Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
*)
Section HaskFlattener.
- Definition getlev {Γ}{κ}(lht:LeveledHaskType Γ κ) : HaskLevel Γ :=
- match lht with t @@ l => l end.
-
- Definition arrange :
- forall {T} (Σ:Tree ??T) (f:T -> bool),
- Arrange Σ (dropT (mkFlags (liftBoolFunc false f) Σ),,( (dropT (mkFlags (liftBoolFunc false (bnot ○ f)) Σ)))).
- intros.
- induction Σ.
- simpl.
- destruct a.
- simpl.
- destruct (f t); simpl.
- apply RuCanL.
- apply RuCanR.
- simpl.
- apply RuCanL.
- simpl in *.
- eapply RComp; [ idtac | apply arrangeSwapMiddle ].
- eapply RComp.
- eapply RLeft.
- apply IHΣ2.
- eapply RRight.
- apply IHΣ1.
- Defined.
-
- Definition arrange' :
- forall {T} (Σ:Tree ??T) (f:T -> bool),
- Arrange (dropT (mkFlags (liftBoolFunc false f) Σ),,( (dropT (mkFlags (liftBoolFunc false (bnot ○ f)) Σ)))) Σ.
- intros.
- induction Σ.
- simpl.
- destruct a.
- simpl.
- destruct (f t); simpl.
- apply RCanL.
- apply RCanR.
- simpl.
- apply RCanL.
- simpl in *.
- eapply RComp; [ apply arrangeSwapMiddle | idtac ].
- eapply RComp.
- eapply RLeft.
- apply IHΣ2.
- eapply RRight.
- apply IHΣ1.
- Defined.
Ltac eqd_dec_refl' :=
match goal with
rewrite <- IHx2 at 2.
reflexivity.
Qed.
-(*
- Lemma drop_lev_lemma' : forall Γ (lev:HaskLevel Γ) x, drop_lev lev (x @@@ lev) = [].
- intros.
- induction x.
- destruct a; simpl; try reflexivity.
- apply drop_lev_lemma.
- simpl.
- change (@drop_lev _ lev (x1 @@@ lev ,, x2 @@@ lev))
- with ((@drop_lev _ lev (x1 @@@ lev)) ,, (@drop_lev _ lev (x2 @@@ lev))).
- simpl.
- rewrite IHx1.
- rewrite IHx2.
- reflexivity.
- Qed.
-*)
+
Ltac drop_simplify :=
match goal with
| [ |- context[@drop_lev ?G ?L [ ?X @@ ?L ] ] ] =>
rewrite (drop_lev_lemma G L X)
-(*
- | [ |- context[@drop_lev ?G ?L [ ?X @@@ ?L ] ] ] =>
- rewrite (drop_lev_lemma' G L X)
-*)
| [ |- context[@drop_lev ?G (?E :: ?L) [ ?X @@ (?E :: ?L) ] ] ] =>
rewrite (drop_lev_lemma_s G L E X)
| [ |- context[@drop_lev ?G ?N (?A,,?B)] ] =>
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 RCanL ].
- eapply nd_comp; [ idtac | eapply nd_rule; apply (@RLet Γ Δ [] ant y x lev) ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | apply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
apply X.
eapply nd_rule.
eapply RArrange.
- apply RuCanL.
- Defined.
-
- Definition postcompose' : ∀ 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 b c @@ lev] |- [@ga_mk Γ ec a c @@ lev] ].
- intros.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
- eapply nd_comp; [ idtac
- | eapply nd_rule; apply (@RLet Γ Δ [@ga_mk _ ec b c @@lev] Σ (@ga_mk _ ec a c) (@ga_mk _ ec a b) lev) ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod.
- apply X.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
- apply ga_comp.
- Defined.
+ apply AuCanR.
+ Defined.
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.
- apply nd_rlecnac.
- eapply nd_comp.
- eapply nd_prod.
+ eapply nd_comp; [ idtac | eapply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
apply nd_id.
- eapply ga_comp.
-
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
-
- apply nd_rule.
- apply RLet.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ apply ga_comp.
Defined.
- Definition precompose' : ∀ Γ Δ ec lev a b c Σ,
- ND Rule [] [ Γ > Δ > Σ |- [@ga_mk Γ ec b c @@ lev] ] ->
- ND Rule [] [ Γ > Δ > Σ,,[@ga_mk Γ ec a b @@ lev] |- [@ga_mk Γ ec a c @@ lev] ].
- intros.
- eapply nd_comp.
- apply X.
- apply precompose.
- Defined.
-
- Definition postcompose : ∀ Γ Δ ec lev a b c,
- ND Rule [] [ Γ > Δ > [] |- [@ga_mk Γ ec a b @@ lev] ] ->
- ND Rule [] [ Γ > Δ > [@ga_mk Γ ec b c @@ lev] |- [@ga_mk Γ ec a c @@ lev] ].
- intros.
- eapply nd_comp.
- apply postcompose'.
- apply X.
- apply nd_rule.
- apply RArrange.
- apply RCanL.
- Defined.
+ 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 ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ALeft; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuAssoc ].
+ apply precompose.
+ 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] ].
+ 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 ].
intros.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
- eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
+ eapply nd_comp; [ idtac | eapply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
- apply X.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanL ].
- apply ga_first.
+ apply nd_id.
+ apply ga_comp.
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] ].
+ 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 ].
intros.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
- eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
+ eapply nd_comp; [ idtac | eapply postcompose_ ].
apply X.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanL ].
- apply ga_second.
Defined.
- Lemma ga_unkappa : ∀ Γ Δ ec l z a b Σ,
- ND Rule
- [Γ > Δ > Σ |- [@ga_mk Γ ec a b @@ l] ]
- [Γ > Δ > Σ,,[@ga_mk Γ ec z a @@ l] |- [@ga_mk Γ ec z b @@ l] ].
+ 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 ].
intros.
- set (ga_comp Γ Δ ec l z a b) as q.
-
- set (@RLet Γ Δ) as q'.
- set (@RLet Γ Δ [@ga_mk _ ec z a @@ l] Σ (@ga_mk _ ec z b) (@ga_mk _ 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.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | apply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
- apply q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply ga_first.
Defined.
- 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] ].
+ 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 ].
intros.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod.
- apply ga_first.
+ eapply nd_comp.
+ apply X.
+ apply first_nd.
+ Defined.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
+ 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 ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | apply RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
- apply postcompose.
- apply ga_uncancell.
- apply precompose.
+ apply nd_id.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply ga_second.
Defined.
- Lemma ga_kappa' : ∀ Γ Δ ec l a b Σ x,
- ND Rule
- [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] ].
- apply (Prelude_error "ga_kappa not supported yet (BIG FIXME)").
+ 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 ].
+ intros.
+ eapply nd_comp.
+ apply X.
+ apply second_nd.
Defined.
+ 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 ].
+ intros.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ apply ga_first.
+
+ eapply nd_comp; [ idtac | eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ apply postcompose.
+ apply ga_uncancell.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ apply precompose.
+ Defined.
+
(* useful for cutting down on the pretty-printed noise
Notation "` x" := (take_lev _ x) (at level 20).
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 _ _ _ _ _
- | RCanR a => let case_RCanR := tt in ga_uncancelr _ _ _ _ _
- | RuCanL a => let case_RuCanL := tt in ga_cancell _ _ _ _ _
- | RuCanR a => let case_RuCanR := tt in ga_cancelr _ _ _ _ _
- | RAssoc a b c => let case_RAssoc := tt in ga_assoc _ _ _ _ _ _ _
- | RCossa a b c => let case_RCossa := tt in ga_unassoc _ _ _ _ _ _ _
- | RExch a b => let case_RExch := tt in ga_swap _ _ _ _ _ _
- | RWeak a => let case_RWeak := tt in ga_drop _ _ _ _ _
- | RCont a => let case_RCont := tt in ga_copy _ _ _ _ _
- | RLeft a b c r' => let case_RLeft := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _)
- | RRight a b c r' => let case_RRight := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _)
- | RComp c b a r1 r2 => let case_RComp := tt in (fun r1' r2' => _) (flatten _ _ r1) (flatten _ _ r2)
+ | AId a => let case_AId := tt in ga_id _ _ _ _ _
+ | ACanL a => let case_ACanL := tt in ga_uncancell _ _ _ _ _
+ | ACanR a => let case_ACanR := tt in ga_uncancelr _ _ _ _ _
+ | AuCanL a => let case_AuCanL := tt in ga_cancell _ _ _ _ _
+ | AuCanR a => let case_AuCanR := tt in ga_cancelr _ _ _ _ _
+ | AAssoc a b c => let case_AAssoc := tt in ga_assoc _ _ _ _ _ _ _
+ | AuAssoc a b c => let case_AuAssoc := tt in ga_unassoc _ _ _ _ _ _ _
+ | AExch a b => let case_AExch := tt in ga_swap _ _ _ _ _ _
+ | AWeak a => let case_AWeak := tt in ga_drop _ _ _ _ _
+ | ACont a => let case_ACont := tt in ga_copy _ _ _ _ _
+ | ALeft a b c r' => let case_ALeft := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_second _ _ _ _ _ _ _)
+ | ARight a b c r' => let case_ARight := tt in flatten _ _ r' ;; boost _ _ _ _ _ (ga_first _ _ _ _ _ _ _)
+ | AComp c b a r1 r2 => let case_AComp := tt in (fun r1' r2' => _) (flatten _ _ r1) (flatten _ _ r2)
end); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
- destruct case_RComp.
+ destruct case_AComp.
set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) a)) as a' in *.
set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) b)) as b' in *.
set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) c)) as c' in *.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ].
- eapply nd_comp; [ idtac | eapply nd_rule; apply
- (@RLet Γ Δ [] [] (@ga_mk _ (v2t ec) a' c') (@ga_mk _ (v2t ec) a' b')) ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanL ].
+ eapply nd_comp; [ idtac | apply
+ (@RLet Γ Δ [] [] (@ga_mk _ (v2t ec) a' b') (@ga_mk _ (v2t ec) a' c')) ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod.
apply r2'.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RuCanL ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; apply
- (@RLet Γ Δ [@ga_mk _ (v2t ec) a' b' @@ _] [] (@ga_mk _ (v2t ec) a' c') (@ga_mk _ (v2t ec) b' c'))].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanL ].
+ eapply nd_comp; [ idtac | apply RLet ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
eapply nd_prod.
apply r1'.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
apply ga_comp.
Defined.
[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.
match r as R in Arrange A B return
Arrange (mapOptionTree (flatten_leveled_type ) (drop_lev _ A))
(mapOptionTree (flatten_leveled_type ) (drop_lev _ B)) with
- | RId a => let case_RId := tt in RId _
- | RCanL a => let case_RCanL := tt in RCanL _
- | RCanR a => let case_RCanR := tt in RCanR _
- | RuCanL a => let case_RuCanL := tt in RuCanL _
- | RuCanR a => let case_RuCanR := tt in RuCanR _
- | RAssoc a b c => let case_RAssoc := tt in RAssoc _ _ _
- | RCossa a b c => let case_RCossa := tt in RCossa _ _ _
- | RExch a b => let case_RExch := tt in RExch _ _
- | RWeak a => let case_RWeak := tt in RWeak _
- | RCont a => let case_RCont := tt in RCont _
- | RLeft a b c r' => let case_RLeft := tt in RLeft _ (flatten _ _ r')
- | RRight a b c r' => let case_RRight := tt in RRight _ (flatten _ _ r')
- | RComp a b c r1 r2 => let case_RComp := tt in RComp (flatten _ _ r1) (flatten _ _ r2)
+ | AId a => let case_AId := tt in AId _
+ | ACanL a => let case_ACanL := tt in ACanL _
+ | ACanR a => let case_ACanR := tt in ACanR _
+ | AuCanL a => let case_AuCanL := tt in AuCanL _
+ | AuCanR a => let case_AuCanR := tt in AuCanR _
+ | AAssoc a b c => let case_AAssoc := tt in AAssoc _ _ _
+ | AuAssoc a b c => let case_AuAssoc := tt in AuAssoc _ _ _
+ | AExch a b => let case_AExch := tt in AExch _ _
+ | AWeak a => let case_AWeak := tt in AWeak _
+ | ACont a => let case_ACont := tt in ACont _
+ | ALeft a b c r' => let case_ALeft := tt in ALeft _ (flatten _ _ r')
+ | ARight a b c r' => let case_ARight := tt in ARight _ (flatten _ _ r')
+ | AComp a b c r1 r2 => let case_AComp := tt in AComp (flatten _ _ r1) (flatten _ _ r2)
end) ant1 ant2 r); clear flatten; repeat take_simplify; repeat drop_simplify; intros.
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.
- apply RId.
- apply RCanL.
- apply RCanR.
- apply RuCanL.
- apply RuCanR.
- apply RAssoc.
- apply RCossa.
- apply RExch.
- apply RWeak.
- apply RCont.
- apply RLeft; auto.
- apply RRight; auto.
- eapply RComp; [ apply IHr1 | apply IHr2 ].
+ apply AId.
+ apply ACanL.
+ apply ACanR.
+ apply AuCanL.
+ apply AuCanR.
+ apply AAssoc.
+ apply AuAssoc.
+ apply AExch. (* TO DO: check for all-leaf trees here *)
+ apply AWeak.
+ apply ACont.
+ apply ALeft; auto.
+ apply ARight; auto.
+ eapply AComp; [ apply IHr1 | apply IHr2 ].
apply flatten_arrangement.
apply r.
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.
- eapply nd_comp.
- Focus 2.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ apply firstify with (c:=[]) in pfa.
+ eapply nd_comp; [ idtac | eapply RLet ].
eapply nd_comp; [ eapply nd_llecnac | idtac ].
- eapply nd_prod.
- apply pfb.
- clear pfb.
- apply postcompose'.
- eapply nd_comp.
+ apply nd_prod.
apply pfa.
clear pfa.
- apply boost.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
- apply precompose'.
+
+ eapply nd_comp; [ idtac | eapply RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
+ eapply nd_comp; [ idtac | eapply postcompose_ ].
apply ga_uncancelr.
- apply nd_id.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply precompose ].
+ apply pfb.
Defined.
Definition arrange_brak : forall Γ Δ ec succ t,
ND Rule
- [Γ > Δ > mapOptionTree (flatten_leveled_type ) (drop_lev (ec :: nil) succ),,
- [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [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]
+ [Γ > Δ > mapOptionTree (flatten_leveled_type ) succ |- [t]@nil].
+
intros.
unfold drop_lev.
- set (@arrange' _ succ (levelMatch (ec::nil))) as q.
+ set (@arrangeUnPartition _ succ (levelMatch (ec::nil))) as q.
set (arrangeMap _ _ flatten_leveled_type q) as y.
eapply nd_comp.
Focus 2.
apply y.
idtac.
clear y q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
simpl.
eapply nd_comp; [ apply nd_llecnac | idtac ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ idtac | eapply RLet ].
apply nd_prod.
Focus 2.
apply nd_id.
apply IHsucc2.
Defined.
- Definition arrange_empty_tree : forall {T}{A}(q:Tree A)(t:Tree ??T),
- t = mapTree (fun _:A => None) q ->
- Arrange t [].
- intros T A q.
- induction q; intros.
- simpl in H.
- rewrite H.
- apply RId.
- simpl in *.
- destruct t; try destruct o; inversion H.
- set (IHq1 _ H1) as x1.
- set (IHq2 _ H2) as x2.
- eapply RComp.
- eapply RRight.
- rewrite <- H1.
- apply x1.
- eapply RComp.
- apply RCanL.
- rewrite <- H2.
- apply x2.
- Defined.
-
-(* Definition unarrange_empty_tree : forall {T}{A}(t:Tree ??T)(q:Tree A),
- t = mapTree (fun _:A => None) q ->
- Arrange [] t.
- Defined.*)
-
- Definition decide_tree_empty : forall {T:Type}(t:Tree ??T),
- sum { q:Tree unit & t = mapTree (fun _ => None) q } unit.
- intro T.
- refine (fix foo t :=
- match t with
- | T_Leaf x => _
- | T_Branch b1 b2 => let b1' := foo b1 in let b2' := foo b2 in _
- end).
- intros.
- destruct x.
- right; apply tt.
- left.
- exists (T_Leaf tt).
- auto.
- destruct b1'.
- destruct b2'.
- destruct s.
- destruct s0.
- subst.
- left.
- exists (x,,x0).
- reflexivity.
- right; auto.
- right; auto.
- Defined.
-
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 ) (drop_lev (ec :: nil) succ),,
- [(@ga_mk _ (v2t ec) [] (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: nil) succ))) @@ nil] |- [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].
intros.
- set (@arrange _ succ (levelMatch (ec::nil))) as q.
+ set (@arrangePartition _ succ (levelMatch (ec::nil))) as q.
set (@drop_lev Γ (ec::nil) succ) as q'.
assert (@drop_lev Γ (ec::nil) succ=q') as H.
reflexivity.
destruct s.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AExch ].
set (fun z z' => @RLet Γ Δ z (mapOptionTree flatten_leveled_type q') t z' nil) as q''.
- eapply nd_comp; [ idtac | eapply nd_rule; apply q'' ].
+ eapply nd_comp; [ idtac | apply RLet ].
clear q''.
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_rule.
apply RArrange.
- eapply RComp; [ idtac | apply RCanR ].
- apply RLeft.
- apply (@arrange_empty_tree _ _ _ _ e).
-
+ eapply AComp; [ idtac | apply ACanR ].
+ apply ALeft.
+ apply (@arrangeCancelEmptyTree _ _ _ _ e).
+
eapply nd_comp.
eapply nd_rule.
eapply (@RVar Γ Δ t nil).
apply nd_rule.
apply RArrange.
- eapply RComp.
- apply RuCanL.
- apply RRight.
- apply RWeak.
+ eapply AComp.
+ apply AuCanR.
+ apply ALeft.
+ apply AWeak.
(*
eapply decide_tree_empty.
simpl.
apply nd_rule.
apply RArrange.
- apply RLeft.
- apply RWeak.
+ apply ALeft.
+ apply AWeak.
simpl.
apply nd_rule.
unfold take_lev.
simpl.
apply RArrange.
- apply RLeft.
- apply RWeak.
+ apply ALeft.
+ apply AWeak.
apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported").
*)
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 unlev_relev : forall {Γ}(t:Tree ??(HaskType Γ ★)) lev, mapOptionTree unlev (t @@@ lev) = t.
intros.
induction t.
reflexivity.
Qed.
- Lemma tree_of_nothing : forall Γ ec t a,
- Arrange (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))) a.
+ Lemma tree_of_nothing : forall Γ ec t,
+ Arrange (mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))) [].
intros.
- induction t; try destruct o; try destruct a0.
+ induction t; try destruct o; try destruct a.
simpl.
drop_simplify.
simpl.
- apply RCanR.
+ apply AId.
simpl.
- apply RCanR.
+ apply AId.
+ eapply AComp; [ idtac | apply ACanL ].
+ eapply AComp; [ idtac | eapply ALeft; apply IHt2 ].
Opaque drop_lev.
simpl.
Transparent drop_lev.
+ idtac.
drop_simplify.
- simpl.
- eapply RComp.
- eapply RComp.
- eapply RAssoc.
- eapply RRight.
+ apply ARight.
apply IHt1.
- apply IHt2.
Defined.
- Lemma tree_of_nothing' : forall Γ ec t a,
- Arrange a (a,,mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))).
+ Lemma tree_of_nothing' : forall Γ ec t,
+ Arrange [] (mapOptionTree flatten_leveled_type (drop_lev(Γ:=Γ) (ec :: nil) (t @@@ (ec :: nil)))).
intros.
- induction t; try destruct o; try destruct a0.
+ induction t; try destruct o; try destruct a.
simpl.
drop_simplify.
simpl.
- apply RuCanR.
+ apply AId.
simpl.
- apply RuCanR.
+ apply AId.
+ eapply AComp; [ apply AuCanL | idtac ].
+ eapply AComp; [ eapply ARight; apply IHt1 | idtac ].
Opaque drop_lev.
simpl.
Transparent drop_lev.
+ idtac.
drop_simplify.
- simpl.
- eapply RComp.
- Focus 2.
- eapply RComp.
- Focus 2.
- eapply RCossa.
- Focus 2.
- eapply RRight.
- apply IHt1.
+ apply ALeft.
apply IHt2.
Defined.
admit.
Qed.
- Definition flatten_proof :
+ Lemma drop_to_nothing : forall (Γ:TypeEnv) Σ (lev:HaskLevel Γ),
+ drop_lev lev (Σ @@@ lev) = mapTree (fun _ => None) (mapTree (fun _ => tt) Σ).
+ intros.
+ induction Σ.
+ destruct a; simpl.
+ drop_simplify.
+ auto.
+ drop_simplify.
+ auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ Qed.
+
+ Definition flatten_skolemized_proof :
forall {h}{c},
ND SRule h c ->
ND Rule (mapOptionTree (flatten_judgment ) h) (mapOptionTree (flatten_judgment ) c).
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 _
| RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ lev => let case_RAppCo := tt in _
| RAbsCo Γ Δ Σ κ σ σ₁ σ₂ lev => let case_RAbsCo := 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 _
- | RJoin Γ p lri m x q => let case_RJoin := tt in _
- | RVoid _ _ => let case_RVoid := tt in _
+ | RCut Γ Δ Σ Σ₁ Σ₁₂ Σ₂ Σ₃ l => let case_RCut := tt in _
+ | RLeft Γ Δ Σ₁ Σ₂ Σ l => let case_RLeft := tt in _
+ | RRight Γ Δ Σ₁ Σ₂ Σ l => let case_RRight := 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.
eapply nd_rule.
eapply RArrange.
simpl.
- apply RCanR.
+ apply ACanR.
apply boost.
simpl.
apply ga_curry.
apply flatten_coercion; auto.
apply (Prelude_error "RCast at level >0; casting inside of code brackets is currently not supported").
- destruct case_RJoin.
- simpl.
- destruct (getjlev x); destruct (getjlev q);
- [ apply nd_rule; apply RJoin | idtac | idtac | 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.
Notation "!<[@]> x" := (mapOptionTree flatten_leveled_type x) (at level 1).
*)
- destruct case_RLet.
+ destruct case_RCut.
simpl.
- destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RLet; auto | idtac ].
- repeat drop_simplify.
- repeat take_simplify.
+ destruct l as [|ec lev]; simpl.
+ apply nd_rule.
+ replace (mapOptionTree flatten_leveled_type (Σ₁₂ @@@ nil)) with (mapOptionTree flatten_type Σ₁₂ @@@ nil).
+ apply RCut.
+ induction Σ₁₂; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ₁₂1.
+ rewrite <- IHΣ₁₂2.
+ reflexivity.
+ simpl; repeat drop_simplify.
+ simpl; repeat take_simplify.
simpl.
-
- eapply nd_comp.
- eapply nd_prod; [ idtac | apply nd_id ].
- eapply boost.
- apply ga_second.
-
- eapply nd_comp.
- eapply nd_prod.
+ set (drop_lev (ec :: lev) (Σ₁₂ @@@ (ec :: lev))) as x1.
+ rewrite take_lemma'.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ rewrite <- mapOptionTree_compose.
+ rewrite <- mapOptionTree_compose.
+ rewrite <- mapOptionTree_compose.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ apply nd_prod.
apply nd_id.
eapply nd_comp.
eapply nd_rule.
eapply RArrange.
- apply RCanR.
- eapply precompose.
+ eapply ALeft.
+ eapply ARight.
+ unfold x1.
+ rewrite drop_to_nothing.
+ apply arrangeCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ₁₂)).
+ admit. (* OK *)
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ALeft; eapply ACanL | idtac ].
+ set (mapOptionTree flatten_type Σ₁₂) as a.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as b.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₂)) as c.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₂)) as d.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ)) as e.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ)) as f.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ simpl.
+ eapply secondify.
+ apply ga_first.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ALeft; eapply AExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuAssoc ].
+ simpl.
+ apply precompose.
- apply nd_rule.
- apply RLet.
+ destruct case_RLeft.
+ simpl.
+ destruct l as [|ec lev].
+ simpl.
+ replace (mapOptionTree flatten_leveled_type (Σ @@@ nil)) with (mapOptionTree flatten_type Σ @@@ nil).
+ apply nd_rule.
+ apply RLeft.
+ induction Σ; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ repeat drop_simplify.
+ rewrite drop_to_nothing.
+ simpl.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply ARight.
+ apply arrangeUnCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ)).
+ admit (* FIXME *).
+ idtac.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanL ].
+ apply boost.
+ take_simplify.
+ simpl.
+ replace (take_lev (ec :: lev) (Σ @@@ (ec :: lev))) with (Σ @@@ (ec::lev)).
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ apply ga_second.
+ rewrite take_lemma'.
+ reflexivity.
+
+ destruct case_RRight.
+ simpl.
+ destruct l as [|ec lev].
+ simpl.
+ replace (mapOptionTree flatten_leveled_type (Σ @@@ nil)) with (mapOptionTree flatten_type Σ @@@ nil).
+ apply nd_rule.
+ apply RRight.
+ induction Σ; try destruct a; auto.
+ simpl.
+ rewrite <- IHΣ1.
+ rewrite <- IHΣ2.
+ reflexivity.
+ repeat drop_simplify.
+ rewrite drop_to_nothing.
+ simpl.
+ eapply nd_comp.
+ Focus 2.
+ eapply nd_rule.
+ eapply RArrange.
+ eapply ALeft.
+ apply arrangeUnCancelEmptyTree with (q:=(mapTree (fun _ : ??(HaskType Γ ★) => tt) Σ)).
+ admit (* FIXME *).
+ idtac.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
+ apply boost.
+ take_simplify.
+ simpl.
+ replace (take_lev (ec :: lev) (Σ @@@ (ec :: lev))) with (Σ @@@ (ec::lev)).
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ apply ga_first.
+ rewrite take_lemma'.
+ reflexivity.
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.
- apply (Prelude_error "LetRec not supported (FIXME)").
+ simpl. destruct lev; simpl.
+ apply nd_rule.
+ set (@RLetRec Γ Δ (mapOptionTree flatten_leveled_type lri) (flatten_type x) (mapOptionTree flatten_type y) nil) as 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.
+ apply (Prelude_error "LetRec not supported inside brackets yet (FIXME)").
destruct case_RCase.
simpl.
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.
set (mapOptionTree (flatten_type ○ unlev)(take_lev (ec :: nil) succ)) as succ_guest.
set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args.
unfold empty_tree.
- eapply nd_comp; [ eapply nd_rule; eapply RArrange; apply tree_of_nothing | idtac ].
- refine (ga_unkappa' Γ Δ (v2t ec) nil _ _ _ _ ;; _).
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ALeft; apply tree_of_nothing | idtac ].
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ACanR | idtac ].
+ refine (ga_unkappa Γ Δ (v2t ec) nil _ _ _ _ ;; _).
+ eapply nd_comp; [ idtac | eapply arrange_brak ].
unfold succ_host.
unfold succ_guest.
- apply arrange_brak.
+ eapply nd_rule.
+ eapply RArrange.
+ apply AExch.
apply (Prelude_error "found Brak at depth >0 indicating 3-level code; only two-level code is currently supported").
destruct case_SEsc.
take_simplify.
drop_simplify.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply tree_of_nothing' ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ALeft; apply tree_of_nothing' ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanR ].
simpl.
rewrite take_lemma'.
rewrite unlev_relev.
set (mapOptionTree flatten_leveled_type (drop_lev (ec :: nil) succ)) as succ_host.
set (mapOptionTree flatten_type (take_arg_types_as_tree t)) as succ_args.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanL ].
+ eapply nd_comp; [ idtac | eapply RLet ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod; [ idtac | eapply boost ].
induction x.
apply ga_id.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
simpl.
apply ga_join.
apply IHx1.
apply IHx2.
(* environment has non-empty leaves *)
- apply ga_kappa'.
+ apply (Prelude_error "ga_kappa not supported yet (BIG FIXME)").
(* nesting too deep *)
apply (Prelude_error "found Esc at depth >0 indicating 3-level code; only two-level code is currently supported").
Defined.
+ Definition flatten_proof :
+ forall {h}{c},
+ ND Rule h c ->
+ ND Rule h c.
+ apply (Prelude_error "sorry, non-skolemized flattening isn't implemented").
+ Defined.
+
Definition skolemize_and_flatten_proof :
forall {h}{c},
ND Rule h c ->
intros.
rewrite mapOptionTree_compose.
rewrite mapOptionTree_compose.
- apply flatten_proof.
+ apply flatten_skolemized_proof.
apply skolemize_proof.
apply X.
Defined.
projects / coq-hetmet.git / blobdiff