YES

We show the termination of the TRS R:

  a__fst(|0|(),Z) -> nil()
  a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
  a__from(X) -> cons(mark(X),from(s(X)))
  a__add(|0|(),X) -> mark(X)
  a__add(s(X),Y) -> s(add(X,Y))
  a__len(nil()) -> |0|()
  a__len(cons(X,Z)) -> s(len(Z))
  mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
  mark(from(X)) -> a__from(mark(X))
  mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
  mark(len(X)) -> a__len(mark(X))
  mark(|0|()) -> |0|()
  mark(s(X)) -> s(X)
  mark(nil()) -> nil()
  mark(cons(X1,X2)) -> cons(mark(X1),X2)
  a__fst(X1,X2) -> fst(X1,X2)
  a__from(X) -> from(X)
  a__add(X1,X2) -> add(X1,X2)
  a__len(X) -> len(X)

-- SCC decomposition.

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

p1: a__fst#(s(X),cons(Y,Z)) -> mark#(Y)
p2: a__from#(X) -> mark#(X)
p3: a__add#(|0|(),X) -> mark#(X)
p4: mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2))
p5: mark#(fst(X1,X2)) -> mark#(X1)
p6: mark#(fst(X1,X2)) -> mark#(X2)
p7: mark#(from(X)) -> a__from#(mark(X))
p8: mark#(from(X)) -> mark#(X)
p9: mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2))
p10: mark#(add(X1,X2)) -> mark#(X1)
p11: mark#(add(X1,X2)) -> mark#(X2)
p12: mark#(len(X)) -> a__len#(mark(X))
p13: mark#(len(X)) -> mark#(X)
p14: mark#(cons(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__fst(|0|(),Z) -> nil()
r2: a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
r3: a__from(X) -> cons(mark(X),from(s(X)))
r4: a__add(|0|(),X) -> mark(X)
r5: a__add(s(X),Y) -> s(add(X,Y))
r6: a__len(nil()) -> |0|()
r7: a__len(cons(X,Z)) -> s(len(Z))
r8: mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
r9: mark(from(X)) -> a__from(mark(X))
r10: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
r11: mark(len(X)) -> a__len(mark(X))
r12: mark(|0|()) -> |0|()
r13: mark(s(X)) -> s(X)
r14: mark(nil()) -> nil()
r15: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r16: a__fst(X1,X2) -> fst(X1,X2)
r17: a__from(X) -> from(X)
r18: a__add(X1,X2) -> add(X1,X2)
r19: a__len(X) -> len(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p13, p14}


-- Reduction pair.

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

p1: a__fst#(s(X),cons(Y,Z)) -> mark#(Y)
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(len(X)) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X2)
p5: mark#(add(X1,X2)) -> mark#(X1)
p6: mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2))
p7: a__add#(|0|(),X) -> mark#(X)
p8: mark#(from(X)) -> mark#(X)
p9: mark#(from(X)) -> a__from#(mark(X))
p10: a__from#(X) -> mark#(X)
p11: mark#(fst(X1,X2)) -> mark#(X2)
p12: mark#(fst(X1,X2)) -> mark#(X1)
p13: mark#(fst(X1,X2)) -> a__fst#(mark(X1),mark(X2))

and R consists of:

r1: a__fst(|0|(),Z) -> nil()
r2: a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
r3: a__from(X) -> cons(mark(X),from(s(X)))
r4: a__add(|0|(),X) -> mark(X)
r5: a__add(s(X),Y) -> s(add(X,Y))
r6: a__len(nil()) -> |0|()
r7: a__len(cons(X,Z)) -> s(len(Z))
r8: mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
r9: mark(from(X)) -> a__from(mark(X))
r10: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
r11: mark(len(X)) -> a__len(mark(X))
r12: mark(|0|()) -> |0|()
r13: mark(s(X)) -> s(X)
r14: mark(nil()) -> nil()
r15: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r16: a__fst(X1,X2) -> fst(X1,X2)
r17: a__from(X) -> from(X)
r18: a__add(X1,X2) -> add(X1,X2)
r19: a__len(X) -> len(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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__fst#_A(x1,x2) = x1 + ((0,1),(0,1)) x2
        s_A(x1) = (1,0)
        cons_A(x1,x2) = x1 + (1,3)
        mark#_A(x1) = ((0,1),(0,1)) x1 + (3,0)
        len_A(x1) = x1 + (0,3)
        add_A(x1,x2) = x1 + x2 + (1,0)
        a__add#_A(x1,x2) = x1 + ((0,1),(0,1)) x2 + (3,0)
        mark_A(x1) = ((0,1),(0,1)) x1 + (3,0)
        |0|_A() = (1,2)
        from_A(x1) = x1 + (1,3)
        a__from#_A(x1) = ((0,1),(0,1)) x1 + (4,0)
        fst_A(x1,x2) = x1 + x2 + (1,1)
        a__fst_A(x1,x2) = x1 + x2 + (2,1)
        nil_A() = (0,0)
        a__from_A(x1) = x1 + (2,3)
        a__add_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (2,0)
        a__len_A(x1) = x1 + (2,3)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        a__fst#_A(x1,x2) = ((1,1),(0,0)) x1
        s_A(x1) = (1,1)
        cons_A(x1,x2) = (4,3)
        mark#_A(x1) = (4,1)
        len_A(x1) = (1,1)
        add_A(x1,x2) = (1,1)
        a__add#_A(x1,x2) = (4,1)
        mark_A(x1) = (2,1)
        |0|_A() = (1,1)
        from_A(x1) = (6,3)
        a__from#_A(x1) = (3,0)
        fst_A(x1,x2) = (1,1)
        a__fst_A(x1,x2) = (3,0)
        nil_A() = (1,1)
        a__from_A(x1) = (5,2)
        a__add_A(x1,x2) = (0,0)
        a__len_A(x1) = (0,0)
    

The next rules are strictly ordered:

  p1, p2, p3, p8, p9, p10, p11, p12, p13

We remove them from the problem.

-- SCC decomposition.

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

p1: mark#(add(X1,X2)) -> mark#(X2)
p2: mark#(add(X1,X2)) -> mark#(X1)
p3: mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2))
p4: a__add#(|0|(),X) -> mark#(X)

and R consists of:

r1: a__fst(|0|(),Z) -> nil()
r2: a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
r3: a__from(X) -> cons(mark(X),from(s(X)))
r4: a__add(|0|(),X) -> mark(X)
r5: a__add(s(X),Y) -> s(add(X,Y))
r6: a__len(nil()) -> |0|()
r7: a__len(cons(X,Z)) -> s(len(Z))
r8: mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
r9: mark(from(X)) -> a__from(mark(X))
r10: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
r11: mark(len(X)) -> a__len(mark(X))
r12: mark(|0|()) -> |0|()
r13: mark(s(X)) -> s(X)
r14: mark(nil()) -> nil()
r15: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r16: a__fst(X1,X2) -> fst(X1,X2)
r17: a__from(X) -> from(X)
r18: a__add(X1,X2) -> add(X1,X2)
r19: a__len(X) -> len(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

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

p1: mark#(add(X1,X2)) -> mark#(X2)
p2: mark#(add(X1,X2)) -> a__add#(mark(X1),mark(X2))
p3: a__add#(|0|(),X) -> mark#(X)
p4: mark#(add(X1,X2)) -> mark#(X1)

and R consists of:

r1: a__fst(|0|(),Z) -> nil()
r2: a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z))
r3: a__from(X) -> cons(mark(X),from(s(X)))
r4: a__add(|0|(),X) -> mark(X)
r5: a__add(s(X),Y) -> s(add(X,Y))
r6: a__len(nil()) -> |0|()
r7: a__len(cons(X,Z)) -> s(len(Z))
r8: mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2))
r9: mark(from(X)) -> a__from(mark(X))
r10: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
r11: mark(len(X)) -> a__len(mark(X))
r12: mark(|0|()) -> |0|()
r13: mark(s(X)) -> s(X)
r14: mark(nil()) -> nil()
r15: mark(cons(X1,X2)) -> cons(mark(X1),X2)
r16: a__fst(X1,X2) -> fst(X1,X2)
r17: a__from(X) -> from(X)
r18: a__add(X1,X2) -> add(X1,X2)
r19: a__len(X) -> len(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

Take the reduction pair:

  lexicographic combination of reduction pairs:
  
    1. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        mark#_A(x1) = x1
        add_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (5,4)
        a__add#_A(x1,x2) = x2 + (1,0)
        mark_A(x1) = ((1,1),(1,0)) x1 + (1,1)
        |0|_A() = (1,1)
        a__fst_A(x1,x2) = ((1,1),(1,0)) x1 + x2 + (6,4)
        nil_A() = (1,1)
        s_A(x1) = ((0,1),(0,1)) x1 + (1,1)
        cons_A(x1,x2) = (1,1)
        fst_A(x1,x2) = ((1,1),(1,0)) x1 + x2 + (5,4)
        a__from_A(x1) = ((1,1),(1,0)) x1 + (4,3)
        from_A(x1) = ((1,1),(1,0)) x1 + (3,3)
        a__add_A(x1,x2) = x1 + ((1,1),(1,0)) x2 + (6,4)
        a__len_A(x1) = (3,2)
        len_A(x1) = (2,1)
    
    2. matrix interpretations:
    
      carrier: N^2
      order: standard order
      interpretations:
        mark#_A(x1) = ((1,1),(1,1)) x1 + (0,1)
        add_A(x1,x2) = ((1,1),(1,1)) x1 + x2 + (1,1)
        a__add#_A(x1,x2) = x2 + (3,0)
        mark_A(x1) = (2,4)
        |0|_A() = (1,6)
        a__fst_A(x1,x2) = x1 + x2 + (0,1)
        nil_A() = (3,8)
        s_A(x1) = (3,6)
        cons_A(x1,x2) = (1,8)
        fst_A(x1,x2) = ((1,1),(1,0)) x1 + ((0,0),(1,1)) x2 + (1,2)
        a__from_A(x1) = (3,5)
        from_A(x1) = ((1,0),(1,1)) x1 + (4,6)
        a__add_A(x1,x2) = x1
        a__len_A(x1) = (0,5)
        len_A(x1) = (1,6)
    

The next rules are strictly ordered:

  p1, p2, p3, p4

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