YES
We show the termination of the TRS R:

  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  quot(|0|(),s(y)) -> |0|()
  quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
  minus(minus(x,y),z) -> minus(x,plus(y,z))
  plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
  plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
  plus(zero(),y) -> y
  plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
  id(x) -> x
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  not(x) -> if(x,false(),true())
  gt(s(x),zero()) -> true()
  gt(zero(),y) -> false()
  gt(s(x),s(y)) -> gt(x,y)


-- SCC decomposition.

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

p1: minus#(s(x),s(y)) -> minus#(x,y)
p2: quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
p3: quot#(s(x),s(y)) -> minus#(x,y)
p4: minus#(minus(x,y),z) -> minus#(x,plus(y,z))
p5: minus#(minus(x,y),z) -> plus#(y,z)
p6: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p7: plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
p8: plus#(s(x),s(y)) -> gt#(x,y)
p9: plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p10: plus#(s(x),s(y)) -> not#(gt(x,y))
p11: plus#(s(x),s(y)) -> id#(x)
p12: plus#(s(x),s(y)) -> id#(y)
p13: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
p14: plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p15: plus#(s(x),x) -> gt#(x,x)
p16: plus#(s(x),x) -> id#(x)
p17: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p18: plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p19: plus#(id(x),s(y)) -> gt#(s(y),y)
p20: not#(x) -> if#(x,false(),true())
p21: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p2}
  {p1, p4}
  {p6, p13, p17}
  {p21}


-- Reduction pair.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1, r2, r5, r12

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            quot#_A(x1,x2) = max{0, x2 - 8}
            s_A(x1) = x1 + 9
            minus_A(x1,x2) = max{5, x1 + 3}
            |0|_A = 13
            plus_A(x1,x2) = max{18, x1 - 105}
            if_A(x1,x2,x3) = max{x1 - 1, x2 + 38, x3 + 29}
            false_A = 99
            quot#!_A(x1,x2) = max{54, x2 + 45}
            s!_A(x1) = x1 + 54
            minus!_A(x1,x2) = 53
            |0|!_A = 5
            plus!_A(x1,x2) = max{1, x1 - 21}
            if!_A(x1,x2,x3) = max{3, x1 - 25}
            false!_A = 53
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              quot#_A(x1,x2) = 0
              s_A(x1) = x1 + 2
              minus_A(x1,x2) = x1 + 1
              |0|_A() = 2
              plus_A(x1,x2) = 1
              if_A(x1,x2,x3) = x3 + 1
              false_A() = 2
              quot#!_A(x1,x2) = x2 + 1
              s!_A(x1) = 7
              minus!_A(x1,x2) = 2
              |0|!_A() = 0
              plus!_A(x1,x2) = 1
              if!_A(x1,x2,x3) = 0
              false!_A() = 3
        
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = [1]
      pi(if) = [2, 3]
      pi(false) = []

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r1

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            minus#_A(x1,x2) = max{0, x1 - 17}
            s_A(x1) = max{19, x1 + 16}
            minus_A(x1,x2) = max{x1 + 15, x2 - 23}
            plus_A(x1,x2) = max{22, x1 + 16, x2 + 16}
            |0|_A = 2
            minus#!_A(x1,x2) = max{0, x1 - 18}
            s!_A(x1) = max{0, x1 - 27}
            minus!_A(x1,x2) = max{0, x1 - 17, x2 - 20}
            plus!_A(x1,x2) = max{0, x1 - 16}
            |0|!_A = 2
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              minus#_A(x1,x2) = x1
              s_A(x1) = x1 + 5
              minus_A(x1,x2) = x1 + x2 + 1
              plus_A(x1,x2) = 2
              |0|_A() = 2
              minus#!_A(x1,x2) = x1 + 5
              s!_A(x1) = 4
              minus!_A(x1,x2) = x1 + 6
              plus!_A(x1,x2) = x1 + 2
              |0|!_A() = 6
        
  
    partial status:
  
      pi(minus#) = []
      pi(s) = [1]
      pi(minus) = [1, 2]
      pi(plus) = [2]
      pi(|0|) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

p1: plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p3: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r10, r11, r12, r13, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = 0
            s_A(x1) = x1 + 15
            if_A(x1,x2,x3) = max{8, x2 + 4, x3}
            gt_A(x1,x2) = max{15, x1 - 31, x2 - 16}
            not_A(x1) = max{16, x1 - 46}
            id_A(x1) = x1 + 11
            true_A = 1
            false_A = 12
            zero_A = 0
            plus#!_A(x1,x2) = max{26, x1 + 14, x2 + 5}
            s!_A(x1) = max{27, x1 + 21}
            if!_A(x1,x2,x3) = 29
            gt!_A(x1,x2) = max{34, x1 - 16, x2 + 30}
            not!_A(x1) = 32
            id!_A(x1) = x1 + 29
            true!_A = 26
            false!_A = 31
            zero!_A = 0
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              plus#_A(x1,x2) = 0
              s_A(x1) = 6
              if_A(x1,x2,x3) = x2 + x3 + 3
              gt_A(x1,x2) = 1
              not_A(x1) = 5
              id_A(x1) = x1 + 1
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 7
              s!_A(x1) = 9
              if!_A(x1,x2,x3) = x3
              gt!_A(x1,x2) = 14
              not!_A(x1) = 8
              id!_A(x1) = 6
              true!_A() = 1
              false!_A() = 1
              zero!_A() = 0
        
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(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: plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
p2: plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r10, r11, r12, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            plus#_A(x1,x2) = 0
            id_A(x1) = max{3, x1 + 1}
            s_A(x1) = 10
            if_A(x1,x2,x3) = max{27, x2 + 10, x3}
            gt_A(x1,x2) = 8
            true_A = 6
            false_A = 2
            zero_A = 24
            plus#!_A(x1,x2) = 7
            id!_A(x1) = 7
            s!_A(x1) = x1 + 24
            if!_A(x1,x2,x3) = max{23, x3 + 21}
            gt!_A(x1,x2) = max{12, x2 + 2}
            true!_A = 25
            false!_A = 1
            zero!_A = 26
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              plus#_A(x1,x2) = 0
              id_A(x1) = x1 + 6
              s_A(x1) = 4
              if_A(x1,x2,x3) = x2 + x3 + 5
              gt_A(x1,x2) = 1
              true_A() = 1
              false_A() = 1
              zero_A() = 1
              plus#!_A(x1,x2) = 0
              id!_A(x1) = 0
              s!_A(x1) = 19
              if!_A(x1,x2,x3) = 6
              gt!_A(x1,x2) = 8
              true!_A() = 0
              false!_A() = 0
              zero!_A() = 1
        
  
    partial status:
  
      pi(plus#) = [1]
      pi(id) = [1]
      pi(s) = [1]
      pi(if) = [2, 3]
      pi(gt) = [2]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1, p2

Then no dependency pair remains.

-- Reduction pair.

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

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

and R consists of:

r1: minus(x,|0|()) -> x
r2: minus(s(x),s(y)) -> minus(x,y)
r3: quot(|0|(),s(y)) -> |0|()
r4: quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
r5: minus(minus(x,y),z) -> minus(x,plus(y,z))
r6: plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r7: plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
r8: plus(zero(),y) -> y
r9: plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
r10: id(x) -> x
r11: if(true(),x,y) -> x
r12: if(false(),x,y) -> y
r13: not(x) -> if(x,false(),true())
r14: gt(s(x),zero()) -> true()
r15: gt(zero(),y) -> false()
r16: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      trivial reduction triple (T x T, T x T, Φ)
  
    base order 2:
  
      lexicographic combination of reduction triples:
      
        1. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          max/plus interpretation on natural numbers:
          
            gt#_A(x1,x2) = max{0, x1 - 17, x2 - 17}
            s_A(x1) = x1 + 16
            gt#!_A(x1,x2) = max{0, x1 - 17}
            s!_A(x1) = max{0, x1 - 16}
        
        2. marked-algebra reduction triple (>=A, >=A!, >A!):
        
          matrix interpretation with standard point-wise order:
          
            carrier: N^1
            order: standard order
            interpretations:
              gt#_A(x1,x2) = x1 + x2
              s_A(x1) = x1 + 2
              gt#!_A(x1,x2) = 1
              s!_A(x1) = 2
        
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.