include "basics/logic.ma".

include "common/AST.ma".
include "common/CostLabel.ma".
include "common/FrontEndOps.ma".
include "common/Registers.ma".

include "ASM/Vector.ma".
include "common/Graphs.ma".

inductive statement : Type[0] ≝
| St_skip : label → statement
| St_cost : costlabel → label → statement
| St_const : ∀t. register → constant t → label → statement
| St_op1 : ∀t',t. unary_operation t t' → register → register → label → statement (* destination source *)
| St_op2 : ∀t',t1,t2. binary_operation t1 t2 t' → register → register → register → label → statement (* destination source1 source2 *)
| St_load : typ → register → register → label → statement
| St_store : typ → register → register → label → statement
| St_call_id : ident → list register → option register → label → statement
| St_call_ptr : register → list register → option register → label → statement
(* We're not using these just now, and they'd make the Traces.ma proofs more difficult.
| St_tailcall_id : ident → list register → statement
| St_tailcall_ptr : register → list register → statement
*)
| St_cond : register → label → label → statement
| St_return : statement
.

definition env_has : list (register × typ) → register → typ → Prop ≝
λl,r,t. Exists ? (λx. 〈r,t〉 = x) l.

definition statement_typed : list (register × typ) → statement → Prop ≝
λe,s. match s with
[ St_const t r _ _ ⇒ env_has e r t
| St_op1 t' t _ r' r _ ⇒ env_has e r' t' ∧ env_has e r t
| St_op2 t' t1 t2 _ r' r1 r2 _ ⇒ env_has e r1 t1 ∧ env_has e r2 t2 ∧ env_has e r' t'
| St_load t' addr dst _ ⇒ env_has e dst t' ∧ env_has e addr ASTptr
| St_store t' addr src _ ⇒ env_has e src t' ∧ env_has e addr ASTptr
| St_call_id id args dst _ ⇒ All ? (λr. ∃t'.env_has e r t') args ∧ (match dst with [ Some r ⇒ ∃t'.env_has e r t' | _ ⇒ True])
| St_call_ptr fnptr args dst _ ⇒ env_has e fnptr ASTptr ∧ All ? (λr. ∃t'.env_has e r t') args ∧ (match dst with [ Some r ⇒ ∃t'.env_has e r t' | _ ⇒ True])
| St_cond r _ _ ⇒ ∃t. env_has e r t
| _ ⇒ True
].

definition labels_P : (label → Prop) → statement → Prop ≝
λP,s. match s with
[ St_skip l ⇒ P l
| St_cost _ l ⇒ P l
| St_const _ _ _ l ⇒ P l
| St_op1 _ _ _ _ _ l ⇒ P l
| St_op2 _ _ _ _ _ _ _ l ⇒ P l
| St_load _ _ _ l ⇒ P l
| St_store _ _ _ l ⇒ P l
| St_call_id _ _ _ l ⇒ P l
| St_call_ptr _ _ _ l ⇒ P l
(*
| St_tailcall_id _ _ ⇒ True
| St_tailcall_ptr _ _ ⇒ True
*)
| St_cond _ l1 l2 ⇒ P l1 ∧ P l2
| St_return ⇒ True
].

lemma labels_P_mp : ∀P,Q. (∀l. P l → Q l) → ∀s.labels_P P s → labels_P Q s.
#P #Q #H * /3/
#r #l #l' * /3/
qed.

definition labels_present : graph statement → statement → Prop ≝
λg,s. labels_P (present ?? g) s.

definition forall_nodes : ∀A.∀P:A → Prop. graph A → Prop ≝
λA,P,g. ∀l,n. lookup ?? g l = Some ? n → P n.

definition graph_closed : graph statement → Prop ≝
  λg. forall_nodes ? (labels_present g) g.
definition graph_typed : list (register × typ) → graph statement → Prop ≝
  λe. forall_nodes ? (statement_typed e).

record internal_function : Type[0] ≝
{ f_labgen    : universe LabelTag
; f_reggen    : universe RegisterTag
; f_result    : option (register × typ)
; f_params    : list (register × typ)
; f_locals    : list (register × typ)
; f_stacksize : nat
; f_graph     : graph statement
; f_closed    : graph_closed f_graph
; f_typed     : graph_typed (f_locals @ f_params) f_graph
; f_entry     : Σl:label. present ?? f_graph l
; f_exit      : Σl:label. present ?? f_graph l
}.

(* Note that the global variables will be initialised by the code in main
   by this stage, so the only initialisation data is the amount of space to
   allocate. *)

definition RTLabs_program ≝ program (λ_.fundef internal_function) nat.