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, r11

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          quot#_A(x1,x2) = x1 + 1
          s_A(x1) = x1 + 6
          minus_A(x1,x2) = x1 + 4
          |0|_A() = 1
          plus_A(x1,x2) = 7
          if_A(x1,x2,x3) = x2 + 8
          true_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          quot#_A(x1,x2) = max{0, x2 - 5}
          s_A(x1) = max{4, x1}
          minus_A(x1,x2) = max{16, x1 + 15}
          |0|_A = 16
          plus_A(x1,x2) = 1
          if_A(x1,x2,x3) = max{x1 - 17, x2 + 5}
          true_A = 27
          quot#!_A(x1,x2) = x2 + 3
          s!_A(x1) = 0
          minus!_A(x1,x2) = 0
          |0|!_A = 1
          plus!_A(x1,x2) = max{1, x1 - 30}
          if!_A(x1,x2,x3) = max{3, x1 - 58, x2 - 28}
          true!_A = 0
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = [1]
      pi(minus) = []
      pi(|0|) = []
      pi(plus) = []
      pi(if) = [2]
      pi(true) = []

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, r8, r11, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus#_A(x1,x2) = x1 + 1
          s_A(x1) = x1
          minus_A(x1,x2) = x1 + 2
          plus_A(x1,x2) = x1 + x2
          |0|_A() = 0
          zero_A() = 2
          if_A(x1,x2,x3) = x2
          true_A() = 1
          gt_A(x1,x2) = x2 + 2
          false_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = x1
          s_A(x1) = max{5, x1}
          minus_A(x1,x2) = x1 + 10
          plus_A(x1,x2) = max{x1, x2}
          |0|_A = 5
          zero_A = 0
          if_A(x1,x2,x3) = max{x1 - 1, x2, x3}
          true_A = 1
          gt_A(x1,x2) = max{7, x1 - 13, x2 - 13}
          false_A = 1
          minus#!_A(x1,x2) = max{14, x1 - 24}
          s!_A(x1) = x1 + 5
          minus!_A(x1,x2) = max{15, x1 + 6, x2 + 13}
          plus!_A(x1,x2) = max{4, x1 - 23}
          |0|!_A = 0
          zero!_A = 7
          if!_A(x1,x2,x3) = max{6, x1 - 8, x2 - 3, x3 + 1}
          true!_A = 6
          gt!_A(x1,x2) = max{6, x1 - 29, x2 - 6}
          false!_A = 2
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(minus) = [1]
      pi(plus) = []
      pi(|0|) = []
      pi(zero) = []
      pi(if) = [2]
      pi(true) = []
      pi(gt) = []
      pi(false) = []

The following rules are removed:

  p2



-- 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(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}


-- 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(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

  r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          minus#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 5
          quot_A(x1,x2) = 7
          |0|_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = 0
          s_A(x1) = max{48, x1 - 13}
          quot_A(x1,x2) = max{15, x1, x2 - 50}
          |0|_A = 16
          minus#!_A(x1,x2) = max{65, x2 - 49}
          s!_A(x1) = 64
          quot!_A(x1,x2) = max{x1 + 15, x2 - 15}
          |0|!_A = 32
  
    partial status:
  
      pi(minus#) = [2]
      pi(s) = []
      pi(quot) = []
      pi(|0|) = []

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),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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = 0
          s_A(x1) = 0
          if_A(x1,x2,x3) = x2 + x3 + 10
          gt_A(x1,x2) = 3
          not_A(x1) = x1 + 15
          id_A(x1) = x1 + 5
          true_A() = 2
          false_A() = 2
          zero_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 37, x2 - 37}
          s_A(x1) = max{27, x1 + 22}
          if_A(x1,x2,x3) = max{12, x2, x3}
          gt_A(x1,x2) = max{63, x1 - 55}
          not_A(x1) = 41
          id_A(x1) = x1 + 12
          true_A = 23
          false_A = 41
          zero_A = 54
          plus#!_A(x1,x2) = max{4, x1 - 12, x2 - 23}
          s!_A(x1) = max{24, x1 + 22}
          if!_A(x1,x2,x3) = max{44, x1 - 57, x2 - 14}
          gt!_A(x1,x2) = max{5, x1 - 15, x2 - 28}
          not!_A(x1) = max{26, x1 - 63}
          id!_A(x1) = max{43, x1 - 12}
          true!_A = 25
          false!_A = 4
          zero!_A = 26
  
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = [1]
      pi(id) = [1]
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p1, p3



-- 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)))

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}


-- 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)))

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

  r11, r12, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          plus#_A(x1,x2) = x1 + 3
          id_A(x1) = x1 + 10
          s_A(x1) = 2
          if_A(x1,x2,x3) = x2 + x3 + 3
          gt_A(x1,x2) = 11
          true_A() = 3
          false_A() = 1
          zero_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          plus#_A(x1,x2) = max{0, x1 - 26, x2 - 17}
          id_A(x1) = x1 + 3
          s_A(x1) = x1 + 3
          if_A(x1,x2,x3) = max{x1 - 1, x2 + 3, x3}
          gt_A(x1,x2) = max{17, x1 - 7, x2 - 4}
          true_A = 5
          false_A = 2
          zero_A = 0
          plus#!_A(x1,x2) = max{8, x1 + 6}
          id!_A(x1) = 0
          s!_A(x1) = 0
          if!_A(x1,x2,x3) = max{10, x1 - 5, x2 - 33, x3 - 38}
          gt!_A(x1,x2) = max{11, x1 - 33, x2 - 1}
          true!_A = 1
          false!_A = 1
          zero!_A = 0
  
    partial status:
  
      pi(plus#) = []
      pi(id) = [1]
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

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: 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          gt#_A(x1,x2) = x2
          s_A(x1) = x1 + 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = 0
          s_A(x1) = 32
          gt#!_A(x1,x2) = 0
          s!_A(x1) = max{96, x1 + 38}
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.