Definition ExprVarResolver Γ := VV -> LeveledHaskType Γ ★.
- Definition ujudg2exprType {Γ}{Δ}(ξ:ExprVarResolver Γ)(j:UJudg Γ Δ) : Type :=
- match j with
- mkUJudg Σ τ => forall vars, Σ = mapOptionTree ξ vars ->
- FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
- end.
-
Definition judg2exprType (j:Judg) : Type :=
match j with
(Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
apply X0.
Defined.
- Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) l1 l2 q,
- update_ξ ξ (app l1 l2) q = update_ξ (update_ξ ξ l2) l1 q.
+ Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) lev l1 l2 q,
+ update_ξ ξ lev (app l1 l2) q = update_ξ (update_ξ ξ lev l2) lev l1 q.
intros.
induction l1.
reflexivity.
reflexivity.
Qed.
- Lemma mapOptionTree_extensional {A}{B}(f g:A->B) : (forall a, f a = g a) -> (forall t, mapOptionTree f t = mapOptionTree g t).
- intros.
- induction t.
- destruct a; auto.
- simpl; rewrite H; auto.
- simpl; rewrite IHt1; rewrite IHt2; auto.
- Qed.
-
Lemma quark {T} (l1:list T) l2 vf :
(In vf (app l1 l2)) <->
(In vf l1) \/ (In vf l2).
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, Σ = mapOptionTree ξ vars ->
+ : forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars ->
FreshM { varstypes : _
- | mapOptionTree (update_ξ(Γ:=Γ) ξ (leaves varstypes)) vars = Σ
- /\ mapOptionTree (update_ξ ξ (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = types }.
+ | mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ
+ /\ mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+ /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
induction types.
intros; destruct a.
refine (bind vf = fresh (leaves vars) ; return _).
apply FreshMon.
destruct vf as [ vf vf_pf ].
- exists [(vf,l)].
+ exists [(vf,h)].
split; auto.
simpl.
- set (helper VV _ vars vf ξ l vf_pf) as q.
+ set (helper VV _ vars vf ξ (h@@lev) vf_pf) as q.
rewrite q.
symmetry; auto.
simpl.
destruct (eqd_dec vf vf); [ idtac | set (n (refl_equal _)) as n'; inversion n' ]; auto.
+ split; auto.
+ apply distinct_cons.
+ intro.
+ inversion H0.
+ apply distinct_nil.
refine (return _).
exists []; auto.
- intros vars Σ ξ pf; refine (bind x2 = IHtypes2 vars Σ ξ pf; _).
+ split.
+ simpl.
+ symmetry; auto.
+ split.
+ simpl.
+ reflexivity.
+ simpl.
+ apply distinct_nil.
+ intros vars Σ ξ lev pf; refine (bind x2 = IHtypes2 vars Σ ξ lev pf; _).
apply FreshMon.
- destruct x2 as [vt2 [pf21 pf22]].
- refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,types2) (update_ξ ξ (leaves vt2)) _; return _).
+ destruct x2 as [vt2 [pf21 [pf22 pfdist]]].
+ refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_ξ ξ lev
+ (leaves vt2)) _ _; return _).
apply FreshMon.
simpl.
rewrite pf21.
destruct x1 as [vt1 [pf11 pf12]].
exists (vt1,,vt2); split; auto.
- set (update_branches Γ ξ (leaves vt1) (leaves vt2)) as q.
+ set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
set (mapOptionTree_extensional _ _ q) as q'.
rewrite q'.
clear q' q.
reflexivity.
simpl.
- set (update_branches Γ ξ (leaves vt1) (leaves vt2)) as q.
+ set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
set (mapOptionTree_extensional _ _ q) as q'.
rewrite q'.
rewrite q'.
rewrite <- mapOptionTree_compose.
rewrite <- mapOptionTree_compose.
rewrite <- mapOptionTree_compose in *.
- rewrite pf12.
+ split.
+ destruct pf12.
+ rewrite H.
inversion pf11.
rewrite <- mapOptionTree_compose.
reflexivity.
+
+ admit.
Defined.
- Lemma fresh_lemma Γ ξ vars Σ Σ'
+ Lemma fresh_lemma Γ ξ vars Σ Σ' lev
: Σ = mapOptionTree ξ vars ->
FreshM { vars' : _
- | mapOptionTree (update_ξ(Γ:=Γ) ξ ((vars',Σ')::nil)) vars = Σ
- /\ mapOptionTree (update_ξ ξ ((vars',Σ')::nil)) [vars'] = [Σ'] }.
+ | mapOptionTree (update_ξ(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ
+ /\ mapOptionTree (update_ξ ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }.
intros.
- set (fresh_lemma' Γ [Σ'] vars Σ ξ H) as q.
+ set (fresh_lemma' Γ [Σ'] vars Σ ξ lev H) as q.
refine (q >>>= fun q' => return _).
apply FreshMon.
clear q.
- destruct q' as [varstypes [pf1 pf2]].
+ destruct q' as [varstypes [pf1 [pf2 pfdist]]].
destruct varstypes; try destruct o; try destruct p; simpl in *.
destruct (eqd_dec v v); [ idtac | set (n (refl_equal _)) as n'; inversion n' ].
inversion pf2; subst.
inversion pf2.
Defined.
- Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
- FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
- intros.
- set (fresh_lemma' Γ Σ [] [] ξ0 (refl_equal _)) as q.
- refine (q >>>= fun q' => return _).
- apply FreshMon.
- clear q.
- destruct q' as [varstypes [pf1 pf2]].
- exists (mapOptionTree (@fst _ _) varstypes).
- exists (update_ξ ξ0 (leaves varstypes)).
- symmetry; auto.
- Defined.
-
- Definition urule2expr : forall Γ Δ h j (r:@URule Γ Δ h j) (ξ:VV -> LeveledHaskType Γ ★),
- ITree _ (ujudg2exprType ξ) h -> ITree _ (ujudg2exprType ξ) j.
-
- refine (fix urule2expr Γ Δ h j (r:@URule Γ Δ h j) ξ {struct r} :
- ITree _ (ujudg2exprType ξ) h -> ITree _ (ujudg2exprType ξ) j :=
- match r as R in URule H C return ITree _ (ujudg2exprType ξ) H -> ITree _ (ujudg2exprType ξ) C with
- | RLeft h c ctx r => let case_RLeft := tt in (fun e => _) (urule2expr _ _ _ _ r)
- | RRight h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ _ r)
- | RCanL t a => let case_RCanL := tt in _
- | RCanR t a => let case_RCanR := tt in _
- | RuCanL t a => let case_RuCanL := tt in _
- | RuCanR t a => let case_RuCanR := tt in _
- | RAssoc t a b c => let case_RAssoc := tt in _
- | RCossa t a b c => let case_RCossa := tt in _
- | RExch t a b => let case_RExch := tt in _
- | RWeak t a => let case_RWeak := tt in _
- | RCont t a => let case_RCont := tt in _
+ Definition ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ : Type :=
+ forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ).
+
+ Definition urule2expr : forall Γ Δ h j t (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
+ ujudg2exprType Γ ξ Δ h t ->
+ ujudg2exprType Γ ξ Δ j t
+ .
+ intros Γ Δ.
+ refine (fix urule2expr h j t (r:@Arrange _ h j) ξ {struct r} :
+ ujudg2exprType Γ ξ Δ h t ->
+ ujudg2exprType Γ ξ Δ j t :=
+ match r as R in Arrange H C return
+ ujudg2exprType Γ ξ Δ H t ->
+ ujudg2exprType Γ ξ Δ C t
+ with
+ | RLeft h c ctx r => let case_RLeft := tt in (fun e => _) (urule2expr _ _ _ r)
+ | RRight h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ r)
+ | RCanL a => let case_RCanL := tt in _
+ | RCanR a => let case_RCanR := tt in _
+ | RuCanL a => let case_RuCanL := tt in _
+ | RuCanR a => let case_RuCanR := tt in _
+ | RAssoc a b c => let case_RAssoc := tt in _
+ | RCossa a b c => let case_RCossa := tt in _
+ | RExch a b => let case_RExch := tt in _
+ | RWeak a => let case_RWeak := tt in _
+ | RCont a => let case_RCont := tt in _
+ | RComp a b c f g => let case_RComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ f) (urule2expr _ _ _ g)
end); clear urule2expr; intros.
destruct case_RCanL.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
- apply (X0 ([],,vars)).
+ simpl; unfold ujudg2exprType; intros.
+ simpl in X.
+ apply (X ([],,vars)).
simpl; rewrite <- H; auto.
destruct case_RCanR.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
- apply (X0 (vars,,[])).
+ simpl; unfold ujudg2exprType; intros.
+ simpl in X.
+ apply (X (vars,,[])).
simpl; rewrite <- H; auto.
destruct case_RuCanL.
- apply ILeaf; simpl; intros.
+ simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
- inversion X.
- simpl in X0.
- apply (X0 vars2); auto.
+ simpl in X.
+ apply (X vars2); auto.
destruct case_RuCanR.
- apply ILeaf; simpl; intros.
+ simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
- inversion X.
- simpl in X0.
- apply (X0 vars1); auto.
+ simpl in X.
+ apply (X vars1); auto.
destruct case_RAssoc.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
+ simpl; unfold ujudg2exprType; intros.
+ simpl in X.
destruct vars; try destruct o; inversion H.
destruct vars1; try destruct o; inversion H.
- apply (X0 (vars1_1,,(vars1_2,,vars2))).
+ apply (X (vars1_1,,(vars1_2,,vars2))).
subst; auto.
destruct case_RCossa.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
+ simpl; unfold ujudg2exprType; intros.
+ simpl in X.
destruct vars; try destruct o; inversion H.
destruct vars2; try destruct o; inversion H.
- apply (X0 ((vars1,,vars2_1),,vars2_2)).
+ apply (X ((vars1,,vars2_1),,vars2_2)).
subst; auto.
+ destruct case_RExch.
+ simpl; unfold ujudg2exprType ; intros.
+ simpl in X.
+ destruct vars; try destruct o; inversion H.
+ apply (X (vars2,,vars1)).
+ inversion H; subst; auto.
+
+ destruct case_RWeak.
+ simpl; unfold ujudg2exprType; intros.
+ simpl in X.
+ apply (X []).
+ auto.
+
+ destruct case_RCont.
+ simpl; unfold ujudg2exprType ; intros.
+ simpl in X.
+ apply (X (vars,,vars)).
+ simpl.
+ rewrite <- H.
+ auto.
+
destruct case_RLeft.
- destruct c; [ idtac | apply no_urules_with_multiple_conclusions in r0; inversion r0; exists c1; exists c2; auto ].
- destruct o; [ idtac | apply INone ].
- destruct u; simpl in *.
- apply ILeaf; simpl; intros.
+ intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
- set (fun q => ileaf (e ξ q)) as r'.
- simpl in r'.
- apply r' with (vars:=vars2).
- clear r' e.
- clear r0.
- induction h0.
- destruct a.
- destruct u.
+ apply (fun q => e ξ q vars2 H2).
+ clear r0 e H2.
simpl in X.
- apply ileaf in X.
- apply ILeaf.
simpl.
- simpl in X.
+ unfold ujudg2exprType.
intros.
apply X with (vars:=vars1,,vars).
- simpl.
rewrite H0.
rewrite H1.
+ simpl.
reflexivity.
- apply INone.
- apply IBranch.
- apply IHh0_1. inversion X; auto.
- apply IHh0_2. inversion X; auto.
- auto.
-
+
destruct case_RRight.
- destruct c; [ idtac | apply no_urules_with_multiple_conclusions in r0; inversion r0; exists c1; exists c2; auto ].
- destruct o; [ idtac | apply INone ].
- destruct u; simpl in *.
- apply ILeaf; simpl; intros.
+ intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
- set (fun q => ileaf (e ξ q)) as r'.
- simpl in r'.
- apply r' with (vars:=vars1).
- clear r' e.
- clear r0.
- induction h0.
- destruct a.
- destruct u.
+ apply (fun q => e ξ q vars1 H1).
+ clear r0 e H2.
simpl in X.
- apply ileaf in X.
- apply ILeaf.
simpl.
- simpl in X.
+ unfold ujudg2exprType.
intros.
apply X with (vars:=vars,,vars2).
- simpl.
rewrite H0.
- rewrite H2.
+ inversion H.
+ simpl.
reflexivity.
- apply INone.
- apply IBranch.
- apply IHh0_1. inversion X; auto.
- apply IHh0_2. inversion X; auto.
- auto.
- destruct case_RExch.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
- destruct vars; try destruct o; inversion H.
- apply (X0 (vars2,,vars1)).
- inversion H; subst; auto.
-
- destruct case_RWeak.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
- apply (X0 []).
- auto.
-
- destruct case_RCont.
- apply ILeaf; simpl; intros.
- inversion X.
- simpl in X0.
- apply (X0 (vars,,vars)).
- simpl.
- rewrite <- H.
- auto.
+ destruct case_RComp.
+ apply e2.
+ apply e1.
+ apply X.
Defined.
- Definition bridge Γ Δ (c:Tree ??(UJudg Γ Δ)) ξ :
- ITree Judg judg2exprType (mapOptionTree UJudg2judg c) -> ITree (UJudg Γ Δ) (ujudg2exprType ξ) c.
- intro it.
- induction c.
- destruct a.
- destruct u; simpl in *.
- apply ileaf in it.
- apply ILeaf.
- simpl in *.
- intros; apply it with (vars:=vars); auto.
- apply INone.
- apply IBranch; [ apply IHc1 | apply IHc2 ]; inversion it; auto.
- Defined.
-
- Definition letrec_helper Γ Δ l varstypes ξ' :
+ Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' :
ITree (LeveledHaskType Γ ★)
(fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t)
(mapOptionTree (ξ' ○ (@fst _ _)) varstypes)
- -> ELetRecBindings Γ Δ ξ' l
- (mapOptionTree (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) ⟩) varstypes).
+ -> ELetRecBindings Γ Δ ξ' l varstypes.
intros.
induction varstypes.
destruct a; simpl in *.
destruct p.
- destruct l0 as [τ l'].
simpl.
apply ileaf in X. simpl in X.
- destruct (eqd_dec (unlev (ξ' v)) τ).
- rewrite <- e.
apply ELR_leaf.
+ rename h into τ.
+ destruct (eqd_dec (unlev (ξ' v)) τ).
+ rewrite <- e.
destruct (ξ' v).
simpl.
destruct (eqd_dec h0 l).
rewrite <- e0.
apply X.
- apply (Prelude_error "level mismatch; should never happen").
- apply (Prelude_error "letrec type mismatch; should never happen").
+ apply (Prelude_error "level mismatch; should never happen").
+ apply (Prelude_error "letrec type mismatch; should never happen").
apply ELR_nil.
+ apply ELR_branch.
+ apply IHvarstypes1; inversion X; auto.
+ apply IHvarstypes2; inversion X; auto.
+ Defined.
- simpl; apply ELR_branch.
- apply IHvarstypes1.
- simpl in X.
- inversion X; auto.
- apply IHvarstypes2.
- simpl in X.
- inversion X; auto.
-
+ 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.
- Lemma leaves_unleaves {T}(t:list T) : leaves (unleaves t) = t.
- induction t; auto.
- simpl.
- rewrite IHt; auto.
- Qed.
+ 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 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))}.
- intros.
+ 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 (map (fun x => x@@(weakL' lev)) (vec2list boundvars))) Σ
- (mapOptionTree weakLT' pcb_freevars)
- (weakLT' ○ ξ) _
- >>>= fun ξvars => _). apply FreshMon.
- rewrite H.
- rewrite <- mapOptionTree_compose.
- reflexivity.
- destruct ξvars as [ exprvars [pf1 pf2 ]].
- set (list2vec (leaves (mapOptionTree (@fst _ _) exprvars))) as exprvars'.
- 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 (weakLT' ○ ξ) (leaves exprvars) ○ (@fst _ _)).
- rewrite <- mapleaves.
- rewrite pf2.
- rewrite leaves_unleaves.
- rewrite vec2list_map_list2vec.
- rewrite vec2list_len.
- reflexivity.
- rewrite <- H' in exprvars'.
- clear H'.
-
- set (@Build_StrongCaseBranchWithVVs VV _ tc _ avars pcb_scb exprvars') 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.
- rewrite <- vec2list_map_list2vec.
- rewrite <- pf1.
- admit.
-
- apply ileaf in X'.
- simpl in X'.
- exists scb.
- unfold weakCK''.
- unfold ξ' in X'.
- apply X'.
- Defined.
+ rewrite mapleaves'.
- Fixpoint treeM {T}(t:Tree ??(FreshM T)) : FreshM (Tree ??T) :=
- match t with
- | T_Leaf None => return []
- | T_Leaf (Some x) => bind x' = x ; return [x']
- | T_Branch b1 b2 => bind b1' = treeM b1 ; bind b2' = treeM b2 ; return (b1',,b2')
- end.
+ admit.
- Lemma itree_mapOptionTree : forall T T' F (f:T->T') t,
- ITree _ F (mapOptionTree f t) ->
- ITree _ (F ○ f) t.
+ exists scb.
+ apply ileaf in q.
+ apply q.
+
+ admit.
+ admit.
+ Defined.
+
+ 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.
intros h j r.
refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with
- | RURule a b c d e => let case_RURule := tt in _
+ | RArrange a b c d e r => let case_RURule := tt in _
| RNote Γ Δ Σ τ l n => let case_RNote := tt in _
| RLit Γ Δ l _ => let case_RLit := tt in _
| RVar Γ Δ σ p => let case_RVar := tt in _
| RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _
| RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
| RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
- | RBindingGroup Γ p lri m x q => let case_RBindingGroup := tt in _
- | REmptyGroup _ _ => let case_REmptyGroup := tt in _
+ | RJoin Γ p lri m x q => let case_RJoin := tt in _
+ | RVoid _ _ => let case_RVoid := tt in _
| RBrak Σ a b c n m => let case_RBrak := tt in _
| REsc Σ a b c n m => let case_REsc := tt in _
| RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
- | RLetRec Γ Δ lri x y => let case_RLetRec := tt in _
+ | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
end); intro X_; try apply ileaf in X_; simpl in X_.
- destruct case_RURule.
- destruct d; try destruct o.
- apply ILeaf; destruct u; simpl; intros.
- set (@urule2expr a b _ _ e ξ) as q.
- set (fun z => ileaf (q z)) as q'.
+ destruct case_RURule.
+ apply ILeaf. simpl. intros.
+ set (@urule2expr a b _ _ e r0 ξ) as q.
+ set (fun z => q z) as q'.
simpl in q'.
apply q' with (vars:=vars).
clear q' q.
- apply bridge.
- apply X_.
+ unfold ujudg2exprType.
+ intros.
+ apply X_ with (vars:=vars0).
+ auto.
auto.
- apply no_urules_with_empty_conclusion in e; inversion e; auto.
- apply no_urules_with_multiple_conclusions in e; inversion e; auto; exists d1; exists d2; auto.
destruct case_RBrak.
apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
destruct case_RLam.
apply ILeaf.
simpl in *; intros.
- refine (fresh_lemma _ ξ vars _ (tx@@x) H >>>= (fun pf => _)).
+ refine (fresh_lemma _ ξ vars _ tx x H >>>= (fun pf => _)).
apply FreshMon.
destruct pf as [ vnew [ pf1 pf2 ]].
- set (update_ξ ξ ((⟨vnew, tx @@ x ⟩) :: nil)) as ξ' in *.
+ set (update_ξ ξ x (((vnew, tx )) :: nil)) as ξ' in *.
refine (X_ ξ' (vars,,[vnew]) _ >>>= _).
apply FreshMon.
simpl.
apply ileaf in X. simpl in X.
apply X.
- destruct case_RBindingGroup.
+ destruct case_RJoin.
apply ILeaf; simpl; intros.
inversion X_.
apply ileaf in X.
apply ILeaf.
simpl in *; intros.
destruct vars; try destruct o; inversion H.
- refine (fresh_lemma _ ξ vars1 _ (σ₂@@p) H1 >>>= (fun pf => _)).
+ refine (fresh_lemma _ ξ vars1 _ σ₂ p H1 >>>= (fun pf => _)).
apply FreshMon.
destruct pf as [ vnew [ pf1 pf2 ]].
- set (update_ξ ξ ((⟨vnew, σ₂ @@ p ⟩) :: nil)) as ξ' in *.
+ set (update_ξ ξ p (((vnew, σ₂ )) :: nil)) as ξ' in *.
inversion X_.
apply ileaf in X.
apply ileaf in X0.
simpl in *.
- refine (X0 ξ vars2 _ >>>= fun X0' => _).
+ refine (X ξ vars2 _ >>>= fun X0' => _).
apply FreshMon.
auto.
- refine (X ξ' (vars1,,[vnew]) _ >>>= fun X1' => _).
+
+ refine (X0 ξ' (vars1,,[vnew]) _ >>>= fun X1' => _).
apply FreshMon.
rewrite H1.
simpl.
rewrite pf1.
rewrite H1.
reflexivity.
+
refine (return _).
apply ILeaf.
apply ileaf in X0'.
apply X0'.
apply X1'.
- destruct case_REmptyGroup.
+ destruct case_RVoid.
apply ILeaf; simpl; intros.
refine (return _).
apply INone.
destruct case_RLetRec.
apply ILeaf; simpl; intros.
- refine (bind ξvars = fresh_lemma' _ y _ _ _ H; _). apply FreshMon.
- destruct ξvars as [ varstypes [ pf1 pf2 ]].
- refine (X_ ((update_ξ ξ (leaves varstypes)))
+ refine (bind ξvars = fresh_lemma' _ y _ _ _ t H; _). apply FreshMon.
+ destruct ξvars as [ varstypes [ pf1[ pf2 pfdist]]].
+ refine (X_ ((update_ξ ξ t (leaves varstypes)))
(vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_. apply FreshMon.
simpl.
rewrite pf2.
rewrite pf1.
auto.
apply ILeaf.
- destruct x as [τ l].
inversion X; subst; clear X.
- (* getting rid of this will require strengthening RLetRec *)
- assert ((mapOptionTree (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) @@ l ⟩) varstypes) = varstypes) as HHH.
- admit.
-
- apply (@ELetRec _ _ _ _ _ _ _ (mapOptionTree (fun x => ((fst x),unlev (snd x))) varstypes));
- rewrite mapleaves; rewrite <- map_compose; simpl;
- [ idtac
- | rewrite <- mapleaves; rewrite HHH; apply (ileaf X0) ].
-
- clear X0.
- rewrite <- mapOptionTree_compose in X1.
- set (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) @@ l ⟩) as ξ' in *.
- rewrite <- mapleaves.
- rewrite HHH.
-
- apply (letrec_helper _ _ _ _ _ X1).
+ apply (@ELetRec _ _ _ _ _ _ _ varstypes).
+ apply (@letrec_helper Γ Δ t varstypes).
+ rewrite <- pf2 in X1.
+ rewrite mapOptionTree_compose.
+ apply X1.
+ apply ileaf in X0.
+ apply X0.
destruct case_RCase.
apply ILeaf; simpl; intros.
subst.
apply ileaf in X0.
simpl in X0.
- set (mapOptionTreeAndFlatten pcb_freevars alts) as Σalts in *.
- refine (bind ξvars = fresh_lemma' _ (Σalts,,Σ) _ _ _ 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.
+ Definition closed2expr : forall c (pn:@ClosedSIND _ Rule c), ITree _ judg2exprType c.
refine ((
- fix closed2expr' j (pn:@ClosedND _ Rule j) {struct pn} : ITree _ judg2exprType j :=
- match pn in @ClosedND _ _ J return ITree _ judg2exprType J with
+ fix closed2expr' j (pn:@ClosedSIND _ Rule j) {struct pn} : ITree _ judg2exprType j :=
+ match pn in @ClosedSIND _ _ J return ITree _ judg2exprType J with
| cnd_weak => let case_nil := tt in INone _ _
| cnd_rule h c cnd' r => let case_rule := tt in rule2expr _ _ r (closed2expr' _ cnd')
| cnd_branch _ _ c1 c2 => let case_branch := tt in IBranch _ _ (closed2expr' _ c1) (closed2expr' _ c2)
end)); clear closed2expr'; intros; subst.
Defined.
+ Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
+ FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
+ intros Γ Σ.
+ induction Σ; intro ξ.
+ destruct a.
+ destruct l as [τ l].
+ set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ clear q.
+ destruct q' as [varstypes [pf1 [pf2 distpf]]].
+ exists (mapOptionTree (@fst _ _) varstypes).
+ exists (update_ξ ξ l (leaves varstypes)).
+ symmetry; auto.
+ refine (return _).
+ exists [].
+ exists ξ; auto.
+ refine (bind f1 = IHΣ1 ξ ; _).
+ apply FreshMon.
+ destruct f1 as [vars1 [ξ1 pf1]].
+ refine (bind f2 = IHΣ2 ξ1 ; _).
+ apply FreshMon.
+ destruct f2 as [vars2 [ξ2 pf22]].
+ refine (return _).
+ exists (vars1,,vars2).
+ exists ξ2.
+ simpl.
+ rewrite pf22.
+ rewrite pf1.
+ admit.
+ Defined.
+
Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
{zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] ->
FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
intro pf.
- set (closedFromSCND _ _ (mkSCND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) cnd_weak) as cnd.
+ set (closedFromSIND _ _ (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) cnd_weak) as cnd.
apply closed2expr in cnd.
apply ileaf in cnd.
simpl in *.
projects / coq-hetmet.git / blobdiff