We consider termination of the STRS with no additional rule schemes:

  Signature: cons  :: nat -> list -> list
             map   :: (nat -> nat) -> list -> list
             merge :: list -> list -> list -> list
             nil   :: list

  Rules: merge(nil, nil, X) -> X
         merge(nil, cons(Y, U), V) -> merge(U, nil, cons(Y, V))
         merge(cons(W, P), X1, Y1) -> merge(X1, P, cons(W, Y1))
         map(G1, nil) -> nil
         map(H1, cons(W1, P1)) -> cons(H1(W1), map(H1, P1))

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

We start by computing the initial DP problem D1 = (P1, R, i, c), where:

  P1. (1) merge#(nil, cons(Y, U), V) => merge#(U, nil, cons(Y, V))
      (2) merge#(cons(W, P), X1, Y1) => merge#(X1, P, cons(W, Y1))
      (3) map#(H1, cons(W1, P1)) => map#(H1, P1)

***** We apply the Graph Processor on D1 = (P1, R, i, c).

We compute a graph approximation with the following edges:

  1: 2 
  2: 1 2 
  3: 3 

There are 2 SCCs.

Processor output: { D2 = (P2, R, i, c) ; D3 = (P3, R, i, c) }, where:

  P2. (1) merge#(nil, cons(Y, U), V) => merge#(U, nil, cons(Y, V))
      (2) merge#(cons(W, P), X1, Y1) => merge#(X1, P, cons(W, Y1))

  P3. (1) map#(H1, cons(W1, P1)) => map#(H1, P1)

***** We apply the Usable Rules Processor on D2 = (P2, R, i, c).

We obtain 0 usable rules (out of 5 rules in the input problem).

Processor output: { D4 = (P2, {}, i, c) }.

***** We apply the Subterm Criterion Processor on D3 = (P3, R, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(W1, P1) |>| P1

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

Processor output: { }.

***** We apply the Usable rules with respect to HORPO Processor on D4 = (P2, {}, i, c).

Constrained HORPO yields:

  merge#3(nil0, cons2(Y, U), V) (>) merge#3(U, nil0, cons2(Y, V))
  merge#3(cons2(W, P), X1, Y1) (>) merge#3(X1, P, cons2(W, Y1))

We do this using the following settings:

* Disregarded arguments:

  merge#3 3 

* Precedence and permutation:

    nil0     { }
  > cons2    { 1 2 }
  = merge#3  { 1 2 }

* Well-founded theory orderings:

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

Processor output: { }.