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Add related_hetero_and_Proper #128

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Dec 3, 2023
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Add related_hetero_and_Proper #128

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28 changes: 20 additions & 8 deletions src/Rewriter/Language/Language.v
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Original file line number Diff line number Diff line change
Expand Up @@ -100,6 +100,26 @@ Module Compilers.
| arrow s d => respectful_hetero _ _ _ _ (@related_hetero _ _ _ R s) (fun _ _ => @related_hetero _ _ _ R d)
end%signature.

Fixpoint related_hetero_and_Proper {skip_base : bool} {base_type} {base_interp1 base_interp2 : base_type -> Type}
(R1 : forall t, relation (base_interp1 t))
(R2 : forall t, relation (base_interp2 t))
(R : forall t, base_interp1 t -> base_interp2 t -> Prop) {t : type base_type}
: interp base_interp1 t -> interp base_interp2 t -> Prop
:= match t return interp base_interp1 t -> interp base_interp2 t -> Prop with
| base t
=> fun f1 f2
=> if skip_base
then R t f1 f2
else Proper (R1 _) f1
/\ Proper (R2 _) f2
/\ R t f1 f2
| arrow s d
=> fun f1 f2
=> Proper (related R1) f1
/\ Proper (related R2) f2
/\ respectful_hetero _ _ _ _ (@related_hetero_and_Proper skip_base _ _ _ R1 R2 R s) (fun _ _ => @related_hetero_and_Proper skip_base _ _ _ R1 R2 R d) f1 f2
end%signature.

Fixpoint related_hetero3 {base_type} {base_interp1 base_interp2 base_interp3 : base_type -> Type}
(R : forall t, base_interp1 t -> base_interp2 t -> base_interp3 t -> Prop) {t : type base_type}
: interp base_interp1 t -> interp base_interp2 t -> interp base_interp3 t -> Prop
Expand Down Expand Up @@ -166,14 +186,6 @@ Module Compilers.

Notation is_not_higher_order := (@is_not_higher_order_than 1).

Lemma eqv_of_is_not_higher_order {base_type base_interp t}
(H : is_not_higher_order t = true)
: forall v, Proper (@related base_type base_interp (fun _ => eq) t) v.
Proof.
cbv [Proper]; induction t; cbn; eauto; try apply HR; repeat intro; cbn in *.
cbv [is_base Proper] in *; break_innermost_match_hyps; cbn in *; subst; try congruence; eauto.
Qed.

Section interpM.
Context {base_type} (M : Type -> Type) (base_interp : base_type -> Type).
(** half-monadic denotation function; denote [type]s into their
Expand Down
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