YES
We show the termination of the TRS R:

  minus(minus(x,y),z) -> minus(x,plus(y,z))
  minus(|0|(),y) -> |0|()
  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  plus(|0|(),y) -> y
  plus(s(x),y) -> plus(x,s(y))
  plus(s(x),y) -> s(plus(y,x))
  zero(s(x)) -> false()
  zero(|0|()) -> true()
  p(s(x)) -> x
  p(|0|()) -> |0|()
  div(x,y) -> quot(x,y,|0|())
  quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
  quot(|0|(),s(y),z) -> z
  ack(|0|(),x) -> s(x)
  ack(|0|(),x) -> plus(x,s(|0|()))
  ack(s(x),|0|()) -> ack(x,s(|0|()))
  ack(s(x),s(y)) -> ack(x,ack(s(x),y))


-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p2: minus#(minus(x,y),z) -> plus#(y,z)
p3: minus#(s(x),s(y)) -> minus#(x,y)
p4: plus#(s(x),y) -> plus#(x,s(y))
p5: plus#(s(x),y) -> plus#(y,x)
p6: div#(x,y) -> quot#(x,y,|0|())
p7: quot#(s(x),s(y),z) -> quot#(minus(p(ack(|0|(),x)),y),s(y),s(z))
p8: quot#(s(x),s(y),z) -> minus#(p(ack(|0|(),x)),y)
p9: quot#(s(x),s(y),z) -> p#(ack(|0|(),x))
p10: quot#(s(x),s(y),z) -> ack#(|0|(),x)
p11: ack#(|0|(),x) -> plus#(x,s(|0|()))
p12: ack#(s(x),|0|()) -> ack#(x,s(|0|()))
p13: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p14: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The estimated dependency graph contains the following SCCs:

  {p7}
  {p1, p3}
  {p12, p13, p14}
  {p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: quot#(s(x),s(y),z) -> quot#(minus(p(ack(|0|(),x)),y),s(y),s(z))

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quot#_A(x1,x2,x3) = x1
          s_A(x1) = x1 + 1
          minus_A(x1,x2) = x1
          p_A(x1) = x1
          ack_A(x1,x2) = x2 + 1
          |0|_A() = 0
          plus_A(x1,x2) = x1 + x2
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          quot_A(x1,x2,x3) = x2 + x3 + 2
  
    base order 2:
  
      precedence:
      
        ack > quot > s = plus = zero > quot# = false > minus = p > |0| > true
      
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = []
      pi(minus) = [1]
      pi(p) = []
      pi(ack) = [2]
      pi(|0|) = []
      pi(plus) = [1, 2]
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(quot) = [3]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p2: minus#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus#_A(x1,x2) = x1 + x2 + 3
          minus_A(x1,x2) = x1 + x2 + 4
          plus_A(x1,x2) = x1 + x2 + 3
          s_A(x1) = x1 + 2
          |0|_A() = 1
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
          ack_A(x1,x2) = x1 + x2 + 6
  
    base order 2:
  
      precedence:
      
        minus# > minus = p > quot > plus > |0| = zero > s = true = ack > false
      
  
    partial status:
  
      pi(minus#) = []
      pi(minus) = []
      pi(plus) = []
      pi(s) = [1]
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []
      pi(ack) = []

The following rules are removed:

  p1



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: minus#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: minus#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r2, r3, r8, r9, r10, r11, r14

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus#_A(x1,x2) = x1 + x2 + 6
          s_A(x1) = x1 + 5
          minus_A(x1,x2) = x1 + x2 + 6
          |0|_A() = 1
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
  
    base order 2:
  
      precedence:
      
        minus > quot > minus# = |0| = zero > s = true > false = p
      
  
    partial status:
  
      pi(minus#) = [1]
      pi(s) = [1]
      pi(minus) = []
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p2: ack#(s(x),s(y)) -> ack#(s(x),y)
p3: ack#(s(x),|0|()) -> ack#(x,s(|0|()))

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r2, r8, r9, r11, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + 2
          s_A(x1) = x1 + 4
          ack_A(x1,x2) = x1 + x2 + 6
          |0|_A() = 1
          minus_A(x1,x2) = x2 + 2
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = 2
          quot_A(x1,x2,x3) = x2 + x3 + 2
  
    base order 2:
  
      precedence:
      
        ack > quot > ack# = p > s = minus = zero > |0| = false > true
      
  
    partial status:
  
      pi(ack#) = [1]
      pi(s) = [1]
      pi(ack) = []
      pi(|0|) = []
      pi(minus) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = [2]

The following rules are removed:

  p3



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p2: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ack#(s(x),s(y)) -> ack#(x,ack(s(x),y))
p2: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r1, r2, r3, r4, r8, r9, r10, r11, r14

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 7
          ack_A(x1,x2) = x1 + x2 + 3
          minus_A(x1,x2) = x1 + 1
          plus_A(x1,x2) = x1 + 3
          |0|_A() = 1
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
  
    base order 2:
  
      precedence:
      
        ack# = p > s = ack = minus = quot > |0| = zero > plus = false = true
      
  
    partial status:
  
      pi(ack#) = [1]
      pi(s) = [1]
      pi(ack) = [1, 2]
      pi(minus) = []
      pi(plus) = [1]
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []

The following rules are removed:

  p1



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ack#(s(x),s(y)) -> ack#(s(x),y)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r2, r3, r8, r9, r10, r11, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          ack#_A(x1,x2) = x1 + x2 + 1
          s_A(x1) = x1 + 3
          minus_A(x1,x2) = x1 + x2 + 4
          |0|_A() = 2
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
          ack_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        minus = p > ack# = |0| = zero > s = true = quot = ack > false
      
  
    partial status:
  
      pi(ack#) = [2]
      pi(s) = []
      pi(minus) = [1, 2]
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []
      pi(ack) = [1, 2]

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: plus#(s(x),y) -> plus#(x,s(y))
p2: plus#(s(x),y) -> plus#(y,x)

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r2, r3, r8, r9, r10, r11, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + x2 + 4
          s_A(x1) = x1 + 3
          minus_A(x1,x2) = x1 + x2 + 5
          |0|_A() = 2
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
          ack_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        minus > quot > plus# > s = |0| = zero = p > false = true = ack
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []
      pi(ack) = [1]

The following rules are removed:

  p2



-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: plus#(s(x),y) -> plus#(x,s(y))

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: plus#(s(x),y) -> plus#(x,s(y))

and R consists of:

r1: minus(minus(x,y),z) -> minus(x,plus(y,z))
r2: minus(|0|(),y) -> |0|()
r3: minus(x,|0|()) -> x
r4: minus(s(x),s(y)) -> minus(x,y)
r5: plus(|0|(),y) -> y
r6: plus(s(x),y) -> plus(x,s(y))
r7: plus(s(x),y) -> s(plus(y,x))
r8: zero(s(x)) -> false()
r9: zero(|0|()) -> true()
r10: p(s(x)) -> x
r11: p(|0|()) -> |0|()
r12: div(x,y) -> quot(x,y,|0|())
r13: quot(s(x),s(y),z) -> quot(minus(p(ack(|0|(),x)),y),s(y),s(z))
r14: quot(|0|(),s(y),z) -> z
r15: ack(|0|(),x) -> s(x)
r16: ack(|0|(),x) -> plus(x,s(|0|()))
r17: ack(s(x),|0|()) -> ack(x,s(|0|()))
r18: ack(s(x),s(y)) -> ack(x,ack(s(x),y))

The set of usable rules consists of

  r2, r8, r9, r10, r11, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 2
          minus_A(x1,x2) = x2 + 5
          |0|_A() = 3
          zero_A(x1) = 1
          false_A() = 0
          true_A() = 0
          p_A(x1) = x1 + 1
          quot_A(x1,x2,x3) = x2 + x3 + 2
          ack_A(x1,x2) = x1 + x2 + 4
  
    base order 2:
  
      precedence:
      
        minus > quot > |0| = zero > plus# = s = true = p = ack > false
      
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(minus) = []
      pi(|0|) = []
      pi(zero) = []
      pi(false) = []
      pi(true) = []
      pi(p) = []
      pi(quot) = []
      pi(ack) = []

The following rules are removed:

  p1

Then no dependency pair remains.