include "ERTLptr/liveness.ma".

inductive decision: Type[0] ≝
  | decision_spill: nat → decision
  | decision_colour: Register → decision.

(* prop_colouring is the non interferece
   prop_colouring 2 and 3 together say that spilled_no is the number of spilled
   registers *)
(* Wilmer: the generation of the destruct principle diverges;
   Ctr-C make the file pass *)
record coloured_graph (after: valuation register_lattice): Type[1] ≝
{ colouring: vertex → decision
; spilled_no: nat
; prop_colouring: ∀l. ∀v1, v2: vertex.v1 ≠v2 → 
  lives v1 (after l) → lives v2 (after l) → colouring v1 ≠ colouring v2
; prop_spilled_no: (*CSC: the exist-guarded premise is just to make the proof more general *)
   ∀v1:vertex. (∃l. bool_to_Prop (lives v1 (after l))) → ∀i. colouring v1 = decision_spill i → i < spilled_no
}.

definition coloured_graph_computer ≝
 ∀globals.
  joint_internal_function ERTLptr globals →
   ∀valuation.
    coloured_graph valuation.