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 RCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; 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 RuCanR.
+ 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 nd_rule; 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 RExch ].
+ 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 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 RLeft; eapply RExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCossa ].
+ 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 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 RLet ].
+ eapply nd_comp; [ apply nd_rlecnac | idtac ].
+ apply nd_prod.
+ apply nd_id.
+ apply ga_comp.
+ 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 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 postcompose_ ].
apply X.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanL ].
- apply ga_first.
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 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.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
+ 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_second.
+ apply nd_id.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanR ].
+ apply ga_first.
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 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.
- 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''.
+ apply X.
+ apply first_nd.
+ Defined.
- idtac.
- clear q'' q'.
- eapply nd_comp.
- apply nd_rlecnac.
+ 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 RCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; 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 RuCanR ].
+ apply ga_second.
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 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; [ idtac | eapply nd_rule; eapply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod.
- apply ga_first.
-
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply nd_prod.
- apply postcompose.
- apply ga_uncancell.
- apply precompose.
+ eapply nd_comp.
+ apply X.
+ apply second_nd.
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)").
- 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 RExch ].
+ 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; [ idtac | eapply nd_rule; 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 RExch ].
+ apply precompose.
+ Defined.
(* useful for cutting down on the pretty-printed noise
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 _ _ _ _ _
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')) ].
+ (@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 RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
eapply nd_prod.
apply r1'.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
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.
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 RuCanR.
apply RAssoc.
apply RCossa.
- apply RExch.
+ apply RExch. (* TO DO: check for all-leaf trees here *)
apply RWeak.
apply RCont.
apply RLeft; auto.
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.
+ apply firstify with (c:=[]) in pfa.
eapply nd_comp; [ idtac | eapply nd_rule; 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 RLet ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanL ].
- apply precompose'.
+ eapply nd_comp; [ idtac | eapply postcompose_ ].
apply ga_uncancelr.
- apply nd_id.
+
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
+ 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.
apply y.
idtac.
clear y q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
simpl.
eapply nd_comp; [ apply nd_llecnac | idtac ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
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.
- unfold drop_lev.
set (@arrange _ 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.
+ unfold drop_lev in H.
+ unfold mkDropFlags in H.
+ rewrite H in q.
+ clear H.
set (arrangeMap _ _ flatten_leveled_type q) as y.
eapply nd_comp.
eapply nd_rule.
eapply RArrange.
apply y.
- idtac.
clear y q.
+ set (mapOptionTree flatten_leveled_type (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ))) as q.
+ destruct (decide_tree_empty q); [ idtac | apply (Prelude_error "escapifying open code not yet supported") ].
+ destruct s.
+
+ simpl.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RExch ].
+ 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 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 nd_comp.
+ eapply nd_rule.
+ eapply (@RVar Γ Δ t nil).
+ apply nd_rule.
+ apply RArrange.
+ eapply RComp.
+ apply RuCanR.
+ apply RLeft.
+ apply RWeak.
+(*
+ eapply decide_tree_empty.
+
+ simpl.
+ set (dropT (mkFlags (liftBoolFunc false (bnot ○ levelMatch (ec :: nil))) succ)) as escapified.
+ destruct (decide_tree_empty escapified).
+
induction succ.
destruct a.
+ unfold drop_lev.
destruct l.
simpl.
unfold mkDropFlags; simpl.
apply RLeft.
apply RWeak.
apply (Prelude_error "escapifying code with multi-leaf antecedents is not supported").
+*)
Defined.
Lemma mapOptionTree_distributes
reflexivity.
Qed.
- 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.
-
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 RId.
simpl.
- apply RCanR.
+ apply RId.
+ eapply RComp; [ idtac | apply RCanL ].
+ eapply RComp; [ idtac | eapply RLeft; apply IHt2 ].
Opaque drop_lev.
simpl.
Transparent drop_lev.
+ idtac.
drop_simplify.
- simpl.
- eapply RComp.
- eapply RComp.
- eapply RAssoc.
- eapply RRight.
+ apply RRight.
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 RId.
simpl.
- apply RuCanR.
+ apply RId.
+ eapply RComp; [ apply RuCanL | idtac ].
+ eapply RComp; [ eapply RRight; 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 RLeft.
apply IHt2.
Defined.
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 _
| 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 _
+ | RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ lev => let case_RWhere := 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.
repeat take_simplify.
simpl.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as Σ₁'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₂)) as Σ₂'.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₁)) as Σ₁''.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₂)) as Σ₂''.
+
eapply nd_comp.
eapply nd_prod; [ idtac | apply nd_id ].
eapply boost.
- apply ga_second.
+ apply (ga_first _ _ _ _ _ _ Σ₂').
- eapply nd_comp.
- eapply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
+ apply nd_prod.
apply nd_id.
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply RCanL | idtac ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch (* okay *)].
+ apply precompose.
+
+ destruct case_RWhere.
+ simpl.
+ destruct lev as [|ec lev]; simpl; [ apply nd_rule; apply RWhere; auto | idtac ].
+ repeat take_simplify.
+ repeat drop_simplify.
+ simpl.
+
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₁)) as Σ₁'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₂)) as Σ₂'.
+ set (mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) Σ₃)) as Σ₃'.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₁)) as Σ₁''.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₂)) as Σ₂''.
+ set (mapOptionTree flatten_leveled_type (drop_lev (ec :: lev) Σ₃)) as Σ₃''.
+
eapply nd_comp.
- eapply nd_rule.
- eapply RArrange.
- apply RCanR.
- eapply precompose.
+ eapply nd_prod; [ eapply nd_id | idtac ].
+ eapply (first_nd _ _ _ _ _ _ Σ₃').
+ eapply nd_comp.
+ eapply nd_prod; [ eapply nd_id | idtac ].
+ eapply (second_nd _ _ _ _ _ _ Σ₁').
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RWhere ].
+ apply nd_prod; [ idtac | apply nd_id ].
+ eapply nd_comp; [ idtac | eapply precompose' ].
apply nd_rule.
- apply RLet.
+ apply RArrange.
+ apply RLeft.
+ apply RCanL.
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 RLeft; apply tree_of_nothing | idtac ].
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply RCanR | 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 RExch.
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 RLeft; apply tree_of_nothing' ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
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 RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ].
eapply nd_comp; [ idtac | eapply nd_rule; 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 RCanL ].
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").