Certification Problem

Input (TPDB TRS_Standard/AProVE_07/thiemann31)

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)
mod(x,0)	→	modZeroErro	(4)
mod(x,s(y))	→	modIter(x,s(y),0,0)	(5)
modIter(x,s(y),z,u)	→	if(le(x,z),x,s(y),z,u)	(6)
if(true,x,y,z,u)	→	u	(7)
if(false,x,y,z,u)	→	if2(le(y,s(u)),x,y,s(z),s(u))	(8)
if2(false,x,y,z,u)	→	modIter(x,y,z,u)	(9)
if2(true,x,y,z,u)	→	modIter(x,y,z,0)	(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)
mod^#(x,s(y))	→	modIter^#(x,s(y),0,0)	(12)
modIter^#(x,s(y),z,u)	→	if^#(le(x,z),x,s(y),z,u)	(13)
modIter^#(x,s(y),z,u)	→	le^#(x,z)	(14)
if^#(false,x,y,z,u)	→	if2^#(le(y,s(u)),x,y,s(z),s(u))	(15)
if^#(false,x,y,z,u)	→	le^#(y,s(u))	(16)
if2^#(false,x,y,z,u)	→	modIter^#(x,y,z,u)	(17)
if2^#(true,x,y,z,u)	→	modIter^#(x,y,z,0)	(18)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.

The 1^st component contains the pair

modIter^#(x,s(y),z,u)	→	if^#(le(x,z),x,s(y),z,u)	(13)
if^#(false,x,y,z,u)	→	if2^#(le(y,s(u)),x,y,s(z),s(u))	(15)
if2^#(false,x,y,z,u)	→	modIter^#(x,y,z,u)	(17)
if2^#(true,x,y,z,u)	→	modIter^#(x,y,z,0)	(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,...,x₅)]	=	-1 · x₁ + 1 · x₂ + -1 · x₄ + -1
[modIter^#(x₁,...,x₄)]	=	1 · x₁ + -1 · x₃ + -1
[if2^#(x₁,...,x₅)]	=	-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)

→

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

(15)

are strictly oriented and the pair(s)

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

→

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

(15)

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

The following constraints are generated for the pairs.

(if^#(false,x86,s(x25),x85,x27)≥c⟹if^#(false,s(x86),s(x25),s(x85),x27)≥c)
if^#(false,s(x89),s(x25),0,x27)≥c
modIter^#(x8,s(x13),x10,x11)≥if^#(le(x8,x10),x8,s(x13),x10,x11)
modIter^#(x16,s(x21),x18,0)≥if^#(le(x16,x18),x16,s(x21),x18,0)
(if^#(false,x86,s(x25),x85,x27) > if2^#(le(s(x25),s(x27)),x86,s(x25),s(x85),s(x27))⟹if^#(false,s(x86),s(x25),s(x85),x27) > if2^#(le(s(x25),s(x27)),s(x86),s(x25),s(s(x85)),s(x27)))
if^#(false,s(x89),s(x25),0,x27) > if2^#(le(s(x25),s(x27)),s(x89),s(x25),s(0),s(x27))
(if2^#(false,x48,s(x100),s(x50),s(x99))≥modIter^#(x48,s(x100),s(x50),s(x99))⟹if2^#(false,x48,s(s(x100)),s(x50),s(s(x99)))≥modIter^#(x48,s(s(x100)),s(x50),s(s(x99))))
if2^#(false,x48,s(s(x103)),s(x50),s(0))≥modIter^#(x48,s(s(x103)),s(x50),s(0))
if2^#(true,x68,0,s(x70),s(x71))≥modIter^#(x68,0,s(x70),0)
if2^#(true,x68,s(0),s(x70),s(x112))≥modIter^#(x68,s(0),s(x70),0)
(if2^#(true,x68,s(x114),s(x70),s(x113))≥modIter^#(x68,s(x114),s(x70),0)⟹if2^#(true,x68,s(s(x114)),s(x70),s(s(x113)))≥modIter^#(x68,s(s(x114)),s(x70),0))

The details are shown below:

For the chain we build the initial constraint
(if^#(le(x24,x26),x24,s(x25),x26,x27)→^*if^#(false,x28,x29,x30,x31)⟹if^#(false,x28,x29,x30,x31)≥c)

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x24,x26)→^*false⟹x24→^*x28⟹s(x25)→^*x29⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x28,x29,x30,x31)≥c)
- Applying Rule "Variable in Equation" allows to substitute x28 by x24 which results in
  (le(x24,x26)→^*false⟹s(x25)→^*x29⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x24,x29,x30,x31)≥c)
- Applying Rule "Variable in Equation" allows to substitute x29 by s(x25) which results in
  (le(x24,x26)→^*false⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x24,s(x25),x30,x31)≥c)
- Applying Rule "Variable in Equation" allows to substitute x30 by x26 which results in
  (le(x24,x26)→^*false⟹x27→^*x31⟹if^#(false,x24,s(x25),x26,x31)≥c)
- Applying Rule "Variable in Equation" allows to substitute x31 by x27 which results in
  (le(x24,x26)→^*false⟹if^#(false,x24,s(x25),x26,x27)≥c)
- Applying Rule "Induction" on le(x24,x26)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if^#(false,0,s(x25),x84,x27)≥c)
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x86,x85)→^*false⟹∀x87 . ∀x88 . (le(x86,x85)→^*false⟹if^#(false,x86,s(x87),x85,x88)≥c)⟹if^#(false,s(x86),s(x25),s(x85),x27)≥c)
    - Applying Rule "Simplify Conditions" results in
      (if^#(false,x86,s(x25),x85,x27)≥c⟹if^#(false,s(x86),s(x25),s(x85),x27)≥c)
    - This constraint is kept as final constraint.
  3. (false→^*false⟹if^#(false,s(x89),s(x25),0,x27)≥c)
    - Applying Rule "Same Constructor" results in
      if^#(false,s(x89),s(x25),0,x27)≥c
    - This constraint is kept as final constraint.
For the chain we build the initial constraint
(modIter^#(x8,x9,x10,x11)→^*modIter^#(x12,s(x13),x14,x15)⟹modIter^#(x12,s(x13),x14,x15)≥if^#(le(x12,x14),x12,s(x13),x14,x15))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (x8→^*x12⟹x9→^*s(x13)⟹x10→^*x14⟹x11→^*x15⟹modIter^#(x12,s(x13),x14,x15)≥if^#(le(x12,x14),x12,s(x13),x14,x15))
- Applying Rule "Variable in Equation" allows to substitute x12 by x8 which results in
  (x9→^*s(x13)⟹x10→^*x14⟹x11→^*x15⟹modIter^#(x8,s(x13),x14,x15)≥if^#(le(x8,x14),x8,s(x13),x14,x15))
- Applying Rule "Variable in Equation" allows to substitute x14 by x10 which results in
  (x9→^*s(x13)⟹x11→^*x15⟹modIter^#(x8,s(x13),x10,x15)≥if^#(le(x8,x10),x8,s(x13),x10,x15))
- Applying Rule "Variable in Equation" allows to substitute x15 by x11 which results in
  (x9→^*s(x13)⟹modIter^#(x8,s(x13),x10,x11)≥if^#(le(x8,x10),x8,s(x13),x10,x11))
- Applying Rule "Variable in Equation" allows to substitute x9 by s(x13) which results in
  modIter^#(x8,s(x13),x10,x11)≥if^#(le(x8,x10),x8,s(x13),x10,x11)
- This constraint is kept as final constraint.
For the chain we build the initial constraint
(modIter^#(x16,x17,x18,0)→^*modIter^#(x20,s(x21),x22,x23)⟹modIter^#(x20,s(x21),x22,x23)≥if^#(le(x20,x22),x20,s(x21),x22,x23))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (x16→^*x20⟹x17→^*s(x21)⟹x18→^*x22⟹0→^*x23⟹modIter^#(x20,s(x21),x22,x23)≥if^#(le(x20,x22),x20,s(x21),x22,x23))
- Applying Rule "Variable in Equation" allows to substitute x20 by x16 which results in
  (x17→^*s(x21)⟹x18→^*x22⟹0→^*x23⟹modIter^#(x16,s(x21),x22,x23)≥if^#(le(x16,x22),x16,s(x21),x22,x23))
- Applying Rule "Variable in Equation" allows to substitute x22 by x18 which results in
  (x17→^*s(x21)⟹0→^*x23⟹modIter^#(x16,s(x21),x18,x23)≥if^#(le(x16,x18),x16,s(x21),x18,x23))
- Applying Rule "Variable in Equation" allows to substitute x23 by 0 which results in
  (x17→^*s(x21)⟹modIter^#(x16,s(x21),x18,0)≥if^#(le(x16,x18),x16,s(x21),x18,0))
- Applying Rule "Variable in Equation" allows to substitute x17 by s(x21) which results in
  modIter^#(x16,s(x21),x18,0)≥if^#(le(x16,x18),x16,s(x21),x18,0)
- This constraint is kept as final constraint.
For the chain we build the initial constraint
(if^#(le(x24,x26),x24,s(x25),x26,x27)→^*if^#(false,x28,x29,x30,x31)⟹if^#(false,x28,x29,x30,x31) > if2^#(le(x29,s(x31)),x28,x29,s(x30),s(x31)))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x24,x26)→^*false⟹x24→^*x28⟹s(x25)→^*x29⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x28,x29,x30,x31) > if2^#(le(x29,s(x31)),x28,x29,s(x30),s(x31)))
- Applying Rule "Variable in Equation" allows to substitute x28 by x24 which results in
  (le(x24,x26)→^*false⟹s(x25)→^*x29⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x24,x29,x30,x31) > if2^#(le(x29,s(x31)),x24,x29,s(x30),s(x31)))
- Applying Rule "Variable in Equation" allows to substitute x29 by s(x25) which results in
  (le(x24,x26)→^*false⟹x26→^*x30⟹x27→^*x31⟹if^#(false,x24,s(x25),x30,x31) > if2^#(le(s(x25),s(x31)),x24,s(x25),s(x30),s(x31)))
- Applying Rule "Variable in Equation" allows to substitute x30 by x26 which results in
  (le(x24,x26)→^*false⟹x27→^*x31⟹if^#(false,x24,s(x25),x26,x31) > if2^#(le(s(x25),s(x31)),x24,s(x25),s(x26),s(x31)))
- Applying Rule "Variable in Equation" allows to substitute x31 by x27 which results in
  (le(x24,x26)→^*false⟹if^#(false,x24,s(x25),x26,x27) > if2^#(le(s(x25),s(x27)),x24,s(x25),s(x26),s(x27)))
- Applying Rule "Induction" on le(x24,x26)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if^#(false,0,s(x25),x84,x27) > if2^#(le(s(x25),s(x27)),0,s(x25),s(x84),s(x27)))
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x86,x85)→^*false⟹∀x87 . ∀x88 . (le(x86,x85)→^*false⟹if^#(false,x86,s(x87),x85,x88) > if2^#(le(s(x87),s(x88)),x86,s(x87),s(x85),s(x88)))⟹if^#(false,s(x86),s(x25),s(x85),x27) > if2^#(le(s(x25),s(x27)),s(x86),s(x25),s(s(x85)),s(x27)))
    - Applying Rule "Simplify Conditions" results in
      (if^#(false,x86,s(x25),x85,x27) > if2^#(le(s(x25),s(x27)),x86,s(x25),s(x85),s(x27))⟹if^#(false,s(x86),s(x25),s(x85),x27) > if2^#(le(s(x25),s(x27)),s(x86),s(x25),s(s(x85)),s(x27)))
    - This constraint is kept as final constraint.
  3. (false→^*false⟹if^#(false,s(x89),s(x25),0,x27) > if2^#(le(s(x25),s(x27)),s(x89),s(x25),s(0),s(x27)))
    - Applying Rule "Same Constructor" results in
      if^#(false,s(x89),s(x25),0,x27) > if2^#(le(s(x25),s(x27)),s(x89),s(x25),s(0),s(x27))
    - This constraint is kept as final constraint.
For the chain we build the initial constraint
(if2^#(le(x49,s(x51)),x48,x49,s(x50),s(x51))→^*if2^#(false,x52,x53,x54,x55)⟹if2^#(false,x52,x53,x54,x55)≥modIter^#(x52,x53,x54,x55))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x49,s(x51))→^*false⟹x48→^*x52⟹x49→^*x53⟹s(x50)→^*x54⟹s(x51)→^*x55⟹if2^#(false,x52,x53,x54,x55)≥modIter^#(x52,x53,x54,x55))
- Applying Rule "Variable in Equation" allows to substitute x52 by x48 which results in
  (le(x49,s(x51))→^*false⟹x49→^*x53⟹s(x50)→^*x54⟹s(x51)→^*x55⟹if2^#(false,x48,x53,x54,x55)≥modIter^#(x48,x53,x54,x55))
- Applying Rule "Variable in Equation" allows to substitute x53 by x49 which results in
  (le(x49,s(x51))→^*false⟹s(x50)→^*x54⟹s(x51)→^*x55⟹if2^#(false,x48,x49,x54,x55)≥modIter^#(x48,x49,x54,x55))
- Applying Rule "Variable in Equation" allows to substitute x54 by s(x50) which results in
  (le(x49,s(x51))→^*false⟹s(x51)→^*x55⟹if2^#(false,x48,x49,s(x50),x55)≥modIter^#(x48,x49,s(x50),x55))
- Applying Rule "Variable in Equation" allows to substitute x55 by s(x51) which results in
  (le(x49,s(x51))→^*false⟹if2^#(false,x48,x49,s(x50),s(x51))≥modIter^#(x48,x49,s(x50),s(x51)))
- Applying Rule "Introduce fresh variable" results in
  (s(x51)→^*x90⟹le(x49,x90)→^*false⟹if2^#(false,x48,x49,s(x50),s(x51))≥modIter^#(x48,x49,s(x50),s(x51)))
- Applying Rule "Induction" on le(x49,x90)→^*false results in the following 3 new constraints.
  1. (true→^*false⟹if2^#(false,x48,0,s(x50),s(x51))≥modIter^#(x48,0,s(x50),s(x51)))
    - Applying Rule "Different Constructors" allows to drop this constraint.
  2. (le(x93,x92)→^*false⟹s(x51)→^*s(x92)⟹∀x94 . ∀x95 . ∀x96 . (le(x93,x92)→^*false⟹s(x94)→^*x92⟹if2^#(false,x95,x93,s(x96),s(x94))≥modIter^#(x95,x93,s(x96),s(x94)))⟹if2^#(false,x48,s(x93),s(x50),s(x51))≥modIter^#(x48,s(x93),s(x50),s(x51)))
    - Applying Rule "Same Constructor" results in
      (le(x93,x92)→^*false⟹x51→^*x92⟹∀x94 . ∀x95 . ∀x96 . (le(x93,x92)→^*false⟹s(x94)→^*x92⟹if2^#(false,x95,x93,s(x96),s(x94))≥modIter^#(x95,x93,s(x96),s(x94)))⟹if2^#(false,x48,s(x93),s(x50),s(x51))≥modIter^#(x48,s(x93),s(x50),s(x51)))
    - Applying Rule "Variable in Equation" allows to substitute x51 by x92 which results in
      (le(x93,x92)→^*false⟹∀x94 . ∀x95 . ∀x96 . (le(x93,x92)→^*false⟹s(x94)→^*x92⟹if2^#(false,x95,x93,s(x96),s(x94))≥modIter^#(x95,x93,s(x96),s(x94)))⟹if2^#(false,x48,s(x93),s(x50),s(x92))≥modIter^#(x48,s(x93),s(x50),s(x92)))
    - Applying Rule "Delete Conditions" results in
      (le(x93,x92)→^*false⟹if2^#(false,x48,s(x93),s(x50),s(x92))≥modIter^#(x48,s(x93),s(x50),s(x92)))
    - Applying Rule "Induction" on le(x93,x92)→^*false results in the following 3 new constraints.
      1. (true→^*false⟹if2^#(false,x48,s(0),s(x50),s(x98))≥modIter^#(x48,s(0),s(x50),s(x98)))
        Applying Rule "Different Constructors" allows to drop this constraint.
      2. (le(x100,x99)→^*false⟹∀x101 . ∀x102 . (le(x100,x99)→^*false⟹if2^#(false,x101,s(x100),s(x102),s(x99))≥modIter^#(x101,s(x100),s(x102),s(x99)))⟹if2^#(false,x48,s(s(x100)),s(x50),s(s(x99)))≥modIter^#(x48,s(s(x100)),s(x50),s(s(x99))))
        
        Applying Rule "Simplify Conditions" results in
        (if2^#(false,x48,s(x100),s(x50),s(x99))≥modIter^#(x48,s(x100),s(x50),s(x99))⟹if2^#(false,x48,s(s(x100)),s(x50),s(s(x99)))≥modIter^#(x48,s(s(x100)),s(x50),s(s(x99))))
        
        This constraint is kept as final constraint.
      3. (false→^*false⟹if2^#(false,x48,s(s(x103)),s(x50),s(0))≥modIter^#(x48,s(s(x103)),s(x50),s(0)))
        
        Applying Rule "Same Constructor" results in
        if2^#(false,x48,s(s(x103)),s(x50),s(0))≥modIter^#(x48,s(s(x103)),s(x50),s(0))
        
        This constraint is kept as final constraint.
  3. (false→^*false⟹s(x51)→^*0⟹if2^#(false,x48,s(x97),s(x50),s(x51))≥modIter^#(x48,s(x97),s(x50),s(x51)))
    - Applying Rule "Same Constructor" results in
      (s(x51)→^*0⟹if2^#(false,x48,s(x97),s(x50),s(x51))≥modIter^#(x48,s(x97),s(x50),s(x51)))
    - Applying Rule "Different Constructors" allows to drop this constraint.
For the chain we build the initial constraint
(if2^#(le(x69,s(x71)),x68,x69,s(x70),s(x71))→^*if2^#(true,x72,x73,x74,x75)⟹if2^#(true,x72,x73,x74,x75)≥modIter^#(x72,x73,x74,0))

which is simplified as follows.
- Applying Rule "Same Constructor" results in
  (le(x69,s(x71))→^*true⟹x68→^*x72⟹x69→^*x73⟹s(x70)→^*x74⟹s(x71)→^*x75⟹if2^#(true,x72,x73,x74,x75)≥modIter^#(x72,x73,x74,0))
- Applying Rule "Variable in Equation" allows to substitute x72 by x68 which results in
  (le(x69,s(x71))→^*true⟹x69→^*x73⟹s(x70)→^*x74⟹s(x71)→^*x75⟹if2^#(true,x68,x73,x74,x75)≥modIter^#(x68,x73,x74,0))
- Applying Rule "Variable in Equation" allows to substitute x73 by x69 which results in
  (le(x69,s(x71))→^*true⟹s(x70)→^*x74⟹s(x71)→^*x75⟹if2^#(true,x68,x69,x74,x75)≥modIter^#(x68,x69,x74,0))
- Applying Rule "Variable in Equation" allows to substitute x74 by s(x70) which results in
  (le(x69,s(x71))→^*true⟹s(x71)→^*x75⟹if2^#(true,x68,x69,s(x70),x75)≥modIter^#(x68,x69,s(x70),0))
- Applying Rule "Variable in Equation" allows to substitute x75 by s(x71) which results in
  (le(x69,s(x71))→^*true⟹if2^#(true,x68,x69,s(x70),s(x71))≥modIter^#(x68,x69,s(x70),0))
- Applying Rule "Introduce fresh variable" results in
  (s(x71)→^*x104⟹le(x69,x104)→^*true⟹if2^#(true,x68,x69,s(x70),s(x71))≥modIter^#(x68,x69,s(x70),0))
- Applying Rule "Induction" on le(x69,x104)→^*true results in the following 3 new constraints.
  1. (true→^*true⟹s(x71)→^*x105⟹if2^#(true,x68,0,s(x70),s(x71))≥modIter^#(x68,0,s(x70),0))
    - Applying Rule "Same Constructor" results in
      (s(x71)→^*x105⟹if2^#(true,x68,0,s(x70),s(x71))≥modIter^#(x68,0,s(x70),0))
    - Applying Rule "Delete Conditions" results in
      if2^#(true,x68,0,s(x70),s(x71))≥modIter^#(x68,0,s(x70),0)
    - This constraint is kept as final constraint.
  2. (le(x107,x106)→^*true⟹s(x71)→^*s(x106)⟹∀x108 . ∀x109 . ∀x110 . (le(x107,x106)→^*true⟹s(x108)→^*x106⟹if2^#(true,x109,x107,s(x110),s(x108))≥modIter^#(x109,x107,s(x110),0))⟹if2^#(true,x68,s(x107),s(x70),s(x71))≥modIter^#(x68,s(x107),s(x70),0))
    - Applying Rule "Same Constructor" results in
      (le(x107,x106)→^*true⟹x71→^*x106⟹∀x108 . ∀x109 . ∀x110 . (le(x107,x106)→^*true⟹s(x108)→^*x106⟹if2^#(true,x109,x107,s(x110),s(x108))≥modIter^#(x109,x107,s(x110),0))⟹if2^#(true,x68,s(x107),s(x70),s(x71))≥modIter^#(x68,s(x107),s(x70),0))
    - Applying Rule "Variable in Equation" allows to substitute x71 by x106 which results in
      (le(x107,x106)→^*true⟹∀x108 . ∀x109 . ∀x110 . (le(x107,x106)→^*true⟹s(x108)→^*x106⟹if2^#(true,x109,x107,s(x110),s(x108))≥modIter^#(x109,x107,s(x110),0))⟹if2^#(true,x68,s(x107),s(x70),s(x106))≥modIter^#(x68,s(x107),s(x70),0))
    - Applying Rule "Delete Conditions" results in
      (le(x107,x106)→^*true⟹if2^#(true,x68,s(x107),s(x70),s(x106))≥modIter^#(x68,s(x107),s(x70),0))
    - Applying Rule "Induction" on le(x107,x106)→^*true results in the following 3 new constraints.
      1. (true→^*true⟹if2^#(true,x68,s(0),s(x70),s(x112))≥modIter^#(x68,s(0),s(x70),0))
        
        Applying Rule "Same Constructor" results in
        if2^#(true,x68,s(0),s(x70),s(x112))≥modIter^#(x68,s(0),s(x70),0)
        
        This constraint is kept as final constraint.
      2. (le(x114,x113)→^*true⟹∀x115 . ∀x116 . (le(x114,x113)→^*true⟹if2^#(true,x115,s(x114),s(x116),s(x113))≥modIter^#(x115,s(x114),s(x116),0))⟹if2^#(true,x68,s(s(x114)),s(x70),s(s(x113)))≥modIter^#(x68,s(s(x114)),s(x70),0))
        
        Applying Rule "Simplify Conditions" results in
        (if2^#(true,x68,s(x114),s(x70),s(x113))≥modIter^#(x68,s(x114),s(x70),0)⟹if2^#(true,x68,s(s(x114)),s(x70),s(s(x113)))≥modIter^#(x68,s(s(x114)),s(x70),0))
        
        This constraint is kept as final constraint.
      3. (false→^*true⟹if2^#(true,x68,s(s(x117)),s(x70),s(0))≥modIter^#(x68,s(s(x117)),s(x70),0))
        Applying Rule "Different Constructors" allows to drop this constraint.
  3. (false→^*true⟹if2^#(true,x68,s(x111),s(x70),s(x71))≥modIter^#(x68,s(x111),s(x70),0))
    - 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.