We consider termination of the LCTRS with only rule scheme Calc:

  Signature: f :: Int -> A
             g :: Int -> A
             h :: Int -> A

  Rules: f(x) -> f(x - 1) | x >= 0
         g(y) -> h(y)
         h(y) -> g(y + 1) | 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, f, c), where:

  P1. (1) f#(x) => f#(x - 1) | x >= 0
      (2) g#(y) => h#(y)
      (3) h#(y) => g#(y + 1) | y <= 0

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

We compute a graph approximation with the following edges:

  1: 1 
  2: 3 
  3: 2 

There are 2 SCCs.

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

  P2. (1) f#(x) => f#(x - 1) | x >= 0

  P3. (1) g#(y) => h#(y)
      (2) h#(y) => g#(y + 1) | y <= 0

***** We apply the Integer Function Processor on D2 = (P2, R, f, c).

We use the following integer mapping:

  J(f#) = arg_1 + 1

We thus have:

  (1) x >= 0 |= x + 1 > x - 1 + 1 (and x + 1 >= 0)

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

Processor output: { }.

***** We apply the Theory Arguments Processor on D3 = (P3, R, f, c).

We use the following theory arguments function:

  g# : [1]
  h# : [1]

Processor output: { D4 = (P4, R, f, c) ; D5 = (P5, R, f, c) }, where:

  P4. (1) g#(y) => h#(y) { y }
      (2) h#(y) => g#(y + 1) | y <= 0

  P5. (1) g#(y) => h#(y)

***** We apply the Reduction Pair [with HORPO] Processor on D4 = (P4, R, f, c).

Constrained HORPO yields:

  g#(y) (>=) h#(y) { y }
  h#(y) (>) g#(y + 1) | y <= 0
  f(x) (>=) f(x - 1) | x >= 0
  g(y) (>=) h(y)
  h(y) (>=) g(y + 1) | y <= 0

We do this using the following settings:

* Disregarded arguments:

  f 1 

* Precedence and permutation:

    g#  { 1 }
  = h#  { 1 }
  > g   { 1 }
  = h   { 1 }
  > f   { }

* Well-founded theory orderings:

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

Processor output: { D6 = (P6, R, f, c) }, where:

  P6. (1) g#(y) => h#(y) { y }

***** We apply the Graph Processor on D5 = (P5, R, f, c).

We compute a graph approximation with the following edges:

  1:

As there are no SCCs, this DP problem is removed.

Processor output: { }.

***** We apply the Graph Processor on D6 = (P6, R, f, c).

We compute a graph approximation with the following edges:

  1:

As there are no SCCs, this DP problem is removed.

Processor output: { }.