The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the following initial DP problem:

  P1. (1) map#(Z, cons(U, V)) => map#(Z, V)
      (2) minus#(s(P), s(X1)) => minus#(P, X1)
      (3) div#(s(U1), s(V1)) => minus#(U1, V1)
      (4) div#(s(U1), s(V1)) => div#(minus(U1, V1), s(V1))

***** We apply the Graph Processor on P1.

Considering the 3 SCCs, this DP problem is split into the following new problems.

  P2. (1) map#(Z, cons(U, V)) => map#(Z, V)

  P3. (1) minus#(s(P), s(X1)) => minus#(P, X1)

  P4. (1) div#(s(U1), s(V1)) => div#(minus(U1, V1), s(V1))

***** We apply the Subterm Criterion Processor on P2.

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(U, V) |>| V

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

***** We apply the Subterm Criterion Processor on P3.

We use the following projection function:

  nu(minus#) = 1

We thus have:

  (1) s(P) |>| P

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

***** We apply the horpo Processor on P4.

Constrained HORPO yields:

  div#(s(U1), s(V1)) (>) div#(minus(U1, V1), s(V1))
  map(F, nil) (>=) nil
  map(Z, cons(U, V)) (>=) cons(Z(U), map(Z, V))
  minus(W, 0) (>=) W
  minus(s(P), s(X1)) (>=) minus(P, X1)
  div(0, s(Y1)) (>=) 0
  div(s(U1), s(V1)) (>=) s(div(minus(U1, V1), s(V1)))

We do this using the following settings:

* Precedence and status (for non-mentioned symbols the precedence is irrelevant and the status is Lex):

  div               (status: Mul_2) >
  s                 (status: Lex)   >
  0                 (status: Lex)   >
  minus             (status: Mul_2) >
  div#              (status: Lex)   >
  cons = map = nil  (status: Lex)

* Well-founded theory orderings:

  [>]_{Bool} = {(true,false)}
  [>]_{Int}  = {(x,y) | x < 1000 /\ x < y }

* Filter:

  cons  disregards argument(s) 1 2 
  div   disregards argument(s) 2 
  minus disregards argument(s) 2 

All dependency pairs were removed.