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Require Import Crypto.Reflection.Syntax.
Require Import Crypto.Reflection.WfProofs.
Require Import Crypto.Util.Tactics Crypto.Util.Sigma Crypto.Util.Prod.
Local Open Scope ctype_scope.
Section language.
Context (base_type_code : Type).
Local Notation flat_type := (flat_type base_type_code).
Local Notation type := (type base_type_code).
Let Tbase := @Tbase base_type_code.
Local Coercion Tbase : base_type_code >-> Syntax.flat_type.
Context (interp_base_type : base_type_code -> Type).
Context (op : flat_type (* input tuple *) -> flat_type (* output type *) -> Type).
Let interp_type := interp_type interp_base_type.
Let interp_flat_type := interp_flat_type_gen interp_base_type.
Context (interp_op : forall src dst, op src dst -> interp_flat_type src -> interp_flat_type dst).
Lemma interpf_SmartVar t v
: Syntax.interpf interp_op (SmartVar (t:=t) v) = v.
Proof.
unfold SmartVar; induction t;
repeat match goal with
| _ => reflexivity
| _ => progress simpl in *
| _ => progress rewrite_hyp *
| _ => rewrite <- surjective_pairing
end.
Qed.
Lemma interpf_SmartConst {t t'} v x x'
(Hin : List.In
(existT (fun t : base_type_code => (exprf base_type_code interp_base_type op (Syntax.Tbase t) * interp_base_type t)%type)
t (x, x'))
(flatten_binding_list (t := t') base_type_code (SmartConst v) v))
: interpf interp_op x = x'.
Proof.
clear -Hin.
induction t'; simpl in *.
{ intuition (inversion_sigma; inversion_prod; subst; eauto). }
{ apply List.in_app_iff in Hin.
intuition (inversion_sigma; inversion_prod; subst; eauto). }
Qed.
Lemma interpf_SmartVarVar {t t'} v x x'
(Hin : List.In
(existT (fun t : base_type_code => (exprf base_type_code interp_base_type op (Syntax.Tbase t) * interp_base_type t)%type)
t (x, x'))
(flatten_binding_list (t := t') base_type_code (SmartVarVar v) v))
: interpf interp_op x = x'.
Proof.
clear -Hin.
induction t'; simpl in *.
{ intuition (inversion_sigma; inversion_prod; subst; eauto). }
{ apply List.in_app_iff in Hin.
intuition (inversion_sigma; inversion_prod; subst; eauto). }
Qed.
Lemma interpf_SmartVarVar_eq {t t'} v v' x x'
(Heq : v = v')
(Hin : List.In
(existT (fun t : base_type_code => (exprf base_type_code interp_base_type op (Syntax.Tbase t) * interp_base_type t)%type)
t (x, x'))
(flatten_binding_list (t := t') base_type_code (SmartVarVar v') v))
: interpf interp_op x = x'.
Proof.
subst; eapply interpf_SmartVarVar; eassumption.
Qed.
End language.
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