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, r13, r15, r16, r22, r23, r24, r25, r26, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g#_A(x1,x2) = max{433, x1 + 284, x2 + 241}
        s_A(x1) = max{233, x1 + 193}
        minus_A(x1,x2) = max{2, x1}
        m_A(x1,x2) = max{106, x1 + 41, x2 + 66}
        n_A(x1,x2) = max{15, x1}
        k_A(x1,x2) = max{50, x1 + 42}
        |0|_A = 14
        p_A(x1,x2) = max{x1 + 211, x2 + 203}
        and_A(x1,x2) = max{1034, x1 - 4, x2 - 4}
        true_A = 2
        f_A(x1) = max{3, x1 - 234}
        h_A(x1) = max{2, x1 - 194}
        false_A = 1
        gt_A(x1,x2) = x2 + 2013
  
    base order 2:
  
      precedence:
      
        minus = m > g# > n = k = and = gt > s > |0| = h > f > p > true = false
      
  
    partial status:
  
      pi(g#) = []
      pi(s) = []
      pi(minus) = [1]
      pi(m) = [1]
      pi(n) = [1]
      pi(k) = []
      pi(|0|) = []
      pi(p) = [1]
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

The following rules are removed:

  p2



-- SCC decomposition.

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)))

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: g#(s(x),s(y)) -> g#(minus(m(x,y),n(x,y)),n(s(x),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, r4, r5, r6, r7, r15, r16, r22, r23, r25, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        g#_A(x1,x2) = max{76, x1 + 1, x2 - 7}
        s_A(x1) = max{95, x1 + 84}
        minus_A(x1,x2) = max{86, x1 + 3}
        m_A(x1,x2) = max{5, x1, x2}
        n_A(x1,x2) = max{6, x1}
        |0|_A = 5
        and_A(x1,x2) = max{94, x1 - 1}
        true_A = 0
        f_A(x1) = 0
        h_A(x1) = max{0, x1 - 96}
        false_A = 0
        gt_A(x1,x2) = max{0, x1 - 488, x2 - 171}
  
    base order 2:
  
      precedence:
      
        n > gt > g# = s = minus = m > |0| = and = h > f = false > true
      
  
    partial status:
  
      pi(g#) = [1]
      pi(s) = []
      pi(minus) = []
      pi(m) = [1, 2]
      pi(n) = [1]
      pi(|0|) = []
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

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: 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, r8, r21, r22, r23, r25, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        f#_A(x1) = x1 + 9
        s_A(x1) = x1 + 11
        h#_A(x1) = max{10, x1 - 1}
        n_A(x1,x2) = max{69, x1 - 42, x2 - 42}
        |0|_A = 0
        k_A(x1,x2) = max{58, x1 - 13, x2 - 95}
        and_A(x1,x2) = max{3, x1 - 11, x2 - 7}
        false_A = 0
        true_A = 0
        f_A(x1) = 12
        h_A(x1) = max{0, x1 - 12}
        gt_A(x1,x2) = max{143, x1 - 400, x2 - 12}
  
    base order 2:
  
      precedence:
      
        n = and > s = k > f# = |0| = f = h = gt > h# = false = true
      
  
    partial status:
  
      pi(f#) = [1]
      pi(s) = [1]
      pi(h#) = []
      pi(n) = []
      pi(|0|) = []
      pi(k) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []
      pi(gt) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: f#(s(x)) -> h#(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 estimated dependency graph contains the following SCCs:

  (no SCCs)

-- 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, r22, r23, r24, r25, r26, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        k#_A(x1,x2) = max{1, x1 - 74}
        s_A(x1) = max{148, x1 + 109}
        minus_A(x1,x2) = max{73, x1}
        |0|_A = 1148
        and_A(x1,x2) = max{x1 - 2, x2 + 281}
        true_A = 147
        f_A(x1) = x1 + 216
        h_A(x1) = x1 + 108
        false_A = 74
        gt_A(x1,x2) = max{245, x1 - 129, x2 - 1149}
  
    base order 2:
  
      precedence:
      
        gt > |0| = and > k# = s > minus = h > f = false > true
      
  
    partial status:
  
      pi(k#) = []
      pi(s) = [1]
      pi(minus) = [1]
      pi(|0|) = []
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(h) = [1]
      pi(false) = []
      pi(gt) = []

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, r8, r22, r23, r24, r25, r26, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        minus#_A(x1,x2) = max{41, x1 + 39}
        s_A(x1) = max{55, x1 + 40}
        n_A(x1,x2) = max{3, x1 - 65, x2 - 56}
        |0|_A = 2
        k_A(x1,x2) = max{54, x1 - 82, x2 - 229}
        and_A(x1,x2) = max{19, x1 - 62, x2 - 42}
        true_A = 1
        f_A(x1) = max{41, x1 - 56}
        h_A(x1) = max{41, x1 - 96}
        false_A = 1
        gt_A(x1,x2) = max{18, x1 - 756, x2 - 386}
  
    base order 2:
  
      precedence:
      
        k > n = and > minus# = s = gt > |0| = f = h > true = false
      
  
    partial status:
  
      pi(minus#) = [1]
      pi(s) = [1]
      pi(n) = []
      pi(|0|) = []
      pi(k) = []
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

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

  r22, r23, r24, r25, r26, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        m#_A(x1,x2) = max{63, x1 + 1, x2 - 1}
        s_A(x1) = max{74, x1 + 65}
        and_A(x1,x2) = max{69, x1 - 659, x2 - 75}
        true_A = 66
        f_A(x1) = max{67, x1 - 727}
        |0|_A = 186
        h_A(x1) = max{67, x1 - 792}
        false_A = 66
        gt_A(x1,x2) = max{68, x1 - 187, x2 - 187}
  
    base order 2:
  
      precedence:
      
        |0| > m# = and > s > f = h = gt > true = false
      
  
    partial status:
  
      pi(m#) = [1]
      pi(s) = []
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(|0|) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

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, r8, r22, r23, r24, r25, r26, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        n#_A(x1,x2) = max{30, x1 + 1, x2 - 1}
        s_A(x1) = max{40, x1 + 32}
        n_A(x1,x2) = max{6, x1 - 55, x2 - 33}
        |0|_A = 0
        k_A(x1,x2) = max{248, x1 - 51, x2 - 73}
        and_A(x1,x2) = max{402, x1 - 8, x2 - 217}
        true_A = 0
        f_A(x1) = max{0, x1 - 398}
        h_A(x1) = max{0, x1 - 430}
        false_A = 0
        gt_A(x1,x2) = max{35, x1 - 41, x2 - 374}
  
    base order 2:
  
      precedence:
      
        k > n# = n = and > s = gt > |0| = f = h > true = false
      
  
    partial status:
  
      pi(n#) = [1]
      pi(s) = [1]
      pi(n) = []
      pi(|0|) = []
      pi(k) = []
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

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, r21, r22, r23, r24, r25, r26, r27, r28, r29

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        p#_A(x1,x2) = max{121, x1 + 77, x2 + 59}
        s_A(x1) = x1 + 62
        if_A(x1,x2,x3) = max{52, x2 + 48, x3}
        gt_A(x1,x2) = max{0, x2 - 124}
        not_A(x1) = x1 + 45
        id_A(x1) = x1
        n_A(x1,x2) = max{15, x1 - 435, x2 - 155}
        |0|_A = 14
        true_A = 0
        false_A = 0
        and_A(x1,x2) = max{63, x1 - 47, x2 - 64}
        f_A(x1) = max{46, x1 - 63}
        h_A(x1) = max{44, x1 - 15}
  
    base order 2:
  
      precedence:
      
        p# > if = gt > not = id > and > s = n > h > |0| = f > true = false
      
  
    partial status:
  
      pi(p#) = []
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = [1]
      pi(n) = []
      pi(|0|) = []
      pi(true) = []
      pi(false) = []
      pi(and) = []
      pi(f) = []
      pi(h) = []

The following rules are removed:

  p3



-- SCC decomposition.

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)))

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, p2}


-- 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)))

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, r8, r15, r16, r17, r18, r19, r20, r22, r25, r27, r28, r29

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        p#_A(x1,x2) = max{39, x1 - 48, x2 - 126}
        s_A(x1) = x1 + 252
        if_A(x1,x2,x3) = max{164, x1 - 935, x2 + 152, x3}
        gt_A(x1,x2) = max{x1 - 484, x2 + 581}
        not_A(x1) = max{1188, x1 + 481}
        id_A(x1) = max{102, x1 + 88}
        n_A(x1,x2) = max{484, x1 - 488, x2 - 491}
        |0|_A = 483
        k_A(x1,x2) = max{486, x1 - 742, x2 - 743}
        minus_A(x1,x2) = max{x1 + 253, x2 - 485}
        true_A = 482
        false_A = 2
        and_A(x1,x2) = max{484, x1 - 238, x2 - 3}
        h_A(x1) = max{1, x1 - 240}
  
    base order 2:
  
      precedence:
      
        not > if > gt > p# = minus = and > k > s = id = n > |0| = h > true = false
      
  
    partial status:
  
      pi(p#) = []
      pi(s) = [1]
      pi(if) = [3]
      pi(gt) = []
      pi(not) = []
      pi(id) = []
      pi(n) = []
      pi(|0|) = []
      pi(k) = []
      pi(minus) = [1]
      pi(true) = []
      pi(false) = []
      pi(and) = []
      pi(h) = []

The following rules are removed:

  p1



-- 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, r8, r21, r22, r23, r25, r27, r28

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        p#_A(x1,x2) = max{6, x1 + 2}
        id_A(x1) = x1 + 7
        s_A(x1) = max{0, x1 - 23}
        if_A(x1,x2,x3) = max{14, x1 - 44, x2 - 91, x3 - 17}
        gt_A(x1,x2) = max{10, x1 - 2, x2 - 25}
        n_A(x1,x2) = max{15, x1 - 2, x2 - 2}
        |0|_A = 1
        k_A(x1,x2) = max{19, x1 - 44, x2 - 5}
        and_A(x1,x2) = max{22, x1 - 139, x2 - 24}
        false_A = 0
        true_A = 0
        f_A(x1) = max{0, x1 - 23}
        h_A(x1) = max{0, x1 - 23}
  
    base order 2:
  
      precedence:
      
        gt = n > p# = s = k = and > if = |0| = f = h > id = false = true
      
  
    partial status:
  
      pi(p#) = [1]
      pi(id) = [1]
      pi(s) = []
      pi(if) = []
      pi(gt) = []
      pi(n) = []
      pi(|0|) = []
      pi(k) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(f) = []
      pi(h) = []

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

  r22, r23, r24, r25, r26, r27, r28

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 + 93, x2 - 1}
        s_A(x1) = max{164, x1 + 94}
        and_A(x1,x2) = max{16, x1 - 96, x2 - 555}
        true_A = 15
        f_A(x1) = max{95, x1 - 872}
        |0|_A = 297
        h_A(x1) = max{95, x1 - 966}
        false_A = 0
        gt_A(x1,x2) = max{16, x1 - 298, x2 - 298}
  
    base order 2:
  
      precedence:
      
        and > gt# = s = gt > f = |0| = h > true = false
      
  
    partial status:
  
      pi(gt#) = [1]
      pi(s) = [1]
      pi(and) = []
      pi(true) = []
      pi(f) = []
      pi(|0|) = []
      pi(h) = []
      pi(false) = []
      pi(gt) = []

The following rules are removed:

  p1

Then no dependency pair remains.