reflexivity.
Qed.
+ Lemma fresh_lemma'' Γ
+ : forall types ξ lev,
+ FreshM { varstypes : _
+ | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+ /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
+ admit.
+ Defined.
+
Lemma fresh_lemma' Γ
: forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars ->
FreshM { varstypes : _
apply IHvarstypes2; inversion X; auto.
Defined.
- Definition case_helper tc Γ Δ lev tbranches avars ξ (Σ:Tree ??VV) :
- forall pcb : ProofCaseBranch tc Γ Δ lev tbranches avars,
- judg2exprType (pcb_judg pcb) ->
- (pcb_freevars pcb) = mapOptionTree ξ Σ ->
- FreshM
- {scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars &
- Expr (sac_Γ scb Γ) (sac_Δ scb Γ avars (weakCK'' Δ))
- (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev))}.
+ Definition unindex_tree {V}{F} : forall {t:Tree ??V}, ITree V F t -> Tree ??{ v:V & F v }.
+ refine (fix rec t it := match it as IT return Tree ??{ v:V & F v } with
+ | INone => T_Leaf None
+ | ILeaf x y => T_Leaf (Some _)
+ | IBranch _ _ b1 b2 => (rec _ b1),,(rec _ b2)
+ end).
+ exists x; auto.
+ Defined.
- intros.
+ Definition fix_indexing X (F:X->Type)(J:X->Type)(t:Tree ??{ x:X & F x })
+ : ITree { x:X & F x } (fun x => J (projT1 x)) t
+ -> ITree X (fun x:X => J x) (mapOptionTree (@projT1 _ _) t).
+ intro it.
+ induction it; simpl in *.
+ apply INone.
+ apply ILeaf.
+ apply f.
+ simpl; apply IBranch; auto.
+ Defined.
+
+ Definition fix2 {X}{F} : Tree ??{ x:X & FreshM (F x) } -> Tree ??(FreshM { x:X & F x }).
+ refine (fix rec t := match t with
+ | T_Leaf None => T_Leaf None
+ | T_Leaf (Some x) => T_Leaf (Some _)
+ | T_Branch b1 b2 => T_Branch (rec b1) (rec b2)
+ end).
+ destruct x as [x fx].
+ refine (bind fx' = fx ; return _).
+ apply FreshMon.
+ exists x.
+ apply fx'.
+ Defined.
+
+ Definition case_helper tc Γ Δ lev tbranches avars ξ :
+ forall pcb:{sac : StrongAltCon & ProofCaseBranch tc Γ Δ lev tbranches avars sac},
+ prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} ->
+ ((fun sac => FreshM
+ { scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac
+ & Expr (sac_Γ sac Γ) (sac_Δ sac Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@ lev)) }) (projT1 pcb)).
+ intro pcb.
+ intro X.
simpl in X.
- destruct pcb.
+ simpl.
+ destruct pcb as [sac pcb].
simpl in *.
- set (sac_types pcb_scb _ avars) as boundvars.
- refine (fresh_lemma' _ (unleaves (vec2list boundvars)) Σ (mapOptionTree weakLT' pcb_freevars) (weakLT' ○ ξ) (weakL' lev) _
- >>>= fun ξvars => _). apply FreshMon.
- rewrite H.
- rewrite <- mapOptionTree_compose.
- reflexivity.
- destruct ξvars as [ exprvars [pf1 [pf2 pfdistinct]]].
- set (list2vec (leaves (mapOptionTree (@fst _ _) exprvars))) as exprvars'.
-
- assert (distinct (vec2list exprvars')) as pfdistinct'.
- unfold exprvars'.
- rewrite vec2list_list2vec.
- apply pfdistinct.
-
- assert (sac_numExprVars pcb_scb = Datatypes.length (leaves (mapOptionTree (@fst _ _) exprvars))) as H'.
- rewrite <- mapOptionTree_compose in pf2.
- simpl in pf2.
- rewrite mapleaves.
- rewrite <- map_preserves_length.
- rewrite map_preserves_length with (f:=
- (@update_ξ VV eqdec_vv (@sac_Γ tc pcb_scb Γ)
- (@weakLT' Γ (@vec2list (@sac_numExTyVars tc pcb_scb) Kind (@sac_ekinds tc pcb_scb)) ★ ○ ξ)
- (@weakL' Γ (@vec2list (@sac_numExTyVars tc pcb_scb) Kind (@sac_ekinds tc pcb_scb)) lev)
- (@leaves (VV * HaskType (@sac_Γ tc pcb_scb Γ) ★) exprvars) ○ @fst VV (HaskType (@sac_Γ tc pcb_scb Γ) ★))).
- rewrite <- mapleaves.
- rewrite pf2.
- rewrite <- mapleaves'.
- rewrite leaves_unleaves.
- rewrite vec2list_map_list2vec.
- rewrite vec2list_len.
- reflexivity.
-
- clear pfdistinct'.
- rewrite <- H' in exprvars'.
- clear H'.
- assert (distinct (vec2list exprvars')) as pfdistinct'.
- admit.
-
- set (@Build_StrongCaseBranchWithVVs VV _ tc _ avars pcb_scb exprvars' pfdistinct') as scb.
- set (scbwv_ξ scb ξ lev) as ξ'.
- refine (X ξ' (Σ,,(unleaves (vec2list exprvars'))) _ >>>= fun X' => return _). apply FreshMon.
+
+ destruct X.
+ destruct s as [vars vars_pf].
+
+ refine (bind localvars = fresh_lemma' _ (unleaves (vec2list (sac_types sac _ avars))) vars
+ (mapOptionTree weakLT' (pcb_freevars pcb)) (weakLT' ○ ξ) (weakL' lev) _ ; _).
+ apply FreshMon.
+ rewrite vars_pf.
+ rewrite <- mapOptionTree_compose.
+ reflexivity.
+ destruct localvars as [localvars [localvars_pf1 [localvars_pf2 localvars_dist ]]].
+ set (mapOptionTree (@fst _ _) localvars) as localvars'.
+
+ set (list2vec (leaves localvars')) as localvars''.
+ cut (length (leaves localvars') = sac_numExprVars sac). intro H''.
+ rewrite H'' in localvars''.
+ cut (distinct (vec2list localvars'')). intro H'''.
+ set (@Build_StrongCaseBranchWithVVs _ _ _ _ avars sac localvars'' H''') as scb.
+
+ refine (bind q = (f (scbwv_ξ scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _).
+ apply FreshMon.
simpl.
- unfold ξ'.
unfold scbwv_ξ.
+ rewrite vars_pf.
+ rewrite <- mapOptionTree_compose.
+ clear localvars_pf1.
simpl.
- admit.
-
- apply ileaf in X'.
- simpl in X'.
- exists scb.
- unfold weakCK''.
- unfold ξ' in X'.
- apply X'.
+ rewrite mapleaves'.
+
+ admit.
+
+ exists scb.
+ apply ileaf in q.
+ apply q.
+
+ admit.
+ admit.
Defined.
Fixpoint treeM {T}(t:Tree ??(FreshM T)) : FreshM (Tree ??T) :=
| T_Branch b1 b2 => bind b1' = treeM b1 ; bind b2' = treeM b2 ; return (b1',,b2')
end.
- Lemma itree_mapOptionTree : forall T T' F (f:T->T') t,
- ITree _ F (mapOptionTree f t) ->
- ITree _ (F ○ f) t.
+ Definition gather_branch_variables
+ Γ Δ (ξ:VV -> LeveledHaskType Γ ★) tc avars tbranches lev (alts:Tree ?? {sac : StrongAltCon &
+ ProofCaseBranch tc Γ Δ lev tbranches avars sac})
+ :
+ forall vars,
+ mapOptionTreeAndFlatten (fun x => pcb_freevars(Γ:=Γ) (projT2 x)) alts = mapOptionTree ξ vars
+ -> ITree Judg judg2exprType (mapOptionTree (fun x => pcb_judg (projT2 x)) alts)
+ -> ITree _ (fun q => prod (judg2exprType (pcb_judg (projT2 q)))
+ { vars' : _ & pcb_freevars (projT2 q) = mapOptionTree ξ vars' })
+ alts.
+ induction alts;
+ intro vars;
+ intro pf;
+ intro source.
+ destruct a; [ idtac | apply INone ].
+ simpl in *.
+ apply ileaf in source.
+ apply ILeaf.
+ destruct s as [sac pcb].
+ simpl in *.
+ split.
intros.
- induction t; try destruct a; simpl in *.
- apply ILeaf.
- inversion X; auto.
- apply INone.
- apply IBranch.
- apply IHt1; inversion X; auto.
- apply IHt2; inversion X; auto.
- Defined.
+ eapply source.
+ apply H.
+ clear source.
+
+ exists vars.
+ auto.
+
+ simpl in pf.
+ destruct vars; try destruct o; simpl in pf; inversion pf.
+ simpl in source.
+ inversion source.
+ subst.
+ apply IBranch.
+ apply (IHalts1 vars1 H0 X); auto.
+ apply (IHalts2 vars2 H1 X0); auto.
+
+ Defined.
+
Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j.
subst.
apply ileaf in X0.
simpl in X0.
- set (mapOptionTreeAndFlatten pcb_freevars alts) as Σalts in *.
- refine (bind ξvars = fresh_lemma' _ (mapOptionTree unlev (Σalts,,Σ)) _ _ _ lev H ; _).
- apply FreshMon.
- destruct vars; try destruct o; inversion H; clear H.
- rename vars1 into varsalts.
- rename vars2 into varsΣ.
- refine ( _ >>>= fun Y => X0 ξ varsΣ _ >>>= fun X => return ILeaf _ (@ECase _ _ _ _ _ _ _ _ _ (ileaf X) Y)); auto.
+ (* body_freevars and alts_freevars are the types of variables in the body and alternatives (respectively) which are free
+ * from the viewpoint just outside the case block -- i.e. not bound by any of the branches *)
+ rename Σ into body_freevars_types.
+ rename vars into all_freevars.
+ rename X0 into body_expr.
+ rename X into alts_exprs.
+
+ destruct all_freevars; try destruct o; inversion H.
+ rename all_freevars2 into body_freevars.
+ rename all_freevars1 into alts_freevars.
+
+ set (gather_branch_variables _ _ _ _ _ _ _ _ _ H1 alts_exprs) as q.
+ set (itmap (fun pcb alt_expr => case_helper tc Γ Δ lev tbranches avars ξ pcb alt_expr) q) as alts_exprs'.
+ apply fix_indexing in alts_exprs'.
+ simpl in alts_exprs'.
+ apply unindex_tree in alts_exprs'.
+ simpl in alts_exprs'.
+ apply fix2 in alts_exprs'.
+ apply treeM in alts_exprs'.
+
+ refine ( alts_exprs' >>>= fun Y =>
+ body_expr ξ _ _
+ >>>= fun X => return ILeaf _ (@ECase _ _ _ _ _ _ _ _ _ (ileaf X) Y)); auto.
apply FreshMon.
- destruct ξvars as [varstypes [pf1 pf2]].
-
- apply treeM.
- apply itree_mapOptionTree in X.
- refine (itree_to_tree (itmap _ X)).
- intros.
- eapply case_helper.
- apply X1.
- instantiate (1:=varsΣ).
- rewrite <- H2.
- admit.
apply FreshMon.
+ apply H2.
Defined.
Definition closed2expr : forall c (pn:@ClosedND _ Rule c), ITree _ judg2exprType c.