YES
We show the termination of the TRS R:

  le(|0|(),y) -> true()
  le(s(x),|0|()) -> false()
  le(s(x),s(y)) -> le(x,y)
  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  mod(|0|(),y) -> |0|()
  mod(s(x),|0|()) -> |0|()
  mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
  if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
  if_mod(false(),s(x),s(y)) -> s(x)


-- SCC decomposition.

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

p1: le#(s(x),s(y)) -> le#(x,y)
p2: minus#(s(x),s(y)) -> minus#(x,y)
p3: mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
p4: mod#(s(x),s(y)) -> le#(y,x)
p5: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
p6: if_mod#(true(),s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: mod(|0|(),y) -> |0|()
r7: mod(s(x),|0|()) -> |0|()
r8: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r9: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r10: if_mod(false(),s(x),s(y)) -> s(x)

The estimated dependency graph contains the following SCCs:

  {p3, p5}
  {p1}
  {p2}


-- Reduction pair.

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

p1: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
p2: mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: mod(|0|(),y) -> |0|()
r7: mod(s(x),|0|()) -> |0|()
r8: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r9: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r10: if_mod(false(),s(x),s(y)) -> s(x)

The set of usable rules consists of

  r1, r2, r3, r4, r5

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        if_mod#_A(x1,x2,x3) = max{1, x1 - 4, x2 - 1}
        true_A = 5
        s_A(x1) = x1 + 8
        mod#_A(x1,x2) = max{7, x1 + 2}
        minus_A(x1,x2) = x1 + 5
        le_A(x1,x2) = x2 + 7
        |0|_A = 0
        false_A = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            if_mod#_A(x1,x2,x3) = x3
            true_A() = 1
            s_A(x1) = x1 + 1
            mod#_A(x1,x2) = x2
            minus_A(x1,x2) = x1 + 9
            le_A(x1,x2) = 1
            |0|_A() = 2
            false_A() = 1
            if_mod#!_A(x1,x2,x3) = 7
            true!_A() = 9
            s!_A(x1) = 8
            mod#!_A(x1,x2) = 6
            minus!_A(x1,x2) = x2 + 6
            le!_A(x1,x2) = x1 + x2 + 3
            |0|!_A() = 6
            false!_A() = 7
  
    partial status:
  
      pi(if_mod#) = []
      pi(true) = []
      pi(s) = [1]
      pi(mod#) = [1]
      pi(minus) = []
      pi(le) = [2]
      pi(|0|) = []
      pi(false) = []

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: le#(s(x),s(y)) -> le#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: mod(|0|(),y) -> |0|()
r7: mod(s(x),|0|()) -> |0|()
r8: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r9: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r10: if_mod(false(),s(x),s(y)) -> s(x)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        le#_A(x1,x2) = max{9, x1 - 9, x2 - 9}
        s_A(x1) = max{8, x1 + 3}
        le_A(x1,x2) = max{12, x1 - 9, x2 - 9}
        |0|_A = 7
        true_A = 1
        false_A = 4
        minus_A(x1,x2) = max{15, x1 + 1}
        mod_A(x1,x2) = max{13, x1 + 3, x2 - 10}
        if_mod_A(x1,x2,x3) = max{14, x1 - 15, x2 + 2, x3 - 9}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            le#_A(x1,x2) = x1 + x2
            s_A(x1) = x1 + 2
            le_A(x1,x2) = x1 + x2 + 1
            |0|_A() = 1
            true_A() = 1
            false_A() = 1
            minus_A(x1,x2) = x1 + x2 + 1
            mod_A(x1,x2) = x1
            if_mod_A(x1,x2,x3) = x2
            le#!_A(x1,x2) = x1 + x2 + 6
            s!_A(x1) = x1 + 5
            le!_A(x1,x2) = x1 + 5
            |0|!_A() = 1
            true!_A() = 0
            false!_A() = 0
            minus!_A(x1,x2) = x1 + 4
            mod!_A(x1,x2) = x1 + x2 + 1
            if_mod!_A(x1,x2,x3) = x2 + x3 + 1
  
    partial status:
  
      pi(le#) = []
      pi(s) = [1]
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(minus) = []
      pi(mod) = [1]
      pi(if_mod) = [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: minus#(s(x),s(y)) -> minus#(x,y)

and R consists of:

r1: le(|0|(),y) -> true()
r2: le(s(x),|0|()) -> false()
r3: le(s(x),s(y)) -> le(x,y)
r4: minus(x,|0|()) -> x
r5: minus(s(x),s(y)) -> minus(x,y)
r6: mod(|0|(),y) -> |0|()
r7: mod(s(x),|0|()) -> |0|()
r8: mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
r9: if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
r10: if_mod(false(),s(x),s(y)) -> s(x)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r10

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = max{0, x1 - 9}
        s_A(x1) = max{8, x1 + 3}
        le_A(x1,x2) = max{12, x1 - 9, x2 - 9}
        |0|_A = 7
        true_A = 1
        false_A = 4
        minus_A(x1,x2) = max{15, x1 + 1}
        mod_A(x1,x2) = max{13, x1 + 3, x2 - 10}
        if_mod_A(x1,x2,x3) = max{14, x1 - 15, x2 + 2, x3 - 9}
  
    base order 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 + x2
            s_A(x1) = x1 + 2
            le_A(x1,x2) = x1 + x2 + 1
            |0|_A() = 1
            true_A() = 1
            false_A() = 1
            minus_A(x1,x2) = x1 + x2 + 1
            mod_A(x1,x2) = x1
            if_mod_A(x1,x2,x3) = x2
            minus#!_A(x1,x2) = x1 + x2 + 6
            s!_A(x1) = x1 + 5
            le!_A(x1,x2) = x1 + 5
            |0|!_A() = 1
            true!_A() = 0
            false!_A() = 0
            minus!_A(x1,x2) = x1 + 4
            mod!_A(x1,x2) = x1 + x2 + 1
            if_mod!_A(x1,x2,x3) = x2 + x3 + 1
  
    partial status:
  
      pi(minus#) = []
      pi(s) = [1]
      pi(le) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(minus) = []
      pi(mod) = [1]
      pi(if_mod) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.