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, r8, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        quot#_A(x1,x2) = max{x1 + 4, x2 - 1}
        s_A(x1) = x1 + 8
        minus_A(x1,x2) = x1
        |0|_A = 0
        plus_A(x1,x2) = max{28, x1 - 2, x2}
        zero_A = 9
        gt_A(x1,x2) = max{7, x1 - 18, x2 - 10}
        true_A = 6
        false_A = 4
  
    base order 2:
  
      precedence:
      
        plus > minus > quot# = s = zero = gt > |0| = true = false
      
  
    partial status:
  
      pi(quot#) = [1]
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(plus) = [2]
      pi(zero) = []
      pi(gt) = []
      pi(true) = []
      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, r3, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = max{92, x1 + 1}
        s_A(x1) = max{97, x1 + 18}
        minus_A(x1,x2) = max{x1 + 9, x2 + 93}
        plus_A(x1,x2) = max{1, x1 - 122, x2 - 94}
        |0|_A = 9
        quot_A(x1,x2) = max{98, x1 - 10, x2 - 116}
        gt_A(x1,x2) = max{72, x1 - 100, x2 - 103}
        zero_A = 72
        true_A = 71
        false_A = 71
  
    base order 2:
  
      precedence:
      
        minus > quot > s = plus = gt = zero > minus# = |0| = true = false
      
  
    partial status:
  
      pi(minus#) = [1]
      pi(s) = [1]
      pi(minus) = [2]
      pi(plus) = []
      pi(|0|) = []
      pi(quot) = []
      pi(gt) = []
      pi(zero) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: 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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: 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, r3, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = x1 + 98
        minus_A(x1,x2) = max{x1 + 99, x2 - 95}
        plus_A(x1,x2) = max{5, x1 - 227, x2 - 96}
        |0|_A = 45
        quot_A(x1,x2) = max{51, x1 - 144, x2 - 284}
        s_A(x1) = max{50, x1 + 44}
        gt_A(x1,x2) = max{4, x1 - 97, x2 - 184}
        zero_A = 96
        true_A = 3
        false_A = 3
  
    base order 2:
  
      precedence:
      
        minus = plus > quot > s = gt = zero > minus# = |0| = true = false
      
  
    partial status:
  
      pi(minus#) = [1]
      pi(minus) = [1]
      pi(plus) = []
      pi(|0|) = []
      pi(quot) = []
      pi(s) = [1]
      pi(gt) = []
      pi(zero) = []
      pi(true) = []
      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: 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

  r3, r10, r11, r12, r13, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{x1 + 82, x2 + 65}
        s_A(x1) = max{86, x1 + 64}
        if_A(x1,x2,x3) = max{x1 - 308, x2 + 60, x3}
        gt_A(x1,x2) = max{168, x1 - 87, x2 - 87}
        not_A(x1) = max{242, x1 - 307}
        id_A(x1) = max{12, x1 + 3}
        quot_A(x1,x2) = max{66, x2 - 268}
        |0|_A = 65
        true_A = 65
        false_A = 0
        zero_A = 66
  
    base order 2:
  
      precedence:
      
        not > if > plus# > gt = zero > s = id = quot = false > |0| = true
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(quot) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p3



-- SCC decomposition.

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)))

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#(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)))

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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{x1 + 16, x2 - 19}
        s_A(x1) = max{186, x1 + 54}
        if_A(x1,x2,x3) = max{18, x1 - 235, x2 + 4, x3}
        gt_A(x1,x2) = max{406, x1 - 229, x2 - 230}
        not_A(x1) = max{201, x1 - 234}
        id_A(x1) = x1 + 3
        quot_A(x1,x2) = max{187, x1 - 94, x2 - 108}
        |0|_A = 79
        true_A = 55
        false_A = 0
        zero_A = 56
  
    base order 2:
  
      precedence:
      
        not > if > quot > gt = zero > plus# = s = id = false > |0| = true
      
  
    partial status:
  
      pi(plus#) = [1]
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(quot) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(zero) = []

The following rules are removed:

  p2



-- SCC decomposition.

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)))

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#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(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, r10, r11, r12, r13, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        plus#_A(x1,x2) = max{4, x1 - 37, x2 - 41}
        s_A(x1) = max{103, x1 + 64}
        if_A(x1,x2,x3) = max{x1 - 93, x2 + 25, x3 + 21}
        gt_A(x1,x2) = max{5, x1 - 96, x2 - 133}
        not_A(x1) = max{28, x1 - 6}
        id_A(x1) = max{19, x1 + 1}
        quot_A(x1,x2) = max{63, x1 - 105, x2 - 168}
        |0|_A = 62
        true_A = 6
        false_A = 0
        zero_A = 7
  
    base order 2:
  
      precedence:
      
        plus# > not > if > s = quot > gt = zero > id = |0| = true = false
      
  
    partial status:
  
      pi(plus#) = []
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(not) = []
      pi(id) = []
      pi(quot) = []
      pi(|0|) = []
      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

  r3, r14, r15

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gt#_A(x1,x2) = max{x1 - 1, x2 + 11}
        s_A(x1) = max{39, x1 + 10}
        quot_A(x1,x2) = max{8, x1 - 51, x2 - 109}
        |0|_A = 7
        gt_A(x1,x2) = max{1, x1 - 50, x2 - 37}
        zero_A = 49
        true_A = 11
        false_A = 0
  
    base order 2:
  
      precedence:
      
        quot > gt# = s = gt = zero > |0| = true = false
      
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = [1]
      pi(quot) = []
      pi(|0|) = []
      pi(gt) = []
      pi(zero) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p1

Then no dependency pair remains.