YES

We show the termination of the TRS R:

  a__nats() -> cons(|0|(),incr(nats()))
  a__pairs() -> cons(|0|(),incr(odds()))
  a__odds() -> a__incr(a__pairs())
  a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
  a__head(cons(X,XS)) -> mark(X)
  a__tail(cons(X,XS)) -> mark(XS)
  mark(nats()) -> a__nats()
  mark(pairs()) -> a__pairs()
  mark(odds()) -> a__odds()
  mark(incr(X)) -> a__incr(mark(X))
  mark(head(X)) -> a__head(mark(X))
  mark(tail(X)) -> a__tail(mark(X))
  mark(|0|()) -> |0|()
  mark(s(X)) -> s(mark(X))
  mark(nil()) -> nil()
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  a__nats() -> nats()
  a__pairs() -> pairs()
  a__odds() -> odds()
  a__incr(X) -> incr(X)
  a__head(X) -> head(X)
  a__tail(X) -> tail(X)

-- SCC decomposition.

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

p1: a__odds#() -> a__incr#(a__pairs())
p2: a__odds#() -> a__pairs#()
p3: a__incr#(cons(X,XS)) -> mark#(X)
p4: a__head#(cons(X,XS)) -> mark#(X)
p5: a__tail#(cons(X,XS)) -> mark#(XS)
p6: mark#(nats()) -> a__nats#()
p7: mark#(pairs()) -> a__pairs#()
p8: mark#(odds()) -> a__odds#()
p9: mark#(incr(X)) -> a__incr#(mark(X))
p10: mark#(incr(X)) -> mark#(X)
p11: mark#(head(X)) -> a__head#(mark(X))
p12: mark#(head(X)) -> mark#(X)
p13: mark#(tail(X)) -> a__tail#(mark(X))
p14: mark#(tail(X)) -> mark#(X)
p15: mark#(s(X)) -> mark#(X)
p16: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p4, p5, p8, p9, p10, p11, p12, p13, p14, p15, p16}


-- Reduction pair.

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

p1: a__odds#() -> a__incr#(a__pairs())
p2: a__incr#(cons(X,XS)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(s(X)) -> mark#(X)
p5: mark#(tail(X)) -> mark#(X)
p6: mark#(tail(X)) -> a__tail#(mark(X))
p7: a__tail#(cons(X,XS)) -> mark#(XS)
p8: mark#(head(X)) -> mark#(X)
p9: mark#(head(X)) -> a__head#(mark(X))
p10: a__head#(cons(X,XS)) -> mark#(X)
p11: mark#(incr(X)) -> mark#(X)
p12: mark#(incr(X)) -> a__incr#(mark(X))
p13: mark#(odds()) -> a__odds#()

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__odds#_A() = (2,1)
          a__incr#_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          a__pairs_A() = (4,1)
          cons_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,0)
          mark#_A(x1) = ((0,1),(0,1)) x1 + (1,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (3,0)
          tail_A(x1) = ((0,1),(0,1)) x1 + (11,6)
          a__tail#_A(x1) = x1 + (1,1)
          mark_A(x1) = ((0,1),(0,1)) x1 + (5,0)
          head_A(x1) = x1 + (2,6)
          a__head#_A(x1) = x1 + (1,1)
          incr_A(x1) = ((0,0),(0,1)) x1
          odds_A() = (1,1)
          a__nats_A() = (4,1)
          |0|_A() = (1,0)
          nats_A() = (1,1)
          a__odds_A() = (5,1)
          a__incr_A(x1) = x1
          a__head_A(x1) = x1 + (3,6)
          a__tail_A(x1) = ((0,1),(0,1)) x1 + (11,6)
          pairs_A() = (1,1)
          nil_A() = (1,1)
  
    precedence:
    
      nats > a__nats > s = mark = a__odds = a__incr = a__head = a__tail > a__pairs = cons > head > tail > a__odds# = a__incr# = mark# = a__tail# > |0| > a__head# > incr = odds > pairs = nil
    
    partial status:
  
      pi(a__odds#) = []
      pi(a__incr#) = []
      pi(a__pairs) = []
      pi(cons) = []
      pi(mark#) = []
      pi(s) = []
      pi(tail) = []
      pi(a__tail#) = [1]
      pi(mark) = []
      pi(head) = [1]
      pi(a__head#) = [1]
      pi(incr) = []
      pi(odds) = []
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__odds) = []
      pi(a__incr) = []
      pi(a__head) = []
      pi(a__tail) = []
      pi(pairs) = []
      pi(nil) = []

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

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

p1: a__odds#() -> a__incr#(a__pairs())
p2: a__incr#(cons(X,XS)) -> mark#(X)
p3: mark#(cons(X1,X2)) -> mark#(X1)
p4: mark#(s(X)) -> mark#(X)
p5: mark#(tail(X)) -> mark#(X)
p6: a__tail#(cons(X,XS)) -> mark#(XS)
p7: mark#(head(X)) -> mark#(X)
p8: mark#(head(X)) -> a__head#(mark(X))
p9: a__head#(cons(X,XS)) -> mark#(X)
p10: mark#(incr(X)) -> mark#(X)
p11: mark#(incr(X)) -> a__incr#(mark(X))
p12: mark#(odds()) -> a__odds#()

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: a__odds#() -> a__incr#(a__pairs())
p2: a__incr#(cons(X,XS)) -> mark#(X)
p3: mark#(odds()) -> a__odds#()
p4: mark#(incr(X)) -> a__incr#(mark(X))
p5: mark#(incr(X)) -> mark#(X)
p6: mark#(head(X)) -> a__head#(mark(X))
p7: a__head#(cons(X,XS)) -> mark#(X)
p8: mark#(head(X)) -> mark#(X)
p9: mark#(tail(X)) -> mark#(X)
p10: mark#(s(X)) -> mark#(X)
p11: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__odds#_A() = (4,3)
          a__incr#_A(x1) = ((0,1),(0,0)) x1 + (1,3)
          a__pairs_A() = (2,2)
          cons_A(x1,x2) = ((0,0),(1,1)) x1 + ((0,1),(1,0)) x2
          mark#_A(x1) = ((1,1),(0,0)) x1 + (1,3)
          odds_A() = (2,2)
          incr_A(x1) = x1
          mark_A(x1) = x1
          head_A(x1) = ((1,1),(1,1)) x1
          a__head#_A(x1) = ((1,1),(0,0)) x1 + (1,3)
          tail_A(x1) = ((0,1),(1,0)) x1 + (2,1)
          s_A(x1) = ((1,1),(0,0)) x1
          a__nats_A() = (1,1)
          |0|_A() = (0,0)
          nats_A() = (1,1)
          a__odds_A() = (2,2)
          a__incr_A(x1) = x1
          a__head_A(x1) = ((1,1),(1,1)) x1
          a__tail_A(x1) = ((0,1),(1,0)) x1 + (2,1)
          pairs_A() = (2,2)
          nil_A() = (1,1)
  
    precedence:
    
      a__odds# > a__incr# = a__pairs = cons = mark# = odds = incr = mark = head = a__head# = tail = s = a__nats = |0| = nats = a__odds = a__incr = a__head = a__tail = pairs = nil
    
    partial status:
  
      pi(a__odds#) = []
      pi(a__incr#) = []
      pi(a__pairs) = []
      pi(cons) = []
      pi(mark#) = []
      pi(odds) = []
      pi(incr) = []
      pi(mark) = []
      pi(head) = []
      pi(a__head#) = []
      pi(tail) = []
      pi(s) = []
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__odds) = []
      pi(a__incr) = []
      pi(a__head) = []
      pi(a__tail) = []
      pi(pairs) = []
      pi(nil) = []

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

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

p1: a__odds#() -> a__incr#(a__pairs())
p2: a__incr#(cons(X,XS)) -> mark#(X)
p3: mark#(incr(X)) -> a__incr#(mark(X))
p4: mark#(incr(X)) -> mark#(X)
p5: mark#(head(X)) -> a__head#(mark(X))
p6: a__head#(cons(X,XS)) -> mark#(X)
p7: mark#(head(X)) -> mark#(X)
p8: mark#(tail(X)) -> mark#(X)
p9: mark#(s(X)) -> mark#(X)
p10: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6, p7, p8, p9, p10}


-- Reduction pair.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(tail(X)) -> mark#(X)
p5: mark#(head(X)) -> mark#(X)
p6: mark#(head(X)) -> a__head#(mark(X))
p7: a__head#(cons(X,XS)) -> mark#(X)
p8: mark#(incr(X)) -> mark#(X)
p9: mark#(incr(X)) -> a__incr#(mark(X))

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__incr#_A(x1) = ((1,1),(1,1)) x1 + (2,0)
          cons_A(x1,x2) = ((0,1),(1,0)) x1 + x2
          mark#_A(x1) = ((1,1),(1,1)) x1 + (2,0)
          s_A(x1) = ((0,1),(1,0)) x1
          tail_A(x1) = ((1,1),(1,1)) x1 + (3,1)
          head_A(x1) = ((1,1),(1,1)) x1 + (1,2)
          a__head#_A(x1) = ((1,1),(1,1)) x1 + (3,1)
          mark_A(x1) = ((1,1),(1,1)) x1
          incr_A(x1) = ((1,1),(1,1)) x1
          a__nats_A() = (2,2)
          |0|_A() = (0,0)
          nats_A() = (1,1)
          a__pairs_A() = (0,0)
          odds_A() = (0,0)
          a__odds_A() = (0,0)
          a__incr_A(x1) = ((1,1),(1,1)) x1
          a__head_A(x1) = ((1,1),(1,1)) x1 + (2,3)
          a__tail_A(x1) = ((1,1),(1,1)) x1 + (4,2)
          pairs_A() = (0,0)
          nil_A() = (0,1)
  
    precedence:
    
      a__incr# = mark# = a__head# > nats > s = mark = |0| = a__pairs = odds = a__odds = a__incr = a__tail > incr > a__nats = pairs = nil > tail > a__head > head > cons
    
    partial status:
  
      pi(a__incr#) = []
      pi(cons) = []
      pi(mark#) = []
      pi(s) = []
      pi(tail) = []
      pi(head) = []
      pi(a__head#) = [1]
      pi(mark) = []
      pi(incr) = []
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__pairs) = []
      pi(odds) = []
      pi(a__odds) = []
      pi(a__incr) = []
      pi(a__head) = []
      pi(a__tail) = []
      pi(pairs) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(head(X)) -> mark#(X)
p5: mark#(head(X)) -> a__head#(mark(X))
p6: a__head#(cons(X,XS)) -> mark#(X)
p7: mark#(incr(X)) -> mark#(X)
p8: mark#(incr(X)) -> a__incr#(mark(X))

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8}


-- Reduction pair.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(incr(X)) -> a__incr#(mark(X))
p3: mark#(incr(X)) -> mark#(X)
p4: mark#(head(X)) -> a__head#(mark(X))
p5: a__head#(cons(X,XS)) -> mark#(X)
p6: mark#(head(X)) -> mark#(X)
p7: mark#(s(X)) -> mark#(X)
p8: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__incr#_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          cons_A(x1,x2) = x1 + x2 + (0,1)
          mark#_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          incr_A(x1) = ((1,0),(0,0)) x1 + (0,2)
          mark_A(x1) = x1 + (0,6)
          head_A(x1) = x1 + (5,6)
          a__head#_A(x1) = ((1,0),(1,0)) x1 + (1,1)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,2)
          a__nats_A() = (1,7)
          |0|_A() = (0,1)
          nats_A() = (1,1)
          a__pairs_A() = (1,6)
          odds_A() = (1,1)
          a__odds_A() = (1,5)
          a__incr_A(x1) = ((1,0),(0,0)) x1 + (0,5)
          a__head_A(x1) = x1 + (5,6)
          a__tail_A(x1) = x1 + (1,6)
          pairs_A() = (1,6)
          tail_A(x1) = x1 + (1,6)
          nil_A() = (1,0)
  
    precedence:
    
      |0| > mark = head = a__pairs = odds = a__odds = a__incr = a__head = a__tail = pairs > a__nats = nats > incr > cons = nil > s > a__incr# = mark# > a__head# > tail
    
    partial status:
  
      pi(a__incr#) = []
      pi(cons) = [1, 2]
      pi(mark#) = []
      pi(incr) = []
      pi(mark) = [1]
      pi(head) = []
      pi(a__head#) = []
      pi(s) = []
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__pairs) = []
      pi(odds) = []
      pi(a__odds) = []
      pi(a__incr) = []
      pi(a__head) = [1]
      pi(a__tail) = [1]
      pi(pairs) = []
      pi(tail) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(incr(X)) -> a__incr#(mark(X))
p3: mark#(incr(X)) -> mark#(X)
p4: a__head#(cons(X,XS)) -> mark#(X)
p5: mark#(head(X)) -> mark#(X)
p6: mark#(s(X)) -> mark#(X)
p7: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p5, p6, p7}


-- Reduction pair.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(head(X)) -> mark#(X)
p5: mark#(incr(X)) -> mark#(X)
p6: mark#(incr(X)) -> a__incr#(mark(X))

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__incr#_A(x1) = ((0,1),(0,0)) x1 + (12,2)
          cons_A(x1,x2) = x1 + x2 + (2,0)
          mark#_A(x1) = ((0,1),(0,0)) x1 + (12,2)
          s_A(x1) = ((0,0),(0,1)) x1 + (3,0)
          head_A(x1) = ((1,1),(0,1)) x1 + (11,1)
          incr_A(x1) = ((0,0),(0,1)) x1 + (3,0)
          mark_A(x1) = ((1,1),(0,1)) x1 + (10,0)
          a__nats_A() = (6,1)
          |0|_A() = (1,0)
          nats_A() = (1,1)
          a__pairs_A() = (7,1)
          odds_A() = (8,1)
          a__odds_A() = (10,1)
          a__incr_A(x1) = ((0,0),(0,1)) x1 + (9,0)
          a__head_A(x1) = ((1,1),(0,1)) x1 + (12,1)
          a__tail_A(x1) = ((1,1),(0,1)) x1 + (9,1)
          pairs_A() = (6,1)
          tail_A(x1) = ((1,1),(0,1)) x1 + (9,1)
          nil_A() = (1,1)
  
    precedence:
    
      mark = a__head = a__tail = tail > a__incr# = mark# = head > nats > odds > |0| > pairs > a__pairs > a__nats > a__odds > s > nil > cons = incr = a__incr
    
    partial status:
  
      pi(a__incr#) = []
      pi(cons) = []
      pi(mark#) = []
      pi(s) = []
      pi(head) = []
      pi(incr) = []
      pi(mark) = []
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__pairs) = []
      pi(odds) = []
      pi(a__odds) = []
      pi(a__incr) = []
      pi(a__head) = []
      pi(a__tail) = []
      pi(pairs) = []
      pi(tail) = []
      pi(nil) = []

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(incr(X)) -> mark#(X)
p5: mark#(incr(X)) -> a__incr#(mark(X))

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(incr(X)) -> a__incr#(mark(X))
p3: mark#(incr(X)) -> mark#(X)
p4: mark#(s(X)) -> mark#(X)
p5: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          a__incr#_A(x1) = x1
          cons_A(x1,x2) = x1 + ((0,1),(1,0)) x2
          mark#_A(x1) = x1
          incr_A(x1) = ((1,1),(1,1)) x1
          mark_A(x1) = ((1,1),(1,1)) x1
          s_A(x1) = x1
          a__nats_A() = (2,2)
          |0|_A() = (0,0)
          nats_A() = (1,1)
          a__pairs_A() = (0,0)
          odds_A() = (0,0)
          a__odds_A() = (0,0)
          a__incr_A(x1) = ((1,1),(1,1)) x1
          a__head_A(x1) = ((1,1),(1,1)) x1 + (2,3)
          a__tail_A(x1) = ((1,1),(1,1)) x1 + (2,3)
          pairs_A() = (0,0)
          head_A(x1) = ((1,1),(1,1)) x1 + (1,2)
          tail_A(x1) = ((1,1),(1,1)) x1 + (1,2)
          nil_A() = (1,1)
  
    precedence:
    
      odds = a__odds > a__head = head > nil > incr = mark = a__pairs = a__incr = a__tail = tail > |0| > a__nats > s > cons = mark# > a__incr# > nats > pairs
    
    partial status:
  
      pi(a__incr#) = [1]
      pi(cons) = [1]
      pi(mark#) = [1]
      pi(incr) = [1]
      pi(mark) = [1]
      pi(s) = [1]
      pi(a__nats) = []
      pi(|0|) = []
      pi(nats) = []
      pi(a__pairs) = []
      pi(odds) = []
      pi(a__odds) = []
      pi(a__incr) = [1]
      pi(a__head) = []
      pi(a__tail) = []
      pi(pairs) = []
      pi(head) = []
      pi(tail) = []
      pi(nil) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: a__incr#(cons(X,XS)) -> mark#(X)
p2: mark#(incr(X)) -> mark#(X)
p3: mark#(s(X)) -> mark#(X)
p4: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4}


-- Reduction pair.

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

p1: mark#(incr(X)) -> mark#(X)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          incr_A(x1) = x1 + (1,1)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,1),(1,1)) x2 + (0,1)
          s_A(x1) = ((0,0),(0,1)) x1 + (3,1)
  
    precedence:
    
      mark# > incr = cons = s
    
    partial status:
  
      pi(mark#) = []
      pi(incr) = [1]
      pi(cons) = [2]
      pi(s) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(incr(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

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

p1: mark#(incr(X)) -> mark#(X)
p2: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((0,1),(0,0)) x1 + (2,2)
          incr_A(x1) = ((0,0),(0,1)) x1 + (1,1)
          s_A(x1) = ((0,1),(0,1)) x1 + (1,1)
  
    precedence:
    
      mark# = incr = s
    
    partial status:
  
      pi(mark#) = []
      pi(incr) = []
      pi(s) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

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

p1: mark#(s(X)) -> mark#(X)

and R consists of:

r1: a__nats() -> cons(|0|(),incr(nats()))
r2: a__pairs() -> cons(|0|(),incr(odds()))
r3: a__odds() -> a__incr(a__pairs())
r4: a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
r5: a__head(cons(X,XS)) -> mark(X)
r6: a__tail(cons(X,XS)) -> mark(XS)
r7: mark(nats()) -> a__nats()
r8: mark(pairs()) -> a__pairs()
r9: mark(odds()) -> a__odds()
r10: mark(incr(X)) -> a__incr(mark(X))
r11: mark(head(X)) -> a__head(mark(X))
r12: mark(tail(X)) -> a__tail(mark(X))
r13: mark(|0|()) -> |0|()
r14: mark(s(X)) -> s(mark(X))
r15: mark(nil()) -> nil()
r16: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r17: a__nats() -> nats()
r18: a__pairs() -> pairs()
r19: a__odds() -> odds()
r20: a__incr(X) -> incr(X)
r21: a__head(X) -> head(X)
r22: a__tail(X) -> tail(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          mark#_A(x1) = ((1,0),(1,0)) x1 + (2,2)
          s_A(x1) = ((1,0),(0,0)) x1 + (1,1)
  
    precedence:
    
      mark# = s
    
    partial status:
  
      pi(mark#) = []
      pi(s) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.