Section HaskStrongToProof.
-(* Whereas RLeft/RRight perform left and right context expansion on a single uniform rule, these functions perform
- * expansion on an entire uniform proof *)
-Definition ext_left {Γ}{Δ}(ctx:Tree ??(LeveledHaskType Γ ★))
- := @nd_map' _ _ _ _ (ext_tree_left ctx) (fun h c r => nd_rule (@RLeft Γ Δ h c ctx r)).
-Definition ext_right {Γ}{Δ}(ctx:Tree ??(LeveledHaskType Γ ★))
- := @nd_map' _ _ _ _ (ext_tree_right ctx) (fun h c r => nd_rule (@RRight Γ Δ h c ctx r)).
-
-Definition pivotContext {Γ}{Δ} a b c τ :
- @ND _ (@URule Γ Δ)
- [ Γ >> Δ > (a,,b),,c |- τ]
- [ Γ >> Δ > (a,,c),,b |- τ].
- set (ext_left a _ _ (nd_rule (@RExch Γ Δ τ c b))) as q.
- simpl in *.
- eapply nd_comp ; [ eapply nd_rule; apply RCossa | idtac ].
- eapply nd_comp ; [ idtac | eapply nd_rule; apply RAssoc ].
- apply q.
+Definition pivotContext {T} a b c : @Arrange T ((a,,b),,c) ((a,,c),,b) :=
+ RComp (RComp (RCossa _ _ _) (RLeft a (RExch c b))) (RAssoc _ _ _).
+
+Definition copyAndPivotContext {T} a b c : @Arrange T ((a,,b),,(c,,b)) ((a,,c),,b).
+ eapply RComp; [ idtac | apply (RLeft (a,,c) (RCont b)) ].
+ eapply RComp; [ idtac | apply RCossa ].
+ eapply RComp; [ idtac | apply (RRight b (pivotContext a b c)) ].
+ apply RAssoc.
Defined.
-
-Definition copyAndPivotContext {Γ}{Δ} a b c τ :
- @ND _ (@URule Γ Δ)
- [ Γ >> Δ > (a,,b),,(c,,b) |- τ]
- [ Γ >> Δ > (a,,c),,b |- τ].
- set (ext_left (a,,c) _ _ (nd_rule (@RCont Γ Δ τ b))) as q.
- simpl in *.
- eapply nd_comp; [ idtac | apply q ].
- clear q.
- eapply nd_comp ; [ idtac | eapply nd_rule; apply RCossa ].
- set (ext_right b _ _ (@pivotContext _ Δ a b c τ)) as q.
- simpl in *.
- eapply nd_comp ; [ idtac | apply q ].
- clear q.
- apply nd_rule.
- apply RAssoc.
- Defined.
-
-
Context {VV:Type}{eqd_vv:EqDecidable VV}.
- (* maintenance of Xi *)
- Fixpoint dropVar (lv:list VV)(v:VV) : ??VV :=
- match lv with
- | nil => Some v
- | v'::lv' => if eqd_dec v v' then None else dropVar lv' v
- end.
+(* maintenance of Xi *)
+Fixpoint dropVar (lv:list VV)(v:VV) : ??VV :=
+ match lv with
+ | nil => Some v
+ | v'::lv' => if eqd_dec v v' then None else dropVar lv' v
+ end.
Fixpoint mapOptionTree' {a b:Type}(f:a->??b)(t:@Tree ??a) : @Tree ??b :=
match t with
reflexivity.
Qed.
-Lemma strip_twice_lemma x y t : stripOutVars x (stripOutVars y t) = stripOutVars (app y x) t.
-(*
- induction x.
- simpl.
+Lemma strip_nil_lemma t : stripOutVars nil t = t.
+ induction t; simpl.
+ unfold stripOutVars.
+ destruct a; reflexivity.
+ rewrite <- IHt1 at 2.
+ rewrite <- IHt2 at 2.
+ reflexivity.
+ Qed.
+
+Lemma strip_swap0_lemma : forall a a0 y t,
+ stripOutVars (a :: a0 :: y) t = stripOutVars (a0 :: a :: y) t.
+ intros.
unfold stripOutVars.
- simpl.
- rewrite mapOptionTree'_compose.
induction t.
- destruct a; try reflexivity.
- simpl.
- destruct (dropVar y v); reflexivity.
- simpl.
- rewrite IHt1.
- rewrite IHt2.
- reflexivity.
- rewrite strip_lemma.
- rewrite IHx.
- rewrite <- strip_lemma.
- rewrite app_comm_cons.
- reflexivity.
-*)
-admit.
+ destruct a1; simpl; [ idtac | reflexivity ].
+ destruct (eqd_dec v a); subst.
+ destruct (eqd_dec a a0); subst.
+ reflexivity.
+ reflexivity.
+ destruct (eqd_dec v a0); subst.
+ reflexivity.
+ reflexivity.
+ simpl in *.
+ rewrite IHt1.
+ rewrite IHt2.
+ reflexivity.
+ Qed.
+
+Lemma strip_swap1_lemma : forall a y t,
+ stripOutVars (a :: nil) (stripOutVars y t) =
+ stripOutVars y (stripOutVars (a :: nil) t).
+ intros.
+ induction y.
+ rewrite strip_nil_lemma.
+ rewrite strip_nil_lemma.
+ reflexivity.
+ rewrite (strip_lemma a0 y (stripOutVars (a::nil) t)).
+ rewrite <- IHy.
+ clear IHy.
+ rewrite <- (strip_lemma a y t).
+ rewrite <- strip_lemma.
+ rewrite <- strip_lemma.
+ apply strip_swap0_lemma.
+ Qed.
+
+Lemma strip_swap_lemma : forall x y t, stripOutVars x (stripOutVars y t) = stripOutVars y (stripOutVars x t).
+ intros; induction t.
+ destruct a; simpl.
+
+ induction x.
+ rewrite strip_nil_lemma.
+ rewrite strip_nil_lemma.
+ reflexivity.
+ rewrite strip_lemma.
+ rewrite (strip_lemma a x [v]).
+ rewrite IHx.
+ clear IHx.
+ apply strip_swap1_lemma.
+ reflexivity.
+ unfold stripOutVars in *.
+ simpl.
+ rewrite IHt1.
+ rewrite IHt2.
+ reflexivity.
Qed.
-Lemma strip_distinct a y : (not (In a (leaves y))) -> stripOutVars (a :: nil) y = y.
+Lemma strip_twice_lemma x y t : stripOutVars x (stripOutVars y t) = stripOutVars (app x y) t.
+ induction x; simpl.
+ apply strip_nil_lemma.
+ rewrite strip_lemma.
+ rewrite IHx.
+ clear IHx.
+ rewrite <- strip_lemma.
+ reflexivity.
+ Qed.
+
+Lemma notin_strip_inert a y : (not (In a (leaves y))) -> stripOutVars (a :: nil) y = y.
intros.
induction y.
destruct a0; try reflexivity.
auto.
Qed.
-Lemma strip_distinct' y : forall x, distinct (app x (leaves y)) -> stripOutVars x y = y.
+Lemma notin_strip_inert' y : forall x, distinct (app x (leaves y)) -> stripOutVars x y = y.
induction x; intros.
simpl in H.
unfold stripOutVars.
set (IHx H3) as qq.
rewrite strip_lemma.
rewrite IHx.
- apply strip_distinct.
+ apply notin_strip_inert.
unfold not; intros.
apply H2.
apply In_both'.
auto.
Qed.
+Lemma dropvar_lemma v v' y : dropVar y v = Some v' -> v=v'.
+ intros.
+ induction y; auto.
+ simpl in H.
+ inversion H.
+ auto.
+ apply IHy.
+ simpl in H.
+ destruct (eqd_dec v a).
+ inversion H.
+ auto.
+ Qed.
+
Lemma updating_stripped_tree_is_inert'
{Γ} lev
(ξ:VV -> LeveledHaskType Γ ★)
lv tree2 :
mapOptionTree (update_ξ ξ lev lv) (stripOutVars (map (@fst _ _) lv) tree2)
= mapOptionTree ξ (stripOutVars (map (@fst _ _) lv) tree2).
+
induction tree2.
- destruct a.
- simpl.
- induction lv.
- reflexivity.
- simpl.
- destruct a.
- simpl.
- set (eqd_dec v v0) as q.
- destruct q.
- auto.
- set (dropVar (map (@fst _ _) lv) v) as b in *.
- destruct b.
- inversion IHlv.
- admit.
- auto.
- reflexivity.
+ destruct a; [ idtac | reflexivity ]; simpl.
+ induction lv; [ reflexivity | idtac ]; simpl.
+ destruct a; simpl.
+ set (eqd_dec v v0) as q; destruct q; auto.
+ set (dropVar (map (@fst _ _) lv) v) as b in *.
+ assert (dropVar (map (@fst _ _) lv) v=b). reflexivity.
+ destruct b; [ idtac | reflexivity ].
+ apply dropvar_lemma in H.
+ subst.
+ inversion IHlv.
+ rewrite H0.
+ clear H0 IHlv.
+ destruct (eqd_dec v0 v1).
+ subst.
+ assert False. apply n. intros; auto. inversion H.
+ reflexivity.
+ unfold stripOutVars in *.
+ simpl.
+ rewrite <- IHtree2_1.
+ rewrite <- IHtree2_2.
+ reflexivity.
+ Qed.
+
+Lemma distinct_bogus : forall {T}a0 (a:list T), distinct (a0::(app a (a0::nil))) -> False.
+ intros; induction a; auto.
+ simpl in H.
+ inversion H; subst.
+ apply H2; auto.
+ unfold In; simpl.
+ left; auto.
+ apply IHa.
+ clear IHa.
+ rewrite <- app_comm_cons in H.
+ inversion H; subst.
+ inversion H3; subst.
+ apply distinct_cons; auto.
+ intros.
+ apply H2.
simpl.
- unfold stripOutVars in *.
- rewrite <- IHtree2_1.
- rewrite <- IHtree2_2.
- reflexivity.
+ right; auto.
Qed.
-Lemma update_ξ_lemma `{EQD_VV:EqDecidable VV} : forall Γ ξ (lev:HaskLevel Γ)(varstypes:Tree ??(VV*_)),
- distinct (map (@fst _ _) (leaves varstypes)) ->
- mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) =
- mapOptionTree (fun t => t@@ lev) (mapOptionTree (@snd _ _) varstypes).
- admit.
+Lemma distinct_swap' : forall {T}a (b:list T), distinct (app b (a::nil)) -> distinct (a::b).
+ intros.
+ apply distinct_cons.
+ induction b; intros; simpl; auto.
+ rewrite <- app_comm_cons in H.
+ inversion H; subst.
+ set (IHb H4) as H4'.
+ apply H4'.
+ inversion H0; subst; auto.
+ apply distinct_bogus in H; inversion H.
+ induction b; intros; simpl; auto.
+ apply distinct_nil.
+ apply distinct_app in H.
+ destruct H; auto.
+ Qed.
+
+Lemma in_both : forall {T}(a b:list T) x, In x (app a b) -> In x a \/ In x b.
+ induction a; intros; simpl; auto.
+ rewrite <- app_comm_cons in H.
+ inversion H.
+ subst.
+ left; left; auto.
+ set (IHa _ _ H0) as H'.
+ destruct H'.
+ left; right; auto.
+ right; auto.
+ Qed.
+
+Lemma distinct_lemma : forall {T} (a b:list T) a0, distinct (app a (a0 :: b)) -> distinct (app a b).
+ intros.
+ induction a; simpl; auto.
+ simpl in H.
+ inversion H; auto.
+ assert (distinct (app a1 b)) as Q.
+ intros.
+ apply IHa.
+ clear IHa.
+ rewrite <- app_comm_cons in H.
+ inversion H; subst; auto.
+ apply distinct_cons; [ idtac | apply Q ].
+ intros.
+ apply in_both in H0.
+ destruct H0.
+ rewrite <- app_comm_cons in H.
+ inversion H; subst; auto.
+ apply H3.
+ apply In_both; auto.
+ rewrite <- app_comm_cons in H.
+ inversion H; subst; auto.
+ apply H3.
+ apply In_both'; auto.
+ simpl.
+ right; auto.
Qed.
+Lemma nil_app : forall {T}(a:list T) x, x::a = (app (x::nil) a).
+ induction a; intros; simpl; auto.
+ Qed.
+Lemma distinct_swap : forall {T}(a b:list T), distinct (app a b) -> distinct (app b a).
+ intros.
+ induction b.
+ rewrite <- app_nil_end in H; auto.
+ rewrite <- app_comm_cons.
+ set (distinct_lemma _ _ _ H) as H'.
+ set (IHb H') as H''.
+ apply distinct_cons; [ idtac | apply H'' ].
+ intros.
+ apply in_both in H0.
+ clear H'' H'.
+ destruct H0.
+ apply distinct_app in H.
+ destruct H.
+ inversion H1; auto.
+ clear IHb.
+ rewrite nil_app in H.
+ rewrite ass_app in H.
+ apply distinct_app in H.
+ destruct H; auto.
+ apply distinct_swap' in H.
+ inversion H; auto.
+ Qed.
+Lemma update_ξ_lemma' `{EQD_VV:EqDecidable VV} Γ ξ (lev:HaskLevel Γ)(varstypes:Tree ??(VV*_)) :
+ forall v1 v2,
+ distinct (map (@fst _ _) (leaves (v1,,(varstypes,,v2)))) ->
+ mapOptionTree (update_ξ ξ lev (leaves (v1,,(varstypes,,v2)))) (mapOptionTree (@fst _ _) varstypes) =
+ mapOptionTree (fun t => t@@ lev) (mapOptionTree (@snd _ _) varstypes).
+ induction varstypes; intros.
+ destruct a; simpl; [ idtac | reflexivity ].
+ destruct p.
+ simpl.
+ simpl in H.
+ induction (leaves v1).
+ simpl; auto.
+ destruct (eqd_dec v v); auto.
+ assert False. apply n. auto. inversion H0.
+ simpl.
+ destruct a.
+ destruct (eqd_dec v0 v); subst; auto.
+ simpl in H.
+ rewrite map_app in H.
+ simpl in H.
+ rewrite nil_app in H.
+ apply distinct_swap in H.
+ rewrite app_ass in H.
+ apply distinct_app in H.
+ destruct H.
+ apply distinct_swap in H0.
+ simpl in H0.
+ inversion H0; subst.
+ assert False.
+ apply H3.
+ simpl; left; auto.
+ inversion H1.
+ apply IHl.
+ simpl in H.
+ inversion H; auto.
+ set (IHvarstypes1 v1 (varstypes2,,v2)) as i1.
+ set (IHvarstypes2 (v1,,varstypes1) v2) as i2.
+ simpl in *.
+ rewrite <- i1.
+ rewrite <- i2.
+ rewrite ass_app.
+ rewrite ass_app.
+ rewrite ass_app.
+ rewrite ass_app.
+ reflexivity.
+ clear i1 i2 IHvarstypes1 IHvarstypes2.
+ repeat rewrite ass_app in *; auto.
+ clear i1 i2 IHvarstypes1 IHvarstypes2.
+ repeat rewrite ass_app in *; auto.
+ Qed.
+Lemma update_ξ_lemma `{EQD_VV:EqDecidable VV} Γ ξ (lev:HaskLevel Γ)(varstypes:Tree ??(VV*_)) :
+ distinct (map (@fst _ _) (leaves varstypes)) ->
+ mapOptionTree (update_ξ ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) =
+ mapOptionTree (fun t => t@@ lev) (mapOptionTree (@snd _ _) varstypes).
+ set (update_ξ_lemma' Γ ξ lev varstypes [] []) as q.
+ simpl in q.
+ rewrite <- app_nil_end in q.
+ apply q.
+ Qed.
Fixpoint expr2antecedent {Γ'}{Δ'}{ξ'}{τ'}(exp:Expr Γ' Δ' ξ' τ') : Tree ??VV :=
match exp as E in Expr Γ Δ ξ τ with
| ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => expr2antecedent e
| ECoApp Γ Δ κ γ σ₁ σ₂ σ ξ l e => expr2antecedent e
| ETyApp Γ Δ κ σ τ ξ l e => expr2antecedent e
- | ELetRec Γ Δ ξ l τ vars branches body =>
+ | ELetRec Γ Δ ξ l τ vars _ branches body =>
let branch_context := eLetRecContext branches
in let all_contexts := (expr2antecedent body),,branch_context
in stripOutVars (leaves (mapOptionTree (@fst _ _ ) vars)) all_contexts
| ECase Γ Δ ξ l tc tbranches atypes e' alts =>
((fix varsfromalts (alts:
- Tree ??{ scb : StrongCaseBranchWithVVs _ _ tc atypes
- & Expr (sac_Γ scb Γ)
- (sac_Δ scb Γ atypes (weakCK'' Δ))
+ Tree ??{ sac : _ & { scb : StrongCaseBranchWithVVs _ _ tc atypes sac
+ & Expr (sac_Γ sac Γ)
+ (sac_Δ sac Γ atypes (weakCK'' Δ))
(scbwv_ξ scb ξ l)
- (weakLT' (tbranches@@l)) }
+ (weakLT' (tbranches@@l)) } }
): Tree ??VV :=
match alts with
| T_Leaf None => []
- | T_Leaf (Some h) => stripOutVars (vec2list (scbwv_exprvars (projT1 h))) (expr2antecedent (projT2 h))
+ | T_Leaf (Some h) => stripOutVars (vec2list (scbwv_exprvars (projT1 (projT2 h)))) (expr2antecedent (projT2 (projT2 h)))
| T_Branch b1 b2 => (varsfromalts b1),,(varsfromalts b2)
end) alts),,(expr2antecedent e')
end
end.
Definition mkProofCaseBranch {Γ}{Δ}{ξ}{l}{tc}{tbranches}{atypes}
- (alt: { scb : StrongCaseBranchWithVVs _ _ tc atypes
- & Expr (sac_Γ scb Γ)
- (sac_Δ scb Γ atypes (weakCK'' Δ))
+(alt : { sac : _ & { scb : StrongCaseBranchWithVVs _ _ tc atypes sac
+ & Expr (sac_Γ sac Γ)
+ (sac_Δ sac Γ atypes (weakCK'' Δ))
(scbwv_ξ scb ξ l)
- (weakLT' (tbranches@@l)) })
- : ProofCaseBranch tc Γ Δ l tbranches atypes.
+ (weakLT' (tbranches@@l)) } })
+ : { sac : _ & ProofCaseBranch tc Γ Δ l tbranches atypes sac }.
+ destruct alt.
+ exists x.
exact
- {| pcb_scb := projT1 alt
- ; pcb_freevars := mapOptionTree ξ (stripOutVars (vec2list (scbwv_exprvars (projT1 alt))) (expr2antecedent (projT2 alt)))
+ {| pcb_freevars := mapOptionTree ξ
+ (stripOutVars (vec2list (scbwv_exprvars (projT1 s)))
+ (expr2antecedent (projT2 s)))
|}.
Defined.
| ELR_leaf Γ Δ ξ lev v t e => [t]
| ELR_branch Γ Δ ξ lev t1 t2 b1 b2 => (eLetRecTypes b1),,(eLetRecTypes b2)
end.
-(*
-Fixpoint eLetRecVars {Γ}{Δ}{ξ}{lev}{τ}(elrb:ELetRecBindings Γ Δ ξ lev τ) : Tree ??VV :=
- match elrb with
- | ELR_nil Γ Δ ξ lev => []
- | ELR_leaf Γ Δ ξ lev v _ _ e => [v]
- | ELR_branch Γ Δ ξ lev t1 t2 b1 b2 => (eLetRecVars b1),,(eLetRecVars b2)
- end.
-
-Fixpoint eLetRecTypesVars {Γ}{Δ}{ξ}{lev}{τ}(elrb:ELetRecBindings Γ Δ ξ lev τ) : Tree ??(VV * HaskType Γ ★):=
- match elrb with
- | ELR_nil Γ Δ ξ lev => []
- | ELR_leaf Γ Δ ξ lev v t _ e => [(v, t)]
- | ELR_branch Γ Δ ξ lev t1 t2 b1 b2 => (eLetRecTypesVars b1),,(eLetRecTypesVars b2)
- end.
-*)
Lemma stripping_nothing_is_inert
{Γ:TypeEnv}
reflexivity.
Qed.
-Definition arrangeContext
+Definition arrangeContext
(Γ:TypeEnv)(Δ:CoercionEnv Γ)
v (* variable to be pivoted, if found *)
ctx (* initial context *)
- τ (* type of succedent *)
(ξ:VV -> LeveledHaskType Γ ★)
:
(* a proof concluding in a context where that variable does not appear *)
- sum (ND (@URule Γ Δ)
- [Γ >> Δ > mapOptionTree ξ ctx |- τ]
- [Γ >> Δ > mapOptionTree ξ (stripOutVars (v::nil) ctx),,[] |- τ])
+ sum (Arrange
+ (mapOptionTree ξ ctx )
+ (mapOptionTree ξ (stripOutVars (v::nil) ctx),,[] ))
(* or a proof concluding in a context where that variable appears exactly once in the left branch *)
- (ND (@URule Γ Δ)
- [Γ >> Δ > mapOptionTree ξ ctx |- τ]
- [Γ >> Δ > mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v]) |- τ]).
+ (Arrange
+ (mapOptionTree ξ ctx )
+ (mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v]) )).
induction ctx.
(* where the leaf is v *)
apply inr.
subst.
- apply nd_rule.
apply RuCanL.
(* where the leaf is NOT v *)
apply inl.
- apply nd_rule.
apply RuCanR.
(* empty leaf *)
destruct case_None.
apply inl; simpl in *.
- apply nd_rule.
apply RuCanR.
(* branch *)
destruct case_Neither.
apply inl.
- eapply nd_comp; [idtac | eapply nd_rule; apply RuCanR ].
- exact (nd_comp
+ eapply RComp; [idtac | apply RuCanR ].
+ exact (RComp
(* order will not matter because these are central as morphisms *)
- (ext_right _ _ _ (nd_comp lpf (nd_rule (@RCanR _ _ _ _))))
- (ext_left _ _ _ (nd_comp rpf (nd_rule (@RCanR _ _ _ _))))).
+ (RRight _ (RComp lpf (RCanR _)))
+ (RLeft _ (RComp rpf (RCanR _)))).
destruct case_Right.
apply inr.
fold (stripOutVars (v::nil)).
- set (ext_right (mapOptionTree ξ ctx2) _ _ (nd_comp lpf (nd_rule (@RCanR _ _ _ _)))) as q.
+ set (RRight (mapOptionTree ξ ctx2) (RComp lpf ((RCanR _)))) as q.
simpl in *.
- eapply nd_comp.
+ eapply RComp.
apply q.
clear q.
clear lpf.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
- eapply nd_comp; [ idtac | eapply nd_rule; apply RAssoc ].
- set (ext_left (mapOptionTree ξ (stripOutVars (v :: nil) ctx1)) [Γ >> Δ>mapOptionTree ξ ctx2 |- τ]
- [Γ >> Δ> (mapOptionTree ξ (stripOutVars (v :: nil) ctx2),,[ξ v]) |- τ]) as qq.
- apply qq.
- clear qq.
+ eapply RComp; [ idtac | apply RAssoc ].
+ apply RLeft.
apply rpf.
destruct case_Left.
apply inr.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
fold (stripOutVars (v::nil)).
- eapply nd_comp; [ idtac | eapply pivotContext ].
- set (nd_comp rpf (nd_rule (@RCanR _ _ _ _ ) ) ) as rpf'.
- set (ext_left ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) _ _ rpf') as qq.
+ eapply RComp; [ idtac | eapply pivotContext ].
+ set (RComp rpf (RCanR _ )) as rpf'.
+ set (RLeft ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) rpf') as qq.
simpl in *.
- eapply nd_comp; [ idtac | apply qq ].
+ eapply RComp; [ idtac | apply qq ].
clear qq rpf' rpf.
- set (ext_right (mapOptionTree ξ ctx2) [Γ >>Δ> mapOptionTree ξ ctx1 |- τ] [Γ >>Δ> (mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v]) |- τ]) as q.
- apply q.
- clear q.
+ apply (RRight (mapOptionTree ξ ctx2)).
apply lpf.
destruct case_Both.
apply inr.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
fold (stripOutVars (v::nil)).
- eapply nd_comp; [ idtac | eapply copyAndPivotContext ].
- exact (nd_comp
- (* order will not matter because these are central as morphisms *)
- (ext_right _ _ _ lpf)
- (ext_left _ _ _ rpf)).
+ eapply RComp; [ idtac | eapply copyAndPivotContext ].
+ (* order will not matter because these are central as morphisms *)
+ exact (RComp (RRight _ lpf) (RLeft _ rpf)).
Defined.
(* same as before, but use RWeak if necessary *)
-Definition arrangeContextAndWeaken v ctx Γ Δ τ ξ :
- ND (@URule Γ Δ)
- [Γ >> Δ>mapOptionTree ξ ctx |- τ]
- [Γ >> Δ>mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v]) |- τ].
- set (arrangeContext Γ Δ v ctx τ ξ) as q.
+Definition arrangeContextAndWeaken
+ (Γ:TypeEnv)(Δ:CoercionEnv Γ)
+ v (* variable to be pivoted, if found *)
+ ctx (* initial context *)
+ (ξ:VV -> LeveledHaskType Γ ★) :
+ Arrange
+ (mapOptionTree ξ ctx )
+ (mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v]) ).
+ set (arrangeContext Γ Δ v ctx ξ) as q.
destruct q; auto.
- eapply nd_comp; [ apply n | idtac ].
- clear n.
- refine (ext_left _ _ _ (nd_rule (RWeak _ _))).
+ eapply RComp; [ apply a | idtac ].
+ refine (RLeft _ (RWeak _)).
Defined.
-Lemma updating_stripped_tree_is_inert {Γ} (ξ:VV -> LeveledHaskType Γ ★) v tree t lev :
- mapOptionTree (update_ξ ξ lev ((v,t)::nil)) (stripOutVars (v :: nil) tree)
- = mapOptionTree ξ (stripOutVars (v :: nil) tree).
- set (@updating_stripped_tree_is_inert' Γ lev ξ ((v,t)::nil)) as p.
- rewrite p.
- simpl.
- reflexivity.
- Qed.
-
-Lemma cheat : forall {T}(a b:list T), distinct (app a b) -> distinct (app b a).
- admit.
- Qed.
-
-Definition arrangeContextAndWeaken'' Γ Δ ξ v : forall ctx z,
+Definition arrangeContextAndWeaken''
+ (Γ:TypeEnv)(Δ:CoercionEnv Γ)
+ v (* variable to be pivoted, if found *)
+ (ξ:VV -> LeveledHaskType Γ ★) : forall ctx,
distinct (leaves v) ->
- ND (@URule Γ Δ)
- [Γ >> Δ>(mapOptionTree ξ ctx) |- z]
- [Γ >> Δ>(mapOptionTree ξ (stripOutVars (leaves v) ctx)),,(mapOptionTree ξ v) |- z].
+ Arrange
+ ((mapOptionTree ξ ctx) )
+ ((mapOptionTree ξ (stripOutVars (leaves v) ctx)),,(mapOptionTree ξ v)).
induction v; intros.
destruct a.
fold (mapOptionTree ξ) in *.
intros.
apply arrangeContextAndWeaken.
+ apply Δ.
unfold mapOptionTree; simpl in *.
intros.
rewrite (@stripping_nothing_is_inert Γ); auto.
- apply nd_rule.
apply RuCanR.
intros.
unfold mapOptionTree in *.
fold (mapOptionTree ξ) in *.
set (mapOptionTree ξ) as X in *.
- set (IHv2 ((stripOutVars (leaves v1) ctx),, v1) z) as IHv2'.
+ set (IHv2 ((stripOutVars (leaves v1) ctx),, v1)) as IHv2'.
unfold stripOutVars in IHv2'.
simpl in IHv2'.
fold (stripOutVars (leaves v2)) in IHv2'.
fold X in IHv2'.
set (distinct_app _ _ _ H) as H'.
destruct H' as [H1 H2].
- set (nd_comp (IHv1 _ _ H1) (IHv2' H2)) as qq.
- eapply nd_comp.
+ set (RComp (IHv1 _ H1) (IHv2' H2)) as qq.
+ eapply RComp.
apply qq.
clear qq IHv2' IHv2 IHv1.
+ rewrite strip_swap_lemma.
rewrite strip_twice_lemma.
-
- rewrite (strip_distinct' v1 (leaves v2)).
- apply nd_rule.
+ rewrite (notin_strip_inert' v1 (leaves v2)).
apply RCossa.
- apply cheat.
+ apply distinct_swap.
auto.
Defined.
-(* IDEA: use multi-conclusion proofs instead *)
+Lemma updating_stripped_tree_is_inert {Γ} (ξ:VV -> LeveledHaskType Γ ★) v tree t lev :
+ mapOptionTree (update_ξ ξ lev ((v,t)::nil)) (stripOutVars (v :: nil) tree)
+ = mapOptionTree ξ (stripOutVars (v :: nil) tree).
+ set (@updating_stripped_tree_is_inert' Γ lev ξ ((v,t)::nil)) as p.
+ rewrite p.
+ simpl.
+ reflexivity.
+ Qed.
+
+(* TODO: use multi-conclusion proofs instead *)
Inductive LetRecSubproofs Γ Δ ξ lev : forall tree, ELetRecBindings Γ Δ ξ lev tree -> Type :=
| lrsp_nil : LetRecSubproofs Γ Δ ξ lev [] (ELR_nil _ _ _ _)
| lrsp_leaf : forall v t e ,
|- (mapOptionTree (@snd _ _) tree) @@@ lev ].
intro X; induction X; intros; simpl in *.
apply nd_rule.
- apply REmptyGroup.
+ apply RVoid.
set (ξ v) as q in *.
destruct q.
simpl in *.
apply n.
- eapply nd_comp; [ idtac | eapply nd_rule; apply RBindingGroup ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RJoin ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod; auto.
Defined.
Lemma letRecSubproofsToND' Γ Δ ξ lev τ tree :
- forall branches body,
- distinct (leaves (mapOptionTree (@fst _ _) tree)) ->
+ forall branches body (dist:distinct (leaves (mapOptionTree (@fst _ _) tree))),
ND Rule [] [Γ > Δ > mapOptionTree (update_ξ ξ lev (leaves tree)) (expr2antecedent body) |- [τ @@ lev]] ->
LetRecSubproofs Γ Δ (update_ξ ξ lev (leaves tree)) lev tree branches ->
- ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent (@ELetRec VV _ Γ Δ ξ lev τ tree branches body)) |- [τ @@ lev]].
+ ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent (@ELetRec VV _ Γ Δ ξ lev τ tree dist branches body)) |- [τ @@ lev]].
(* NOTE: how we interpret stuff here affects the order-of-side-effects *)
intro branches.
intro pf.
intro lrsp.
- rewrite mapleaves in disti.
- set (@update_ξ_lemma _ Γ ξ lev tree disti) as ξlemma.
+ assert (distinct (leaves (mapOptionTree (@fst _ _) tree))) as disti'.
+ apply disti.
+ rewrite mapleaves in disti'.
+
+ set (@update_ξ_lemma _ Γ ξ lev tree disti') as ξlemma.
rewrite <- mapOptionTree_compose in ξlemma.
set ((update_ξ ξ lev (leaves tree))) as ξ' in *.
eapply nd_comp; [ idtac | eapply nd_rule; apply z ].
clear z.
- set (@arrangeContextAndWeaken'' Γ Δ ξ' pctx (expr2antecedent body,,eLetRecContext branches)) as q'.
+ set (@arrangeContextAndWeaken'' Γ Δ pctx ξ' (expr2antecedent body,,eLetRecContext branches)) as q'.
unfold passback in *; clear passback.
unfold pctx in *; clear pctx.
- eapply UND_to_ND in q'.
+ set (q' disti) as q''.
unfold ξ' in *.
set (@updating_stripped_tree_is_inert' Γ lev ξ (leaves tree)) as zz.
rewrite <- mapleaves in zz.
- rewrite zz in q'.
+ rewrite zz in q''.
clear zz.
clear ξ'.
Opaque stripOutVars.
simpl.
- rewrite <- mapOptionTree_compose in q'.
+ rewrite <- mapOptionTree_compose in q''.
rewrite <- ξlemma.
- eapply nd_comp; [ idtac | apply q' ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply (RArrange _ _ _ _ _ q'') ].
clear q'.
+ clear q''.
simpl.
set (letRecSubproofsToND _ _ _ _ _ branches lrsp) as q.
- eapply nd_comp; [ idtac | eapply nd_rule; apply RBindingGroup ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RJoin ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod; auto.
rewrite ξlemma.
apply q.
- clear q'.
-
- rewrite <- mapleaves in disti.
- apply disti.
Defined.
-Lemma scbwv_coherent {tc}{Γ}{atypes:IList _ (HaskType Γ) _} :
- forall scb:StrongCaseBranchWithVVs _ _ tc atypes,
+Lemma scbwv_coherent {tc}{Γ}{atypes:IList _ (HaskType Γ) _}{sac} :
+ forall scb:StrongCaseBranchWithVVs _ _ tc atypes sac,
forall l ξ,
vec2list (vec_map (scbwv_ξ scb ξ l) (scbwv_exprvars scb)) =
- vec2list (vec_map (fun t => t @@ weakL' l) (sac_types (scbwv_sac scb) _ atypes)).
+ vec2list (vec_map (fun t => t @@ weakL' l) (sac_types sac _ atypes)).
intros.
unfold scbwv_ξ.
unfold scbwv_varstypes.
set (@update_ξ_lemma _ _ (weakLT' ○ ξ) (weakL' l)
- (unleaves (vec2list (vec_zip (scbwv_exprvars scb) (sac_types (scbwv_sac scb) Γ atypes))))
+ (unleaves (vec2list (vec_zip (scbwv_exprvars scb) (sac_types sac Γ atypes))))
) as q.
rewrite <- mapleaves' in q.
rewrite <- mapleaves' in q.
apply scbwv_exprvars_distinct.
Qed.
-Lemma case_lemma : forall Γ Δ ξ l tc tbranches atypes e alts',
- mapOptionTree ξ (expr2antecedent (ECase Γ Δ ξ l tc tbranches atypes e alts'))
- = ((mapOptionTreeAndFlatten pcb_freevars (mapOptionTree mkProofCaseBranch alts')),,mapOptionTree ξ (expr2antecedent e)).
+
+Lemma case_lemma : forall Γ Δ ξ l tc tbranches atypes e
+ (alts':Tree
+ ??{sac : StrongAltCon &
+ {scb : StrongCaseBranchWithVVs VV eqd_vv tc atypes sac &
+ Expr (sac_Γ sac Γ) (sac_Δ sac Γ atypes (weakCK'' Δ))
+ (scbwv_ξ scb ξ l) (weakLT' (tbranches @@ l))}}),
+
+ (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x))
+ (mapOptionTree mkProofCaseBranch alts'))
+ ,,
+ mapOptionTree ξ (expr2antecedent e) =
+ mapOptionTree ξ
+ (expr2antecedent (ECase Γ Δ ξ l tc tbranches atypes e alts')).
intros.
simpl.
Ltac hack := match goal with [ |- ?A,,?B = ?C,,?D ] => assert (A=C) end.
hack.
induction alts'.
- simpl.
- destruct a.
+ destruct a; simpl.
+ destruct s; simpl.
unfold mkProofCaseBranch.
- simpl.
reflexivity.
reflexivity.
simpl.
reflexivity.
Qed.
-
Definition expr2proof :
forall Γ Δ ξ τ (e:Expr Γ Δ ξ τ),
ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [τ]].
| ELam Γ Δ ξ t1 t2 lev v e => let case_ELam := tt in (fun e' => _) (expr2proof _ _ _ _ e)
| ELet Γ Δ ξ tv t v lev ev ebody => let case_ELet := tt in
(fun pf_let pf_body => _) (expr2proof _ _ _ _ ev) (expr2proof _ _ _ _ ebody)
- | ELetRec Γ Δ ξ lev t tree branches ebody =>
+ | ELetRec Γ Δ ξ lev t tree disti branches ebody =>
let ξ' := update_ξ ξ lev (leaves tree) in
let case_ELetRec := tt in (fun e' subproofs => _) (expr2proof _ _ _ _ ebody)
((fix subproofs Γ'' Δ'' ξ'' lev'' (tree':Tree ??(VV * HaskType Γ'' ★))
| ECase Γ Δ ξ l tc tbranches atypes e alts' =>
let dcsp :=
((fix mkdcsp (alts:
- Tree ??{ scb : StrongCaseBranchWithVVs _ _ tc atypes
- & Expr (sac_Γ scb Γ)
- (sac_Δ scb Γ atypes (weakCK'' Δ))
+ Tree ??{ sac : _ & { scb : StrongCaseBranchWithVVs _ _ tc atypes sac
+ & Expr (sac_Γ sac Γ)
+ (sac_Δ sac Γ atypes (weakCK'' Δ))
(scbwv_ξ scb ξ l)
- (weakLT' (tbranches@@l)) })
- : ND Rule [] (mapOptionTree (pcb_judg ○ mkProofCaseBranch) alts) :=
- match alts as ALTS return ND Rule [] (mapOptionTree (pcb_judg ○ mkProofCaseBranch) ALTS) with
+ (weakLT' (tbranches@@l)) } })
+ : ND Rule [] (mapOptionTree (fun x => pcb_judg (projT2 (mkProofCaseBranch x))) alts) :=
+ match alts as ALTS return ND Rule []
+ (mapOptionTree (fun x => pcb_judg (projT2 (mkProofCaseBranch x))) ALTS) with
| T_Leaf None => let case_nil := tt in _
| T_Branch b1 b2 => let case_branch := tt in (fun b1' b2' => _) (mkdcsp b1) (mkdcsp b2)
| T_Leaf (Some x) =>
- match x as X return ND Rule [] [(pcb_judg ○ mkProofCaseBranch) X] with
- existT scbx ex =>
+ match x as X return ND Rule [] [pcb_judg (projT2 (mkProofCaseBranch X))] with
+ existT sac (existT scbx ex) =>
(fun e' => let case_leaf := tt in _) (expr2proof _ _ _ _ ex)
end
end) alts')
unfold mapOptionTree; simpl; fold (mapOptionTree ξ).
eapply nd_comp; [ idtac | eapply nd_rule; apply RLam ].
set (update_ξ ξ lev ((v,t1)::nil)) as ξ'.
- set (arrangeContextAndWeaken v (expr2antecedent e) Γ Δ [t2 @@ lev] ξ') as pfx.
- apply UND_to_ND in pfx.
- unfold mapOptionTree in pfx; simpl in pfx; fold (mapOptionTree ξ) in pfx.
+ set (arrangeContextAndWeaken Γ Δ v (expr2antecedent e) ξ') as pfx.
+ eapply RArrange in pfx.
+ unfold mapOptionTree in pfx; simpl in pfx.
unfold ξ' in pfx.
- fold (mapOptionTree (update_ξ ξ lev ((v,t1)::nil))) in pfx.
rewrite updating_stripped_tree_is_inert in pfx.
unfold update_ξ in pfx.
destruct (eqd_dec v v).
- eapply nd_comp; [ idtac | apply pfx ].
+ eapply nd_comp; [ idtac | apply (nd_rule pfx) ].
clear pfx.
apply e'.
assert False.
destruct case_ELet; intros; simpl in *.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod; [ idtac | apply pf_let].
- clear pf_let.
- eapply nd_comp; [ apply pf_body | idtac ].
- clear pf_body.
+ apply nd_prod.
+ apply pf_let.
+ clear pf_let.
+ eapply nd_comp; [ apply pf_body | idtac ].
+ clear pf_body.
fold (@mapOptionTree VV).
fold (mapOptionTree ξ).
set (update_ξ ξ v ((lev,tv)::nil)) as ξ'.
- set (arrangeContextAndWeaken lev (expr2antecedent ebody) Γ Δ [t@@v] ξ') as n.
+ set (arrangeContextAndWeaken Γ Δ lev (expr2antecedent ebody) ξ') as n.
unfold mapOptionTree in n; simpl in n; fold (mapOptionTree ξ') in n.
unfold ξ' in n.
rewrite updating_stripped_tree_is_inert in n.
destruct (eqd_dec lev lev).
unfold ξ'.
unfold update_ξ.
- apply UND_to_ND in n.
- apply n.
+ eapply RArrange in n.
+ apply (nd_rule n).
assert False. apply n0; auto. inversion H.
destruct case_EEsc.
clear o x alts alts' e.
eapply nd_comp; [ apply e' | idtac ].
clear e'.
- set (existT
- (fun scb : StrongCaseBranchWithVVs VV eqd_vv tc atypes =>
- Expr (sac_Γ scb Γ) (sac_Δ scb Γ atypes (weakCK'' Δ))
- (scbwv_ξ scb ξ l) (weakLT' (tbranches @@ l))) scbx ex) as stuff.
- set (fun q r x1 x2 y1 y2 => @UND_to_ND q r [q >> r > x1 |- y1] [q >> r > x2 |- y2]).
- simpl in n.
- apply n.
- clear n.
-
+ apply nd_rule.
+ apply RArrange.
+ simpl.
rewrite mapleaves'.
simpl.
rewrite <- mapOptionTree_compose.
replace (stripOutVars (vec2list (scbwv_exprvars scbx))) with
(stripOutVars (leaves (unleaves (vec2list (scbwv_exprvars scbx))))).
apply q.
+ apply (sac_Δ sac Γ atypes (weakCK'' Δ)).
rewrite leaves_unleaves.
apply (scbwv_exprvars_distinct scbx).
rewrite leaves_unleaves.
apply b2'.
destruct case_ECase.
- rewrite case_lemma.
+ set (@RCase Γ Δ l tc) as q.
+ rewrite <- case_lemma.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RCase ].
eapply nd_comp; [ apply nd_llecnac | idtac ]; apply nd_prod.
rewrite <- mapOptionTree_compose.
unfold ξ'1 in *.
clear ξ'1.
apply letRecSubproofsToND'.
- admit.
apply e'.
apply subproofs.
projects / coq-hetmet.git / blobdiff