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Require Import Coq.NArith.BinNat.
Require Import Coq.Strings.Ascii.
Require Import Coq.Strings.String.
From stdpp Require Import fin_sets gmap option.
From mininix Require Import expr maptools.
Open Scope string_scope.
Open Scope N_scope.
Open Scope Z_scope.
Open Scope nat_scope.
Definition nonrecs : gmap string b_rhs → gmap string expr :=
omap (λ b, match b with B_Nonrec e => Some e | _ => None end).
Lemma nonrecs_nonrec_fmap bs : nonrecs (B_Nonrec <$> bs) = bs.
Proof.
induction bs using map_ind; [done|].
unfold nonrecs.
rewrite fmap_insert.
rewrite omap_insert_Some with (y := x); try done.
by f_equal.
Qed.
Lemma rec_subst_nonrec_fmap bs : rec_subst (B_Nonrec <$> bs) = bs.
Proof.
unfold rec_subst.
rewrite <-map_fmap_compose.
unfold compose.
by rewrite map_fmap_id.
Qed.
Definition ascii_ltb (c1 c2 : ascii) : bool :=
bool_decide (N_of_ascii c1 < N_of_ascii c2)%N.
Fixpoint string_lt (s1 s2 : string) : bool :=
match s1, s2 with
| _, EmptyString => false
| EmptyString, String _ _ => true
| String c1 s1', String c2 s2' => ascii_ltb c1 c2 || string_lt s1' s2'
end.
Fixpoint expr_eq (e1 e2 : expr) : bool :=
match e1, e2 with
| V_Null, V_Null => false
| V_Bool b1, V_Bool b2 => bool_decide (b1 = b2)
| V_Int n1, V_Int n2 => bool_decide (n1 = n2)
| V_Str s1, V_Str s2 => bool_decide (s1 = s2)
| V_Attrset bs1, V_Attrset bs2 => bool_decide (dom bs1 = dom bs2) &&
map_fold (λ x e1' acc, match bs2 !! x with
| None => false
| Some e2' => acc && expr_eq e1' e2'
end) true bs1
| _, _ => false
end.
Definition prelude : gmap string expr :=
{[ "true" := X_V true;
"false" := X_V false;
"null" := X_V V_Null;
"builtins" := X_Attrset
{[ "true" := B_Nonrec true;
"false" := B_Nonrec false;
"null" := B_Nonrec V_Null
]}
]}.
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