YES
We show the termination of the TRS R:

  g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
  n(|0|(),y) -> |0|()
  n(x,|0|()) -> |0|()
  n(s(x),s(y)) -> s(n(x,y))
  m(|0|(),y) -> y
  m(x,|0|()) -> x
  m(s(x),s(y)) -> s(m(x,y))
  k(|0|(),s(y)) -> |0|()
  k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
  t(x) -> p(x,x)
  p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
  p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
  p(|0|(),y) -> y
  p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
  minus(x,|0|()) -> x
  minus(s(x),s(y)) -> minus(x,y)
  id(x) -> x
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  not(x) -> if(x,false(),true())
  and(x,false()) -> false()
  and(true(),true()) -> true()
  f(|0|()) -> true()
  f(s(x)) -> h(x)
  h(|0|()) -> false()
  h(s(x)) -> f(x)
  gt(s(x),|0|()) -> true()
  gt(|0|(),y) -> false()
  gt(s(x),s(y)) -> gt(x,y)


-- SCC decomposition.

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

p1: g#(s(x),s(y)) -> if#(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
p2: g#(s(x),s(y)) -> and#(f(s(x)),f(s(y)))
p3: g#(s(x),s(y)) -> f#(s(x))
p4: g#(s(x),s(y)) -> f#(s(y))
p5: g#(s(x),s(y)) -> t#(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|())))))
p6: g#(s(x),s(y)) -> g#(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))
p7: g#(s(x),s(y)) -> k#(minus(m(x,y),n(x,y)),s(s(|0|())))
p8: g#(s(x),s(y)) -> minus#(m(x,y),n(x,y))
p9: g#(s(x),s(y)) -> m#(x,y)
p10: g#(s(x),s(y)) -> n#(x,y)
p11: g#(s(x),s(y)) -> k#(n(s(x),s(y)),s(s(|0|())))
p12: g#(s(x),s(y)) -> n#(s(x),s(y))
p13: g#(s(x),s(y)) -> g#(minus(m(x,y),n(x,y)),n(s(x),s(y)))
p14: n#(s(x),s(y)) -> n#(x,y)
p15: m#(s(x),s(y)) -> m#(x,y)
p16: k#(s(x),s(y)) -> k#(minus(x,y),s(y))
p17: k#(s(x),s(y)) -> minus#(x,y)
p18: t#(x) -> p#(x,x)
p19: p#(s(x),s(y)) -> p#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p20: p#(s(x),s(y)) -> if#(gt(x,y),x,y)
p21: p#(s(x),s(y)) -> gt#(x,y)
p22: p#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
p23: p#(s(x),s(y)) -> not#(gt(x,y))
p24: p#(s(x),s(y)) -> id#(x)
p25: p#(s(x),s(y)) -> id#(y)
p26: p#(s(x),x) -> p#(if(gt(x,x),id(x),id(x)),s(x))
p27: p#(s(x),x) -> if#(gt(x,x),id(x),id(x))
p28: p#(s(x),x) -> gt#(x,x)
p29: p#(s(x),x) -> id#(x)
p30: p#(id(x),s(y)) -> p#(x,if(gt(s(y),y),y,s(y)))
p31: p#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
p32: p#(id(x),s(y)) -> gt#(s(y),y)
p33: minus#(s(x),s(y)) -> minus#(x,y)
p34: not#(x) -> if#(x,false(),true())
p35: f#(s(x)) -> h#(x)
p36: h#(s(x)) -> f#(x)
p37: gt#(s(x),s(y)) -> gt#(x,y)

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The estimated dependency graph contains the following SCCs:

  {p6, p13}
  {p35, p36}
  {p16}
  {p33}
  {p15}
  {p14}
  {p19, p26, p30}
  {p37}


-- Reduction pair.

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

p1: g#(s(x),s(y)) -> g#(minus(m(x,y),n(x,y)),n(s(x),s(y)))
p2: g#(s(x),s(y)) -> g#(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3, r4, r5, r6, r7, r8, r9, r15, r16, r21, r22, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g#_A(x1,x2) = max{x1 + 68, x2 - 8}
        s_A(x1) = x1 + 139
        minus_A(x1,x2) = x1
        m_A(x1,x2) = max{x1 + 71, x2 + 1}
        n_A(x1,x2) = x1 + 5
        k_A(x1,x2) = x1 + 61
        |0|_A = 2
        and_A(x1,x2) = max{2, x1 - 889, x2 - 889}
        false_A = 1
        true_A = 1
        gt_A(x1,x2) = max{1668, x1 + 277, x2 - 139}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            g#_A(x1,x2) = 0
            s_A(x1) = x1 + 1
            minus_A(x1,x2) = x1
            m_A(x1,x2) = x1 + x2 + 1
            n_A(x1,x2) = x1 + x2 + 1
            k_A(x1,x2) = x1 + 2
            |0|_A() = 1
            and_A(x1,x2) = x2
            false_A() = 2
            true_A() = 4
            gt_A(x1,x2) = x2 + 2
            g#!_A(x1,x2) = x2 + 1
            s!_A(x1) = 5
            minus!_A(x1,x2) = 5
            m!_A(x1,x2) = x1 + x2 + 4
            n!_A(x1,x2) = x1 + x2 + 4
            k!_A(x1,x2) = 7
            |0|!_A() = 1
            and!_A(x1,x2) = 6
            false!_A() = 0
            true!_A() = 0
            gt!_A(x1,x2) = 4
  
    partial status:
  
      pi(g#) = []
      pi(s) = [1]
      pi(minus) = [1]
      pi(m) = []
      pi(n) = [1]
      pi(k) = []
      pi(|0|) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(gt) = []

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: f#(s(x)) -> h#(x)
p2: h#(s(x)) -> f#(x)

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = max{20, x1 - 20}
        s_A(x1) = max{62, x1 + 40}
        h#_A(x1) = max{41, x1 - 41}
        n_A(x1,x2) = x2 + 270
        |0|_A = 63
        p_A(x1,x2) = max{97, x1 + 29, x2}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            f#_A(x1) = x1
            s_A(x1) = x1 + 6
            h#_A(x1) = x1
            n_A(x1,x2) = x1 + x2 + 1
            |0|_A() = 8
            p_A(x1,x2) = x2 + 2
            f#!_A(x1) = 14
            s!_A(x1) = x1 + 12
            h#!_A(x1) = 13
            n!_A(x1,x2) = 11
            |0|!_A() = 10
            p!_A(x1,x2) = x1 + x2 + 1
  
    partial status:
  
      pi(f#) = []
      pi(s) = []
      pi(h#) = []
      pi(n) = [2]
      pi(|0|) = []
      pi(p) = [2]

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: k#(s(x),s(y)) -> k#(minus(x,y),s(y))

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r3, r15, r16, r23

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        k#_A(x1,x2) = max{0, x2 - 1}
        s_A(x1) = x1 + 51
        minus_A(x1,x2) = max{49, x1, x2 + 48}
        n_A(x1,x2) = max{x1 + 45, x2 + 145}
        |0|_A = 52
        f_A(x1) = max{14, x1 - 53}
        true_A = 13
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            k#_A(x1,x2) = 0
            s_A(x1) = x1 + 7
            minus_A(x1,x2) = x1 + 6
            n_A(x1,x2) = x2
            |0|_A() = 1
            f_A(x1) = 3
            true_A() = 3
            k#!_A(x1,x2) = x1
            s!_A(x1) = 8
            minus!_A(x1,x2) = 11
            n!_A(x1,x2) = 7
            |0|!_A() = 9
            f!_A(x1) = 7
            true!_A() = 0
  
    partial status:
  
      pi(k#) = []
      pi(s) = []
      pi(minus) = [1, 2]
      pi(n) = [2]
      pi(|0|) = []
      pi(f) = []
      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)

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = 1
        s_A(x1) = x1 + 24
        n_A(x1,x2) = max{171, x2 - 5}
        |0|_A = 44
  
    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 + 4
            n_A(x1,x2) = x1 + x2 + 3
            |0|_A() = 2
            minus#!_A(x1,x2) = x1 + x2 + 21
            s!_A(x1) = x1 + 20
            n!_A(x1,x2) = 22
            |0|!_A() = 21
  
    partial status:
  
      pi(minus#) = []
      pi(s) = []
      pi(n) = []
      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: m#(s(x),s(y)) -> m#(x,y)

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        m#_A(x1,x2) = 1
        s_A(x1) = 0
        n_A(x1,x2) = max{50, x1 - 27}
        |0|_A = 49
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            m#_A(x1,x2) = x1 + x2
            s_A(x1) = x1 + 1
            n_A(x1,x2) = x1 + x2 + 1
            |0|_A() = 3
            m#!_A(x1,x2) = x1 + x2 + 9
            s!_A(x1) = 8
            n!_A(x1,x2) = 10
            |0|!_A() = 9
  
    partial status:
  
      pi(m#) = []
      pi(s) = []
      pi(n) = []
      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: n#(s(x),s(y)) -> n#(x,y)

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        n#_A(x1,x2) = x1 + 17
        s_A(x1) = max{40, x1 + 16}
        n_A(x1,x2) = x2 + 294
        |0|_A = 43
        p_A(x1,x2) = max{140, x1 - 62, x2 + 102}
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        matrix interpretation with standard point-wise order:
        
          carrier: N^1
          order: standard order
          interpretations:
            n#_A(x1,x2) = x1 + x2
            s_A(x1) = x1 + 2
            n_A(x1,x2) = x1 + x2 + 3
            |0|_A() = 6
            p_A(x1,x2) = x2 + 3
            n#!_A(x1,x2) = x1 + x2 + 6
            s!_A(x1) = x1 + 5
            n!_A(x1,x2) = 7
            |0|!_A() = 7
            p!_A(x1,x2) = x1 + x2 + 8
  
    partial status:
  
      pi(n#) = [1]
      pi(s) = []
      pi(n) = [2]
      pi(|0|) = []
      pi(p) = [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: p#(s(x),s(y)) -> p#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
p2: p#(id(x),s(y)) -> p#(x,if(gt(s(y),y),y,s(y)))
p3: p#(s(x),x) -> p#(if(gt(x,x),id(x),id(x)),s(x))

and R consists of:

r1: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3, r17, r18, r19, r20, r27, r28, r29

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        p#_A(x1,x2) = max{32, x1 + 3, x2 - 26}
        s_A(x1) = max{41, x1 + 30}
        if_A(x1,x2,x3) = max{25, x1 - 9, x2, x3}
        gt_A(x1,x2) = max{30, x1 - 74}
        not_A(x1) = max{40, x1 - 1}
        id_A(x1) = max{28, x1 + 21}
        n_A(x1,x2) = x2 + 104
        |0|_A = 73
        true_A = 7
        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:
            p#_A(x1,x2) = 0
            s_A(x1) = x1 + 4
            if_A(x1,x2,x3) = x2 + x3 + 1
            gt_A(x1,x2) = x1 + 2
            not_A(x1) = 10
            id_A(x1) = x1 + 1
            n_A(x1,x2) = x1 + x2 + 1
            |0|_A() = 2
            true_A() = 6
            false_A() = 3
            p#!_A(x1,x2) = x1 + 3
            s!_A(x1) = 9
            if!_A(x1,x2,x3) = 17
            gt!_A(x1,x2) = x2 + 6
            not!_A(x1) = 18
            id!_A(x1) = 7
            n!_A(x1,x2) = x1 + x2 + 17
            |0|!_A() = 8
            true!_A() = 0
            false!_A() = 0
  
    partial status:
  
      pi(p#) = [1]
      pi(s) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(not) = []
      pi(id) = []
      pi(n) = [2]
      pi(|0|) = []
      pi(true) = []
      pi(false) = []

The following rules are removed:

  p1, p2, p3

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: g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(|0|()))),k(n(s(x),s(y)),s(s(|0|()))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))
r2: n(|0|(),y) -> |0|()
r3: n(x,|0|()) -> |0|()
r4: n(s(x),s(y)) -> s(n(x,y))
r5: m(|0|(),y) -> y
r6: m(x,|0|()) -> x
r7: m(s(x),s(y)) -> s(m(x,y))
r8: k(|0|(),s(y)) -> |0|()
r9: k(s(x),s(y)) -> s(k(minus(x,y),s(y)))
r10: t(x) -> p(x,x)
r11: p(s(x),s(y)) -> s(s(p(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
r12: p(s(x),x) -> p(if(gt(x,x),id(x),id(x)),s(x))
r13: p(|0|(),y) -> y
r14: p(id(x),s(y)) -> s(p(x,if(gt(s(y),y),y,s(y))))
r15: minus(x,|0|()) -> x
r16: minus(s(x),s(y)) -> minus(x,y)
r17: id(x) -> x
r18: if(true(),x,y) -> x
r19: if(false(),x,y) -> y
r20: not(x) -> if(x,false(),true())
r21: and(x,false()) -> false()
r22: and(true(),true()) -> true()
r23: f(|0|()) -> true()
r24: f(s(x)) -> h(x)
r25: h(|0|()) -> false()
r26: h(s(x)) -> f(x)
r27: gt(s(x),|0|()) -> true()
r28: gt(|0|(),y) -> false()
r29: gt(s(x),s(y)) -> gt(x,y)

The set of usable rules consists of

  r2, r3, r13

Take the reduction pair:

  generalized weighted path order (>=MG, >G)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        gt#_A(x1,x2) = max{x1 + 7, x2 - 35}
        s_A(x1) = max{34, x1 + 6}
        n_A(x1,x2) = max{x1 + 148, x2 + 154}
        |0|_A = 23
        p_A(x1,x2) = max{36, x2 + 6}
  
    base order 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 + 4
            n_A(x1,x2) = x1 + x2 + 3
            |0|_A() = 11
            p_A(x1,x2) = x2 + 3
            gt#!_A(x1,x2) = x1 + x2 + 41
            s!_A(x1) = 40
            n!_A(x1,x2) = 14
            |0|!_A() = 14
            p!_A(x1,x2) = x1 + 35
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = []
      pi(n) = [1, 2]
      pi(|0|) = []
      pi(p) = [2]

The following rules are removed:

  p1

Then no dependency pair remains.