YES
We show the termination of the TRS R:

  cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
  cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
  cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
  gr(|0|(),x) -> false()
  gr(s(x),|0|()) -> true()
  gr(s(x),s(y)) -> gr(x,y)
  p(|0|()) -> |0|()
  p(s(x)) -> x


-- SCC decomposition.

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

p1: cond1#(true(),x,y) -> cond2#(gr(y,|0|()),x,y)
p2: cond1#(true(),x,y) -> gr#(y,|0|())
p3: cond2#(true(),x,y) -> cond2#(gr(y,|0|()),x,p(y))
p4: cond2#(true(),x,y) -> gr#(y,|0|())
p5: cond2#(true(),x,y) -> p#(y)
p6: cond2#(false(),x,y) -> cond1#(gr(x,|0|()),p(x),y)
p7: cond2#(false(),x,y) -> gr#(x,|0|())
p8: cond2#(false(),x,y) -> p#(x)
p9: gr#(s(x),s(y)) -> gr#(x,y)

and R consists of:

r1: cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
r2: cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
r3: cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
r4: gr(|0|(),x) -> false()
r5: gr(s(x),|0|()) -> true()
r6: gr(s(x),s(y)) -> gr(x,y)
r7: p(|0|()) -> |0|()
r8: p(s(x)) -> x

The estimated dependency graph contains the following SCCs:

  {p1, p3, p6}
  {p9}


-- Reduction pair.

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

p1: cond1#(true(),x,y) -> cond2#(gr(y,|0|()),x,y)
p2: cond2#(false(),x,y) -> cond1#(gr(x,|0|()),p(x),y)
p3: cond2#(true(),x,y) -> cond2#(gr(y,|0|()),x,p(y))

and R consists of:

r1: cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
r2: cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
r3: cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
r4: gr(|0|(),x) -> false()
r5: gr(s(x),|0|()) -> true()
r6: gr(s(x),s(y)) -> gr(x,y)
r7: p(|0|()) -> |0|()
r8: p(s(x)) -> x

The set of usable rules consists of

  r4, r5, r6, r7, r8

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        cond1#_A(x1,x2,x3) = max{x1 - 18, x2 + 4}
        true_A = 28
        cond2#_A(x1,x2,x3) = max{9, x2 + 3}
        gr_A(x1,x2) = max{17, x1 - 2, x2 + 10}
        |0|_A = 1
        false_A = 16
        p_A(x1) = max{4, x1 - 2}
        s_A(x1) = x1 + 31
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            cond1#_A(x1,x2,x3) = 0
            true_A() = 3
            cond2#_A(x1,x2,x3) = 0
            gr_A(x1,x2) = x2 + 1
            |0|_A() = 2
            false_A() = 1
            p_A(x1) = x1 + 1
            s_A(x1) = x1 + 3
            cond1#!_A(x1,x2,x3) = x3 + 3
            true!_A() = 3
            cond2#!_A(x1,x2,x3) = 2
            gr!_A(x1,x2) = x1
            |0|!_A() = 1
            false!_A() = 1
            p!_A(x1) = 3
            s!_A(x1) = 4
  
    partial status:
  
      pi(cond1#) = []
      pi(true) = []
      pi(cond2#) = []
      pi(gr) = []
      pi(|0|) = []
      pi(false) = []
      pi(p) = []
      pi(s) = [1]

The following rules are removed:

  p1, p2



-- SCC decomposition.

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

p1: cond2#(true(),x,y) -> cond2#(gr(y,|0|()),x,p(y))

and R consists of:

r1: cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
r2: cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
r3: cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
r4: gr(|0|(),x) -> false()
r5: gr(s(x),|0|()) -> true()
r6: gr(s(x),s(y)) -> gr(x,y)
r7: p(|0|()) -> |0|()
r8: p(s(x)) -> x

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: cond2#(true(),x,y) -> cond2#(gr(y,|0|()),x,p(y))

and R consists of:

r1: cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
r2: cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
r3: cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
r4: gr(|0|(),x) -> false()
r5: gr(s(x),|0|()) -> true()
r6: gr(s(x),s(y)) -> gr(x,y)
r7: p(|0|()) -> |0|()
r8: p(s(x)) -> x

The set of usable rules consists of

  r4, r5, r7, r8

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        cond2#_A(x1,x2,x3) = max{15, x1 + 2, x3 - 1}
        true_A = 14
        gr_A(x1,x2) = max{11, x1 - 4}
        |0|_A = 5
        p_A(x1) = max{6, x1 - 2}
        false_A = 4
        s_A(x1) = x1 + 19
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            cond2#_A(x1,x2,x3) = 0
            true_A() = 1
            gr_A(x1,x2) = x1 + x2
            |0|_A() = 1
            p_A(x1) = x1 + 2
            false_A() = 1
            s_A(x1) = x1 + 1
            cond2#!_A(x1,x2,x3) = 8
            true!_A() = 2
            gr!_A(x1,x2) = x2 + 4
            |0|!_A() = 1
            p!_A(x1) = 7
            false!_A() = 0
            s!_A(x1) = 3
  
    partial status:
  
      pi(cond2#) = [1]
      pi(true) = []
      pi(gr) = []
      pi(|0|) = []
      pi(p) = []
      pi(false) = []
      pi(s) = []

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

and R consists of:

r1: cond1(true(),x,y) -> cond2(gr(y,|0|()),x,y)
r2: cond2(true(),x,y) -> cond2(gr(y,|0|()),x,p(y))
r3: cond2(false(),x,y) -> cond1(gr(x,|0|()),p(x),y)
r4: gr(|0|(),x) -> false()
r5: gr(s(x),|0|()) -> true()
r6: gr(s(x),s(y)) -> gr(x,y)
r7: p(|0|()) -> |0|()
r8: p(s(x)) -> x

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gr#_A(x1,x2) = 0
        s_A(x1) = max{4, x1 + 3}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            gr#_A(x1,x2) = x1 + x2
            s_A(x1) = x1 + 1
            gr#!_A(x1,x2) = x1
            s!_A(x1) = 1
  
    partial status:
  
      pi(gr#) = []
      pi(s) = [1]

The following rules are removed:

  p1

Then no dependency pair remains.