YES

We show the termination of the TRS R:

  fst(|0|(),Z) -> nil()
  fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
  from(X) -> cons(X,n__from(n__s(X)))
  add(|0|(),X) -> X
  add(s(X),Y) -> s(n__add(activate(X),Y))
  len(nil()) -> |0|()
  len(cons(X,Z)) -> s(n__len(activate(Z)))
  fst(X1,X2) -> n__fst(X1,X2)
  from(X) -> n__from(X)
  s(X) -> n__s(X)
  add(X1,X2) -> n__add(X1,X2)
  len(X) -> n__len(X)
  activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
  activate(n__from(X)) -> from(activate(X))
  activate(n__s(X)) -> s(X)
  activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
  activate(n__len(X)) -> len(activate(X))
  activate(X) -> X

-- SCC decomposition.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p3: add#(s(X),Y) -> s#(n__add(activate(X),Y))
p4: add#(s(X),Y) -> activate#(X)
p5: len#(cons(X,Z)) -> s#(n__len(activate(Z)))
p6: len#(cons(X,Z)) -> activate#(Z)
p7: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p8: activate#(n__fst(X1,X2)) -> activate#(X1)
p9: activate#(n__fst(X1,X2)) -> activate#(X2)
p10: activate#(n__from(X)) -> from#(activate(X))
p11: activate#(n__from(X)) -> activate#(X)
p12: activate#(n__s(X)) -> s#(X)
p13: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p14: activate#(n__add(X1,X2)) -> activate#(X1)
p15: activate#(n__add(X1,X2)) -> activate#(X2)
p16: activate#(n__len(X)) -> len#(activate(X))
p17: activate#(n__len(X)) -> activate#(X)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p6, p7, p8, p9, p11, p13, p14, p15, p16, p17}


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__len(X)) -> activate#(X)
p3: activate#(n__len(X)) -> len#(activate(X))
p4: len#(cons(X,Z)) -> activate#(Z)
p5: activate#(n__add(X1,X2)) -> activate#(X2)
p6: activate#(n__add(X1,X2)) -> activate#(X1)
p7: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p8: add#(s(X),Y) -> activate#(X)
p9: activate#(n__from(X)) -> activate#(X)
p10: activate#(n__fst(X1,X2)) -> activate#(X2)
p11: activate#(n__fst(X1,X2)) -> activate#(X1)
p12: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p13: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((1,0),(1,0)) x1 + ((1,0),(1,0)) x2 + (7,0)
          s_A(x1) = ((1,0),(0,0)) x1 + (0,2)
          cons_A(x1,x2) = ((0,0),(0,1)) x1 + ((1,0),(0,0)) x2 + (0,1)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (5,0)
          n__len_A(x1) = ((1,0),(1,1)) x1 + (7,0)
          len#_A(x1) = ((1,0),(1,0)) x1 + (6,0)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,3)
          n__add_A(x1,x2) = x1 + x2 + (4,0)
          add#_A(x1,x2) = ((1,0),(1,0)) x1 + (6,0)
          n__from_A(x1) = x1 + (6,1)
          n__fst_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (6,1)
          fst_A(x1,x2) = ((1,0),(0,0)) x1 + x2 + (6,3)
          |0|_A() = (1,0)
          nil_A() = (2,0)
          from_A(x1) = x1 + (6,1)
          n__s_A(x1) = ((1,0),(0,0)) x1
          add_A(x1,x2) = x1 + x2 + (4,1)
          len_A(x1) = ((1,0),(1,1)) x1 + (7,2)
  
    precedence:
    
      activate = len > n__len > s = n__s = add > cons = fst = nil = from > n__from > n__fst > |0| > fst# = activate# = n__add = add# > len#
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__len) = []
      pi(len#) = []
      pi(activate) = []
      pi(n__add) = []
      pi(add#) = []
      pi(n__from) = []
      pi(n__fst) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(len) = []

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: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__len(X)) -> activate#(X)
p3: activate#(n__len(X)) -> len#(activate(X))
p4: activate#(n__add(X1,X2)) -> activate#(X2)
p5: activate#(n__add(X1,X2)) -> activate#(X1)
p6: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p7: add#(s(X),Y) -> activate#(X)
p8: activate#(n__from(X)) -> activate#(X)
p9: activate#(n__fst(X1,X2)) -> activate#(X2)
p10: activate#(n__fst(X1,X2)) -> activate#(X1)
p11: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p12: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p4, p5, p6, p7, p8, p9, p10, p11, p12}


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)
p6: activate#(n__from(X)) -> activate#(X)
p7: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p8: add#(s(X),Y) -> activate#(X)
p9: activate#(n__add(X1,X2)) -> activate#(X1)
p10: activate#(n__add(X1,X2)) -> activate#(X2)
p11: activate#(n__len(X)) -> activate#(X)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (5,2)
          s_A(x1) = ((0,0),(0,1)) x1 + (8,0)
          cons_A(x1,x2) = ((0,0),(0,1)) x2 + (9,0)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (4,2)
          n__fst_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + (10,2)
          activate_A(x1) = ((1,1),(0,1)) x1 + (7,0)
          n__from_A(x1) = ((0,0),(0,1)) x1 + (3,1)
          n__add_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (9,7)
          add#_A(x1,x2) = ((0,1),(0,0)) x1 + (10,2)
          n__len_A(x1) = ((0,1),(0,1)) x1 + (10,1)
          fst_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,1),(0,1)) x2 + (11,2)
          |0|_A() = (1,1)
          nil_A() = (0,0)
          from_A(x1) = ((0,0),(0,1)) x1 + (10,1)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (8,0)
          add_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (9,7)
          len_A(x1) = ((0,1),(0,1)) x1 + (18,1)
  
    precedence:
    
      activate = n__len = len > s > n__add = nil = add > cons = fst = |0| = from > n__fst > n__from > fst# = activate# = add# > n__s
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__fst) = []
      pi(activate) = []
      pi(n__from) = []
      pi(n__add) = []
      pi(add#) = []
      pi(n__len) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(len) = []

The next rules are strictly ordered:

  p11

We remove them from the problem.

-- SCC decomposition.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)
p6: activate#(n__from(X)) -> activate#(X)
p7: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p8: add#(s(X),Y) -> activate#(X)
p9: activate#(n__add(X1,X2)) -> activate#(X1)
p10: activate#(n__add(X1,X2)) -> activate#(X2)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__add(X1,X2)) -> activate#(X2)
p3: activate#(n__add(X1,X2)) -> activate#(X1)
p4: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p5: add#(s(X),Y) -> activate#(X)
p6: activate#(n__from(X)) -> activate#(X)
p7: activate#(n__fst(X1,X2)) -> activate#(X2)
p8: activate#(n__fst(X1,X2)) -> activate#(X1)
p9: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p10: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,0),(1,0)) x2 + (2,2)
          s_A(x1) = ((0,1),(1,0)) x1
          cons_A(x1,x2) = ((1,0),(0,0)) x2 + (0,1)
          activate#_A(x1) = ((1,0),(1,0)) x1 + (1,2)
          n__add_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,3)
          add#_A(x1,x2) = ((1,1),(1,1)) x1 + ((0,0),(0,1)) x2 + (2,2)
          activate_A(x1) = x1
          n__from_A(x1) = ((1,1),(0,0)) x1 + (1,2)
          n__fst_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(1,1)) x2 + (3,2)
          fst_A(x1,x2) = ((1,1),(0,1)) x1 + ((1,0),(1,1)) x2 + (3,2)
          |0|_A() = (1,0)
          nil_A() = (0,0)
          from_A(x1) = ((1,1),(0,0)) x1 + (1,2)
          n__s_A(x1) = ((0,1),(1,0)) x1
          add_A(x1,x2) = ((1,1),(1,1)) x1 + ((1,1),(1,1)) x2 + (3,3)
          len_A(x1) = (2,2)
          n__len_A(x1) = (2,2)
  
    precedence:
    
      cons = n__from = n__fst = fst = from > fst# = activate# = add# > n__add = activate = add > s = |0| = nil = n__s = len = n__len
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__add) = [1, 2]
      pi(add#) = []
      pi(activate) = [1]
      pi(n__from) = []
      pi(n__fst) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [1, 2]
      pi(len) = []
      pi(n__len) = []

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: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__add(X1,X2)) -> activate#(X2)
p3: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p4: add#(s(X),Y) -> activate#(X)
p5: activate#(n__from(X)) -> activate#(X)
p6: activate#(n__fst(X1,X2)) -> activate#(X2)
p7: activate#(n__fst(X1,X2)) -> activate#(X1)
p8: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p9: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9}


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)
p6: activate#(n__from(X)) -> activate#(X)
p7: activate#(n__add(X1,X2)) -> add#(activate(X1),activate(X2))
p8: add#(s(X),Y) -> activate#(X)
p9: activate#(n__add(X1,X2)) -> activate#(X2)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (5,3)
          s_A(x1) = ((0,0),(0,1)) x1 + (2,0)
          cons_A(x1,x2) = ((0,0),(0,1)) x2 + (3,0)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (0,3)
          n__fst_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + (4,6)
          activate_A(x1) = ((1,1),(0,1)) x1
          n__from_A(x1) = ((0,0),(0,1)) x1 + (0,5)
          n__add_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (1,3)
          add#_A(x1,x2) = ((0,1),(0,0)) x1 + (2,3)
          fst_A(x1,x2) = ((0,1),(0,1)) x1 + ((0,0),(0,1)) x2 + (4,6)
          |0|_A() = (1,1)
          nil_A() = (2,2)
          from_A(x1) = ((0,0),(0,1)) x1 + (4,5)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (2,0)
          add_A(x1,x2) = ((0,0),(0,1)) x1 + x2 + (3,3)
          len_A(x1) = x1
          n__len_A(x1) = x1
  
    precedence:
    
      activate = n__from = from = len > s = n__s = add > n__add = add# > fst# = activate# > n__len > |0| > cons = n__fst = fst = nil
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__fst) = []
      pi(activate) = []
      pi(n__from) = []
      pi(n__add) = [2]
      pi(add#) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(len) = []
      pi(n__len) = []

The next rules are strictly ordered:

  p7

We remove them from the problem.

-- SCC decomposition.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)
p6: activate#(n__from(X)) -> activate#(X)
p7: add#(s(X),Y) -> activate#(X)
p8: activate#(n__add(X1,X2)) -> activate#(X2)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__add(X1,X2)) -> activate#(X2)
p3: activate#(n__from(X)) -> activate#(X)
p4: activate#(n__fst(X1,X2)) -> activate#(X2)
p5: activate#(n__fst(X1,X2)) -> activate#(X1)
p6: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p7: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (4,1)
          s_A(x1) = ((0,0),(0,1)) x1 + (1,0)
          cons_A(x1,x2) = ((0,0),(1,1)) x2
          activate#_A(x1) = ((0,1),(0,0)) x1 + (3,1)
          n__add_A(x1,x2) = x1 + x2 + (2,1)
          n__from_A(x1) = x1 + (0,1)
          n__fst_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + (0,2)
          activate_A(x1) = ((1,1),(0,1)) x1 + (1,0)
          fst_A(x1,x2) = ((0,0),(0,1)) x1 + ((0,0),(0,1)) x2 + (2,2)
          |0|_A() = (1,0)
          nil_A() = (1,0)
          from_A(x1) = x1 + (0,1)
          n__s_A(x1) = ((0,0),(0,1)) x1
          add_A(x1,x2) = x1 + x2 + (2,1)
          len_A(x1) = ((0,0),(0,1)) x1 + (2,0)
          n__len_A(x1) = ((0,0),(0,1)) x1 + (2,0)
  
    precedence:
    
      activate > len > n__len > fst# = activate# > from > s = n__add = |0| = add > n__from > n__s > cons = n__fst = fst = nil
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__add) = [1]
      pi(n__from) = []
      pi(n__fst) = []
      pi(activate) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [1, 2]
      pi(len) = []
      pi(n__len) = []

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: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__from(X)) -> activate#(X)
p3: activate#(n__fst(X1,X2)) -> activate#(X2)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p6: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

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


-- Reduction pair.

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

p1: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)
p6: activate#(n__from(X)) -> activate#(X)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((0,1),(0,0)) x1 + ((0,1),(0,0)) x2 + (2,3)
          s_A(x1) = ((0,0),(0,1)) x1 + (2,0)
          cons_A(x1,x2) = ((0,0),(0,1)) x2 + (3,0)
          activate#_A(x1) = ((0,1),(0,0)) x1 + (1,3)
          n__fst_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (4,2)
          activate_A(x1) = x1 + (5,0)
          n__from_A(x1) = x1 + (3,1)
          fst_A(x1,x2) = ((1,1),(0,1)) x1 + ((0,1),(0,1)) x2 + (4,2)
          |0|_A() = (2,2)
          nil_A() = (1,1)
          from_A(x1) = x1 + (3,1)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (1,0)
          add_A(x1,x2) = x2 + (6,0)
          n__add_A(x1,x2) = x2 + (6,0)
          len_A(x1) = ((0,1),(0,0)) x1 + (4,2)
          n__len_A(x1) = ((0,1),(0,0)) x1 + (4,2)
  
    precedence:
    
      activate = len > n__len > fst > fst# = activate# = n__fst > s = add = n__add > cons = n__from = from > n__s > |0| = nil
    
    partial status:
  
      pi(fst#) = []
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__fst) = [1]
      pi(activate) = []
      pi(n__from) = []
      pi(fst) = [1]
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(n__add) = [2]
      pi(len) = []
      pi(n__len) = []

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: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)
p5: activate#(n__fst(X1,X2)) -> activate#(X2)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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: fst#(s(X),cons(Y,Z)) -> activate#(X)
p2: activate#(n__fst(X1,X2)) -> activate#(X2)
p3: activate#(n__fst(X1,X2)) -> activate#(X1)
p4: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p5: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          fst#_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (2,1)
          s_A(x1) = ((0,0),(0,1)) x1
          cons_A(x1,x2) = x2
          activate#_A(x1) = ((0,1),(0,1)) x1 + (1,1)
          n__fst_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (3,2)
          activate_A(x1) = x1
          fst_A(x1,x2) = ((0,1),(0,1)) x1 + ((1,1),(1,1)) x2 + (3,2)
          |0|_A() = (2,3)
          nil_A() = (1,3)
          from_A(x1) = ((0,1),(0,0)) x1
          n__from_A(x1) = ((0,1),(0,0)) x1
          n__s_A(x1) = ((0,0),(0,1)) x1
          add_A(x1,x2) = ((1,1),(1,1)) x2 + (1,1)
          n__add_A(x1,x2) = ((1,1),(1,1)) x2 + (1,1)
          len_A(x1) = ((0,1),(0,1)) x1
          n__len_A(x1) = ((0,1),(0,1)) x1
  
    precedence:
    
      activate# = activate = from = n__from = add > n__add > len = n__len > cons = n__fst = fst > fst# > s = |0| = nil = n__s
    
    partial status:
  
      pi(fst#) = [2]
      pi(s) = []
      pi(cons) = []
      pi(activate#) = []
      pi(n__fst) = []
      pi(activate) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__from) = []
      pi(n__s) = []
      pi(add) = []
      pi(n__add) = [2]
      pi(len) = []
      pi(n__len) = []

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: activate#(n__fst(X1,X2)) -> activate#(X2)
p2: activate#(n__fst(X1,X2)) -> activate#(X1)
p3: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p4: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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: activate#(n__fst(X1,X2)) -> activate#(X2)
p2: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p3: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p4: activate#(n__fst(X1,X2)) -> activate#(X1)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1
          n__fst_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (8,1)
          fst#_A(x1,x2) = ((1,0),(1,0)) x2 + (1,0)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,8)
          s_A(x1) = x1 + (0,3)
          cons_A(x1,x2) = ((0,0),(1,0)) x1 + ((1,0),(0,0)) x2
          fst_A(x1,x2) = x1 + ((1,0),(1,1)) x2 + (8,1)
          |0|_A() = (2,1)
          nil_A() = (1,0)
          from_A(x1) = ((1,0),(1,0)) x1 + (0,2)
          n__from_A(x1) = ((1,0),(1,0)) x1 + (0,1)
          n__s_A(x1) = x1 + (0,3)
          add_A(x1,x2) = x2 + (3,7)
          n__add_A(x1,x2) = x2 + (3,4)
          len_A(x1) = (3,4)
          n__len_A(x1) = (3,1)
  
    precedence:
    
      activate = |0| = from = n__from = len = n__len > nil > add = n__add > fst > n__fst > activate# = fst# = s = n__s > cons
    
    partial status:
  
      pi(activate#) = []
      pi(n__fst) = []
      pi(fst#) = []
      pi(activate) = []
      pi(s) = []
      pi(cons) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(n__add) = []
      pi(len) = []
      pi(n__len) = []

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: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p2: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p3: activate#(n__fst(X1,X2)) -> activate#(X1)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

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

p1: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p2: fst#(s(X),cons(Y,Z)) -> activate#(Z)
p3: activate#(n__fst(X1,X2)) -> activate#(X1)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((1,0),(1,0)) x1 + (2,4)
          n__fst_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,4)
          fst#_A(x1,x2) = ((1,0),(1,0)) x2 + (3,4)
          activate_A(x1) = ((1,0),(1,1)) x1 + (0,1)
          s_A(x1) = ((1,0),(1,0)) x1 + (0,3)
          cons_A(x1,x2) = ((1,0),(0,0)) x2 + (0,1)
          fst_A(x1,x2) = x1 + ((1,0),(0,0)) x2 + (2,4)
          |0|_A() = (2,2)
          nil_A() = (1,1)
          from_A(x1) = (1,2)
          n__from_A(x1) = (1,1)
          n__s_A(x1) = ((1,0),(0,0)) x1 + (0,3)
          add_A(x1,x2) = ((1,0),(1,1)) x2 + (1,4)
          n__add_A(x1,x2) = ((1,0),(1,1)) x2 + (1,4)
          len_A(x1) = ((0,0),(1,0)) x1 + (3,6)
          n__len_A(x1) = ((0,0),(1,0)) x1 + (3,2)
  
    precedence:
    
      activate# = n__fst = fst# = activate = fst = add = n__add > |0| > len > n__len > s > n__s > nil = from = n__from > cons
    
    partial status:
  
      pi(activate#) = []
      pi(n__fst) = []
      pi(fst#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(cons) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__from) = []
      pi(n__s) = []
      pi(add) = [2]
      pi(n__add) = [2]
      pi(len) = []
      pi(n__len) = []

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: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p2: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))
p2: fst#(s(X),cons(Y,Z)) -> activate#(Z)

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> 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

Take the reduction pair:

  weighted path order
  
    base order:
  
      matrix interpretations:
      
        carrier: N^2
        order: standard order
        interpretations:
          activate#_A(x1) = ((0,1),(0,1)) x1 + (1,1)
          n__fst_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (3,5)
          fst#_A(x1,x2) = ((0,1),(0,1)) x2 + (2,5)
          activate_A(x1) = ((1,1),(0,1)) x1 + (3,0)
          s_A(x1) = ((0,0),(0,1)) x1 + (9,0)
          cons_A(x1,x2) = ((0,0),(0,1)) x2 + (9,0)
          fst_A(x1,x2) = x1 + ((1,1),(0,1)) x2 + (4,5)
          |0|_A() = (0,0)
          nil_A() = (1,1)
          from_A(x1) = (9,1)
          n__from_A(x1) = (9,1)
          n__s_A(x1) = ((0,0),(0,1)) x1 + (8,0)
          add_A(x1,x2) = x1 + x2 + (8,5)
          n__add_A(x1,x2) = x1 + x2 + (7,5)
          len_A(x1) = (10,1)
          n__len_A(x1) = (10,1)
  
    precedence:
    
      add = len = n__len > activate = s = fst = from = n__from = n__s > activate# = n__fst = fst# > cons = n__add > |0| = nil
    
    partial status:
  
      pi(activate#) = []
      pi(n__fst) = []
      pi(fst#) = []
      pi(activate) = [1]
      pi(s) = []
      pi(cons) = []
      pi(fst) = []
      pi(|0|) = []
      pi(nil) = []
      pi(from) = []
      pi(n__from) = []
      pi(n__s) = []
      pi(add) = []
      pi(n__add) = []
      pi(len) = []
      pi(n__len) = []

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: activate#(n__fst(X1,X2)) -> fst#(activate(X1),activate(X2))

and R consists of:

r1: fst(|0|(),Z) -> nil()
r2: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z)))
r3: from(X) -> cons(X,n__from(n__s(X)))
r4: add(|0|(),X) -> X
r5: add(s(X),Y) -> s(n__add(activate(X),Y))
r6: len(nil()) -> |0|()
r7: len(cons(X,Z)) -> s(n__len(activate(Z)))
r8: fst(X1,X2) -> n__fst(X1,X2)
r9: from(X) -> n__from(X)
r10: s(X) -> n__s(X)
r11: add(X1,X2) -> n__add(X1,X2)
r12: len(X) -> n__len(X)
r13: activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2))
r14: activate(n__from(X)) -> from(activate(X))
r15: activate(n__s(X)) -> s(X)
r16: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2))
r17: activate(n__len(X)) -> len(activate(X))
r18: activate(X) -> X

The estimated dependency graph contains the following SCCs:

  (no SCCs)