(*********************************************************************************************************************************)
-(* HaskFlattener: *)
+(* HaskFlattener: *)
(* *)
(* The Flattening Functor. *)
(* *)
(* lemma: if a proof from no hypotheses contains no Brak's or Esc's, then the context contains no variables at level!=0 *)
- (* In a tree of types, replace any type at level "lev" with None *)
- Definition drop_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
- mapTree (fun t => match t with
- | Some (ttype @@ tlev) => if eqd_dec tlev lev then None else t
- | _ => t
- end) tt.
+ Definition minus' n m :=
+ match m with
+ | 0 => n
+ | _ => match n with
+ | 0 => 0
+ | _ => n - m
+ end
+ end.
+
+ Ltac eqd_dec_refl' :=
+ match goal with
+ | [ |- context[@eqd_dec ?T ?V ?X ?X] ] =>
+ destruct (@eqd_dec T V X X) as [eqd_dec1 | eqd_dec2];
+ [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ]
+ end.
+
(* The opposite: replace any type which is NOT at level "lev" with None *)
Definition take_lev {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) :=
mapTree (fun t => match t with
end) tt.
(* In a tree of types, replace any type at depth "lev" or greater None *)
- Definition drop_depth {Γ}(n:nat)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
+ Definition drop_depth {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(LeveledHaskType Γ ★) :=
mapTree (fun t => match t with
- | Some (ttype @@ tlev) => if eqd_dec (length tlev) n then None else t
+ | Some (ttype @@ tlev) => if eqd_dec tlev lev then None else t
| _ => t
end) tt.
- (* delete from the tree any type which is NOT at level "lev" *)
+ Lemma drop_depth_lemma : forall Γ (lev:HaskLevel Γ) x, drop_depth lev [x @@ lev] = [].
+ intros; simpl.
+ Opaque eqd_dec.
+ unfold drop_depth.
+ simpl.
+ Transparent eqd_dec.
+ eqd_dec_refl'.
+ auto.
+ Qed.
+
+ Lemma drop_depth_lemma_s : forall Γ (lev:HaskLevel Γ) ec x, drop_depth (ec::lev) [x @@ (ec :: lev)] = [].
+ intros; simpl.
+ Opaque eqd_dec.
+ unfold drop_depth.
+ simpl.
+ Transparent eqd_dec.
+ eqd_dec_refl'.
+ auto.
+ Qed.
+
+ Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [x].
+ intros; simpl.
+ Opaque eqd_dec.
+ unfold take_lev.
+ simpl.
+ Transparent eqd_dec.
+ eqd_dec_refl'.
+ auto.
+ Qed.
+
+ Ltac drop_simplify :=
+ match goal with
+ | [ |- context[@drop_depth ?G ?L [ ?X @@ ?L ] ] ] =>
+ rewrite (drop_depth_lemma G L X)
+ | [ |- context[@drop_depth ?G (?E :: ?L) [ ?X @@ (?E :: ?L) ] ] ] =>
+ rewrite (drop_depth_lemma_s G L E X)
+ | [ |- context[@drop_depth ?G ?N (?A,,?B)] ] =>
+ change (@drop_depth G N (A,,B)) with ((@drop_depth G N A),,(@drop_depth G N B))
+ | [ |- context[@drop_depth ?G ?N (T_Leaf None)] ] =>
+ change (@drop_depth G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★)))
+ end.
+
+ Ltac take_simplify :=
+ match goal with
+ | [ |- context[@take_lev ?G ?L [ ?X @@ ?L ] ] ] =>
+ rewrite (take_lemma G L X)
+ | [ |- context[@take_lev ?G ?N (?A,,?B)] ] =>
+ change (@take_lev G N (A,,B)) with ((@take_lev G N A),,(@take_lev G N B))
+ | [ |- context[@take_lev ?G ?N (T_Leaf None)] ] =>
+ change (@take_lev G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★)))
+ end.
- Fixpoint take_lev' {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : ??(Tree ??(HaskType Γ ★)) :=
+ Fixpoint reduceTree {T}(unit:T)(merge:T -> T -> T)(tt:Tree ??T) : T :=
match tt with
- | T_Leaf None => Some (T_Leaf None) (* FIXME: unsure of this; it ends up on both sides *)
- | T_Branch b1 b2 =>
- let b1' := take_lev' lev b1 in
- let b2' := take_lev' lev b2 in
- match b1' with
- | None => b2'
- | Some b1'' => match b2' with
- | None => Some b1''
- | Some b2'' => Some (b1'',,b2'')
- end
- end
- | T_Leaf (Some (ttype@@tlev)) =>
- if eqd_dec tlev lev
- then Some [ttype]
- else None
+ | T_Leaf None => unit
+ | T_Leaf (Some x) => x
+ | T_Branch b1 b2 => merge (reduceTree unit merge b1) (reduceTree unit merge b2)
end.
- Context (ga' : forall TV, TV ★ -> Tree ??(RawHaskType TV ★) -> Tree ??(RawHaskType TV ★) -> RawHaskType TV ★).
+ Set Printing Width 130.
- Definition ga : forall Γ, HaskTyVar Γ ★ -> Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> HaskType Γ ★
- := fun Γ ec ant suc =>
- fun TV ite =>
- ga'
- TV
- (ec TV ite)
- (mapOptionTree (fun x => x TV ite) ant)
- (mapOptionTree (fun x => x TV ite) suc).
+ Context {unitTy : forall TV, RawHaskType TV ★ }.
+ Context {prodTy : forall TV, RawHaskType TV (★ ⇛ ★ ⇛ ★) }.
+ Context {gaTy : forall TV, RawHaskType TV (★ ⇛ ★ ⇛ ★ ⇛ ★)}.
- Implicit Arguments ga [ [Γ] ].
- Notation "a ~~~~> b" := (@ga _ _ a b) (at level 20).
+ Definition ga_tree := fun TV tr => reduceTree (unitTy TV) (fun t1 t2 => TApp (TApp (prodTy TV) t1) t2) tr.
+ Definition ga' := fun TV ec ant' suc' => TApp (TApp (TApp (gaTy TV) ec) (ga_tree TV ant')) (ga_tree TV suc').
+ Definition ga {Γ} : HaskType Γ ★ -> Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> HaskType Γ ★ :=
+ fun ec ant suc =>
+ fun TV ite =>
+ let ant' := mapOptionTree (fun x => x TV ite) ant in
+ let suc' := mapOptionTree (fun x => x TV ite) suc in
+ ga' TV (ec TV ite) ant' suc'.
- Context (unit_type : forall TV, RawHaskType TV ★).
+ Class garrow :=
+ { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a a @@ l] ]
+ ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,[]) a @@ l] ]
+ ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ([],,a) a @@ l] ]
+ ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,[]) @@ l] ]
+ ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a ([],,a) @@ l] ]
+ ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ]
+ ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ]
+ ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,b) (b,,a) @@ l] ]
+ ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a [] @@ l] ]
+ ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,a) @@ l] ]
+ ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l] |- [@ga Γ ec (a,,x) (b,,x) @@ l] ]
+ ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l] |- [@ga Γ ec (x,,a) (x,,b) @@ l] ]
+ ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec [] [literalType lit] @@ l] ]
+ ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga Γ ec (a,,[b]) [c] @@ l] |- [@ga Γ ec a [b ---> c] @@ l] ]
+ ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ l],,[@ga Γ ec b c @@ l] |- [@ga Γ ec a c @@ l] ]
+ ; ga_apply : ∀ Γ Δ ec l a a' b c, ND Rule []
+ [Γ > Δ > [@ga Γ ec a [b ---> c] @@ l],,[@ga Γ ec a' [b] @@ l] |- [@ga Γ ec (a,,a') [c] @@ l] ]
+ ; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule
+ [Γ > Δ > Σ,,[@ga Γ ec [] a @@ l] |- [@ga Γ ec [] b @@ l] ]
+ [Γ > Δ > Σ |- [@ga Γ ec a b @@ l] ]
+ }.
+ Context `(gar:garrow).
+
+ Notation "a ~~~~> b" := (@ga _ _ a b) (at level 20).
(*
* The story:
* - free variables of type t at a level lev deeper than the succedent become garrows c () t
* - free variables at the level of the succedent become
*)
- Fixpoint flatten_rawtype {TV}{κ}(depth:nat)(exp: RawHaskType TV κ) : RawHaskType TV κ :=
+ Fixpoint garrowfy_raw_codetypes {TV}{κ}(depth:nat)(exp: RawHaskType TV κ) : RawHaskType TV κ :=
match exp with
| TVar _ x => TVar x
- | TAll _ y => TAll _ (fun v => flatten_rawtype depth (y v))
- | TApp _ _ x y => TApp (flatten_rawtype depth x) (flatten_rawtype depth y)
+ | TAll _ y => TAll _ (fun v => garrowfy_raw_codetypes depth (y v))
+ | TApp _ _ x y => TApp (garrowfy_raw_codetypes depth x) (garrowfy_raw_codetypes depth y)
| TCon tc => TCon tc
- | TCoerc _ t1 t2 t => TCoerc (flatten_rawtype depth t1) (flatten_rawtype depth t2) (flatten_rawtype depth t)
+ | TCoerc _ t1 t2 t => TCoerc (garrowfy_raw_codetypes depth t1) (garrowfy_raw_codetypes depth t2)
+ (garrowfy_raw_codetypes depth t)
| TArrow => TArrow
| TCode v e => match depth with
- | O => match v return RawHaskType TV ★ with
- | TVar ★ ec => ga' TV ec [] [flatten_rawtype depth e]
- | _ => unit_type TV
- end
- | (S depth') => TCode v (flatten_rawtype depth' e)
+ | O => ga' TV v [] [(*garrowfy_raw_codetypes depth*) e]
+ | (S depth') => TCode v (garrowfy_raw_codetypes depth' e)
end
- | TyFunApp tfc lt => TyFunApp tfc (flatten_rawtype_list _ depth lt)
+ | TyFunApp tfc lt => TyFunApp tfc (garrowfy_raw_codetypes_list _ depth lt)
end
- with flatten_rawtype_list {TV}(lk:list Kind)(depth:nat)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk :=
+ with garrowfy_raw_codetypes_list {TV}(lk:list Kind)(depth:nat)(exp:@RawHaskTypeList TV lk) : @RawHaskTypeList TV lk :=
match exp in @RawHaskTypeList _ LK return @RawHaskTypeList TV LK with
| TyFunApp_nil => TyFunApp_nil
- | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (flatten_rawtype depth t) (flatten_rawtype_list _ depth rest)
+ | TyFunApp_cons κ kl t rest => TyFunApp_cons _ _ (garrowfy_raw_codetypes depth t) (garrowfy_raw_codetypes_list _ depth rest)
end.
+ Definition garrowfy_code_types {Γ}{κ}(n:nat)(ht:HaskType Γ κ) : HaskType Γ κ :=
+ fun TV ite =>
+ garrowfy_raw_codetypes n (ht TV ite).
+ Definition garrowfy_leveled_code_types {Γ}(n:nat)(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
+ match ht with htt @@ htlev => garrowfy_code_types (minus' n (length htlev)) htt @@ htlev end.
- Inductive AllT {T:Type}{P:T->Prop} : Tree ??T -> Prop :=
- | allt_none : AllT []
- | allt_some : forall t, P t -> AllT [t]
- | allt_branch : forall b1 b2, AllT b1 -> AllT b2 -> AllT (b1,,b2)
- .
- Implicit Arguments AllT [[T]].
-
- Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ => Γ end.
-
- Definition getSuc (j:Judg) : Tree ??(LeveledHaskType (getΓ j) ★) :=
- match j as J return Tree ??(LeveledHaskType (getΓ J) ★) with
- Γ > _ > _ |- s => s
- end.
+ Axiom literal_types_unchanged : forall n Γ l, garrowfy_code_types n (literalType l) = literalType(Γ:=Γ) l.
- Definition getlev {Γ}{κ}(lht:LeveledHaskType Γ κ) :=
- match lht with t@@l => l end.
+ Axiom flatten_coercion : forall n Γ Δ κ (σ τ:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)),
+ HaskCoercion Γ Δ (garrowfy_code_types n σ ∼∼∼ garrowfy_code_types n τ).
(* This tries to assign a single level to the entire succedent of a judgment. If the succedent has types from different
* levels (should not happen) it just picks one; if the succedent has no non-None leaves (also should not happen) it
* picks nil *)
+ Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ => Γ end.
+ Definition getSuc (j:Judg) : Tree ??(LeveledHaskType (getΓ j) ★) :=
+ match j as J return Tree ??(LeveledHaskType (getΓ J) ★) with Γ > _ > _ |- s => s end.
Fixpoint getjlev {Γ}(tt:Tree ??(LeveledHaskType Γ ★)) : HaskLevel Γ :=
match tt with
| T_Leaf None => nil
end
end.
- (* an XJudg is a Judg for which the SUCCEDENT types all come from the same level, and that level is no deeper than "n" *)
- (* even the empty succedent [] has a level... *)
- Definition QJudg (n:nat)(j:Judg) := le (length (getjlev (getSuc j))) n.
-
-(* Definition qjudg2judg {n}(qj:QJudg n) : Judg := projT1 qj.*)
-
- Inductive QRule : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
- mkQRule : forall n h c,
- Rule h c ->
- ITree _ (QJudg n) h ->
- ITree _ (QJudg n) c ->
- QRule n h c.
-
- Definition QND n := ND (QRule n).
-
- (*
- * Any type "t" at a level with length "n" becomes (g () t)
- * Any type "<[t]>" at a level with length "n-1" becomes (g () t)
- *)
-
- Definition flatten_type {Γ}(n:nat)(ht:HaskType Γ ★) : HaskType Γ ★ :=
- fun TV ite =>
- flatten_rawtype n (ht TV ite).
-
- Definition minus' n m :=
- match m with
- | 0 => n
- | _ => n - m
- end.
-
- (* to do: integrate this more cleanly with qjudg *)
- Definition flatten_leveled_type {Γ}(n:nat)(ht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
- match ht with
- htt @@ htlev =>
- flatten_type (minus' n (length htlev)) htt @@ htlev
- end.
+ Definition v2t {Γ}(ec:HaskTyVar Γ ★) := fun TV ite => TVar (ec TV ite).
- Definition flatten_qjudg_ok (n:nat)(j:Judg) : Judg :=
- match j with
+ (* "n" is the maximum depth remaining AFTER flattening *)
+ Definition flatten_judgment (n:nat)(j:Judg) :=
+ match j as J return Judg with
Γ > Δ > ant |- suc =>
- let ant' := mapOptionTree (flatten_leveled_type n) ant in
- let suc' := mapOptionTree (flatten_leveled_type n) suc
- in (Γ > Δ > ant' |- suc')
- end.
-
- Definition take_lev'' {Γ}(lev:HaskLevel Γ)(tt:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★) :=
- match (take_lev' lev tt) with
- | None => []
- | Some x => x
- end.
-
- Definition flatten_qjudg : forall (n:nat), Judg -> Judg.
- refine (fun {n}{j} =>
- match j as J return Judg with
- Γ > Δ > ant |- suc =>
- match getjlev suc with
- | nil => flatten_qjudg_ok n j
- | (ec::lev') => if eqd_dec (length lev') n
- then let ant_host := drop_depth (S n) ant in
- let ant_guest := take_lev (ec::lev') ant in (* FIXME: I want take_lev''!!!!! *)
- (Γ > Δ > ant_host |- [ga ec ant_guest (mapOptionTree unlev suc) @@ lev'])
- else flatten_qjudg_ok n j
- end
- end).
- Defined.
-
- Axiom literal_types_unchanged : forall n Γ l, flatten_type n (literalType l) = literalType(Γ:=Γ) l.
-
- Lemma drop_depth_lemma : forall Γ (lev:HaskLevel Γ) x, drop_depth (length lev) [x @@ lev] = [].
- admit.
- Defined.
-
- Lemma drop_depth_lemma_s : forall Γ (lev:HaskLevel Γ) ec x, drop_depth (S (length lev)) [x @@ (ec :: lev)] = [].
- admit.
- Defined.
-
- Ltac drop_simplify :=
- match goal with
- | [ |- context[@drop_depth ?G (length ?L) [ ?X @@ ?L ] ] ] =>
- rewrite (drop_depth_lemma G L X)
- | [ |- context[@drop_depth ?G (S (length ?L)) [ ?X @@ (?E :: ?L) ] ] ] =>
- rewrite (drop_depth_lemma_s G L E X)
- | [ |- context[@drop_depth ?G ?N (?A,,?B)] ] =>
- change (@drop_depth G N (A,,B)) with ((@drop_depth G N A),,(@drop_depth G N B))
- | [ |- context[@drop_depth ?G ?N (T_Leaf None)] ] =>
- change (@drop_depth G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★)))
- end.
-
- Lemma take_lemma : forall Γ (lev:HaskLevel Γ) x, take_lev lev [x @@ lev] = [x].
- admit.
- Defined.
-
- Ltac take_simplify :=
- match goal with
- | [ |- context[@take_lev ?G ?L [ ?X @@ ?L ] ] ] =>
- rewrite (take_lemma G L X)
- | [ |- context[@take_lev ?G ?N (?A,,?B)] ] =>
- change (@take_lev G N (A,,B)) with ((@take_lev G N A),,(@take_lev G N B))
- | [ |- context[@take_lev ?G ?N (T_Leaf None)] ] =>
- change (@take_lev G N (T_Leaf (@None (LeveledHaskType G ★)))) with (T_Leaf (@None (LeveledHaskType G ★)))
+ match (match getjlev suc with
+ | nil => inl _ tt
+ | (ec::lev') => if eqd_dec (length lev') n
+ (* If the judgment's level is the deepest in the proof, flatten it by turning
+ * all antecedent variables at this level into None's, garrowfying any other
+ * antecedent variables (from other levels) at the same depth, and turning the
+ * succedent into a garrow type *)
+ then inr _ (Γ > Δ > mapOptionTree (garrowfy_leveled_code_types n) (drop_depth (ec::lev') ant)
+ |- [ga (v2t ec) (take_lev (ec::lev') ant) (mapOptionTree unlev suc) @@ lev'])
+ else inl _ tt
+ end) with
+
+ (* otherwise, just garrowfy all code types of the specified depth, throughout the judgment *)
+ | inl tt => Γ > Δ > mapOptionTree (garrowfy_leveled_code_types n) ant |- mapOptionTree (garrowfy_leveled_code_types n) suc
+ | inr r => r
+ end
end.
- Class garrow :=
- { ga_id : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a a @@ lev] ]
- ; ga_comp : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev],,[@ga Γ ec b c @@ lev] |- [@ga Γ ec a c @@ lev] ]
- ; ga_cancelr : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,[]) a @@ lev] ]
- ; ga_cancell : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ([],,a) a @@ lev] ]
- ; ga_uncancelr : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,[]) @@ lev] ]
- ; ga_uncancell : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a ([],,a) @@ lev] ]
- ; ga_assoc : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec ((a,,b),,c) (a,,(b,,c)) @@ lev] ]
- ; ga_unassoc : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,(b,,c)) ((a,,b),,c) @@ lev] ]
- ; ga_swap : ∀ Γ Δ ec lev a b , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec (a,,b) (b,,a) @@ lev] ]
- ; ga_drop : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a [] @@ lev] ]
- ; ga_copy : ∀ Γ Δ ec lev a , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec a (a,,a) @@ lev] ]
- ; ga_first : ∀ Γ Δ ec lev a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev] |- [@ga Γ ec (a,,x) (b,,x) @@ lev] ]
- ; ga_second : ∀ Γ Δ ec lev a b x, ND Rule [] [Γ > Δ > [@ga Γ ec a b @@ lev] |- [@ga Γ ec (x,,a) (x,,b) @@ lev] ]
- ; ga_lit : ∀ Γ Δ ec lev lit , ND Rule [] [Γ > Δ > [] |- [@ga Γ ec [] [literalType lit] @@ lev] ]
- ; ga_curry : ∀ Γ Δ ec lev a b c, ND Rule [] [Γ > Δ > [@ga Γ ec (a,,[b]) [c] @@ lev] |- [@ga Γ ec a [b ---> c] @@ lev] ]
- ; ga_apply : ∀ Γ Δ ec lev a a' b c, ND Rule [] [Γ > Δ >
- [@ga Γ ec a [b ---> c] @@ lev],,[@ga Γ ec a' [b] @@ lev]
- |-
- [@ga Γ ec (a,,a') [c] @@ lev] ]
- }.
-
- Context (gar:garrow).
-
Definition boost : forall Γ Δ ant x y,
ND Rule [] [ Γ > Δ > x |- y ] ->
ND Rule [ Γ > Δ > ant |- x ] [ Γ > Δ > ant |- y ].
admit.
Defined.
+
Definition postcompose : ∀ Γ Δ ec lev a b c,
ND Rule [] [ Γ > Δ > [] |- [@ga Γ ec a b @@ lev] ] ->
ND Rule [] [ Γ > Δ > [@ga Γ ec b c @@ lev] |- [@ga Γ ec a c @@ lev] ].
admit.
Defined.
- Definition precompose : ∀ Γ Δ lev a b c x,
- ND Rule [] [ Γ > Δ > a |- x,,[b @@ lev] ] ->
- ND Rule [] [ Γ > Δ > [b @@ lev] |- [c @@ lev] ] ->
- ND Rule [] [ Γ > Δ > a,,x |- [c @@ lev] ].
+
+ Definition seq : ∀ Γ Δ lev a b,
+ ND Rule [] [ Γ > Δ > [] |- [a @@ lev] ] ->
+ ND Rule [] [ Γ > Δ > [] |- [b @@ lev] ] ->
+ ND Rule [] [ Γ > Δ > [] |- [a @@ lev],,[b @@ lev] ].
admit.
Defined.
- Set Printing Width 130.
+ Lemma ga_unkappa : ∀ Γ Δ ec l a b Σ,
+ ND Rule
+ [Γ > Δ > Σ |- [@ga Γ ec a b @@ l] ]
+ [Γ > Δ > Σ,,[@ga Γ ec [] a @@ l] |- [@ga Γ ec [] b @@ l] ].
+ intros.
+ set (ga_comp Γ Δ ec l [] a b) as q.
+
+ set (@RLet Γ Δ) as q'.
+ set (@RLet Γ Δ [@ga _ ec [] a @@ l] Σ (@ga _ ec [] b) (@ga _ ec a b) l) as q''.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ apply RExch.
+
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ apply q''.
+
+ idtac.
+ clear q'' q'.
+ eapply nd_comp.
+ apply nd_rlecnac.
+ apply nd_prod.
+ apply nd_id.
+ apply q.
+ Defined.
+(*
+ Notation "` x" := (take_lev _ x) (at level 20).
+ Notation "`` x" := (mapOptionTree unlev x) (at level 20).
+ Notation "``` x" := (drop_depth _ x) (at level 20).
+*)
Definition garrowfy_arrangement' :
forall Γ (Δ:CoercionEnv Γ)
(ec:HaskTyVar Γ ★) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2),
- ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev] ].
+ ND Rule [] [Γ > Δ > [] |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev] ].
intros Γ Δ ec lev.
refine (fix garrowfy ant1 ant2 (r:Arrange ant1 ant2):
- ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev]] :=
+ ND Rule [] [Γ > Δ > [] |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant2) (take_lev (ec :: lev) ant1) @@ lev]] :=
match r as R in Arrange A B return
- ND Rule [] [Γ > Δ > [] |- [@ga _ ec (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ lev]]
+ ND Rule [] [Γ > Δ > [] |- [@ga _ (v2t ec) (take_lev (ec :: lev) B) (take_lev (ec :: lev) A) @@ lev]]
with
| RCanL a => let case_RCanL := tt in ga_uncancell _ _ _ _ _
| RCanR a => let case_RCanR := tt in ga_uncancelr _ _ _ _ _
end); clear garrowfy; repeat take_simplify; repeat drop_simplify; intros.
destruct case_RComp.
- (*
- set (ga_comp Γ Δ ec lev (``c) (``b) (``a)) as q.
- eapply precompose.
- Focus 2.
- apply q.
refine ( _ ;; boost _ _ _ _ _ (ga_comp _ _ _ _ _ _ _)).
- apply precompose.
- Focus 2.
- eapply ga_comp.
- *)
- admit.
+ apply seq.
+ apply r2'.
+ apply r1'.
Defined.
Definition garrowfy_arrangement :
forall Γ (Δ:CoercionEnv Γ) n
(ec:HaskTyVar Γ ★) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2) succ,
- (*ec :: lev = getjlev succ ->*)
- (* H0 : left (Datatypes.length lev ≠ n) e = Peano_dec.eq_nat_dec (Datatypes.length lev) n *)
ND Rule
- [Γ > Δ > drop_depth n ant1 |- [@ga _ ec (take_lev (ec :: lev) ant1) (mapOptionTree unlev succ) @@ lev]]
- [Γ > Δ > drop_depth n ant2 |- [@ga _ ec (take_lev (ec :: lev) ant2) (mapOptionTree unlev succ) @@ lev]].
+ [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth n ant1)
+ |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant1) (mapOptionTree unlev succ) @@ lev]]
+ [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth n ant2)
+ |- [@ga _ (v2t ec) (take_lev (ec :: lev) ant2) (mapOptionTree unlev succ) @@ lev]].
intros.
refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (garrowfy_arrangement' Γ Δ ec lev ant1 ant2 r)))).
apply nd_rule.
apply RArrange.
refine ((fix garrowfy ant1 ant2 (r:Arrange ant1 ant2) :=
- match r as R in Arrange A B return Arrange (drop_depth _ A) (drop_depth _ B) with
+ match r as R in Arrange A B return
+ Arrange (mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth _ A))
+ (mapOptionTree (garrowfy_leveled_code_types ((length lev))) (drop_depth _ B)) with
| RCanL a => let case_RCanL := tt in RCanL _
| RCanR a => let case_RCanR := tt in RCanR _
| RuCanL a => let case_RuCanL := tt in RuCanL _
Definition flatten_arrangement :
forall n Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2),
- ND Rule (mapOptionTree (flatten_qjudg n) [Γ > Δ > ant1 |- succ])
- (mapOptionTree (flatten_qjudg n) [Γ > Δ > ant2 |- succ]).
+ ND Rule (mapOptionTree (flatten_judgment n) [Γ > Δ > ant1 |- succ])
+ (mapOptionTree (flatten_judgment n) [Γ > Δ > ant2 |- succ]).
intros.
simpl.
set (getjlev succ) as succ_lev.
eapply RComp; [ apply IHr1 | apply IHr2 ].
Defined.
- Lemma flatten_coercion : forall n Γ Δ σ τ (γ:HaskCoercion Γ Δ (σ ∼∼∼ τ)),
- HaskCoercion Γ Δ (flatten_type n σ ∼∼∼ flatten_type n τ).
+ Definition arrange_brak : forall Γ Δ ec succ t,
+ ND Rule
+ [Γ > Δ >
+ mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec :: nil) succ),,
+ [(@ga _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |-
+ [(@ga _ (v2t ec) [] [t]) @@ nil]]
+ [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types 0) succ |- [(@ga _ (v2t ec) [] [t]) @@ nil]].
admit.
Defined.
- Ltac eqd_dec_refl' :=
- match goal with
- | [ |- context[@eqd_dec ?T ?V ?X ?X] ] =>
- destruct (@eqd_dec T V X X) as [eqd_dec1 | eqd_dec2];
- [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ]
- end.
+ Definition arrange_esc : forall Γ Δ ec succ t,
+ ND Rule
+ [Γ > Δ > mapOptionTree (garrowfy_leveled_code_types 0) succ |- [(@ga _ (v2t ec) [] [t]) @@ nil]]
+ [Γ > Δ >
+ mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec :: nil) succ),,
+ [(@ga _ (v2t ec) [] (take_lev (ec :: nil) succ)) @@ nil] |- [(@ga _ (v2t ec) [] [t]) @@ nil]].
+ admit.
+ Defined.
+ Lemma mapOptionTree_distributes
+ : forall T R (a b:Tree ??T) (f:T->R),
+ mapOptionTree f (a,,b) = (mapOptionTree f a),,(mapOptionTree f b).
+ reflexivity.
+ Qed.
-(*
- Lemma foog : forall Γ Δ,
- ND Rule
- ( [ Γ > Δ > Σ₁ |- a ],,[ Γ > Δ > Σ₂ |- b ] )
- [ Γ > Δ > Σ₁,,Σ₂ |- a,,b ]
-*)
+ Lemma garrowfy_commutes_with_substT :
+ forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★) (τ:HaskType Γ κ),
+ garrowfy_code_types n (substT σ τ) = substT (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v))
+ (garrowfy_code_types n τ).
+ admit.
+ Qed.
- Notation "` x" := (take_lev _ x) (at level 20).
- Notation "`` x" := (mapOptionTree unlev x) (at level 20).
- Notation "``` x" := (drop_depth _ x) (at level 20).
+ Lemma garrowfy_commutes_with_HaskTAll :
+ forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
+ garrowfy_code_types n (HaskTAll κ σ) = HaskTAll κ (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v)).
+ admit.
+ Qed.
- Definition flatten :
+ Lemma garrowfy_commutes_with_HaskTApp :
+ forall n κ Γ (Δ:CoercionEnv Γ) (σ:∀ TV, InstantiatedTypeEnv TV Γ → TV κ → RawHaskType TV ★),
+ garrowfy_code_types n (HaskTApp (weakF σ) (FreshHaskTyVar κ)) =
+ HaskTApp (weakF (fun TV ite v => garrowfy_raw_codetypes n (σ TV ite v))) (FreshHaskTyVar κ).
+ admit.
+ Qed.
+
+ Lemma garrowfy_commutes_with_weakLT : forall (Γ:TypeEnv) κ n t,
+ garrowfy_leveled_code_types n (weakLT(Γ:=Γ)(κ:=κ) t) = weakLT(Γ:=Γ)(κ:=κ) (garrowfy_leveled_code_types n t).
+ admit.
+ Qed.
+
+ Definition flatten_proof :
forall n {h}{c},
- QND (S n) h c ->
- ND Rule (mapOptionTree (flatten_qjudg n) h) (mapOptionTree (flatten_qjudg n) c).
+ ND Rule h c ->
+ ND Rule (mapOptionTree (flatten_judgment n) h) (mapOptionTree (flatten_judgment n) c).
intros.
eapply nd_map'; [ idtac | apply X ].
clear h c X.
intros.
simpl in *.
- inversion X.
- subst.
- refine (match X0 as R in Rule H C with
+ refine (match X as R in Rule H C with
| RArrange Γ Δ a b x d => let case_RArrange := tt in _
| RNote Γ Δ Σ τ l n => let case_RNote := tt in _
| RLit Γ Δ l _ => let case_RLit := tt in _
| RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ lev => let case_RLet := 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 _
+ | RBrak Γ Δ t ec succ lev => let case_RBrak := tt in _
+ | REsc Γ Δ t ec succ lev => 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 t => let case_RLetRec := tt in _
- end); clear X X0 X1 X2 h c.
+ end); clear X h c.
destruct case_RArrange.
apply (flatten_arrangement n Γ Δ a b x d).
destruct case_RBrak.
- admit.
+ simpl.
+ destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n).
+ destruct lev.
+ simpl.
+ simpl.
+ destruct n.
+ change ([garrowfy_code_types 0 (<[ ec |- t ]>) @@ nil])
+ with ([ga (v2t ec) [] [t] @@ nil]).
+ refine (ga_unkappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) [t]
+ (mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec::nil) succ)) ;; _).
+ apply arrange_brak.
+ inversion e.
+ apply (Prelude_error "found Brak at depth >0").
+ apply (Prelude_error "found Brak at depth >0").
destruct case_REsc.
- admit.
+ simpl.
+ destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n).
+ destruct lev.
+ simpl.
+ destruct n.
+ change ([garrowfy_code_types 0 (<[ ec |- t ]>) @@ nil])
+ with ([ga (v2t ec) [] [t] @@ nil]).
+ refine (_ ;; ga_kappa Γ Δ (v2t ec) nil (take_lev (ec::nil) succ) [t]
+ (mapOptionTree (garrowfy_leveled_code_types 0) (drop_depth (ec::nil) succ))).
+ apply arrange_esc.
+ inversion e.
+ apply (Prelude_error "found Esc at depth >0").
+ apply (Prelude_error "found Esc at depth >0").
destruct case_RNote.
simpl.
apply RLit.
destruct case_RVar.
+ Opaque flatten_judgment.
simpl.
- destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RVar | idtac ].
- destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RVar ]; simpl.
- rewrite <- e.
- clear e n.
- repeat drop_simplify.
+ Transparent flatten_judgment.
+ idtac.
+ unfold flatten_judgment.
+ unfold getjlev.
+ destruct lev.
+ apply nd_rule. apply RVar.
+ destruct (eqd_dec (Datatypes.length lev) n).
+
+ repeat drop_simplify.
repeat take_simplify.
- apply ga_id.
+ simpl.
+ apply ga_id.
+
+ apply nd_rule.
+ apply RVar.
destruct case_RGlobal.
simpl.
apply (Prelude_error "found RGlobal at depth >0").
destruct case_RLam.
+ Opaque drop_depth.
+ Opaque take_lev.
simpl.
destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RLam; auto | idtac ].
destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RLam; auto ]; simpl.
eapply nd_comp.
eapply nd_rule.
eapply RArrange.
+ simpl.
apply RCanR.
apply boost.
apply ga_curry.
simpl.
destruct lev as [|ec lev]. simpl. apply nd_rule.
- replace (flatten_type n (tx ---> te)) with ((flatten_type n tx) ---> (flatten_type n te)).
+ replace (garrowfy_code_types n (tx ---> te)) with ((garrowfy_code_types n tx) ---> (garrowfy_code_types n te)).
apply RApp.
- unfold flatten_type.
+ unfold garrowfy_code_types.
simpl.
reflexivity.
apply boost.
apply ga_apply.
- replace (flatten_type (minus' n (length (ec::lev))) (tx ---> te))
- with ((flatten_type (minus' n (length (ec::lev))) tx) ---> (flatten_type (minus' n (length (ec::lev))) te)).
+ replace (garrowfy_code_types (minus' n (length (ec::lev))) (tx ---> te))
+ with ((garrowfy_code_types (minus' n (length (ec::lev))) tx) --->
+ (garrowfy_code_types (minus' n (length (ec::lev))) te)).
apply nd_rule.
apply RApp.
- unfold flatten_type.
+ unfold garrowfy_code_types.
simpl.
reflexivity.
-
+(*
+ Notation "` x" := (take_lev _ x) (at level 20).
+ Notation "`` x" := (mapOptionTree unlev x) (at level 20).
+ Notation "``` x" := ((drop_depth _ x)) (at level 20).
+ Notation "!<[]> x" := (garrowfy_code_types _ x) (at level 1).
+ Notation "!<[@]>" := (garrowfy_leveled_code_types _) (at level 1).
+*)
destruct case_RLet.
simpl.
destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RLet; auto | idtac ].
destruct (Peano_dec.eq_nat_dec (Datatypes.length lev) n); [ idtac | apply nd_rule; apply RLet; auto ]; simpl.
repeat drop_simplify.
repeat take_simplify.
- admit. (* FIXME *)
+ rename σ₁ into a.
+ rename σ₂ into b.
+ rewrite mapOptionTree_distributes.
+ rewrite mapOptionTree_distributes.
+ set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_depth (ec :: lev) Σ₁)) as A.
+ set (take_lev (ec :: lev) Σ₁) as A'.
+ set (mapOptionTree (garrowfy_leveled_code_types (S n)) (drop_depth (ec :: lev) Σ₂)) as B.
+ set (take_lev (ec :: lev) Σ₂) as B'.
+ simpl.
+
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RLet.
+
+ apply nd_prod.
+
+ apply boost.
+ apply ga_second.
+
+ eapply nd_comp.
+ Focus 2.
+ eapply boost.
+ apply ga_comp.
+
+ eapply nd_comp.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply RCanR.
+
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply RExch.
+ idtac.
+
+ eapply nd_comp.
+ apply nd_llecnac.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ apply RJoin.
+ apply nd_prod.
+
+ eapply nd_rule.
+ eapply RVar.
+
+ apply nd_id.
destruct case_RVoid.
simpl.
destruct case_RAppT.
simpl. destruct lev; simpl.
- admit.
- admit.
+ rewrite garrowfy_commutes_with_HaskTAll.
+ rewrite garrowfy_commutes_with_substT.
+ apply nd_rule.
+ apply RAppT.
+ apply Δ.
+ apply Δ.
+ apply (Prelude_error "AppT at depth>0").
destruct case_RAbsT.
simpl. destruct lev; simpl.
- admit.
- admit.
+ rewrite garrowfy_commutes_with_HaskTAll.
+ rewrite garrowfy_commutes_with_HaskTApp.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RAbsT ].
+ simpl.
+ set (mapOptionTree (garrowfy_leveled_code_types n) (mapOptionTree (weakLT(κ:=κ)) Σ)) as a.
+ set (mapOptionTree (weakLT(κ:=κ)) (mapOptionTree (garrowfy_leveled_code_types n) Σ)) as q'.
+ assert (a=q').
+ unfold a.
+ unfold q'.
+ clear a q'.
+ induction Σ.
+ destruct a.
+ simpl.
+ rewrite garrowfy_commutes_with_weakLT.
+ reflexivity.
+ reflexivity.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ rewrite H.
+ apply nd_id.
+ apply Δ.
+ apply Δ.
+ apply (Prelude_error "AbsT at depth>0").
destruct case_RAppCo.
simpl. destruct lev; simpl.
- admit.
- admit.
+ unfold garrowfy_code_types.
+ simpl.
+ apply nd_rule.
+ apply RAppCo.
+ apply flatten_coercion.
+ apply γ.
+ apply (Prelude_error "AppCo at depth>0").
destruct case_RAbsCo.
simpl. destruct lev; simpl.
- admit.
- admit.
+ unfold garrowfy_code_types.
+ simpl.
+ apply (Prelude_error "AbsCo not supported (FIXME)").
+ apply (Prelude_error "AbsCo at depth>0").
destruct case_RLetRec.
- simpl.
- admit.
+ rename t into lev.
+ apply (Prelude_error "LetRec not supported (FIXME)").
destruct case_RCase.
simpl.
- admit.
+ apply (Prelude_error "Case not supported (FIXME)").
Defined.
- Lemma flatten_qjudg_qjudg : forall {n}{j} (q:QJudg (S n) j), QJudg n (flatten_qjudg n j).
- admit.
- (*
- intros.
- destruct q.
- destruct a.
- unfold flatten_qjudg.
- destruct j.
- destruct (eqd_dec (Datatypes.length x) (S n)).
- destruct x.
- inversion e.
- exists x; split.
- simpl in e.
- inversion e.
- auto.
- simpl in *.
- apply allt_some.
- simpl.
- auto.
- unfold QJudg.
- exists x.
- split; auto.
- clear a.
- Set Printing Implicit.
- simpl.
- set (length x) as x'.
- assert (x'=length x).
- reflexivity.
- simpl in *.
- rewrite <- H.
- Unset Printing Implicit.
- idtac.
- omega.
- simpl in *.
- induction t0.
- destruct a0; simpl in *.
- apply allt_some.
- inversion a.
- subst.
- destruct l0.
- simpl.
- auto.
- apply allt_none.
- simpl in *.
- inversion a; subst.
- apply allt_branch.
- apply IHt0_1; auto.
- apply IHt0_2; auto.
- *)
- Defined.
+ (* to do: establish some metric on judgments (max length of level of any succedent type, probably), show how to
+ * calculate it, and show that the flattening procedure above drives it down by one *)
(*
Instance FlatteningFunctor {Γ}{Δ}{ec} : Functor (JudgmentsL (PCF Γ Δ ec)) (TypesL (SystemFCa Γ Δ)) (obact) :=
{ fmor := FlatteningFunctor_fmor }.
- Admitted.
Definition ReificationFunctor Γ Δ : Functor (JudgmentsL _ _ (PCF n Γ Δ)) SystemFCa' (mapOptionTree brakifyJudg).
refine {| fmor := ReificationFunctor_fmor Γ Δ |}; unfold hom; unfold ob; simpl ; intros.
- Admitted.
Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
refine {| plsmme_pl := PCF n Γ Δ |}.
- admit.
Defined.
Definition SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
refine {| plsmme_pl := SystemFCa n Γ Δ |}.
- admit.
Defined.
Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
- admit.
Defined.
(* 5.1.4 *)
Definition PCF_SystemFCa_two_level n Γ Δ : TwoLevelLanguage (PCF_SMME n Γ Δ) (SystemFCa_SMME (S n) Γ Δ).
- admit.
Defined.
*)
(* ... and the retraction exists *)
End HaskFlattener.
-
-
-
-
-
-
-
-
-(*
-
- Old flattening code; ignore this - just to remind me how I used to do it
-
- (*
- * Here it is, what you've all been waiting for! When reading this,
- * it might help to have the definition for "Inductive ND" (see
- * NaturalDeduction.v) handy as a cross-reference.
- *)
- Hint Constructors Rule_Flat.
- Definition FlatteningFunctor_fmor
- : forall h c,
- (ND (PCFRule Γ Δ ec) h c) ->
- ((obact h)====>(obact c)).
-
- set (@nil (HaskTyVar Γ ★)) as lev.
-
- unfold hom; unfold ob; unfold ehom; simpl; unfold pmon_I; unfold obact; intros.
-
- induction X; simpl.
-
- (* the proof from no hypotheses of no conclusions (nd_id0) becomes RVoid *)
- apply nd_rule; apply (org_fc Γ Δ [] [(_,_)] (RVoid _ _)). apply Flat_RVoid.
-
- (* the proof from hypothesis X of conclusion X (nd_id1) becomes RVar *)
- apply nd_rule; apply (org_fc _ _ [] [(_,_)] (RVar _ _ _ _)). apply Flat_RVar.
-
- (* the proof from hypothesis X of no conclusions (nd_weak) becomes RWeak;;RVoid *)
- eapply nd_comp;
- [ idtac
- | eapply nd_rule
- ; eapply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RWeak _)))
- ; auto ].
- eapply nd_rule.
- eapply (org_fc _ _ [] [(_,_)] (RVoid _ _)); auto. apply Flat_RVoid.
- apply Flat_RArrange.
-
- (* the proof from hypothesis X of two identical conclusions X,,X (nd_copy) becomes RVar;;RJoin;;RCont *)
- eapply nd_comp; [ idtac | eapply nd_rule; eapply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCont _))) ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- set (snd_initial(SequentND:=pl_snd(ProgrammingLanguage:=SystemFCa Γ Δ))
- (mapOptionTree (guestJudgmentAsGArrowType) h @@@ lev)) as q.
- eapply nd_comp.
- eapply nd_prod.
- apply q.
- apply q.
- apply nd_rule.
- eapply (org_fc _ _ ([(_,_)],,[(_,_)]) [(_,_)] (RJoin _ _ _ _ _ _ )).
- destruct h; simpl.
- destruct o.
- simpl.
- apply Flat_RJoin.
- apply Flat_RJoin.
- apply Flat_RJoin.
- apply Flat_RArrange.
-
- (* nd_prod becomes nd_llecnac;;nd_prod;;RJoin *)
- eapply nd_comp.
- apply (nd_llecnac ;; nd_prod IHX1 IHX2).
- apply nd_rule.
- eapply (org_fc _ _ ([(_,_)],,[(_,_)]) [(_,_)] (RJoin _ _ _ _ _ _ )).
- apply (Flat_RJoin Γ Δ (mapOptionTree guestJudgmentAsGArrowType h1 @@@ nil)
- (mapOptionTree guestJudgmentAsGArrowType h2 @@@ nil)
- (mapOptionTree guestJudgmentAsGArrowType c1 @@@ nil)
- (mapOptionTree guestJudgmentAsGArrowType c2 @@@ nil)).
-
- (* nd_comp becomes pl_subst (aka nd_cut) *)
- eapply nd_comp.
- apply (nd_llecnac ;; nd_prod IHX1 IHX2).
- clear IHX1 IHX2 X1 X2.
- apply (@snd_cut _ _ _ _ (pl_snd(ProgrammingLanguage:=SystemFCa Γ Δ))).
-
- (* nd_cancell becomes RVar;;RuCanL *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RuCanL _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- (* nd_cancelr becomes RVar;;RuCanR *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RuCanR _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- (* nd_llecnac becomes RVar;;RCanL *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCanL _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- (* nd_rlecnac becomes RVar;;RCanR *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCanR _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- (* nd_assoc becomes RVar;;RAssoc *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RAssoc _ _ _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- (* nd_cossa becomes RVar;;RCossa *)
- eapply nd_comp;
- [ idtac | eapply nd_rule; apply (org_fc _ _ [(_,_)] [(_,_)] (RArrange _ _ _ _ _ (RCossa _ _ _))) ].
- apply (snd_initial(SequentND:=pl_cnd(ProgrammingLanguage:=(SystemFCa Γ Δ)))).
- apply Flat_RArrange.
-
- destruct r as [r rp].
- rename h into h'.
- rename c into c'.
- rename r into r'.
-
- refine (match rp as R in @Rule_PCF _ _ _ H C _
- return
- ND (OrgR Γ Δ) []
- [sequent (mapOptionTree guestJudgmentAsGArrowType H @@@ nil)
- (mapOptionTree guestJudgmentAsGArrowType C @@@ nil)]
- with
- | PCF_RArrange h c r q => let case_RURule := tt in _
- | PCF_RLit lit => let case_RLit := tt in _
- | PCF_RNote Σ τ n => let case_RNote := tt in _
- | PCF_RVar σ => let case_RVar := tt in _
- | PCF_RLam Σ tx te => let case_RLam := tt in _
- | PCF_RApp Σ tx te p => let case_RApp := tt in _
- | PCF_RLet Σ σ₁ σ₂ p => let case_RLet := tt in _
- | PCF_RJoin b c d e => let case_RJoin := tt in _
- | PCF_RVoid => let case_RVoid := tt in _
- (*| PCF_RCase T κlen κ θ l x => let case_RCase := tt in _*)
- (*| PCF_RLetRec Σ₁ τ₁ τ₂ lev => let case_RLetRec := tt in _*)
- end); simpl in *.
- clear rp h' c' r'.
-
- rewrite (unlev_lemma h (ec::nil)).
- rewrite (unlev_lemma c (ec::nil)).
- destruct case_RURule.
- refine (match q as Q in Arrange H C
- return
- H=(h @@@ (ec :: nil)) ->
- C=(c @@@ (ec :: nil)) ->
- ND (OrgR Γ Δ) []
- [sequent
- [ga_type (ga_rep (mapOptionTree unlev H)) (ga_rep r) @@ nil ]
- [ga_type (ga_rep (mapOptionTree unlev C)) (ga_rep r) @@ nil ] ]
- with
- | RLeft a b c r => let case_RLeft := tt in _
- | RRight a b c r => let case_RRight := tt in _
- | RCanL b => let case_RCanL := tt in _
- | RCanR b => let case_RCanR := tt in _
- | RuCanL b => let case_RuCanL := tt in _
- | RuCanR b => let case_RuCanR := tt in _
- | RAssoc b c d => let case_RAssoc := tt in _
- | RCossa b c d => let case_RCossa := tt in _
- | RExch b c => let case_RExch := tt in _
- | RWeak b => let case_RWeak := tt in _
- | RCont b => let case_RCont := tt in _
- | RComp a b c f g => let case_RComp := tt in _
- end (refl_equal _) (refl_equal _));
- intros; simpl in *;
- subst;
- try rewrite <- unlev_lemma in *.
-
- destruct case_RCanL.
- apply magic.
- apply ga_uncancell.
-
- destruct case_RCanR.
- apply magic.
- apply ga_uncancelr.
-
- destruct case_RuCanL.
- apply magic.
- apply ga_cancell.
-
- destruct case_RuCanR.
- apply magic.
- apply ga_cancelr.
-
- destruct case_RAssoc.
- apply magic.
- apply ga_assoc.
-
- destruct case_RCossa.
- apply magic.
- apply ga_unassoc.
-
- destruct case_RExch.
- apply magic.
- apply ga_swap.
-
- destruct case_RWeak.
- apply magic.
- apply ga_drop.
-
- destruct case_RCont.
- apply magic.
- apply ga_copy.
-
- destruct case_RLeft.
- apply magic.
- (*apply ga_second.*)
- admit.
-
- destruct case_RRight.
- apply magic.
- (*apply ga_first.*)
- admit.
-
- destruct case_RComp.
- apply magic.
- (*apply ga_comp.*)
- admit.
-
- destruct case_RLit.
- apply ga_lit.
-
- (* hey cool, I figured out how to pass CoreNote's through... *)
- destruct case_RNote.
- eapply nd_comp.
- eapply nd_rule.
- eapply (org_fc _ _ [] [(_,_)] (RVar _ _ _ _)) . auto.
- apply Flat_RVar.
- apply nd_rule.
- apply (org_fc _ _ [(_,_)] [(_,_)] (RNote _ _ _ _ _ n)). auto.
- apply Flat_RNote.
-
- destruct case_RVar.
- apply ga_id.
-
- (*
- * class GArrow g (**) u => GArrowApply g (**) u (~>) where
- * ga_applyl :: g (x**(x~>y) ) y
- * ga_applyr :: g ( (x~>y)**x) y
- *
- * class GArrow g (**) u => GArrowCurry g (**) u (~>) where
- * ga_curryl :: g (x**y) z -> g x (y~>z)
- * ga_curryr :: g (x**y) z -> g y (x~>z)
- *)
- destruct case_RLam.
- (* GArrowCurry.ga_curry *)
- admit.
-
- destruct case_RApp.
- (* GArrowApply.ga_apply *)
- admit.
-
- destruct case_RLet.
- admit.
-
- destruct case_RVoid.
- apply ga_id.
-
- destruct case_RJoin.
- (* this assumes we want effects to occur in syntactically-left-to-right order *)
- admit.
- Defined.
-*)
\ No newline at end of file
+Implicit Arguments garrow [ ].