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, r23

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:
          g#_A(x1,x2) = 14
          s_A(x1) = x1 + 13
          minus_A(x1,x2) = x1
          m_A(x1,x2) = x1 + x2 + 4
          n_A(x1,x2) = x1 + x2 + 12
          k_A(x1,x2) = x1
          |0|_A() = 2
          f_A(x1) = 14
          true_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          g#_A(x1,x2) = 0
          s_A(x1) = x1 + 117
          minus_A(x1,x2) = x1
          m_A(x1,x2) = max{x1 + 90, x2 + 1}
          n_A(x1,x2) = max{x1 + 15, x2 - 22}
          k_A(x1,x2) = x1 + 3
          |0|_A = 14
          f_A(x1) = max{43, x1 - 97}
          true_A = 16
          g#!_A(x1,x2) = max{68, x1 - 29, x2 - 48}
          s!_A(x1) = 25
          minus!_A(x1,x2) = x2 + 85
          m!_A(x1,x2) = max{25, x1 - 93}
          n!_A(x1,x2) = max{24, x2 - 97}
          k!_A(x1,x2) = max{25, x1 - 133, x2 - 95}
          |0|!_A = 23
          f!_A(x1) = max{1266, x1 - 72}
          true!_A = 16
  
    partial status:
  
      pi(g#) = []
      pi(s) = []
      pi(minus) = [1]
      pi(m) = [1, 2]
      pi(n) = [1, 2]
      pi(k) = [1]
      pi(|0|) = []
      pi(f) = []
      pi(true) = []

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

  (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:
          f#_A(x1) = x1
          s_A(x1) = x1 + 40
          h#_A(x1) = x1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          f#_A(x1) = max{0, x1 - 96}
          s_A(x1) = max{52, x1 + 43}
          h#_A(x1) = max{0, x1 - 53}
          f#!_A(x1) = max{1, x1 - 97}
          s!_A(x1) = 0
          h#!_A(x1) = max{2, x1 - 53}
  
    partial status:
  
      pi(f#) = []
      pi(s) = []
      pi(h#) = [1]

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

  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:
          k#_A(x1,x2) = x1 + x2 + 1
          s_A(x1) = x1 + 7
          minus_A(x1,x2) = x1
          |0|_A() = 4
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          k#_A(x1,x2) = max{0, x1 - 63, x2 - 74}
          s_A(x1) = max{62, x1 + 51}
          minus_A(x1,x2) = max{52, x1 + 51, x2 - 650}
          |0|_A = 24
          k#!_A(x1,x2) = max{21, x1 - 63, x2 - 64}
          s!_A(x1) = 21
          minus!_A(x1,x2) = max{9, x2 - 52}
          |0|!_A = 22
  
    partial status:
  
      pi(k#) = [2]
      pi(s) = []
      pi(minus) = [1]
      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: 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

  (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:
          minus#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          minus#_A(x1,x2) = max{0, x1 - 56, x2 - 56}
          s_A(x1) = x1 + 55
          minus#!_A(x1,x2) = max{143, x1 - 56, x2 - 56}
          s!_A(x1) = max{142, x1 + 137}
  
    partial status:
  
      pi(minus#) = [2]
      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: 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:
  
      matrix interpretation with standard point-wise order:
      
        carrier: N^1
        order: standard order
        interpretations:
          m#_A(x1,x2) = x1 + x2 + 9
          s_A(x1) = x1 + 8
          n_A(x1,x2) = x2 + 3
          |0|_A() = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          m#_A(x1,x2) = max{0, x1 - 10, x2 - 10}
          s_A(x1) = x1 + 9
          n_A(x1,x2) = max{87, x1 + 78}
          |0|_A = 5
          m#!_A(x1,x2) = max{1, x1 - 10, x2 - 77}
          s!_A(x1) = max{0, x1 - 78}
          n!_A(x1,x2) = max{x1 + 37, x2 - 6}
          |0|!_A = 39
  
    partial status:
  
      pi(m#) = [2]
      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

  (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:
          n#_A(x1,x2) = x1 + x2
          s_A(x1) = x1 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          n#_A(x1,x2) = max{0, x1 - 21, x2 - 21}
          s_A(x1) = x1 + 20
          n#!_A(x1,x2) = max{5, x1 - 21, x2 - 21}
          s!_A(x1) = 4
  
    partial status:
  
      pi(n#) = [2]
      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: 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

  r13, r17, r18, r19, r20, r23, r27, r28, r29

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:
          p#_A(x1,x2) = 3
          s_A(x1) = 37
          if_A(x1,x2,x3) = x2 + x3 + 5
          gt_A(x1,x2) = 2
          not_A(x1) = 8
          id_A(x1) = x1 + 4
          p_A(x1,x2) = x1 + x2 + 18
          |0|_A() = 36
          true_A() = 1
          false_A() = 1
          f_A(x1) = 2
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          p#_A(x1,x2) = 0
          s_A(x1) = max{266, x1 + 257}
          if_A(x1,x2,x3) = max{x1 - 4, x2 + 112, x3}
          gt_A(x1,x2) = max{265, x1 - 508, x2 - 260}
          not_A(x1) = max{261, x1 - 4}
          id_A(x1) = x1 + 118
          p_A(x1,x2) = max{18, x1 - 776, x2 + 8}
          |0|_A = 259
          true_A = 1
          false_A = 1
          f_A(x1) = max{298, x1 - 267}
          p#!_A(x1,x2) = max{1, x1 - 231, x2 - 259}
          s!_A(x1) = max{295, x1 + 8}
          if!_A(x1,x2,x3) = max{2191, x1 + 388, x2 + 2188, x3 - 119}
          gt!_A(x1,x2) = max{297, x1 - 478, x2 - 267}
          not!_A(x1) = max{2190, x1 + 793}
          id!_A(x1) = max{296, x1 + 27}
          p!_A(x1,x2) = x1 + 28
          |0|!_A = 92
          true!_A = 91
          false!_A = 91
          f!_A(x1) = max{296, x1 - 525}
  
    partial status:
  
      pi(p#) = []
      pi(s) = []
      pi(if) = [2]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(p) = [1, 2]
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(f) = []

The following rules are removed:

  p1, p3



-- SCC decomposition.

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

p1: p#(id(x),s(y)) -> p#(x,if(gt(s(y),y),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 estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: p#(id(x),s(y)) -> p#(x,if(gt(s(y),y),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

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

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:
          p#_A(x1,x2) = x1 + 8
          id_A(x1) = x1 + 9
          s_A(x1) = 7
          if_A(x1,x2,x3) = x2 + x3 + 13
          gt_A(x1,x2) = 2
          n_A(x1,x2) = x1 + 15
          |0|_A() = 2
          true_A() = 1
          false_A() = 1
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          p#_A(x1,x2) = max{0, x1 - 193, x2 - 172}
          id_A(x1) = x1 + 15
          s_A(x1) = max{20, x1 + 16}
          if_A(x1,x2,x3) = max{171, x1 - 3, x2 + 15, x3}
          gt_A(x1,x2) = max{18, x1 - 22, x2 - 22}
          n_A(x1,x2) = max{16, x1 - 190, x2 - 37}
          |0|_A = 16
          true_A = 18
          false_A = 18
          p#!_A(x1,x2) = max{175, x1 - 244, x2 + 4}
          id!_A(x1) = max{177, x1 - 245}
          s!_A(x1) = max{177, x1 + 16}
          if!_A(x1,x2,x3) = max{315, x2 + 161, x3 + 19}
          gt!_A(x1,x2) = max{176, x1 - 248, x2 - 229}
          n!_A(x1,x2) = max{315, x1 - 21, x2 - 21}
          |0|!_A = 18
          true!_A = 17
          false!_A = 17
  
    partial status:
  
      pi(p#) = [1]
      pi(id) = [1]
      pi(s) = []
      pi(if) = [2, 3]
      pi(gt) = []
      pi(n) = [1]
      pi(|0|) = []
      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: 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

  (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) = x1 + x2 + 8
          s_A(x1) = x1 + 7
  
    base order 2:
  
      marked-algebra reduction triple (>=A, >=A!, >A!):
      
        max/plus interpretation on natural numbers:
        
          gt#_A(x1,x2) = max{0, x1 - 17, x2 - 17}
          s_A(x1) = max{27, x1 + 16}
          gt#!_A(x1,x2) = max{21, x1 - 28, x2 - 28}
          s!_A(x1) = 20
  
    partial status:
  
      pi(gt#) = [2]
      pi(s) = []

The following rules are removed:

  p1

Then no dependency pair remains.