^*false⟹∀x108 . ∀x109 . ∀x110 . (le(x107,x106)→^*false⟹if^#(false,x107,s(x108),x106,x109,x110)≥c)⟹if^#(false,s(x107),s(x31),s(x106),x33,x34)≥c)
    - Applying Rule "Simplify Conditions" results in
      (if^#(false,x107,s(x31),x106,x33,x34)≥c⟹if^#(false,s(x107),s(x31),s(x106),x33,x34)≥c)
    - This constraint is kept as final constraint.
  3. (false→^*false⟹if^#(false,s(x111),s(x31),0,x33,x34)≥c)
    - Applying Rule "Same Constructor" results in
      if^#(false,s(x111),s(x31),0,x33,x34)≥c
    - This constraint is kept as final constraint.
For the chain we build the initial constraint
(quotIter^#(x10,x11,x12,x13,x14)→^*quotIter^#(x15,s(x16),x17,x18,x19)⟹quotIter^#(x15,s(x16),x17,x18,x19)≥if^#(le(x15,x17),x15,s(x16),x17,x18,x19))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (x10→^*x15⟹x11→^*s(x16)⟹x12→^*x17⟹x13→^*x18⟹x14→^*x19⟹quotIter^#(x15,s(x16),x17,x18,x19)≥if^#(le(x15,x17),x15,s(x16),x17,x18,x19))
- Applying Rule "Variable in Equation" allows to substitute x15 by x10 which results in
  (x11→^*s(x16)⟹x12→^*x17⟹x13→^*x18⟹x14→^*x19⟹quotIter^#(x10,s(x16),x17,x18,x19)≥if^#(le(x10,x17),x10,s(x16),x17,x18,x19))
- Applying Rule "Variable in Equation" allows to substitute x17 by x12 which results in
  (x11→^*s(x16)⟹x13→^*x18⟹x14→^*x19⟹quotIter^#(x10,s(x16),x12,x18,x19)≥if^#(le(x10,x12),x10,s(x16),x12,x18,x19))
- Applying Rule "Variable in Equation" allows to substitute x18 by x13 which results in
  (x11→^*s(x16)⟹x14→^*x19⟹quotIter^#(x10,s(x16),x12,x13,x19)≥if^#(le(x10,x12),x10,s(x16),x12,x13,x19))
- Applying Rule "Variable in Equation" allows to substitute x19 by x14 which results in
  (x11→^*s(x16)⟹quotIter^#(x10,s(x16),x12,x13,x14)≥if^#(le(x10,x12),x10,s(x16),x12,x13,x14))
- Applying Rule "Variable in Equation" allows to substitute x11 by s(x16) which results in
  quotIter^#(x10,s(x16),x12,x13,x14)≥if^#(le(x10,x12),x10,s(x16),x12,x13,x14)
- This constraint is kept as final constraint.
For the chain we build the initial constraint
(quotIter^#(x20,x21,x22,0,s(x24))→^*quotIter^#(x25,s(x26),x27,x28,x29)⟹quotIter^#(x25,s(x26),x27,x28,x29)≥if^#(le(x25,x27),x25,s(x26),x27,x28,x29))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (x20→^*x25⟹x21→^*s(x26)⟹x22→^*x27⟹0→^*x28⟹s(x24)→^*x29⟹quotIter^#(x25,s(x26),x27,x28,x29)≥if^#(le(x25,x27),x25,s(x26),x27,x28,x29))
- Applying Rule "Variable in Equation" allows to substitute x25 by x20 which results in
  (x21→^*s(x26)⟹x22→^*x27⟹0→^*x28⟹s(x24)→^*x29⟹quotIter^#(x20,s(x26),x27,x28,x29)≥if^#(le(x20,x27),x20,s(x26),x27,x28,x29))
- Applying Rule "Variable in Equation" allows to substitute x27 by x22 which results in
  (x21→^*s(x26)⟹0→^*x28⟹s(x24)→^*x29⟹quotIter^#(x20,s(x26),x22,x28,x29)≥if^#(le(x20,x22),x20,s(x26),x22,x28,x29))
- Applying Rule "Variable in Equation" allows to substitute x28 by 0 which results in
  (x21→^*s(x26)⟹s(x24)→^*x29⟹quotIter^#(x20,s(x26),x22,0,x29)≥if^#(le(x20,x22),x20,s(x26),x22,0,x29))
- Applying Rule "Variable in Equation" allows to substitute x29 by s(x24) which results in
  (x21→^*s(x26)⟹quotIter^#(x20,s(x26),x22,0,s(x24))≥if^#(le(x20,x22),x20,s(x26),x22,0,s(x24)))
- Applying Rule "Variable in Equation" allows to substitute x21 by s(x26) which results in
  quotIter^#(x20,s(x26),x22,0,s(x24))≥if^#(le(x20,x22),x20,s(x26),x22,0,s(x24))
- This constraint is kept as final constraint.
For the chain we build the initial constraint
(if^#(le(x30,x32),x30,s(x31),x32,x33,x34)→^*if^#(false,x35,x36,x37,x38,x39)⟹if^#(false,x35,x36,x37,x38,x39) > if2^#(le(x36,s(x38)),x35,x36,s(x37),s(x38),x39))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x30,x32)→^*false⟹x30→^*x35⟹s(x31)→^*x36⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x35,x36,x37,x38,x39) > if2^#(le(x36,s(x38)),x35,x36,s(x37),s(x38),x39))
- Applying Rule "Variable in Equation" allows to substitute x35 by x30 which results in
  (le(x30,x32)→^*false⟹s(x31)→^*x36⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,x36,x37,x38,x39) > if2^#(le(x36,s(x38)),x30,x36,s(x37),s(x38),x39))
- Applying Rule "Variable in Equation" allows to substitute x36 by s(x31) which results in
  (le(x30,x32)→^*false⟹x32→^*x37⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,s(x31),x37,x38,x39) > if2^#(le(s(x31),s(x38)),x30,s(x31),s(x37),s(x38),x39))
- Applying Rule "Variable in Equation" allows to substitute x37 by x32 which results in
  (le(x30,x32)→^*false⟹x33→^*x38⟹x34→^*x39⟹if^#(false,x30,s(x31),x32,x38,x39) > if2^#(le(s(x31),s(x38)),x30,s(x31),s(x32),s(x38),x39))
- Applying Rule "Variable in Equation" allows to substitute x38 by x33 which results in
  (le(x30,x32)→^*false⟹x34→^*x39⟹if^#(false,x30,s(x31),x32,x33,x39) > if2^#(le(s(x31),s(x33)),x30,s(x31),s(x32),s(x33),x39))
- Applying Rule "Variable in Equation" allows to substitute x39 by x34 which results in
  (le(x30,x32)→^*false⟹if^#(false,x30,s(x31),x32,x33,x34) > if2^#(le(s(x31),s(x33)),x30,s(x31),s(x32),s(x33),x34))
- Applying Rule "Induction" on le(x30,x32)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if^#(false,0,s(x31),x105,x33,x34) > if2^#(le(s(x31),s(x33)),0,s(x31),s(x105),s(x33),x34))
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x107,x106)→^*false⟹∀x108 . ∀x109 . ∀x110 . (le(x107,x106)→^*false⟹if^#(false,x107,s(x108),x106,x109,x110) > if2^#(le(s(x108),s(x109)),x107,s(x108),s(x106),s(x109),x110))⟹if^#(false,s(x107),s(x31),s(x106),x33,x34) > if2^#(le(s(x31),s(x33)),s(x107),s(x31),s(s(x106)),s(x33),x34))
    - Applying Rule "Simplify Conditions" results in
      (if^#(false,x107,s(x31),x106,x33,x34) > if2^#(le(s(x31),s(x33)),x107,s(x31),s(x106),s(x33),x34)⟹if^#(false,s(x107),s(x31),s(x106),x33,x34) > if2^#(le(s(x31),s(x33)),s(x107),s(x31),s(s(x106)),s(x33),x34))
    - This constraint is kept as final constraint.
  3. (false→^*false⟹if^#(false,s(x111),s(x31),0,x33,x34) > if2^#(le(s(x31),s(x33)),s(x111),s(x31),s(0),s(x33),x34))
    - Applying Rule "Same Constructor" results in
      if^#(false,s(x111),s(x31),0,x33,x34) > if2^#(le(s(x31),s(x33)),s(x111),s(x31),s(0),s(x33),x34)
    - This constraint is kept as final constraint.
For the chain we build the initial constraint
(if2^#(le(x61,s(x63)),x60,x61,s(x62),s(x63),x64)→^*if2^#(false,x65,x66,x67,x68,x69)⟹if2^#(false,x65,x66,x67,x68,x69)≥quotIter^#(x65,x66,x67,x68,x69))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x61,s(x63))→^*false⟹x60→^*x65⟹x61→^*x66⟹s(x62)→^*x67⟹s(x63)→^*x68⟹x64→^*x69⟹if2^#(false,x65,x66,x67,x68,x69)≥quotIter^#(x65,x66,x67,x68,x69))
- Applying Rule "Variable in Equation" allows to substitute x65 by x60 which results in
  (le(x61,s(x63))→^*false⟹x61→^*x66⟹s(x62)→^*x67⟹s(x63)→^*x68⟹x64→^*x69⟹if2^#(false,x60,x66,x67,x68,x69)≥quotIter^#(x60,x66,x67,x68,x69))
- Applying Rule "Variable in Equation" allows to substitute x66 by x61 which results in
  (le(x61,s(x63))→^*false⟹s(x62)→^*x67⟹s(x63)→^*x68⟹x64→^*x69⟹if2^#(false,x60,x61,x67,x68,x69)≥quotIter^#(x60,x61,x67,x68,x69))
- Applying Rule "Variable in Equation" allows to substitute x67 by s(x62) which results in
  (le(x61,s(x63))→^*false⟹s(x63)→^*x68⟹x64→^*x69⟹if2^#(false,x60,x61,s(x62),x68,x69)≥quotIter^#(x60,x61,s(x62),x68,x69))
- Applying Rule "Variable in Equation" allows to substitute x68 by s(x63) which results in
  (le(x61,s(x63))→^*false⟹x64→^*x69⟹if2^#(false,x60,x61,s(x62),s(x63),x69)≥quotIter^#(x60,x61,s(x62),s(x63),x69))
- Applying Rule "Variable in Equation" allows to substitute x69 by x64 which results in
  (le(x61,s(x63))→^*false⟹if2^#(false,x60,x61,s(x62),s(x63),x64)≥quotIter^#(x60,x61,s(x62),s(x63),x64))
- Applying Rule "Introduce fresh variable" results in
  (s(x63)→^*x112⟹le(x61,x112)→^*false⟹if2^#(false,x60,x61,s(x62),s(x63),x64)≥quotIter^#(x60,x61,s(x62),s(x63),x64))
- Applying Rule "Induction" on le(x61,x112)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if2^#(false,x60,0,s(x62),s(x63),x64)≥quotIter^#(x60,0,s(x62),s(x63),x64))
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x115,x114)→^*false⟹s(x63)→^*s(x114)⟹∀x116 . ∀x117 . ∀x118 . ∀x119 . (le(x115,x114)→^*false⟹s(x116)→^*x114⟹if2^#(false,x117,x115,s(x118),s(x116),x119)≥quotIter^#(x117,x115,s(x118),s(x116),x119))⟹if2^#(false,x60,s(x115),s(x62),s(x63),x64)≥quotIter^#(x60,s(x115),s(x62),s(x63),x64))
    - Applying Rule "Same Constructor" results in
      (le(x115,x114)→^*false⟹x63→^*x114⟹∀x116 . ∀x117 . ∀x118 . ∀x119 . (le(x115,x114)→^*false⟹s(x116)→^*x114⟹if2^#(false,x117,x115,s(x118),s(x116),x119)≥quotIter^#(x117,x115,s(x118),s(x116),x119))⟹if2^#(false,x60,s(x115),s(x62),s(x63),x64)≥quotIter^#(x60,s(x115),s(x62),s(x63),x64))
    - Applying Rule "Variable in Equation" allows to substitute x63 by x114 which results in
      (le(x115,x114)→^*false⟹∀x116 . ∀x117 . ∀x118 . ∀x119 . (le(x115,x114)→^*false⟹s(x116)→^*x114⟹if2^#(false,x117,x115,s(x118),s(x116),x119)≥quotIter^#(x117,x115,s(x118),s(x116),x119))⟹if2^#(false,x60,s(x115),s(x62),s(x114),x64)≥quotIter^#(x60,s(x115),s(x62),s(x114),x64))
    - Applying Rule "Delete Conditions" results in
      (le(x115,x114)→^*false⟹if2^#(false,x60,s(x115),s(x62),s(x114),x64)≥quotIter^#(x60,s(x115),s(x62),s(x114),x64))
    - Applying Rule "Induction" on le(x115,x114)→^*false results in the following 3 new constraints.
      1. (true→^*false⟹if2^#(false,x60,s(0),s(x62),s(x121),x64)≥quotIter^#(x60,s(0),s(x62),s(x121),x64))
        Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x123,x122)→^*false⟹∀x124 . ∀x125 . ∀x126 . (le(x123,x122)→^*false⟹if2^#(false,x124,s(x123),s(x125),s(x122),x126)≥quotIter^#(x124,s(x123),s(x125),s(x122),x126))⟹if2^#(false,x60,s(s(x123)),s(x62),s(s(x122)),x64)≥quotIter^#(x60,s(s(x123)),s(x62),s(s(x122)),x64))
        
        Applying Rule "Simplify Conditions" results in
        (if2^#(false,x60,s(x123),s(x62),s(x122),x64)≥quotIter^#(x60,s(x123),s(x62),s(x122),x64)⟹if2^#(false,x60,s(s(x123)),s(x62),s(s(x122)),x64)≥quotIter^#(x60,s(s(x123)),s(x62),s(s(x122)),x64))
        
        This constraint is kept as final constraint.
      3. (false→^*false⟹if2^#(false,x60,s(s(x127)),s(x62),s(0),x64)≥quotIter^#(x60,s(s(x127)),s(x62),s(0),x64))
        
        Applying Rule "Same Constructor" results in
        if2^#(false,x60,s(s(x127)),s(x62),s(0),x64)≥quotIter^#(x60,s(s(x127)),s(x62),s(0),x64)
        
        This constraint is kept as final constraint.
  3. (false→^*false⟹s(x63)→^*0⟹if2^#(false,x60,s(x120),s(x62),s(x63),x64)≥quotIter^#(x60,s(x120),s(x62),s(x63),x64))
    - Applying Rule "Same Constructor" results in
      (s(x63)→^*0⟹if2^#(false,x60,s(x120),s(x62),s(x63),x64)≥quotIter^#(x60,s(x120),s(x62),s(x63),x64))
    - Applying Rule "Different Constructors" allows to drop this constraint.
For the chain we build the initial constraint
(if2^#(le(x86,s(x88)),x85,x86,s(x87),s(x88),x89)→^*if2^#(true,x90,x91,x92,x93,x94)⟹if2^#(true,x90,x91,x92,x93,x94)≥quotIter^#(x90,x91,x92,0,s(x94)))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x86,s(x88))→^*true⟹x85→^*x90⟹x86→^*x91⟹s(x87)→^*x92⟹s(x88)→^*x93⟹x89→^*x94⟹if2^#(true,x90,x91,x92,x93,x94)≥quotIter^#(x90,x91,x92,0,s(x94)))
- Applying Rule "Variable in Equation" allows to substitute x90 by x85 which results in
  (le(x86,s(x88))→^*true⟹x86→^*x91⟹s(x87)→^*x92⟹s(x88)→^*x93⟹x89→^*x94⟹if2^#(true,x85,x91,x92,x93,x94)≥quotIter^#(x85,x91,x92,0,s(x94)))
- Applying Rule "Variable in Equation" allows to substitute x91 by x86 which results in
  (le(x86,s(x88))→^*true⟹s(x87)→^*x92⟹s(x88)→^*x93⟹x89→^*x94⟹if2^#(true,x85,x86,x92,x93,x94)≥quotIter^#(x85,x86,x92,0,s(x94)))
- Applying Rule "Variable in Equation" allows to substitute x92 by s(x87) which results in
  (le(x86,s(x88))→^*true⟹s(x88)→^*x93⟹x89→^*x94⟹if2^#(true,x85,x86,s(x87),x93,x94)≥quotIter^#(x85,x86,s(x87),0,s(x94)))
- Applying Rule "Variable in Equation" allows to substitute x93 by s(x88) which results in
  (le(x86,s(x88))→^*true⟹x89→^*x94⟹if2^#(true,x85,x86,s(x87),s(x88),x94)≥quotIter^#(x85,x86,s(x87),0,s(x94)))
- Applying Rule "Variable in Equation" allows to substitute x94 by x89 which results in
  (le(x86,s(x88))→^*true⟹if2^#(true,x85,x86,s(x87),s(x88),x89)≥quotIter^#(x85,x86,s(x87),0,s(x89)))
- Applying Rule "Introduce fresh variable" results in
  (s(x88)→^*x128⟹le(x86,x128)→^*true⟹if2^#(true,x85,x86,s(x87),s(x88),x89)≥quotIter^#(x85,x86,s(x87),0,s(x89)))
- Applying Rule "Induction" on le(x86,x128)→^*true results in the following 3 new constraints.
  1. (true→^*true⟹s(x88)→^*x129⟹if2^#(true,x85,0,s(x87),s(x88),x89)≥quotIter^#(x85,0,s(x87),0,s(x89)))
    - Applying Rule "Same Constructor" results in
      (s(x88)→^*x129⟹if2^#(true,x85,0,s(x87),s(x88),x89)≥quotIter^#(x85,0,s(x87),0,s(x89)))
    - Applying Rule "Delete Conditions" results in
      if2^#(true,x85,0,s(x87),s(x88),x89)≥quotIter^#(x85,0,s(x87),0,s(x89))
    - This constraint is kept as final constraint.
  2. (le(x131,x130)→^*true⟹s(x88)→^*s(x130)⟹∀x132 . ∀x133 . ∀x134 . ∀x135 . (le(x131,x130)→^*true⟹s(x132)→^*x130⟹if2^#(true,x133,x131,s(x134),s(x132),x135)≥quotIter^#(x133,x131,s(x134),0,s(x135)))⟹if2^#(true,x85,s(x131),s(x87),s(x88),x89)≥quotIter^#(x85,s(x131),s(x87),0,s(x89)))
    - Applying Rule "Same Constructor" results in
      (le(x131,x130)→^*true⟹x88→^*x130⟹∀x132 . ∀x133 . ∀x134 . ∀x135 . (le(x131,x130)→^*true⟹s(x132)→^*x130⟹if2^#(true,x133,x131,s(x134),s(x132),x135)≥quotIter^#(x133,x131,s(x134),0,s(x135)))⟹if2^#(true,x85,s(x131),s(x87),s(x88),x89)≥quotIter^#(x85,s(x131),s(x87),0,s(x89)))
    - Applying Rule "Variable in Equation" allows to substitute x88 by x130 which results in
      (le(x131,x130)→^*true⟹∀x132 . ∀x133 . ∀x134 . ∀x135 . (le(x131,x130)→^*true⟹s(x132)→^*x130⟹if2^#(true,x133,x131,s(x134),s(x132),x135)≥quotIter^#(x133,x131,s(x134),0,s(x135)))⟹if2^#(true,x85,s(x131),s(x87),s(x130),x89)≥quotIter^#(x85,s(x131),s(x87),0,s(x89)))
    - Applying Rule "Delete Conditions" results in
      (le(x131,x130)→^*true⟹if2^#(true,x85,s(x131),s(x87),s(x130),x89)≥quotIter^#(x85,s(x131),s(x87),0,s(x89)))
    - Applying Rule "Induction" on le(x131,x130)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹if2^#(true,x85,s(0),s(x87),s(x137),x89)≥quotIter^#(x85,s(0),s(x87),0,s(x89)))
        
        Applying Rule "Same Constructor" results in
        if2^#(true,x85,s(0),s(x87),s(x137),x89)≥quotIter^#(x85,s(0),s(x87),0,s(x89))
        
        This constraint is kept as final constraint.
      2. (le(x139,x138)→^*true⟹∀x140 . ∀x141 . ∀x142 . (le(x139,x138)→^*true⟹if2^#(true,x140,s(x139),s(x141),s(x138),x142)≥quotIter^#(x140,s(x139),s(x141),0,s(x142)))⟹if2^#(true,x85,s(s(x139)),s(x87),s(s(x138)),x89)≥quotIter^#(x85,s(s(x139)),s(x87),0,s(x89)))
        
        Applying Rule "Simplify Conditions" results in
        (if2^#(true,x85,s(x139),s(x87),s(x138),x89)≥quotIter^#(x85,s(x139),s(x87),0,s(x89))⟹if2^#(true,x85,s(s(x139)),s(x87),s(s(x138)),x89)≥quotIter^#(x85,s(s(x139)),s(x87),0,s(x89)))
        
        This constraint is kept as final constraint.
      3. (false→^*true⟹if2^#(true,x85,s(s(x143)),s(x87),s(0),x89)≥quotIter^#(x85,s(s(x143)),s(x87),0,s(x89)))
        Applying Rule "Different Constructors" allows to drop this constraint.
  3. (false→^*true⟹if2^#(true,x85,s(x136),s(x87),s(x88),x89)≥quotIter^#(x85,s(x136),s(x87),0,s(x89)))
    - Applying Rule "Different Constructors" allows to drop this constraint.

We remove those pairs which are strictly decreasing and bounded.

1.1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.

The 2^nd component contains the pair
le^#(s(x),s(y)) → le^#(x,y) (11)

1.1.1.2 Usable Rules Processor

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

There are no rules.

1.1.1.2.1 Innermost Lhss Removal Processor

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

There are no lhss.

1.1.1.2.1.1 Size-Change Termination

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

le^#(s(x),s(y)) → le^#(x,y) (11)

1 > 1

2 > 2

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