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

  Signature: 0     :: nat
             false :: bool
             h     :: (nat -> bool) -> (nat -> nat) -> nat -> nat
             if    :: bool -> nat -> nat -> nat
             s     :: nat -> nat
             true  :: bool

  Rules: if(true, X, Y) -> X
         if(false, X, Y) -> Y
         h(F, G, 0) -> 0
         h(F, G, s(Y)) -> G(h(F, G, if(F(Y), Y, 0)))

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) h#(F, G, s(Y)) => if#(F(Y), Y, 0)
      (2) h#(F, G, s(Y)) => h#(F, G, if(F(Y), Y, 0))

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

We compute a graph approximation with the following edges:

  1:
  2: 1 2 

There is only one SCC, so all DPs not inside the SCC can be removed.

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

  P2. (1) h#(F, G, s(Y)) => h#(F, G, if(F(Y), Y, 0))

***** We apply the Usable rules with respect to an argument wiltering [with HORPO] Processor on D2 = (P2, R, i, c).

Constrained HORPO yields:

  h#3(F, G, s1(Y)) (>) h#3(F, G, if3($, Y, 00))
  if3(true0, X, Y) (>=) X
  if3(false0, X, Y) (>=) Y

We do this using the following settings:

* Disregarded arguments:

  h#3 2 
  if3 1 

* Precedence and permutation:

    $       { }
  = h3      { 2 } 1
  = s1      { 1 }
  > $       { }
  = 00      { }
  = false0  { }
  = h#3     { }   1 3
  = if3     { 2 } 3
  = true0   { }

* Well-founded theory orderings:

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

Processor output: { }.