Definition curry {Γ}{Δ}{a}{s}{Σ}{lev} :
ND Rule
- [ Γ > Δ > Σ |- [a ---> s @@ lev ] ]
- [ Γ > Δ > [a @@ lev],,Σ |- [ s @@ lev ] ].
+ [ Γ > Δ > Σ |- [a ---> s ]@lev ]
+ [ Γ > Δ > [a @@ lev],,Σ |- [ s ]@lev ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RApp ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
Defined.
Definition fToC1 {Γ}{Δ}{a}{s}{lev} :
- ND Rule [] [ Γ > Δ > [ ] |- [a ---> s @@ lev ] ] ->
- ND Rule [] [ Γ > Δ > [a @@ lev] |- [ s @@ lev ] ].
+ ND Rule [] [ Γ > Δ > [ ] |- [a ---> s ]@lev ] ->
+ ND Rule [] [ Γ > Δ > [a @@ lev] |- [ s ]@lev ].
intro pf.
eapply nd_comp.
apply pf.
Defined.
Definition fToC2 {Γ}{Δ}{a1}{a2}{s}{lev} :
- ND Rule [] [ Γ > Δ > [] |- [a1 ---> (a2 ---> s) @@ lev ] ] ->
- ND Rule [] [ Γ > Δ > [a1 @@ lev],,[a2 @@ lev] |- [ s @@ lev ] ].
+ ND Rule [] [ Γ > Δ > [] |- [a1 ---> (a2 ---> s) ]@lev ] ->
+ ND Rule [] [ Γ > Δ > [a1 @@ lev],,[a2 @@ lev] |- [ s ]@lev ].
intro pf.
eapply nd_comp.
eapply pf.
Defined.
Definition mkGlob2 {Γ}{Δ}{l}{κ₁}{κ₂}(cv:CoreVar)(f:HaskType Γ κ₁ -> HaskType Γ κ₂ -> HaskType Γ ★) x y
- : ND Rule [] [ Γ > Δ > [] |- [f x y @@ l] ].
+ : ND Rule [] [ Γ > Δ > [] |- [f x y ]@l ].
apply nd_rule.
refine (@RGlobal Γ Δ l
{| glob_wv := coreVarToWeakExprVarOrError cv
Defined.
Definition mkGlob3 {Γ}{Δ}{l}{κ₁}{κ₂}{κ₃}(cv:CoreVar)(f:HaskType Γ κ₁ -> HaskType Γ κ₂ -> HaskType Γ κ₃ -> HaskType Γ ★) x y z
- : ND Rule [] [ Γ > Δ > [] |- [f x y z @@ l] ].
+ : ND Rule [] [ Γ > Δ > [] |- [f x y z ]@l ].
apply nd_rule.
refine (@RGlobal Γ Δ l
{| glob_wv := coreVarToWeakExprVarOrError cv
Defined.
Definition mkGlob4 {Γ}{Δ}{l}{κ₁}{κ₂}{κ₃}{κ₄}(cv:CoreVar)(f:HaskType Γ κ₁ -> HaskType Γ κ₂ -> HaskType Γ κ₃ -> HaskType Γ κ₄ -> HaskType Γ ★) x y z q
- : ND Rule [] [ Γ > Δ > [] |- [f x y z q @@ l] ].
+ : ND Rule [] [ Γ > Δ > [] |- [f x y z q ] @l].
apply nd_rule.
refine (@RGlobal Γ Δ l
{| glob_wv := coreVarToWeakExprVarOrError cv
Definition hetmet_unflatten' := coreVarToWeakExprVarOrError hetmet_unflatten.
Definition hetmet_flattened_id' := coreVarToWeakExprVarOrError hetmet_flattened_id.
- Definition coreToCoreExpr' (ce:@CoreExpr CoreVar) : ???(@CoreExpr CoreVar) :=
- addErrorMessage ("input CoreSyn: " +++ toString ce)
- (addErrorMessage ("input CoreType: " +++ toString (coreTypeOfCoreExpr ce)) (
- coreExprToWeakExpr ce >>= fun we =>
+ Definition coreToCoreExpr' (cex:@CoreExpr CoreVar) : ???(@CoreExpr CoreVar) :=
+ addErrorMessage ("input CoreSyn: " +++ toString cex)
+ (addErrorMessage ("input CoreType: " +++ toString (coreTypeOfCoreExpr cex)) (
+ coreExprToWeakExpr cex >>= fun we =>
addErrorMessage ("WeakExpr: " +++ toString we)
((addErrorMessage ("CoreType of WeakExpr: " +++ toString (coreTypeOfCoreExpr (weakExprToCoreExpr we)))
((weakTypeOfWeakExpr we) >>= fun t =>
(let haskProof := skolemize_and_flatten_proof hetmet_flatten' hetmet_unflatten'
hetmet_flattened_id' my_ga (@expr2proof _ _ _ _ _ _ e)
in (* insert HaskProof-to-HaskProof manipulations here *)
- OK ((@proof2expr nat _ FreshNat _ _ _ _ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
+ OK ((@proof2expr nat _ FreshNat _ _ (flatten_type τ@@nil) _ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
>>= fun e' =>
(snd e') >>= fun e'' =>
strongExprToWeakExpr hetmet_brak' hetmet_esc'
Axiom globals_do_not_have_code_types : forall (Γ:TypeEnv) (g:Global Γ) v,
flatten_type (g v) = g v.
- (* 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
- | T_Leaf (Some (_ @@ lev)) => lev
- | T_Branch b1 b2 =>
- match getjlev b1 with
- | nil => getjlev b2
- | lev => lev
- end
- end.
-
(* "n" is the maximum depth remaining AFTER flattening *)
Definition flatten_judgment (j:Judg) :=
match j as J return Judg with
- Γ > Δ > ant |- suc =>
- match getjlev suc with
- | nil => Γ > Δ > mapOptionTree flatten_leveled_type ant
- |- mapOptionTree flatten_leveled_type suc
-
- | (ec::lev') => Γ > Δ > mapOptionTree flatten_leveled_type (drop_lev (ec::lev') ant)
- |- [ga_mk (v2t ec)
- (mapOptionTree (flatten_type ○ unlev) (take_lev (ec::lev') ant))
- (mapOptionTree (flatten_type ○ unlev) suc )
- @@ nil] (* we know the level of all of suc *)
- end
+ | Γ > Δ > ant |- suc @ nil => Γ > Δ > mapOptionTree flatten_leveled_type ant
+ |- mapOptionTree flatten_type suc @ nil
+ | Γ > Δ > ant |- suc @ (ec::lev') => Γ > Δ > mapOptionTree flatten_leveled_type (drop_lev (ec::lev') ant)
+ |- [ga_mk (v2t ec)
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec::lev') ant))
+ (mapOptionTree flatten_type suc )
+ ] @ nil
end.
Class garrow :=
- { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a @@ l] ]
- ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a @@ l] ]
- ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a @@ l] ]
- ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) @@ l] ]
- ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) @@ l] ]
- ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) @@ l] ]
- ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) @@ l] ]
- ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) @@ l] ]
- ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] @@ l] ]
- ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) @@ l] ]
- ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (a,,x) (b,,x) @@ l] ]
- ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l] |- [@ga_mk Γ ec (x,,a) (x,,b) @@ l] ]
- ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] @@ l] ]
- ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] @@ l] ]
- ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c @@ l] ]
+ { ga_id : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a a ]@l ]
+ ; ga_cancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,[]) a ]@l ]
+ ; ga_cancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ([],,a) a ]@l ]
+ ; ga_uncancelr : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,[]) ]@l ]
+ ; ga_uncancell : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a ([],,a) ]@l ]
+ ; ga_assoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec ((a,,b),,c) (a,,(b,,c)) ]@l ]
+ ; ga_unassoc : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,(b,,c)) ((a,,b),,c) ]@l ]
+ ; ga_swap : ∀ Γ Δ ec l a b , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec (a,,b) (b,,a) ]@l ]
+ ; ga_drop : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a [] ]@l ]
+ ; ga_copy : ∀ Γ Δ ec l a , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec a (a,,a) ]@l ]
+ ; ga_first : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@l] |- [@ga_mk Γ ec (a,,x) (b,,x) ]@l ]
+ ; ga_second : ∀ Γ Δ ec l a b x, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@l] |- [@ga_mk Γ ec (x,,a) (x,,b) ]@l ]
+ ; ga_lit : ∀ Γ Δ ec l lit , ND Rule [] [Γ > Δ > [] |- [@ga_mk Γ ec [] [literalType lit] ]@l ]
+ ; ga_curry : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec (a,,[b]) [c] @@ l] |- [@ga_mk Γ ec a [b ---> c] ]@ l ]
+ ; ga_comp : ∀ Γ Δ ec l a b c, ND Rule [] [Γ > Δ > [@ga_mk Γ ec a b @@ l],,[@ga_mk Γ ec b c @@ l] |- [@ga_mk Γ ec a c ]@l ]
; ga_apply : ∀ Γ Δ ec l a a' b c,
- ND Rule [] [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] @@ l] ]
+ ND Rule [] [Γ > Δ > [@ga_mk Γ ec a [b ---> c] @@ l],,[@ga_mk Γ ec a' [b] @@ l] |- [@ga_mk Γ ec (a,,a') [c] ]@l ]
; ga_kappa : ∀ Γ Δ ec l a b Σ, ND Rule
- [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b @@ l] ]
- [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
+ [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec [] b ]@l ]
+ [Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@l ]
}.
Context `(gar:garrow).
Notation "a ~~~~> b" := (@ga_mk _ _ a b) (at level 20).
Definition boost : forall Γ Δ ant x y {lev},
- ND Rule [] [ Γ > Δ > [x@@lev] |- [y@@lev] ] ->
- ND Rule [ Γ > Δ > ant |- [x@@lev] ] [ Γ > Δ > ant |- [y@@lev] ].
+ ND Rule [] [ Γ > Δ > [x@@lev] |- [y]@lev ] ->
+ ND Rule [ Γ > Δ > ant |- [x]@lev ] [ Γ > Δ > ant |- [y]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
Definition precompose Γ Δ ec : forall a x y z lev,
ND Rule
- [ Γ > Δ > a |- [@ga_mk _ ec y z @@ lev] ]
- [ Γ > Δ > a,,[@ga_mk _ ec x y @@ lev] |- [@ga_mk _ ec x z @@ lev] ].
+ [ Γ > Δ > a |- [@ga_mk _ ec y z ]@lev ]
+ [ Γ > Δ > a,,[@ga_mk _ ec x y @@ lev] |- [@ga_mk _ ec x z ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
Definition precompose' Γ Δ ec : forall a b x y z lev,
ND Rule
- [ Γ > Δ > a,,b |- [@ga_mk _ ec y z @@ lev] ]
- [ Γ > Δ > a,,([@ga_mk _ ec x y @@ lev],,b) |- [@ga_mk _ ec x z @@ lev] ].
+ [ Γ > Δ > a,,b |- [@ga_mk _ ec y z ]@lev ]
+ [ Γ > Δ > a,,([@ga_mk _ ec x y @@ lev],,b) |- [@ga_mk _ ec x z ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RLeft; eapply RExch ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCossa ].
Definition postcompose_ Γ Δ ec : forall a x y z lev,
ND Rule
- [ Γ > Δ > a |- [@ga_mk _ ec x y @@ lev] ]
- [ Γ > Δ > a,,[@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z @@ lev] ].
+ [ Γ > Δ > a |- [@ga_mk _ ec x y ]@lev ]
+ [ Γ > Δ > a,,[@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
Defined.
Definition postcompose Γ Δ ec : forall x y z lev,
- ND Rule [] [ Γ > Δ > [] |- [@ga_mk _ ec x y @@ lev] ] ->
- ND Rule [] [ Γ > Δ > [@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z @@ lev] ].
+ ND Rule [] [ Γ > Δ > [] |- [@ga_mk _ ec x y ]@lev ] ->
+ ND Rule [] [ Γ > Δ > [@ga_mk _ ec y z @@ lev] |- [@ga_mk _ ec x z ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
eapply nd_comp; [ idtac | eapply postcompose_ ].
Defined.
Definition first_nd : ∀ Γ Δ ec lev a b c Σ,
- ND Rule [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ]
- [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) @@ lev] ].
+ ND Rule [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ]
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
Defined.
Definition firstify : ∀ Γ Δ ec lev a b c Σ,
- ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ] ->
- ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) @@ lev] ].
+ ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ] ->
+ ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (a,,c) (b,,c) ]@lev ].
intros.
eapply nd_comp.
apply X.
Definition second_nd : ∀ Γ Δ ec lev a b c Σ,
ND Rule
- [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ]
- [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) @@ lev] ].
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ]
+ [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) ]@lev ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
Defined.
Definition secondify : ∀ Γ Δ ec lev a b c Σ,
- ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ lev] ] ->
- ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) @@ lev] ].
+ ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec a b ]@lev ] ->
+ ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec (c,,a) (c,,b) ]@lev ].
intros.
eapply nd_comp.
apply X.
Lemma ga_unkappa : ∀ Γ Δ ec l a b Σ x,
ND Rule
- [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b @@ l] ]
- [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b @@ l] ].
+ [Γ > Δ > Σ |- [@ga_mk Γ ec (a,,x) b ]@l ]
+ [Γ > Δ > Σ,,[@ga_mk Γ ec [] a @@ l] |- [@ga_mk Γ ec x b ]@l ].
intros.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
forall Γ (Δ:CoercionEnv Γ)
(ec:HaskTyVar Γ ECKind) (lev:HaskLevel Γ) (ant1 ant2:Tree ??(LeveledHaskType Γ ★)) (r:Arrange ant1 ant2),
ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec) (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
- (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil] ].
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) ]@nil ].
intros Γ Δ ec lev.
refine (fix flatten ant1 ant2 (r:Arrange ant1 ant2):
ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec)
(mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
- (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) @@ nil]] :=
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1)) ]@nil] :=
match r as R in Arrange A B return
ND Rule [] [Γ > Δ > [] |- [@ga_mk _ (v2t ec)
(mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) B))
- (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) A)) @@ nil]]
+ (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) A)) ]@nil]
with
| RId a => let case_RId := tt in ga_id _ _ _ _ _
| RCanL a => let case_RCanL := tt in ga_uncancell _ _ _ _ _
[Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant1)
|- [@ga_mk _ (v2t ec)
(mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant1))
- (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]]
+ (mapOptionTree (flatten_type ) succ) ]@nil]
[Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev n ant2)
|- [@ga_mk _ (v2t ec)
(mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) ant2))
- (mapOptionTree (flatten_type ○ unlev) succ) @@ nil]].
+ (mapOptionTree (flatten_type ) succ) ]@nil].
intros.
refine ( _ ;; (boost _ _ _ _ _ (postcompose _ _ _ _ _ _ _ (flatten_arrangement' Γ Δ ec lev ant1 ant2 r)))).
apply nd_rule.
Defined.
Definition flatten_arrangement'' :
- forall Γ Δ ant1 ant2 succ (r:Arrange ant1 ant2),
- ND Rule (mapOptionTree (flatten_judgment ) [Γ > Δ > ant1 |- succ])
- (mapOptionTree (flatten_judgment ) [Γ > Δ > ant2 |- succ]).
+ forall Γ Δ ant1 ant2 succ l (r:Arrange ant1 ant2),
+ ND Rule (mapOptionTree (flatten_judgment ) [Γ > Δ > ant1 |- succ @ l])
+ (mapOptionTree (flatten_judgment ) [Γ > Δ > ant2 |- succ @ l]).
intros.
simpl.
- set (getjlev succ) as succ_lev.
- assert (succ_lev=getjlev succ).
- reflexivity.
-
- destruct succ_lev.
+ destruct l.
apply nd_rule.
apply RArrange.
induction r; simpl.
Defined.
Definition ga_join Γ Δ Σ₁ Σ₂ a b ec :
- ND Rule [] [Γ > Δ > Σ₁ |- [@ga_mk _ ec [] a @@ nil]] ->
- ND Rule [] [Γ > Δ > Σ₂ |- [@ga_mk _ ec [] b @@ nil]] ->
- ND Rule [] [Γ > Δ > Σ₁,,Σ₂ |- [@ga_mk _ ec [] (a,,b) @@ nil]].
+ ND Rule [] [Γ > Δ > Σ₁ |- [@ga_mk _ ec [] a ]@nil] ->
+ ND Rule [] [Γ > Δ > Σ₂ |- [@ga_mk _ ec [] b ]@nil] ->
+ ND Rule [] [Γ > Δ > Σ₁,,Σ₂ |- [@ga_mk _ ec [] (a,,b) ]@nil].
intro pfa.
intro pfb.
apply secondify with (c:=a) in pfb.
ND Rule
[Γ > Δ >
[(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil],,
- mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t @@ nil]]
- [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]].
+ mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t]@nil]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t]@nil].
intros.
unfold drop_lev.
Definition arrange_esc : forall Γ Δ ec succ t,
ND Rule
- [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t @@ nil]]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t]@nil]
[Γ > Δ >
[(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil],,
- mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t @@ nil]].
+ mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ) |- [t]@nil].
intros.
set (@arrange _ succ (levelMatch (ec::nil))) as q.
set (@drop_lev Γ (ec::nil) succ) as q'.
destruct case_SFlat.
refine (match r as R in Rule H C with
- | RArrange Γ Δ a b x d => let case_RArrange := tt in _
+ | RArrange Γ Δ a b x l 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 _
| RVar Γ Δ σ lev => let case_RVar := tt in _
| RApp Γ Δ Σ₁ Σ₂ tx te lev => let case_RApp := tt in _
| RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ lev => let case_RLet := tt in _
| RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := tt in _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := tt in _
+ | RJoin Γ p lri m x q l => let case_RJoin := tt in _
+ | RVoid _ _ l => let case_RVoid := 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 _
end); clear X h c.
destruct case_RArrange.
- apply (flatten_arrangement'' Γ Δ a b x d).
+ apply (flatten_arrangement'' Γ Δ a b x _ d).
destruct case_RBrak.
apply (Prelude_error "found unskolemized Brak rule; this shouldn't happen").
Transparent flatten_judgment.
idtac.
unfold flatten_judgment.
- unfold getjlev.
destruct lev.
apply nd_rule. apply RVar.
repeat drop_simplify.
destruct case_RJoin.
simpl.
- destruct (getjlev x); destruct (getjlev q);
- [ apply nd_rule; apply RJoin | idtac | idtac | idtac ];
+ destruct l;
+ [ apply nd_rule; apply RJoin | idtac ];
apply (Prelude_error "RJoin at depth >0").
destruct case_RApp.
simpl.
- destruct lev as [|ec lev]. simpl. apply nd_rule.
- unfold flatten_leveled_type at 4.
- unfold flatten_leveled_type at 2.
+ destruct lev as [|ec lev].
+ unfold flatten_type at 1.
simpl.
- replace (flatten_type (tx ---> te))
- with (flatten_type tx ---> flatten_type te).
+ apply nd_rule.
apply RApp.
- reflexivity.
repeat drop_simplify.
repeat take_simplify.
destruct case_RVoid.
simpl.
apply nd_rule.
+ destruct l.
apply RVoid.
+ apply (Prelude_error "RVoid at level >0").
destruct case_RAppT.
simpl. destruct lev; simpl.
destruct case_RAbsT.
simpl. destruct lev; simpl.
- unfold flatten_leveled_type at 4.
- unfold flatten_leveled_type at 2.
- simpl.
rewrite flatten_commutes_with_HaskTAll.
rewrite flatten_commutes_with_HaskTApp.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RAbsT ].
destruct case_RAppCo.
simpl. destruct lev; simpl.
- unfold flatten_leveled_type at 4.
- unfold flatten_leveled_type at 2.
unfold flatten_type.
simpl.
apply nd_rule.
destruct case_RLetRec.
rename t into lev.
simpl. destruct lev; simpl.
- replace (getjlev (y @@@ nil)) with (nil: (HaskLevel Γ)).
- replace (mapOptionTree flatten_leveled_type (y @@@ nil))
- with ((mapOptionTree flatten_type y) @@@ nil).
- unfold flatten_leveled_type at 2.
- simpl.
- unfold flatten_leveled_type at 3.
- simpl.
apply nd_rule.
set (@RLetRec Γ Δ (mapOptionTree flatten_leveled_type lri) (flatten_type x) (mapOptionTree flatten_type y) nil) as q.
- simpl in q.
+ replace (mapOptionTree flatten_leveled_type (y @@@ nil)) with (mapOptionTree flatten_type y @@@ nil).
apply q.
induction y; try destruct a; auto.
simpl.
rewrite IHy1.
rewrite IHy2.
reflexivity.
- induction y; try destruct a; auto.
- simpl.
- rewrite <- IHy1.
- rewrite <- IHy2.
- reflexivity.
apply (Prelude_error "LetRec not supported inside brackets yet (FIXME)").
destruct case_RCase.
rewrite mapOptionTree_compose.
rewrite unlev_relev.
rewrite <- mapOptionTree_compose.
- unfold flatten_leveled_type at 4.
simpl.
rewrite krunk.
set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host.
forall Γ:TypeEnv,
forall Δ:CoercionEnv Γ,
Tree ??(LeveledHaskType Γ ★) ->
- Tree ??(LeveledHaskType Γ ★) ->
+ Tree ??(HaskType Γ ★) ->
+ HaskLevel Γ ->
Judg.
- Notation "Γ > Δ > a '|-' s" := (mkJudg Γ Δ a s) (at level 52, Δ at level 50, a at level 52, s at level 50).
+ Notation "Γ > Δ > a '|-' s '@' l" := (mkJudg Γ Δ a s l) (at level 52, Δ at level 50, a at level 52, s at level 50, l at level 50).
(* information needed to define a case branch in a HaskProof *)
Record ProofCaseBranch {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc} :=
; pcb_judg := sac_gamma sac Γ > sac_delta sac Γ avars (map weakCK' Δ)
> (mapOptionTree weakLT' pcb_freevars),,(unleaves (map (fun t => t@@weakL' lev)
(vec2list (sac_types sac Γ avars))))
- |- [weakLT' (branchtype @@ lev)]
+ |- [weakT' branchtype ] @ weakL' lev
}.
Implicit Arguments ProofCaseBranch [ ].
(* Figure 3, production $\vdash_E$, all rules *)
Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
-| RArrange : ∀ Γ Δ Σ₁ Σ₂ Σ, Arrange Σ₁ Σ₂ -> Rule [Γ > Δ > Σ₁ |- Σ ] [Γ > Δ > Σ₂ |- Σ ]
+| RArrange : ∀ Γ Δ Σ₁ Σ₂ Σ l, Arrange Σ₁ Σ₂ -> Rule [Γ > Δ > Σ₁ |- Σ @l] [Γ > Δ > Σ₂ |- Σ @l]
(* λ^α rules *)
-| RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t @@ (v::l) ]] [Γ > Δ > Σ |- [<[v|-t]> @@l]]
-| REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> @@ l]] [Γ > Δ > Σ |- [t @@ (v::l)]]
+| RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t]@(v::l) ] [Γ > Δ > Σ |- [<[v|-t]> ] @l]
+| REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> ] @l] [Γ > Δ > Σ |- [t]@(v::l) ]
(* Part of GHC, but not explicitly in System FC *)
-| RNote : ∀ Γ Δ Σ τ l, Note -> Rule [Γ > Δ > Σ |- [τ @@ l]] [Γ > Δ > Σ |- [τ @@l]]
-| RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v @@l]]
+| RNote : ∀ Γ Δ Σ τ l, Note -> Rule [Γ > Δ > Σ |- [τ ] @l] [Γ > Δ > Σ |- [τ ] @l]
+| RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v ] @l]
(* SystemFC rules *)
-| RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ @@l]]
-| RGlobal : forall Γ Δ l (g:Global Γ) v, Rule [ ] [Γ>Δ> [] |- [g v @@l]]
-| RLam : forall Γ Δ Σ (tx:HaskType Γ ★) te l, Rule [Γ>Δ> Σ,,[tx@@l]|- [te@@l] ] [Γ>Δ> Σ |- [tx--->te @@l]]
+| RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ ] @l]
+| RGlobal : forall Γ Δ l (g:Global Γ) v, Rule [ ] [Γ>Δ> [] |- [g v ] @l]
+| RLam : forall Γ Δ Σ (tx:HaskType Γ ★) te l, Rule [Γ>Δ> Σ,,[tx@@l]|- [te] @l] [Γ>Δ> Σ |- [tx--->te ] @l]
| RCast : forall Γ Δ Σ (σ₁ σ₂:HaskType Γ ★) l,
- HaskCoercion Γ Δ (σ₁∼∼∼σ₂) -> Rule [Γ>Δ> Σ |- [σ₁@@l] ] [Γ>Δ> Σ |- [σ₂ @@l]]
+ HaskCoercion Γ Δ (σ₁∼∼∼σ₂) -> Rule [Γ>Δ> Σ |- [σ₁] @l] [Γ>Δ> Σ |- [σ₂ ] @l]
-| RJoin : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ , Rule ([Γ > Δ > Σ₁ |- τ₁ ],,[Γ > Δ > Σ₂ |- τ₂ ]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ ]
+| RJoin : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ l, Rule ([Γ > Δ > Σ₁ |- τ₁ @l],,[Γ > Δ > Σ₂ |- τ₂ @l]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ @l ]
(* order is important here; we want to be able to skolemize without introducing new RExch'es *)
-| RApp : ∀ Γ Δ Σ₁ Σ₂ tx te l, Rule ([Γ>Δ> Σ₁ |- [tx--->te @@l]],,[Γ>Δ> Σ₂ |- [tx@@l]]) [Γ>Δ> Σ₁,,Σ₂ |- [te @@l]]
+| RApp : ∀ Γ Δ Σ₁ Σ₂ tx te l, Rule ([Γ>Δ> Σ₁ |- [tx--->te]@l],,[Γ>Δ> Σ₂ |- [tx]@l]) [Γ>Δ> Σ₁,,Σ₂ |- [te]@l]
-| RLet : ∀ Γ Δ Σ₁ Σ₂ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁ |- [σ₁@@l]],,[Γ>Δ> [σ₁@@l],,Σ₂ |- [σ₂@@l] ]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₂ @@l]]
-| RWhere : ∀ Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁,,([σ₁@@l],,Σ₃) |- [σ₂@@l] ],,[Γ>Δ> Σ₂ |- [σ₁@@l]]) [Γ>Δ> Σ₁,,(Σ₂,,Σ₃) |- [σ₂ @@l]]
+| RLet : ∀ Γ Δ Σ₁ Σ₂ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁ |- [σ₁]@l],,[Γ>Δ> [σ₁@@l],,Σ₂ |- [σ₂]@l ]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₂ ]@l]
+| RWhere : ∀ Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₁,,([σ₁@@l],,Σ₃) |- [σ₂]@l ],,[Γ>Δ> Σ₂ |- [σ₁]@l]) [Γ>Δ> Σ₁,,(Σ₂,,Σ₃) |- [σ₂ ]@l]
-| RVoid : ∀ Γ Δ , Rule [] [Γ > Δ > [] |- [] ]
+| RVoid : ∀ Γ Δ l, Rule [] [Γ > Δ > [] |- [] @l ]
-| RAppT : forall Γ Δ Σ κ σ (τ:HaskType Γ κ) l, Rule [Γ>Δ> Σ |- [HaskTAll κ σ @@l]] [Γ>Δ> Σ |- [substT σ τ @@l]]
+| RAppT : forall Γ Δ Σ κ σ (τ:HaskType Γ κ) l, Rule [Γ>Δ> Σ |- [HaskTAll κ σ]@l] [Γ>Δ> Σ |- [substT σ τ]@l]
| RAbsT : ∀ Γ Δ Σ κ σ l,
- Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) @@ (weakL l)]]
- [Γ>Δ > Σ |- [HaskTAll κ σ @@ l]]
+ Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) ]@(weakL l)]
+ [Γ>Δ > Σ |- [HaskTAll κ σ ]@l]
| RAppCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ l,
- Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂ ⇒ σ@@l]] [Γ>Δ> Σ |- [σ @@l]]
+ Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂ ⇒ σ]@l] [Γ>Δ> Σ |- [σ ]@l]
| RAbsCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) σ l,
- Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ @@ l]]
- [Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ @@l]]
+ Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ ]@l]
+ [Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ ]@l]
-| RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- (τ₂,,[τ₁])@@@lev ] [Γ > Δ > Σ₁ |- [τ₁@@lev] ]
+| RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- (τ₂,,[τ₁]) @lev ] [Γ > Δ > Σ₁ |- [τ₁] @lev]
| RCase : forall Γ Δ lev tc Σ avars tbranches
(alts:Tree ??{ sac : @StrongAltCon tc & @ProofCaseBranch tc Γ Δ lev tbranches avars sac }),
Rule
((mapOptionTree (fun x => pcb_judg (projT2 x)) alts),,
- [Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
- [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches @@ lev ] ]
+ [Γ > Δ > Σ |- [ caseType tc avars ] @lev])
+ [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches ] @ lev]
.
(* A rule is considered "flat" if it is neither RBrak nor REsc *)
(* TODO: change this to (if RBrak/REsc -> False) *)
Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
-| Flat_RArrange : ∀ Γ Δ h c r a , Rule_Flat (RArrange Γ Δ h c r a)
+| Flat_RArrange : ∀ Γ Δ h c r a l , Rule_Flat (RArrange Γ Δ h c r a l)
| Flat_RNote : ∀ Γ Δ Σ τ l n , Rule_Flat (RNote Γ Δ Σ τ l n)
| Flat_RLit : ∀ Γ Δ Σ τ , Rule_Flat (RLit Γ Δ Σ τ )
| Flat_RVar : ∀ Γ Δ σ l, Rule_Flat (RVar Γ Δ σ l)
| Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 )
| Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
| Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
-| Flat_RJoin : ∀ q a b c d e , Rule_Flat (RJoin q a b c d e)
-| Flat_RVoid : ∀ q a , Rule_Flat (RVoid q a)
+| Flat_RJoin : ∀ q a b c d e l, Rule_Flat (RJoin q a b c d e l)
+| Flat_RVoid : ∀ q a l, Rule_Flat (RVoid q a l)
| Flat_RCase : ∀ Σ Γ T κlen κ θ l x , Rule_Flat (RCase Σ Γ T κlen κ θ l x)
| Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev, Rule_Flat (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
Definition judgmentToRawLatexMath (j:Judg) : LatexMath :=
match match j return VarNameStoreM LatexMath with
- | mkJudg Γ Δ Σ₁ Σ₂ =>
+ | mkJudg Γ Δ Σ₁ Σ₂ l =>
bind Σ₁' = treeM (mapOptionTree ltypeToLatexMath Σ₁)
- ; bind Σ₂' = treeM (mapOptionTree ltypeToLatexMath Σ₂)
+ ; bind Σ₂' = treeM (mapOptionTree (fun t => ltypeToLatexMath (t@@l)) Σ₂)
; return treeToLatexMath Σ₁' +++ (rawLatexMath "\vdash") +++ treeToLatexMath Σ₂'
end with
varNameStoreM f => fst (f (varNameStore 0 0 0))
Fixpoint nd_ruleToRawLatexMath {h}{c}(r:Rule h c) : string :=
match r with
- | RArrange _ _ _ _ _ r => nd_uruleToRawLatexMath r
+ | RArrange _ _ _ _ _ _ r => nd_uruleToRawLatexMath r
| RNote _ _ _ _ _ _ => "Note"
| RLit _ _ _ _ => "Lit"
| RVar _ _ _ _ => "Var"
| RApp _ _ _ _ _ _ _ => "App"
| RLet _ _ _ _ _ _ _ => "Let"
| RWhere _ _ _ _ _ _ _ _ => "Where"
- | RJoin _ _ _ _ _ _ => "RJoin"
+ | RJoin _ _ _ _ _ _ _ => "RJoin"
| RLetRec _ _ _ _ _ _ => "LetRec"
| RCase _ _ _ _ _ _ _ _ => "Case"
| RBrak _ _ _ _ _ _ => "Brak"
| REsc _ _ _ _ _ _ => "Esc"
- | RVoid _ _ => "RVoid"
+ | RVoid _ _ _ => "RVoid"
end.
Fixpoint nd_hideURule {T}{h}{c}(r:@Arrange T h c) : bool :=
end.
Fixpoint nd_hideRule {h}{c}(r:Rule h c) : bool :=
match r with
- | RArrange _ _ _ _ _ r => nd_hideURule r
- | RVoid _ _ => true
- | RJoin _ _ _ _ _ _ => true
+ | RArrange _ _ _ _ _ _ r => nd_hideURule r
+ | RVoid _ _ _ => true
+ | RJoin _ _ _ _ _ _ _ => true
| _ => false
end.
Definition judg2exprType (j:Judg) : Type :=
match j with
- (Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
- FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
+ (Γ > Δ > Σ |- τ @ l) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
+ FreshM (ITree _ (fun t => Expr Γ Δ ξ (t @@ l)) τ)
end.
Definition justOne Γ Δ ξ τ : ITree _ (fun t => Expr Γ Δ ξ t) [τ] -> Expr Γ Δ ξ τ.
inversion pf2.
Defined.
- 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 ujudg2exprType Γ (ξ:ExprVarResolver Γ)(Δ:CoercionEnv Γ) Σ τ l : Type :=
+ forall vars, Σ = mapOptionTree ξ vars -> FreshM (ITree _ (fun t => Expr Γ Δ ξ (t@@l)) 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
+ Definition urule2expr : forall Γ Δ h j t l (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
+ ujudg2exprType Γ ξ Δ h t l ->
+ ujudg2exprType Γ ξ Δ j t l
.
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 :=
+ refine (fix urule2expr h j t l (r:@Arrange _ h j) ξ {struct r} :
+ ujudg2exprType Γ ξ Δ h t l ->
+ ujudg2exprType Γ ξ Δ j t l :=
match r as R in Arrange H C return
- ujudg2exprType Γ ξ Δ H t ->
- ujudg2exprType Γ ξ Δ C t
+ ujudg2exprType Γ ξ Δ H t l ->
+ ujudg2exprType Γ ξ Δ C t l
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)
+ | 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)
| RId a => let case_RId := tt in _
| RCanL a => let case_RCanL := tt in _
| RCanR a => let case_RCanR := 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)
+ | RComp a b c f g => let case_RComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ _ f) (urule2expr _ _ _ _ g)
end); clear urule2expr; intros.
destruct case_RId.
Defined.
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)
+ ITree (HaskType Γ ★)
+ (fun t : HaskType Γ ★ => Expr Γ Δ ξ' (t @@ l))
+ (mapOptionTree (unlev ○ ξ' ○ (@fst _ _)) varstypes)
-> ELetRecBindings Γ Δ ξ' l varstypes.
intros.
induction varstypes.
simpl.
destruct (eqd_dec h0 l).
rewrite <- e0.
+ simpl in X.
+ subst.
apply X.
apply (Prelude_error "level mismatch; should never happen").
apply (Prelude_error "letrec type mismatch; should never happen").
intros h j r.
refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with
- | RArrange a b c d e r => let case_RURule := tt in _
+ | RArrange a b c d e l 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 _
| 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 _
| RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ p => let case_RWhere := tt in _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := tt in _
+ | RJoin Γ p lri m x q l => let case_RJoin := tt in _
+ | RVoid _ _ l => 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 _
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.
+ set (@urule2expr a b _ _ e l r0 ξ) as q.
unfold ujudg2exprType.
+ unfold ujudg2exprType in q.
+ apply q with (vars:=vars).
intros.
apply X_ with (vars:=vars0).
auto.
apply (@ELetRec _ _ _ _ _ _ _ varstypes).
auto.
apply (@letrec_helper Γ Δ t varstypes).
- rewrite <- pf2 in X0.
rewrite mapOptionTree_compose.
- apply X0.
+ rewrite mapOptionTree_compose.
+ rewrite pf2.
+ replace ((mapOptionTree unlev (y @@@ t))) with y.
+ apply X0.
+ clear pf1 X0 X1 pfdist pf2 vars varstypes.
+ induction y; try destruct a; auto.
+ rewrite IHy1 at 1.
+ rewrite IHy2 at 1.
+ reflexivity.
apply ileaf in X1.
+ simpl in X1.
apply X1.
destruct case_RCase.
Defined.
Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
- {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] ->
+ {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [unlev τ] @ getlev τ] ->
FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
intro pf.
set (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) as cnd.
auto.
refine (return OK _).
exists ξ.
- apply (ileaf it).
+ apply ileaf in it.
+ simpl in it.
+ destruct τ.
+ apply it.
apply INone.
Defined.
end.
(* rules of skolemized proofs *)
- 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
- | T_Leaf (Some (_ @@ lev)) => lev
- | T_Branch b1 b2 =>
- match getjlev b1 with
- | nil => getjlev b2
- | lev => lev
- end
- end.
+ Definition getΓ (j:Judg) := match j with Γ > _ > _ |- _ @ _ => Γ end.
Fixpoint take_trustme {Γ}
(n:nat)
| SFlat : forall h c, Rule h c -> SRule h c
| SBrak : forall Γ Δ t ec Σ l,
SRule
- [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t @@ (ec::l) ]]
- [Γ > Δ > Σ |- [<[ec |- t]> @@ l ]]
+ [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
+ [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
| SEsc : forall Γ Δ t ec Σ l,
SRule
- [Γ > Δ > Σ |- [<[ec |- t]> @@ l ]]
- [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t @@ (ec::l) ]]
+ [Γ > Δ > Σ |- [<[ec |- t]> ] @l]
+ [Γ > Δ > Σ,,(take_arg_types_as_tree t @@@ (ec::l)) |- [ drop_arg_types_as_tree t ] @ (ec::l)]
.
Definition take_arg_types_as_tree' {Γ}(lt:LeveledHaskType Γ ★) :=
Definition skolemize_judgment (j:Judg) : Judg :=
match j with
- Γ > Δ > Σ₁ |- Σ₂ =>
- match getjlev Σ₂ with
- | nil => j
- | lev => Γ > Δ > Σ₁,,(mapOptionTreeAndFlatten take_arg_types_as_tree' Σ₂) |- mapOptionTree drop_arg_types_as_tree' Σ₂
- end
+ | Γ > Δ > Σ₁ |- Σ₂ @ nil => j
+ | Γ > Δ > Σ₁ |- Σ₂ @ lev =>
+ Γ > Δ > Σ₁,,(mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂ @@@ lev) |- mapOptionTree drop_arg_types_as_tree Σ₂ @ lev
end.
Definition check_hof : forall {Γ}(t:HaskType Γ ★),
intros.
refine (match X as R in Rule H C with
- | RArrange Γ Δ a b x d => let case_RArrange := tt in _
+ | RArrange Γ Δ a b x l 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 _
| RVar Γ Δ σ lev => let case_RVar := tt in _
| RApp Γ Δ Σ₁ Σ₂ tx te lev => let case_RApp := tt in _
| RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ lev => let case_RLet := tt in _
| RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := tt in _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := tt in _
+ | RJoin Γ p lri m x q l => let case_RJoin := tt in _
+ | RVoid _ _ l => let case_RVoid := 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 _
destruct case_RArrange.
simpl.
- destruct (getjlev x).
+ destruct l.
apply nd_rule.
apply SFlat.
apply RArrange.
destruct case_RJoin.
simpl.
- destruct (getjlev x).
- destruct (getjlev q).
+ destruct l.
apply nd_rule.
apply SFlat.
apply RJoin.
apply (Prelude_error "found RJoin at level >0").
- apply (Prelude_error "found RJoin at level >0").
destruct case_RApp.
simpl.
destruct case_RVoid.
simpl.
+ destruct l.
+ apply nd_rule.
+ apply SFlat.
+ apply RVoid.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RuCanL ].
apply nd_rule.
apply SFlat.
apply RVoid.
destruct case_RLetRec.
simpl.
destruct t.
- destruct (getjlev (y @@@ nil)).
apply nd_rule.
apply SFlat.
apply (@RLetRec Γ Δ lri x y nil).
apply (Prelude_error "RLetRec at depth>0").
- apply (Prelude_error "RLetRec at depth>0").
destruct case_RCase.
simpl.
apply (Prelude_error "CASE: BIG FIXME").
Defined.
+
Transparent take_arg_types_as_tree.
End HaskSkolemizer.
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 ,
- (ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [t@@lev]]) ->
+ (ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [t]@lev]) ->
LetRecSubproofs Γ Δ ξ lev [(v, t)] (ELR_leaf _ _ _ _ _ t e)
| lrsp_cons : forall t1 t2 b1 b2,
LetRecSubproofs Γ Δ ξ lev t1 b1 ->
Lemma letRecSubproofsToND Γ Δ ξ lev tree branches :
LetRecSubproofs Γ Δ ξ lev tree branches ->
ND Rule [] [ Γ > Δ > mapOptionTree ξ (eLetRecContext branches)
- |- (mapOptionTree (@snd _ _) tree) @@@ lev ].
+ |- (mapOptionTree (@snd _ _) tree) @ lev ].
intro X; induction X; intros; simpl in *.
apply nd_rule.
apply RVoid.
Lemma letRecSubproofsToND' Γ Δ ξ lev τ tree :
forall branches body (dist:distinct (leaves (mapOptionTree (@fst _ _) tree))),
- ND Rule [] [Γ > Δ > mapOptionTree (update_xi ξ lev (leaves tree)) (expr2antecedent body) |- [τ @@ lev]] ->
+ ND Rule [] [Γ > Δ > mapOptionTree (update_xi ξ lev (leaves tree)) (expr2antecedent body) |- [τ ]@ lev] ->
LetRecSubproofs Γ Δ (update_xi ξ lev (leaves tree)) lev tree branches ->
- ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent (@ELetRec VV _ Γ Δ ξ lev τ tree dist 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.
simpl.
rewrite <- mapOptionTree_compose in q''.
rewrite <- ξlemma.
- eapply nd_comp; [ idtac | eapply nd_rule; apply (RArrange _ _ _ _ _ q'') ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply (RArrange _ _ _ _ _ _ q'') ].
clear q'.
clear q''.
simpl.
eapply nd_comp; [ idtac | eapply nd_rule; apply RJoin ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod; auto.
- rewrite ξlemma.
- apply q.
Defined.
Lemma scbwv_coherent {tc}{Γ}{atypes:IList _ (HaskType Γ) _}{sac} :
Definition expr2proof :
forall Γ Δ ξ τ (e:Expr Γ Δ ξ τ),
- ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [τ]].
+ ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [unlev τ] @ getlev τ].
refine (fix expr2proof Γ' Δ' ξ' τ' (exp:Expr Γ' Δ' ξ' τ') {struct exp}
- : ND Rule [] [Γ' > Δ' > mapOptionTree ξ' (expr2antecedent exp) |- [τ']] :=
+ : ND Rule [] [Γ' > Δ' > mapOptionTree ξ' (expr2antecedent exp) |- [unlev τ'] @ getlev τ'] :=
match exp as E in Expr Γ Δ ξ τ with
| EGlobal Γ Δ ξ g v lev => let case_EGlobal := tt in _
| EVar Γ Δ ξ ev => let case_EVar := tt in _
Notation "t @@@ l" := (mapOptionTree (fun t' => t' @@ l) t) (at level 20).
Notation "'<[' a '|-' t ']>'" := (@HaskBrak _ a t).
+Definition getlev {Γ}(lt:LeveledHaskType Γ ★) := match lt with _ @@ l => l end.
+
Definition unlev {Γ}{κ}(lht:LeveledHaskType Γ κ) :=
match lht with t@@l => t end.
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