-{-# OPTIONS_GHC -XModalTypes -fcoqpass -dcore-lint #-}
+{-# OPTIONS_GHC -XModalTypes -fflatten -funsafe-skolemize -dcore-lint #-}
module Demo (demo) where
make demo
open .build/test.pdf
-all:
+#sanity += BiGArrow.hs
+#sanity += CircuitExample.hs
+sanity += CommandSyntaxExample.hs
+sanity += DotProduct.hs
+sanity += GArrowTutorial.hs
+sanity += GArrowVerilog.hs
+sanity += ImmutableHeap.hs
+sanity += IsomorphismForCodeTypes.hs
+sanity += LambdaCalculusInterpreter.hs
+sanity += TypeSafeRun.hs
+sanity += Unflattening.hs
+
+sanity:
../../../inplace/bin/ghc-stage2 -dcore-lint -fforce-recomp -fcoqpass -ddump-coqpass -ddump-to-file \
- `ls *.hs | grep -v Regex | grep -v Unify.hs | grep -v GArrowTikZ.hs ` +RTS -K500M
- ../../../inplace/bin/ghc-stage2 -dcore-lint -fforce-recomp \
- RegexMatcher.hs Unify.hs GArrowTikZ.hs
+ $(sanity) \
+ +RTS -K500M
+
demo:
mkdir -p .build
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import HaskKinds.
Require Import HaskLiterals.
OK (eol+++eol+++eol+++
"\begin{preview}"+++eol+++
"$\displaystyle "+++
- toString (nd_ml_toLatexMath (@expr2proof _ _ _ _ _ _ e))+++
+ toString (nd_ml_toLatexMath (@expr2proof _ _ _ _ _ _ _ e))+++
" $"+++eol+++
"\end{preview}"+++eol+++eol+++eol)
)))))))).
ND Rule
[ 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 RArrange; eapply AExch ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RApp ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
intro pf.
eapply nd_comp.
apply pf.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply ACanR ].
apply curry.
Defined.
eapply nd_comp.
eapply nd_rule.
eapply RArrange.
- eapply RCanR.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
+ eapply ACanR.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
apply curry.
Defined.
Section coqPassCoreToCore.
Context
+ (do_flatten : bool)
+ (do_skolemize : bool)
(hetmet_brak : CoreVar)
(hetmet_esc : CoreVar)
(hetmet_flatten : CoreVar)
((weakExprToStrongExpr Γ Δ φ ψ ξ (fun _ => true) τ nil we) >>= fun e =>
(addErrorMessage ("HaskStrong...")
- (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 _ _ (flatten_type τ@@nil) _ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
- >>= fun e' =>
- (snd e') >>= fun e'' =>
- strongExprToWeakExpr hetmet_brak' hetmet_esc'
- mkWeakTypeVar mkWeakCoerVar mkWeakExprVar uniqueSupply
- (projT2 e'') INil
- >>= fun q =>
- OK (weakExprToCoreExpr q)
- )))))))))).
+ (if do_skolemize
+ then
+ (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 _ _ (flatten_type τ) nil _
+ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
+ >>= fun e' => (snd e') >>= fun e'' =>
+ strongExprToWeakExpr hetmet_brak' hetmet_esc'
+ mkWeakTypeVar mkWeakCoerVar mkWeakExprVar uniqueSupply
+ (projT2 e'') INil
+ >>= fun q => OK (weakExprToCoreExpr q))
+ else (if do_flatten
+ then
+ (let haskProof := flatten_proof (*hetmet_flatten' hetmet_unflatten'
+ hetmet_flattened_id' my_ga*) (@expr2proof _ _ _ _ _ _ _ e)
+ in (* insert HaskProof-to-HaskProof manipulations here *)
+ OK ((@proof2expr nat _ FreshNat _ _ τ nil _
+ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
+ >>= fun e' => (snd e') >>= fun e'' =>
+ strongExprToWeakExpr hetmet_brak' hetmet_esc'
+ mkWeakTypeVar mkWeakCoerVar mkWeakExprVar uniqueSupply
+ (projT2 e'') INil
+ >>= fun q => OK (weakExprToCoreExpr q))
+ else
+ (let haskProof := @expr2proof _ _ _ _ _ _ _ e
+ in (* insert HaskProof-to-HaskProof manipulations here *)
+ OK ((@proof2expr nat _ FreshNat _ _ τ nil _
+ (fun _ => Prelude_error "unbound unique") _ haskProof) O)
+ >>= fun e' => (snd e') >>= fun e'' =>
+ strongExprToWeakExpr hetmet_brak' hetmet_esc'
+ mkWeakTypeVar mkWeakCoerVar mkWeakExprVar uniqueSupply
+ (projT2 e'') INil
+ >>= fun q => OK (weakExprToCoreExpr q))))
+ ))))))))).
Definition coreToCoreExpr (ce:@CoreExpr CoreVar) : (@CoreExpr CoreVar) :=
match coreToCoreExpr' ce with
End coqPassCoreToCore.
Definition coqPassCoreToCore
+ (do_flatten : bool)
+ (do_skolemize : bool)
(hetmet_brak : CoreVar)
(hetmet_esc : CoreVar)
(hetmet_flatten : CoreVar)
(hetmet_pga_curryl : CoreVar)
(hetmet_pga_curryr : CoreVar) : list (@CoreBind CoreVar) :=
coqPassCoreToCore'
+ do_flatten
+ do_skolemize
hetmet_brak
hetmet_esc
hetmet_flatten
end.
Definition treeDecomposition {D T:Type} (mapLeaf:T->D) (mergeBranches:D->D->D) :=
forall d:D, { tt:Tree T & d = treeReduce mapLeaf mergeBranches tt }.
+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.
Fixpoint reduceTree {T}(unit:T)(merge:T -> T -> T)(tt:Tree ??T) : T :=
match tt with
end
end.
+Definition takeT' {T}{Σ:Tree ??T}(tf:TreeFlags Σ) : Tree ??T :=
+ match takeT tf with
+ | None => []
+ | Some x => x
+ end.
+
(* lift a function T->bool to a function (option T)->bool by yielding (None |-> b) *)
Definition liftBoolFunc {T}(b:bool)(f:T -> bool) : ??T -> bool :=
fun t =>
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)] ] =>
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; eapply RArrange; eapply ACanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply X.
eapply nd_rule.
eapply RArrange.
- apply RuCanR.
+ apply AuCanR.
Defined.
Definition precompose Γ Δ ec : forall a x y z lev,
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
apply ga_comp.
Defined.
[ 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 ].
+ 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.
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; eapply RArrange; eapply ACanL ].
eapply nd_comp; [ idtac | eapply postcompose_ ].
apply X.
Defined.
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; eapply RArrange; eapply ACanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
apply ga_first.
Defined.
[ 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; eapply RArrange; eapply ACanR ].
eapply nd_comp; [ idtac | eapply nd_rule; apply RLet ].
eapply nd_comp; [ apply nd_rlecnac | idtac ].
apply nd_prod.
apply nd_id.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AuCanR ].
apply ga_second.
Defined.
[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 RArrange; eapply AExch ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply RLet ].
eapply nd_comp; [ apply nd_llecnac | idtac ].
apply nd_prod.
apply nd_prod.
apply postcompose.
apply ga_uncancell.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
apply precompose.
Defined.
(mapOptionTree (flatten_type ○ unlev) (take_lev (ec :: lev) B))
(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; eapply RArrange; apply ACanL ].
eapply nd_comp; [ idtac | eapply nd_rule; 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 RuCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; apply RCanL ].
+ 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 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 ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
apply ga_comp.
Defined.
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.
apply nd_rule.
apply RArrange.
induction r; simpl.
- apply RId.
- apply RCanL.
- apply RCanR.
- apply RuCanL.
- apply RuCanR.
- apply RAssoc.
- apply RCossa.
- apply RExch. (* TO DO: check for all-leaf trees here *)
- 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.
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 ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
eapply nd_comp; [ idtac | eapply postcompose_ ].
apply ga_uncancelr.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RExch ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
eapply nd_comp; [ idtac | eapply precompose ].
apply pfb.
Defined.
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 RExch ].
+ 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 ].
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]
[(@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 RLet ].
clear q''.
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 RuCanR.
- apply RLeft.
- 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.
simpl.
drop_simplify.
simpl.
- apply RId.
+ apply AId.
simpl.
- apply RId.
- eapply RComp; [ idtac | apply RCanL ].
- eapply RComp; [ idtac | eapply RLeft; apply IHt2 ].
+ apply AId.
+ eapply AComp; [ idtac | apply ACanL ].
+ eapply AComp; [ idtac | eapply ALeft; apply IHt2 ].
Opaque drop_lev.
simpl.
Transparent drop_lev.
idtac.
drop_simplify.
- apply RRight.
+ apply ARight.
apply IHt1.
Defined.
simpl.
drop_simplify.
simpl.
- apply RId.
+ apply AId.
simpl.
- apply RId.
- eapply RComp; [ apply RuCanL | idtac ].
- eapply RComp; [ eapply RRight; apply IHt1 | idtac ].
+ apply AId.
+ eapply AComp; [ apply AuCanL | idtac ].
+ eapply AComp; [ eapply ARight; apply IHt1 | idtac ].
Opaque drop_lev.
simpl.
Transparent drop_lev.
idtac.
drop_simplify.
- apply RLeft.
+ 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).
| 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 _
+ | RCut Γ Δ 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 _
| 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 _
eapply nd_rule.
eapply RArrange.
simpl.
- apply RCanR.
+ apply ACanR.
apply boost.
simpl.
apply ga_curry.
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 *)].
+ eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply ACanL | idtac ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch (* okay *)].
apply precompose.
destruct case_RWhere.
eapply nd_comp; [ idtac | eapply precompose' ].
apply nd_rule.
apply RArrange.
- apply RLeft.
- apply RCanL.
+ apply ALeft.
+ apply ACanL.
+
+ destruct case_RCut.
+ simpl.
+ 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.
+ set (drop_lev (ec :: lev) (Σ₁₂ @@@ (ec :: lev))) as x1.
+ rewrite take_lemma'.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite mapOptionTree_compose.
+ rewrite unlev_relev.
+ 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.
+ 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 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.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RCut ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply nd_prod.
+ simpl.
+ eapply ga_first.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply AExch ].
+ simpl.
+ apply precompose.
+
+ 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.
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; eapply RLeft; apply tree_of_nothing | idtac ].
- eapply nd_comp; [ eapply nd_rule; eapply RArrange; eapply RCanR | idtac ].
+ 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.
eapply nd_rule.
eapply RArrange.
- apply RExch.
+ 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; eapply RLeft; apply tree_of_nothing' ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply RCanR ].
+ 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 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 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 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 RCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply RArrange; eapply ACanL ].
simpl.
apply ga_join.
apply IHx1.
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.
apply nd_id.
eapply nd_rule.
set (@org_fc) as ofc.
- set (RArrange Γ Δ _ _ _ (RuCanL [l0])) as rule.
- apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [_])).
+ set (RArrange Γ Δ _ _ _ (AuCanL [l0])) as rule.
+ apply org_fc with (r:=RArrange _ _ _ _ _ (AuCanL [_])).
auto.
- eapply nd_comp; [ idtac | eapply nd_rule; apply org_fc with (r:=RArrange _ _ _ _ _ (RCanL _)) ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply org_fc with (r:=RArrange _ _ _ _ _ (ACanL _)) ].
apply nd_rule.
destruct l.
destruct l0.
; cnd_expand_right := fun a b c => SystemFCa_right c a b }.
(*
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ ((RArrange _ _ _ _ _ (RCossa _ _ _)))).
+ apply (org_fc _ _ _ _ ((RArrange _ _ _ _ _ (AuAssoc _ _ _)))).
auto.
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RAssoc _ _ _))); auto.
+ apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (AAssoc _ _ _))); auto.
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanL _))); auto.
+ apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (ACanL _))); auto.
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanR _))); auto.
+ apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (ACanR _))); auto.
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanL _))); auto.
+ apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (AuCanL _))); auto.
intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanR _))); auto.
+ apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (AuCanR _))); auto.
*)
admit.
admit.
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import HaskKinds.
}.
Implicit Arguments ProofCaseBranch [ ].
-(* Figure 3, production $\vdash_E$, Uniform rules *)
-Inductive Arrange {T} : Tree ??T -> Tree ??T -> Type :=
-| RId : forall a , Arrange a a
-| RCanL : forall a , Arrange ( [],,a ) ( a )
-| RCanR : forall a , Arrange ( a,,[] ) ( a )
-| RuCanL : forall a , Arrange ( a ) ( [],,a )
-| RuCanR : forall a , Arrange ( a ) ( a,,[] )
-| RAssoc : forall a b c , Arrange (a,,(b,,c) ) ((a,,b),,c )
-| RCossa : forall a b c , Arrange ((a,,b),,c ) ( a,,(b,,c) )
-| RExch : forall a b , Arrange ( (b,,a) ) ( (a,,b) )
-| RWeak : forall a , Arrange ( [] ) ( a )
-| RCont : forall a , Arrange ( (a,,a) ) ( a )
-| RLeft : forall {h}{c} x , Arrange h c -> Arrange ( x,,h ) ( x,,c)
-| RRight : forall {h}{c} x , Arrange h c -> Arrange ( h,,x ) ( c,,x)
-| RComp : forall {a}{b}{c}, Arrange a b -> Arrange b c -> Arrange a c
-.
-
(* Figure 3, production $\vdash_E$, all rules *)
Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
| 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 *)
+(* order is important here; we want to be able to skolemize without introducing new AExch'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]
| 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]
+| RCut : ∀ Γ Δ Σ₁ Σ₁₂ Σ₂ Σ₃ l, Rule ([Γ>Δ> Σ₁ |- Σ₁₂ @l],,[Γ>Δ> (Σ₁₂@@@l),,Σ₂ |- Σ₃@l ]) [Γ>Δ> Σ₁,,Σ₂ |- Σ₃@l]
+| RLeft : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> (Σ@@@l),,Σ₁ |- Σ,,Σ₂@l]
+| RRight : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> Σ₁,,(Σ@@@l) |- Σ₂,,Σ@l]
+
| 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]
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
+ destruct X0; destruct s; inversion e.
+ destruct X0; destruct s; inversion e.
+ destruct X0; destruct s; inversion e.
Qed.
Lemma systemfc_all_rules_one_conclusion : forall h c1 c2 (r:Rule h (c1,,c2)), False.
auto.
Qed.
-(* "Arrange" objects are parametric in the type of the leaves of the tree *)
-Definition arrangeMap :
- forall {T} (Σ₁ Σ₂:Tree ??T) {R} (f:T -> R),
- Arrange Σ₁ Σ₂ ->
- Arrange (mapOptionTree f Σ₁) (mapOptionTree f Σ₂).
- intros.
- induction X; 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 IHX1 | apply IHX2 ].
- Defined.
-
-(* a frequently-used Arrange *)
-Definition arrangeSwapMiddle {T} (a b c d:Tree ??T) :
- Arrange ((a,,b),,(c,,d)) ((a,,c),,(b,,d)).
- eapply RComp.
- apply RCossa.
- eapply RComp.
- eapply RLeft.
- eapply RComp.
- eapply RAssoc.
- eapply RRight.
- apply RExch.
- eapply RComp.
- eapply RLeft.
- eapply RCossa.
- eapply RAssoc.
- Defined.
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import HaskKinds.
Fixpoint nd_uruleToRawLatexMath {T}{h}{c}(r:@Arrange T h c) : string :=
match r with
- | RLeft _ _ _ r => nd_uruleToRawLatexMath r
- | RRight _ _ _ r => nd_uruleToRawLatexMath r
- | RId _ => "Id"
- | RCanL _ => "CanL"
- | RCanR _ => "CanR"
- | RuCanL _ => "uCanL"
- | RuCanR _ => "uCanR"
- | RAssoc _ _ _ => "Assoc"
- | RCossa _ _ _ => "Cossa"
- | RExch _ _ => "Exch"
- | RWeak _ => "Weak"
- | RCont _ => "Cont"
- | RComp _ _ _ _ _ => "Comp" (* FIXME: do a better job here *)
+ | ALeft _ _ _ r => nd_uruleToRawLatexMath r
+ | ARight _ _ _ r => nd_uruleToRawLatexMath r
+ | AId _ => "Id"
+ | ACanL _ => "CanL"
+ | ACanR _ => "CanR"
+ | AuCanL _ => "uCanL"
+ | AuCanR _ => "uCanR"
+ | AAssoc _ _ _ => "Assoc"
+ | AuAssoc _ _ _ => "Cossa"
+ | AExch _ _ => "Exch"
+ | AWeak _ => "Weak"
+ | ACont _ => "Cont"
+ | AComp _ _ _ _ _ => "Comp" (* FIXME: do a better job here *)
end.
Fixpoint nd_ruleToRawLatexMath {h}{c}(r:Rule h c) : string :=
| RAbsCo _ _ _ _ _ _ _ _ => "AbsCo"
| RApp _ _ _ _ _ _ _ => "App"
| RLet _ _ _ _ _ _ _ => "Let"
+ | RCut _ _ _ _ _ _ _ => "Cut"
+ | RLeft _ _ _ _ _ _ => "Left"
+ | RRight _ _ _ _ _ _ => "Right"
| RWhere _ _ _ _ _ _ _ _ => "Where"
| RJoin _ _ _ _ _ _ _ => "RJoin"
| RLetRec _ _ _ _ _ _ => "LetRec"
Fixpoint nd_hideURule {T}{h}{c}(r:@Arrange T h c) : bool :=
match r with
- | RLeft _ _ _ r => nd_hideURule r
- | RRight _ _ _ r => nd_hideURule r
- | RCanL _ => true
- | RCanR _ => true
- | RuCanL _ => true
- | RuCanR _ => true
- | RAssoc _ _ _ => true
- | RCossa _ _ _ => true
- | RExch (T_Leaf None) b => true
- | RExch a (T_Leaf None) => true
- | RWeak (T_Leaf None) => true
- | RCont (T_Leaf None) => true
- | RComp _ _ _ _ _ => false (* FIXME: do better *)
+ | ALeft _ _ _ r => nd_hideURule r
+ | ARight _ _ _ r => nd_hideURule r
+ | ACanL _ => true
+ | ACanR _ => true
+ | AuCanL _ => true
+ | AuCanR _ => true
+ | AAssoc _ _ _ => true
+ | AuAssoc _ _ _ => true
+ | AExch (T_Leaf None) b => true
+ | AExch a (T_Leaf None) => true
+ | AWeak (T_Leaf None) => true
+ | ACont (T_Leaf None) => true
+ | AComp _ _ _ _ _ => false (* FIXME: do better *)
| _ => false
end.
Fixpoint nd_hideRule {h}{c}(r:Rule h c) : bool :=
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Init.Specif.
Definition judg2exprType (j:Judg) : Type :=
match j with
(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)
+ 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.
+ Definition justOne Γ Δ ξ τ l : ITree _ (fun t => Expr Γ Δ ξ t l) [τ] -> Expr Γ Δ ξ τ l.
intros.
inversion X; auto.
Defined.
Defined.
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).
+ 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 l (r:@Arrange _ h j) (ξ:VV -> LeveledHaskType Γ ★),
ujudg2exprType Γ ξ Δ h t l ->
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)
- | RId a => let case_RId := tt in _
- | RCanL a => let case_RCanL := tt in _
- | RCanR a => let case_RCanR := tt in _
- | RuCanL a => let case_RuCanL := tt in _
- | RuCanR a => let case_RuCanR := tt in _
- | RAssoc a b c => let case_RAssoc := tt in _
- | RCossa a b c => let case_RCossa := 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)
+ | ALeft h c ctx r => let case_ALeft := tt in (fun e => _) (urule2expr _ _ _ _ r)
+ | ARight h c ctx r => let case_ARight := tt in (fun e => _) (urule2expr _ _ _ _ r)
+ | AId a => let case_AId := tt in _
+ | ACanL a => let case_ACanL := tt in _
+ | ACanR a => let case_ACanR := tt in _
+ | AuCanL a => let case_AuCanL := tt in _
+ | AuCanR a => let case_AuCanR := tt in _
+ | AAssoc a b c => let case_AAssoc := tt in _
+ | AuAssoc a b c => let case_AuAssoc := tt in _
+ | AExch a b => let case_AExch := tt in _
+ | AWeak a => let case_AWeak := tt in _
+ | ACont a => let case_ACont := tt in _
+ | AComp a b c f g => let case_AComp := tt in (fun e1 e2 => _) (urule2expr _ _ _ _ f) (urule2expr _ _ _ _ g)
end); clear urule2expr; intros.
- destruct case_RId.
+ destruct case_AId.
apply X.
- destruct case_RCanL.
+ destruct case_ACanL.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X ([],,vars)).
simpl; rewrite <- H; auto.
- destruct case_RCanR.
+ destruct case_ACanR.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X (vars,,[])).
simpl; rewrite <- H; auto.
- destruct case_RuCanL.
+ destruct case_AuCanL.
simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
simpl in X.
apply (X vars2); auto.
- destruct case_RuCanR.
+ destruct case_AuCanR.
simpl; unfold ujudg2exprType; intros.
destruct vars; try destruct o; inversion H.
simpl in X.
apply (X vars1); auto.
- destruct case_RAssoc.
+ destruct case_AAssoc.
simpl; unfold ujudg2exprType; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X (vars1_1,,(vars1_2,,vars2))).
subst; auto.
- destruct case_RCossa.
+ destruct case_AuAssoc.
simpl; unfold ujudg2exprType; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X ((vars1,,vars2_1),,vars2_2)).
subst; auto.
- destruct case_RExch.
+ destruct case_AExch.
simpl; unfold ujudg2exprType ; intros.
simpl in X.
destruct vars; try destruct o; inversion H.
apply (X (vars2,,vars1)).
inversion H; subst; auto.
- destruct case_RWeak.
+ destruct case_AWeak.
simpl; unfold ujudg2exprType; intros.
simpl in X.
apply (X []).
auto.
- destruct case_RCont.
+ destruct case_ACont.
simpl; unfold ujudg2exprType ; intros.
simpl in X.
apply (X (vars,,vars)).
rewrite <- H.
auto.
- destruct case_RLeft.
+ destruct case_ALeft.
intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
apply (fun q => e ξ q vars2 H2).
simpl.
reflexivity.
- destruct case_RRight.
+ destruct case_ARight.
intro vars; unfold ujudg2exprType; intro H.
destruct vars; try destruct o; inversion H.
apply (fun q => e ξ q vars1 H1).
simpl.
reflexivity.
- destruct case_RComp.
+ destruct case_AComp.
apply e2.
apply e1.
apply X.
Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' :
ITree (HaskType Γ ★)
- (fun t : HaskType Γ ★ => Expr Γ Δ ξ' (t @@ l))
+ (fun t : HaskType Γ ★ => Expr Γ Δ ξ' t l)
(mapOptionTree (unlev ○ ξ' ○ (@fst _ _)) varstypes)
-> ELetRecBindings Γ Δ ξ' l varstypes.
intros.
prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} ->
((fun sac => FreshM
{ scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac
- & Expr (sac_gamma sac Γ) (sac_delta sac Γ avars (weakCK'' Δ)) (scbwv_xi scb ξ lev) (weakLT' (tbranches @@ lev)) }) (projT1 pcb)).
+ & Expr (sac_gamma sac Γ) (sac_delta sac Γ avars (weakCK'' Δ)) (scbwv_xi scb ξ lev)
+ (weakT' tbranches) (weakL' lev) }) (projT1 pcb)).
intro pcb.
intro X.
simpl in X.
Defined.
+ Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
+ FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
+ intros Γ Σ.
+ induction Σ; intro ξ.
+ destruct a.
+ destruct l as [τ l].
+ set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ clear q.
+ destruct q' as [varstypes [pf1 [pf2 distpf]]].
+ exists (mapOptionTree (@fst _ _) varstypes).
+ exists (update_xi ξ l (leaves varstypes)).
+ symmetry; auto.
+ refine (return _).
+ exists [].
+ exists ξ; auto.
+ refine (bind f1 = IHΣ1 ξ ; _).
+ apply FreshMon.
+ destruct f1 as [vars1 [ξ1 pf1]].
+ refine (bind f2 = IHΣ2 ξ1 ; _).
+ apply FreshMon.
+ destruct f2 as [vars2 [ξ2 pf22]].
+ refine (return _).
+ exists (vars1,,vars2).
+ exists ξ2.
+ simpl.
+ rewrite pf22.
+ rewrite pf1.
+ admit. (* freshness assumption *)
+ Defined.
+
+ Definition rlet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p :
+ forall (X_ : ITree Judg judg2exprType
+ ([Γ > Δ > Σ₁ |- [σ₁] @ p],, [Γ > Δ > [σ₁ @@ p],, Σ₂ |- [σ₂] @ p])),
+ ITree Judg judg2exprType [Γ > Δ > Σ₁,, Σ₂ |- [σ₂] @ p].
+ intros.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+
+ refine (fresh_lemma _ ξ _ _ σ₁ p H2 >>>= (fun pf => _)).
+ apply FreshMon.
+
+ destruct pf as [ vnew [ pf1 pf2 ]].
+ set (update_xi ξ p (((vnew, σ₁ )) :: nil)) as ξ' in *.
+ inversion X_.
+ apply ileaf in X.
+ apply ileaf in X0.
+ simpl in *.
+
+ refine (X ξ vars1 _ >>>= fun X0' => _).
+ apply FreshMon.
+ simpl.
+ auto.
+
+ refine (X0 ξ' ([vnew],,vars2) _ >>>= fun X1' => _).
+ apply FreshMon.
+ simpl.
+ rewrite pf2.
+ rewrite pf1.
+ reflexivity.
+ apply FreshMon.
+
+ apply ILeaf.
+ apply ileaf in X1'.
+ apply ileaf in X0'.
+ simpl in *.
+ apply ELet with (ev:=vnew)(tv:=σ₁).
+ apply X0'.
+ apply X1'.
+ Defined.
+
+ Definition vartree Γ Δ Σ lev ξ :
+ forall vars, Σ @@@ lev = mapOptionTree ξ vars ->
+ ITree (HaskType Γ ★) (fun t : HaskType Γ ★ => Expr Γ Δ ξ t lev) Σ.
+ induction Σ; intros.
+ destruct a.
+ intros; simpl in *.
+ apply ILeaf.
+ destruct vars; try destruct o; inversion H.
+ set (EVar Γ Δ ξ v) as q.
+ rewrite <- H1 in q.
+ apply q.
+ intros.
+ apply INone.
+ intros.
+ destruct vars; try destruct o; inversion H.
+ apply IBranch.
+ eapply IHΣ1.
+ apply H1.
+ eapply IHΣ2.
+ apply H2.
+ Defined.
+
+
+ Definition rdrop Γ Δ Σ₁ Σ₁₂ a lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a,,Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *.
+ intros.
+ set (X ξ vars H) as q.
+ simpl in q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ inversion q'.
+ apply X0.
+ Defined.
+
+ Definition rdrop' Γ Δ Σ₁ Σ₁₂ a lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂,,a @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- a @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *.
+ intros.
+ set (X ξ vars H) as q.
+ simpl in q.
+ refine (q >>>= fun q' => return _).
+ apply FreshMon.
+ inversion q'.
+ auto.
+ Defined.
+
+ Definition rdrop'' Γ Δ Σ₁ Σ₁₂ lev :
+ ITree Judg judg2exprType [Γ > Δ > [],,Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ eapply X with (vars:=[],,vars).
+ rewrite H; reflexivity.
+ Defined.
+
+ Definition rdrop''' Γ Δ a Σ₁ Σ₁₂ lev :
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > a,,Σ₁ |- Σ₁₂ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ eapply X with (vars:=vars2).
+ auto.
+ Defined.
+
+ Definition rassoc Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars2; try destruct o; inversion H2.
+ apply X with (vars:=(vars1,,vars2_1),,vars2_2).
+ subst; reflexivity.
+ Defined.
+
+ Definition rassoc' Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > (a,,(b,,c)) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars1; try destruct o; inversion H1.
+ apply X with (vars:=vars1_1,,(vars1_2,,vars2)).
+ subst; reflexivity.
+ Defined.
+
+ Definition swapr Γ Δ Σ₁ a b c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,b),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > ((b,,a),,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ destruct vars1; try destruct o; inversion H1.
+ apply X with (vars:=(vars1_2,,vars1_1),,vars2).
+ subst; reflexivity.
+ Defined.
+
+ Definition rdup Γ Δ Σ₁ a c lev :
+ ITree Judg judg2exprType [Γ > Δ > ((a,,a),,c) |- Σ₁ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > (a,,c) |- Σ₁ @ lev].
+ intros.
+ apply ileaf in X.
+ apply ILeaf.
+ simpl in *; intros.
+ destruct vars; try destruct o; inversion H.
+ apply X with (vars:=(vars1,,vars1),,vars2). (* is this allowed? *)
+ subst; reflexivity.
+ Defined.
+
+ (* holy cow this is ugly *)
+ Definition rcut Γ Δ Σ₃ lev Σ₁₂ :
+ forall Σ₁ Σ₂,
+ ITree Judg judg2exprType [Γ > Δ > Σ₁ |- Σ₁₂ @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁₂ @@@ lev,,Σ₂ |- [Σ₃] @ lev] ->
+ ITree Judg judg2exprType [Γ > Δ > Σ₁,,Σ₂ |- [Σ₃] @ lev].
+
+ induction Σ₁₂.
+ intros.
+ destruct a.
+
+ eapply rlet.
+ apply IBranch.
+ apply X.
+ apply X0.
+
+ simpl in X0.
+ apply rdrop'' in X0.
+ apply rdrop'''.
+ apply X0.
+
+ intros.
+ simpl in X0.
+ apply rassoc in X0.
+ set (IHΣ₁₂1 _ _ (rdrop _ _ _ _ _ _ X) X0) as q.
+ set (IHΣ₁₂2 _ (Σ₁,,Σ₂) (rdrop' _ _ _ _ _ _ X)) as q'.
+ apply rassoc' in q.
+ apply swapr in q.
+ apply rassoc in q.
+ set (q' q) as q''.
+ apply rassoc' in q''.
+ apply rdup in q''.
+ apply q''.
+ Defined.
Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j.
| RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := 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 _
+ | RCut Γ Δ 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 _
| RWhere Γ Δ Σ₁ Σ₂ Σ₃ σ₁ σ₂ p => 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 _
destruct case_RVar.
apply ILeaf; simpl; intros; refine (return ILeaf _ _).
- destruct vars; simpl in H; inversion H; destruct o. inversion H1. rewrite H2.
- apply EVar.
+ destruct vars; simpl in H; inversion H; destruct o. inversion H1.
+ set (@EVar _ _ _ Δ ξ v) as q.
+ rewrite <- H2 in q.
+ simpl in q.
+ apply q.
inversion H.
destruct case_RGlobal.
apply (EApp _ _ _ _ _ _ q1' q2').
destruct case_RLet.
- apply ILeaf.
- simpl in *; intros.
- destruct vars; try destruct o; inversion H.
-
- refine (fresh_lemma _ ξ _ _ σ₁ p H2 >>>= (fun pf => _)).
- apply FreshMon.
-
- destruct pf as [ vnew [ pf1 pf2 ]].
- set (update_xi ξ p (((vnew, σ₁ )) :: nil)) as ξ' in *.
- inversion X_.
- apply ileaf in X.
- apply ileaf in X0.
- simpl in *.
-
- refine (X ξ vars1 _ >>>= fun X0' => _).
- apply FreshMon.
- simpl.
- auto.
-
- refine (X0 ξ' ([vnew],,vars2) _ >>>= fun X1' => _).
- apply FreshMon.
- simpl.
- rewrite pf2.
- rewrite pf1.
- reflexivity.
- apply FreshMon.
-
- apply ILeaf.
- apply ileaf in X1'.
- apply ileaf in X0'.
- simpl in *.
- apply ELet with (ev:=vnew)(tv:=σ₁).
- apply X0'.
- apply X1'.
+ eapply rlet.
+ apply X_.
destruct case_RWhere.
apply ILeaf.
apply X1'.
apply X0'.
+ destruct case_RCut.
+ inversion X_.
+ subst.
+ clear X_.
+ induction Σ₃.
+ destruct a.
+ subst.
+ eapply rcut.
+ apply X.
+ apply X0.
+
+ apply ILeaf.
+ simpl.
+ intros.
+ refine (return _).
+ apply INone.
+ set (IHΣ₃1 (rdrop _ _ _ _ _ _ X0)) as q1.
+ set (IHΣ₃2 (rdrop' _ _ _ _ _ _ X0)) as q2.
+ apply ileaf in q1.
+ apply ileaf in q2.
+ simpl in *.
+ apply ILeaf.
+ simpl.
+ intros.
+ refine (q1 _ _ H >>>= fun q1' => q2 _ _ H >>>= fun q2' => return _).
+ apply FreshMon.
+ apply FreshMon.
+ apply IBranch; auto.
+
+ destruct case_RLeft.
+ apply ILeaf.
+ simpl; intros.
+ destruct vars; try destruct o; inversion H.
+ refine (X_ _ _ H2 >>>= fun X' => return _).
+ apply FreshMon.
+ apply IBranch.
+ eapply vartree.
+ apply H1.
+ apply X'.
+
+ destruct case_RRight.
+ apply ILeaf.
+ simpl; intros.
+ destruct vars; try destruct o; inversion H.
+ refine (X_ _ _ H1 >>>= fun X' => return _).
+ apply FreshMon.
+ apply IBranch.
+ apply X'.
+ eapply vartree.
+ apply H2.
+
destruct case_RVoid.
apply ILeaf; simpl; intros.
refine (return _).
| scnd_branch _ _ _ c1 c2 => let case_branch := tt in fun q => IBranch _ _ (closed2expr _ _ c1 q) (closed2expr _ _ c2 q)
end.
- Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
- FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
- intros Γ Σ.
- induction Σ; intro ξ.
- destruct a.
- destruct l as [τ l].
- set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
- refine (q >>>= fun q' => return _).
- apply FreshMon.
- clear q.
- destruct q' as [varstypes [pf1 [pf2 distpf]]].
- exists (mapOptionTree (@fst _ _) varstypes).
- exists (update_xi ξ l (leaves varstypes)).
- symmetry; auto.
- refine (return _).
- exists [].
- exists ξ; auto.
- refine (bind f1 = IHΣ1 ξ ; _).
- apply FreshMon.
- destruct f1 as [vars1 [ξ1 pf1]].
- refine (bind f2 = IHΣ2 ξ1 ; _).
- apply FreshMon.
- destruct f2 as [vars2 [ξ2 pf22]].
- refine (return _).
- exists (vars1,,vars2).
- exists ξ2.
- simpl.
- rewrite pf22.
- rewrite pf1.
- admit.
- Defined.
-
- Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
- {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [unlev τ] @ getlev τ] ->
- FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
+ Definition proof2expr Γ Δ τ l Σ (ξ0: VV -> LeveledHaskType Γ ★)
+ {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ] @ l] ->
+ FreshM (???{ ξ : _ & Expr Γ Δ ξ τ l}).
intro pf.
set (mkSIND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) as cnd.
apply closed2expr in cnd.
exists ξ.
apply ileaf in it.
simpl in it.
- destruct τ.
apply it.
apply INone.
Defined.
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
rewrite <- e.
simpl.
- apply RId.
+ apply AId.
unfold take_arg_types_as_tree.
Opaque take_arg_types_as_tree.
simpl.
destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
simpl.
replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
- apply RCanR.
+ apply ACanR.
apply phoas_extensionality.
reflexivity.
apply (Prelude_error "should not be possible").
destruct (eqd_dec ([tx],,take_arg_types_as_tree te) (take_arg_types_as_tree (tx ---> te))).
rewrite <- e.
simpl.
- apply RId.
+ apply AId.
unfold take_arg_types_as_tree.
Opaque take_arg_types_as_tree.
simpl.
destruct (count_arg_types (te (fun _ : Kind => unit) (ite_unit Γ))).
simpl.
replace (tx) with (fun (TV : Kind → Type) (ite : InstantiatedTypeEnv TV Γ) => tx TV ite).
- apply RuCanR.
+ apply AuCanR.
apply phoas_extensionality.
reflexivity.
apply (Prelude_error "should not be possible").
| 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 _
+ | RCut Γ Δ 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 _
| 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 _
apply nd_rule.
apply SFlat.
apply RArrange.
- apply RRight.
+ apply ARight.
apply d.
destruct case_RBrak.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply RuCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; apply AuCanL ].
apply nd_rule.
apply SFlat.
apply RLit.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
apply nd_rule.
apply SFlat.
apply RVar.
rewrite H.
rewrite H0.
simpl.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AuCanR ].
apply nd_rule.
apply SFlat.
apply RGlobal.
apply nd_rule.
apply SFlat.
apply RArrange.
- eapply RComp.
- eapply RCossa.
- eapply RLeft.
+ eapply AComp.
+ eapply AuAssoc.
+ eapply ALeft.
apply take_arrange.
destruct case_RCast.
rewrite H0.
simpl.
eapply nd_comp.
- eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
+ eapply nd_prod; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
eapply nd_rule.
eapply SFlat.
eapply RArrange.
- eapply RLeft.
+ eapply ALeft.
eapply take_unarrange.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
eapply nd_rule; eapply SFlat; apply RWhere.
destruct case_RLet.
rewrite H0.
eapply nd_comp.
- eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR | eapply nd_id ].
+ eapply nd_prod; [ eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR | eapply nd_id ].
set (@RLet Γ Δ Σ₁ (Σ₂,,(take_arg_types_as_tree σ₂ @@@ (h::lev))) σ₁ (drop_arg_types_as_tree σ₂) (h::lev)) as q.
- eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply RAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; apply SFlat; eapply RArrange; apply AAssoc ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply q ].
apply nd_prod.
apply nd_id.
apply nd_rule.
eapply SFlat.
eapply RArrange.
- eapply RCossa.
+ eapply AuAssoc.
destruct case_RWhere.
simpl.
rewrite H0.
eapply nd_comp.
- eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RCanR ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RAssoc ].
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RLeft; eapply RAssoc ].
+ eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AAssoc ].
eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RWhere ].
apply nd_prod; [ idtac | eapply nd_id ].
eapply nd_rule; apply SFlat; eapply RArrange.
- eapply RComp.
- eapply RCossa.
- apply RLeft.
- eapply RCossa.
+ eapply AComp.
+ eapply AuAssoc.
+ apply ALeft.
+ eapply AuAssoc.
+
+ destruct case_RCut.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RCut | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₃) as Σ₃''.
+ set (mapOptionTree drop_arg_types_as_tree Σ₃) as Σ₃'''.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₁₂) as Σ₁₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ₁₂) as Σ₁₂'''.
+ destruct (decide_tree_empty Σ₁₂''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
+ destruct (eqd_dec Σ₁₂ Σ₁₂'''); [ idtac | apply (Prelude_error "used RCut on a variable with function type") ].
+ rewrite <- e.
+ eapply nd_comp.
+ eapply nd_prod; [ apply nd_id | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RCut ].
+ apply nd_prod.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ACanR ].
+ apply nd_rule.
+ apply SFlat.
+ apply RArrange.
+ apply ALeft.
+ destruct s.
+ eapply arrangeCancelEmptyTree with (q:=x).
+ rewrite e0.
+ admit. (* FIXME, but not serious *)
+ apply nd_id.
+
+ destruct case_RLeft.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RLeft | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
+ set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
+ destruct (decide_tree_empty (Σ' @@@ (h::l)));
+ [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
+ destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RLeft on a variable with function type") ].
+ rewrite <- e.
+ clear Σ'' e.
+ destruct s.
+ set (arrangeUnCancelEmptyTree _ _ e) as q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ARight; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanL; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ apply nd_rule.
+ eapply SFlat.
+ eapply RLeft.
+
+ destruct case_RRight.
+ simpl; destruct l; [ apply nd_rule; apply SFlat; apply RRight | idtac ].
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ₂) as Σ₂'.
+ set (mapOptionTreeAndFlatten take_arg_types_as_tree Σ) as Σ'.
+ set (mapOptionTree drop_arg_types_as_tree Σ₂) as Σ₂''.
+ set (mapOptionTree drop_arg_types_as_tree Σ) as Σ''.
+ destruct (decide_tree_empty (Σ' @@@ (h::l)));
+ [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
+ destruct (eqd_dec Σ Σ''); [ idtac | apply (Prelude_error "used RRight on a variable with function type") ].
+ rewrite <- e.
+ clear Σ'' e.
+ destruct s.
+ set (arrangeUnCancelEmptyTree _ _ e) as q.
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply ALeft; eapply q ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AuCanR ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AAssoc ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply ALeft; eapply AExch ]. (* yuck *)
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuAssoc ].
+ eapply nd_rule.
+ eapply SFlat.
+ apply RRight.
destruct case_RVoid.
simpl.
apply nd_rule.
apply SFlat.
apply RVoid.
- eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply RuCanL ].
+ eapply nd_comp; [ idtac | eapply nd_rule; eapply SFlat; eapply RArrange; eapply AuCanL ].
apply nd_rule.
apply SFlat.
apply RVoid.
}.
Implicit Arguments StrongCaseBranchWithVVs [[Γ]].
- Inductive Expr : forall Γ (Δ:CoercionEnv Γ), (VV -> LeveledHaskType Γ ★) -> LeveledHaskType Γ ★ -> Type :=
+ Inductive Expr : forall Γ (Δ:CoercionEnv Γ), (VV -> LeveledHaskType Γ ★) -> HaskType Γ ★ -> HaskLevel Γ -> Type :=
(* an "EGlobal" is any variable which is free in the expression which was passed to -fcoqpass (ie bound outside it) *)
- | EGlobal: forall Γ Δ ξ (g:Global Γ) v lev, Expr Γ Δ ξ (g v @@ lev)
+ | EGlobal: forall Γ Δ ξ (g:Global Γ) v lev, Expr Γ Δ ξ (g v) lev
- | EVar : ∀ Γ Δ ξ ev, Expr Γ Δ ξ (ξ ev)
- | ELit : ∀ Γ Δ ξ lit l, Expr Γ Δ ξ (literalType lit@@l)
- | EApp : ∀ Γ Δ ξ t1 t2 l, Expr Γ Δ ξ (t2--->t1 @@ l) -> Expr Γ Δ ξ (t2 @@ l) -> Expr Γ Δ ξ (t1 @@ l)
- | ELam : ∀ Γ Δ ξ t1 t2 l ev, Expr Γ Δ (update_xi ξ l ((ev,t1)::nil)) (t2@@l) -> Expr Γ Δ ξ (t1--->t2@@l)
- | ELet : ∀ Γ Δ ξ tv t l ev,Expr Γ Δ ξ (tv@@l)->Expr Γ Δ (update_xi ξ l ((ev,tv)::nil))(t@@l) -> Expr Γ Δ ξ (t@@l)
- | EEsc : ∀ Γ Δ ξ ec t l, Expr Γ Δ ξ (<[ ec |- t ]> @@ l) -> Expr Γ Δ ξ (t @@ (ec::l))
- | EBrak : ∀ Γ Δ ξ ec t l, Expr Γ Δ ξ (t @@ (ec::l)) -> Expr Γ Δ ξ (<[ ec |- t ]> @@ l)
- | ECast : forall Γ Δ ξ t1 t2 (γ:HaskCoercion Γ Δ (t1 ∼∼∼ t2)) l,
- Expr Γ Δ ξ (t1 @@ l) -> Expr Γ Δ ξ (t2 @@ l)
- | ENote : ∀ Γ Δ ξ t, Note -> Expr Γ Δ ξ t -> Expr Γ Δ ξ t
- | ETyApp : ∀ Γ Δ κ σ τ ξ l, Expr Γ Δ ξ (HaskTAll κ σ @@ l) -> Expr Γ Δ ξ (substT σ τ @@ l)
- | ECoLam : forall Γ Δ κ σ (σ₁ σ₂:HaskType Γ κ) ξ l,
- Expr Γ (σ₁∼∼∼σ₂::Δ) ξ (σ @@ l) -> Expr Γ Δ ξ (σ₁∼∼σ₂ ⇒ σ @@ l)
- | ECoApp : forall Γ Δ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ ξ l,
- Expr Γ Δ ξ (σ₁ ∼∼ σ₂ ⇒ σ @@ l) -> Expr Γ Δ ξ (σ @@l)
+ | EVar : ∀ Γ Δ ξ ev, Expr Γ Δ ξ (unlev (ξ ev)) (getlev (ξ ev))
+ | ELit : ∀ Γ Δ ξ lit l, Expr Γ Δ ξ (literalType lit) l
+ | EApp : ∀ Γ Δ ξ t1 t2 l, Expr Γ Δ ξ (t2--->t1) l -> Expr Γ Δ ξ t2 l -> Expr Γ Δ ξ (t1) l
+ | ELam : ∀ Γ Δ ξ t1 t2 l ev, Expr Γ Δ (update_xi ξ l ((ev,t1)::nil)) t2 l -> Expr Γ Δ ξ (t1--->t2) l
+ | ELet : ∀ Γ Δ ξ tv t l ev,Expr Γ Δ ξ tv l ->Expr Γ Δ (update_xi ξ l ((ev,tv)::nil)) t l -> Expr Γ Δ ξ t l
+ | EEsc : ∀ Γ Δ ξ ec t l, Expr Γ Δ ξ (<[ ec |- t ]>) l -> Expr Γ Δ ξ t (ec::l)
+ | EBrak : ∀ Γ Δ ξ ec t l, Expr Γ Δ ξ t (ec::l) -> Expr Γ Δ ξ (<[ ec |- t ]>) l
+ | ECast : forall Γ Δ ξ t1 t2 (γ:HaskCoercion Γ Δ (t1 ∼∼∼ t2)) l, Expr Γ Δ ξ t1 l -> Expr Γ Δ ξ t2 l
+ | ENote : ∀ Γ Δ ξ t l, Note -> Expr Γ Δ ξ t l -> Expr Γ Δ ξ t l
+ | ETyApp : ∀ Γ Δ κ σ τ ξ l, Expr Γ Δ ξ (HaskTAll κ σ) l -> Expr Γ Δ ξ (substT σ τ) l
+ | ECoLam : forall Γ Δ κ σ (σ₁ σ₂:HaskType Γ κ) ξ l, Expr Γ (σ₁∼∼∼σ₂::Δ) ξ σ l -> Expr Γ Δ ξ (σ₁∼∼σ₂ ⇒ σ) l
+ | ECoApp : forall Γ Δ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ ξ l, Expr Γ Δ ξ (σ₁ ∼∼ σ₂ ⇒ σ) l -> Expr Γ Δ ξ σ l
| ETyLam : ∀ Γ Δ ξ κ σ l,
- Expr (κ::Γ) (weakCE Δ) (weakLT○ξ) (HaskTApp (weakF σ) (FreshHaskTyVar _)@@(weakL l))-> Expr Γ Δ ξ (HaskTAll κ σ @@ l)
+ Expr (κ::Γ) (weakCE Δ) (weakLT○ξ) (HaskTApp (weakF σ) (FreshHaskTyVar _)) (weakL l)-> Expr Γ Δ ξ (HaskTAll κ σ) l
| ECase : forall Γ Δ ξ l tc tbranches atypes,
- Expr Γ Δ ξ (caseType tc atypes @@ l) ->
+ Expr Γ Δ ξ (caseType tc atypes) l ->
Tree ??{ sac : _
& { scb : StrongCaseBranchWithVVs tc atypes sac
& Expr (sac_gamma sac Γ)
(sac_delta sac Γ atypes (weakCK'' Δ))
(scbwv_xi scb ξ l)
- (weakLT' (tbranches@@l)) } } ->
- Expr Γ Δ ξ (tbranches @@ l)
+ (weakT' tbranches)
+ (weakL' l) } } ->
+ Expr Γ Δ ξ tbranches l
| ELetRec : ∀ Γ Δ ξ l τ vars,
distinct (leaves (mapOptionTree (@fst _ _) vars)) ->
let ξ' := update_xi ξ l (leaves vars) in
- ELetRecBindings Γ Δ ξ' l vars ->
- Expr Γ Δ ξ' (τ@@l) ->
- Expr Γ Δ ξ (τ@@l)
+ ELetRecBindings Γ Δ ξ' l vars ->
+ Expr Γ Δ ξ' τ l ->
+ Expr Γ Δ ξ τ l
(* can't avoid having an additional inductive: it is a tree of Expr's, each of whose ξ depends on the type of the entire tree *)
with ELetRecBindings : ∀ Γ, CoercionEnv Γ -> (VV -> LeveledHaskType Γ ★) -> HaskLevel Γ -> Tree ??(VV*HaskType Γ ★) -> Type :=
| ELR_nil : ∀ Γ Δ ξ l , ELetRecBindings Γ Δ ξ l []
- | ELR_leaf : ∀ Γ Δ ξ l v t, Expr Γ Δ ξ (t @@ l) -> ELetRecBindings Γ Δ ξ l [(v,t)]
+ | ELR_leaf : ∀ Γ Δ ξ l v t, Expr Γ Δ ξ t l -> ELetRecBindings Γ Δ ξ l [(v,t)]
| ELR_branch : ∀ Γ Δ ξ l t1 t2, ELetRecBindings Γ Δ ξ l t1 -> ELetRecBindings Γ Δ ξ l t2 -> ELetRecBindings Γ Δ ξ l (t1,,t2)
.
Context {ToStringVV:ToString VV}.
Context {ToLatexVV:ToLatex VV}.
- Fixpoint exprToString {Γ}{Δ}{ξ}{τ}(exp:Expr Γ Δ ξ τ) : string :=
+ Fixpoint exprToString {Γ}{Δ}{ξ}{τ}{l}(exp:Expr Γ Δ ξ τ l) : string :=
match exp with
| EVar Γ' _ ξ' ev => "var."+++ toString ev
| EGlobal Γ' _ ξ' g v _ => "global." +++ toString (g:CoreVar)
| EApp Γ' _ _ _ _ _ e1 e2 => "("+++exprToString e1+++")("+++exprToString e2+++")"
| EEsc Γ' _ _ ec t _ e => "~~("+++exprToString e+++")"
| EBrak Γ' _ _ ec t _ e => "<["+++exprToString e+++"]>"
- | ENote _ _ _ _ n e => "note."+++exprToString e
+ | ENote _ _ _ _ n _ e => "note."+++exprToString e
| ETyApp Γ Δ κ σ τ ξ l e => "("+++exprToString e+++")@("+++toString τ+++")"
| ECoApp Γ Δ κ σ₁ σ₂ γ σ ξ l e => "("+++exprToString e+++")~(co)"
| ECast Γ Δ ξ t1 t2 γ l e => "cast ("+++exprToString e+++"):t2"
| ECase Γ Δ ξ l tc branches atypes escrut alts => "case " +++ exprToString escrut +++ " of FIXME"
| ELetRec _ _ _ _ _ vars vu elrb e => "letrec FIXME in " +++ exprToString e
end.
- Instance ExprToString Γ Δ ξ τ : ToString (Expr Γ Δ ξ τ) := { toString := exprToString }.
+ Instance ExprToString Γ Δ ξ τ l : ToString (Expr Γ Δ ξ τ l) := { toString := exprToString }.
End HaskStrong.
Implicit Arguments StrongCaseBranchWithVVs [[Γ]].
Require Import Preamble.
Require Import General.
Require Import NaturalDeduction.
+Require Import NaturalDeductionContext.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
Require Import Coq.Init.Specif.
Require Import HaskProof.
Section HaskStrongToProof.
-
-Definition pivotContext {T} a b c : @Arrange T ((a,,b),,c) ((a,,c),,b) :=
- RComp (RComp (RCossa _ _ _) (RLeft a (RExch c b))) (RAssoc _ _ _).
-
-Definition pivotContext' {T} a b c : @Arrange T (a,,(b,,c)) (b,,(a,,c)) :=
- RComp (RComp (RAssoc _ _ _) (RRight c (RExch b a))) (RCossa _ _ _).
-
-Definition copyAndPivotContext {T} a b c : @Arrange T ((a,,b),,(c,,b)) ((a,,c),,b).
- eapply RComp; [ idtac | apply (RLeft (a,,c) (RCont b)) ].
- eapply RComp; [ idtac | apply RCossa ].
- eapply RComp; [ idtac | apply (RRight b (pivotContext a b c)) ].
- apply RAssoc.
- Defined.
-
+
Context {VV:Type}{eqd_vv:EqDecidable VV}.
(* maintenance of Xi *)
apply q.
Qed.
-Fixpoint expr2antecedent {Γ'}{Δ'}{ξ'}{τ'}(exp:Expr Γ' Δ' ξ' τ') : Tree ??VV :=
- match exp as E in Expr Γ Δ ξ τ with
+Fixpoint expr2antecedent {Γ'}{Δ'}{ξ'}{τ'}{l'}(exp:Expr Γ' Δ' ξ' τ' l') : Tree ??VV :=
+ match exp as E in Expr Γ Δ ξ τ l with
| EGlobal Γ Δ ξ _ _ _ => []
| EVar Γ Δ ξ ev => [ev]
| ELit Γ Δ ξ lit lev => []
| EEsc Γ Δ ξ ec t lev e => expr2antecedent e
| EBrak Γ Δ ξ ec t lev e => expr2antecedent e
| ECast Γ Δ ξ γ t1 t2 lev e => expr2antecedent e
- | ENote Γ Δ ξ t n e => expr2antecedent e
+ | ENote Γ Δ ξ t l n e => expr2antecedent e
| ETyLam Γ Δ ξ κ σ l e => expr2antecedent e
| ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => expr2antecedent e
| ECoApp Γ Δ κ γ σ₁ σ₂ σ ξ l e => expr2antecedent e
& Expr (sac_gamma sac Γ)
(sac_delta sac Γ atypes (weakCK'' Δ))
(scbwv_xi scb ξ l)
- (weakLT' (tbranches@@l)) } }
+ (weakT' tbranches) (weakL' l)} }
): Tree ??VV :=
match alts with
| T_Leaf None => []
& Expr (sac_gamma sac Γ)
(sac_delta sac Γ atypes (weakCK'' Δ))
(scbwv_xi scb ξ l)
- (weakLT' (tbranches@@l)) } })
+ (weakT' tbranches) (weakL' l) } })
: { sac : _ & ProofCaseBranch tc Γ Δ l tbranches atypes sac }.
destruct alt.
exists x.
(* where the leaf is v *)
apply inr.
subst.
- apply RuCanR.
+ apply AuCanR.
(* where the leaf is NOT v *)
apply inl.
- apply RuCanL.
+ apply AuCanL.
(* empty leaf *)
destruct case_None.
apply inl; simpl in *.
- apply RuCanR.
+ apply AuCanR.
(* branch *)
refine (
destruct case_Neither.
apply inl.
simpl.
- eapply RComp; [idtac | apply RuCanL ].
- exact (RComp
+ eapply AComp; [idtac | apply AuCanL ].
+ exact (AComp
(* order will not matter because these are central as morphisms *)
- (RRight _ (RComp lpf (RCanL _)))
- (RLeft _ (RComp rpf (RCanL _)))).
+ (ARight _ (AComp lpf (ACanL _)))
+ (ALeft _ (AComp rpf (ACanL _)))).
destruct case_Right.
apply inr.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
fold (stripOutVars (v::nil)).
- eapply RComp; [ idtac | eapply pivotContext' ].
- eapply RComp.
- eapply RRight.
- eapply RComp.
+ eapply AComp; [ idtac | eapply pivotContext' ].
+ eapply AComp.
+ eapply ARight.
+ eapply AComp.
apply lpf.
- apply RCanL.
- eapply RLeft.
+ apply ACanL.
+ eapply ALeft.
apply rpf.
destruct case_Left.
apply inr.
fold (stripOutVars (v::nil)).
simpl.
- eapply RComp.
- eapply RLeft.
- eapply RComp.
+ eapply AComp.
+ eapply ALeft.
+ eapply AComp.
apply rpf.
simpl.
- eapply RCanL.
- eapply RComp; [ idtac | eapply RCossa ].
- eapply RRight.
+ eapply ACanL.
+ eapply AComp; [ idtac | eapply AuAssoc ].
+ eapply ARight.
apply lpf.
destruct case_Both.
apply inr.
simpl.
- eapply RComp; [ idtac | eapply RRight; eapply RCont ].
- eapply RComp; [ eapply RRight; eapply lpf | idtac ].
- eapply RComp; [ eapply RLeft; eapply rpf | idtac ].
+ eapply AComp; [ idtac | eapply ARight; eapply ACont ].
+ eapply AComp; [ eapply ARight; eapply lpf | idtac ].
+ eapply AComp; [ eapply ALeft; eapply rpf | idtac ].
clear lpf rpf.
simpl.
apply arrangeSwapMiddle.
(* where the leaf is v *)
apply inr.
subst.
- apply RuCanL.
+ apply AuCanL.
(* where the leaf is NOT v *)
apply inl.
- apply RuCanR.
+ apply AuCanR.
(* empty leaf *)
destruct case_None.
apply inl; simpl in *.
- apply RuCanR.
+ apply AuCanR.
(* branch *)
refine (
destruct case_Neither.
apply inl.
- eapply RComp; [idtac | apply RuCanR ].
- exact (RComp
+ eapply AComp; [idtac | apply AuCanR ].
+ exact (AComp
(* order will not matter because these are central as morphisms *)
- (RRight _ (RComp lpf (RCanR _)))
- (RLeft _ (RComp rpf (RCanR _)))).
+ (ARight _ (AComp lpf (ACanR _)))
+ (ALeft _ (AComp rpf (ACanR _)))).
destruct case_Right.
apply inr.
fold (stripOutVars (v::nil)).
- set (RRight (mapOptionTree ξ ctx2) (RComp lpf ((RCanR _)))) as q.
+ set (ARight (mapOptionTree ξ ctx2) (AComp lpf ((ACanR _)))) as q.
simpl in *.
- eapply RComp.
+ eapply AComp.
apply q.
clear q.
clear lpf.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
- eapply RComp; [ idtac | apply RAssoc ].
- apply RLeft.
+ eapply AComp; [ idtac | apply AAssoc ].
+ apply ALeft.
apply rpf.
destruct case_Left.
apply inr.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
fold (stripOutVars (v::nil)).
- eapply RComp; [ idtac | eapply pivotContext ].
- set (RComp rpf (RCanR _ )) as rpf'.
- set (RLeft ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) rpf') as qq.
+ eapply AComp; [ idtac | eapply pivotContext ].
+ set (AComp rpf (ACanR _ )) as rpf'.
+ set (ALeft ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) rpf') as qq.
simpl in *.
- eapply RComp; [ idtac | apply qq ].
+ eapply AComp; [ idtac | apply qq ].
clear qq rpf' rpf.
- apply (RRight (mapOptionTree ξ ctx2)).
+ apply (ARight (mapOptionTree ξ ctx2)).
apply lpf.
destruct case_Both.
apply inr.
unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
fold (stripOutVars (v::nil)).
- eapply RComp; [ idtac | eapply copyAndPivotContext ].
+ eapply AComp; [ idtac | eapply copyAndPivotContext ].
(* order will not matter because these are central as morphisms *)
- exact (RComp (RRight _ lpf) (RLeft _ rpf)).
+ exact (AComp (ARight _ lpf) (ALeft _ rpf)).
Defined.
-(* same as before, but use RWeak if necessary *)
+(* same as before, but use AWeak if necessary *)
Definition factorContextLeftAndWeaken
(Γ:TypeEnv)(Δ:CoercionEnv Γ)
v (* variable to be pivoted, if found *)
(mapOptionTree ξ ([v],,(stripOutVars (v::nil) ctx)) ).
set (factorContextLeft Γ Δ v ctx ξ) as q.
destruct q; auto.
- eapply RComp; [ apply a | idtac ].
- refine (RRight _ (RWeak _)).
+ eapply AComp; [ apply a | idtac ].
+ refine (ARight _ (AWeak _)).
Defined.
Definition factorContextLeftAndWeaken''
unfold mapOptionTree; simpl in *.
intros.
rewrite (@stripping_nothing_is_inert Γ); auto.
- apply RuCanL.
+ apply AuCanL.
intros.
unfold mapOptionTree in *.
simpl in *.
unfold stripOutVars in q.
rewrite q in IHv1'.
clear q.
- eapply RComp; [ idtac | eapply RAssoc ].
- eapply RComp; [ idtac | eapply IHv1' ].
+ eapply AComp; [ idtac | eapply AAssoc ].
+ eapply AComp; [ idtac | eapply IHv1' ].
clear IHv1'.
apply IHv2; auto.
auto.
auto.
Defined.
-(* same as before, but use RWeak if necessary *)
+(* same as before, but use AWeak if necessary *)
Definition factorContextRightAndWeaken
(Γ:TypeEnv)(Δ:CoercionEnv Γ)
v (* variable to be pivoted, if found *)
(mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v]) ).
set (factorContextRight Γ Δ v ctx ξ) as q.
destruct q; auto.
- eapply RComp; [ apply a | idtac ].
- refine (RLeft _ (RWeak _)).
+ eapply AComp; [ apply a | idtac ].
+ refine (ALeft _ (AWeak _)).
Defined.
Definition factorContextRightAndWeaken''
unfold mapOptionTree; simpl in *.
intros.
rewrite (@stripping_nothing_is_inert Γ); auto.
- apply RuCanR.
+ apply AuCanR.
intros.
unfold mapOptionTree in *.
simpl in *.
fold X in IHv2'.
set (distinct_app _ _ _ H) as H'.
destruct H' as [H1 H2].
- set (RComp (IHv1 _ H1) (IHv2' H2)) as qq.
- eapply RComp.
+ set (AComp (IHv1 _ H1) (IHv2' H2)) as qq.
+ eapply AComp.
apply qq.
clear qq IHv2' IHv2 IHv1.
rewrite strip_swap_lemma.
rewrite strip_twice_lemma.
rewrite (notin_strip_inert' v1 (leaves v2)).
- apply RCossa.
+ apply AuAssoc.
apply distinct_swap.
auto.
Defined.
??{sac : StrongAltCon &
{scb : StrongCaseBranchWithVVs VV eqd_vv tc atypes sac &
Expr (sac_gamma sac Γ) (sac_delta sac Γ atypes (weakCK'' Δ))
- (scbwv_xi scb ξ l) (weakLT' (tbranches @@ l))}}),
+ (scbwv_xi scb ξ l) (weakT' tbranches) (weakL' l)}}),
(mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x))
(mapOptionTree mkProofCaseBranch alts'))
Qed.
Definition expr2proof :
- forall Γ Δ ξ τ (e:Expr Γ Δ ξ τ),
- ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [unlev τ] @ getlev τ].
+ forall Γ Δ ξ τ l (e:Expr Γ Δ ξ τ l),
+ ND Rule [] [Γ > Δ > mapOptionTree ξ (expr2antecedent e) |- [τ] @ l].
- refine (fix expr2proof Γ' Δ' ξ' τ' (exp:Expr Γ' Δ' ξ' τ') {struct exp}
- : ND Rule [] [Γ' > Δ' > mapOptionTree ξ' (expr2antecedent exp) |- [unlev τ'] @ getlev τ'] :=
- match exp as E in Expr Γ Δ ξ τ with
+ refine (fix expr2proof Γ' Δ' ξ' τ' l' (exp:Expr Γ' Δ' ξ' τ' l') {struct exp}
+ : ND Rule [] [Γ' > Δ' > mapOptionTree ξ' (expr2antecedent exp) |- [τ'] @ l'] :=
+ match exp as E in Expr Γ Δ ξ τ l with
| EGlobal Γ Δ ξ g v lev => let case_EGlobal := tt in _
| EVar Γ Δ ξ ev => let case_EVar := tt in _
| ELit Γ Δ ξ lit lev => let case_ELit := tt in _
| EApp Γ Δ ξ t1 t2 lev e1 e2 => let case_EApp := tt in
- (fun e1' e2' => _) (expr2proof _ _ _ _ e1) (expr2proof _ _ _ _ e2)
- | ELam Γ Δ ξ t1 t2 lev v e => let case_ELam := tt in (fun e' => _) (expr2proof _ _ _ _ e)
+ (fun e1' e2' => _) (expr2proof _ _ _ _ _ e1) (expr2proof _ _ _ _ _ e2)
+ | ELam Γ Δ ξ t1 t2 lev v e => let case_ELam := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
| ELet Γ Δ ξ tv t v lev ev ebody => let case_ELet := tt in
- (fun pf_let pf_body => _) (expr2proof _ _ _ _ ev) (expr2proof _ _ _ _ ebody)
+ (fun pf_let pf_body => _) (expr2proof _ _ _ _ _ ev) (expr2proof _ _ _ _ _ ebody)
| ELetRec Γ Δ ξ lev t tree disti branches ebody =>
let ξ' := update_xi ξ lev (leaves tree) in
- let case_ELetRec := tt in (fun e' subproofs => _) (expr2proof _ _ _ _ ebody)
+ let case_ELetRec := tt in (fun e' subproofs => _) (expr2proof _ _ _ _ _ ebody)
((fix subproofs Γ'' Δ'' ξ'' lev'' (tree':Tree ??(VV * HaskType Γ'' ★))
(branches':ELetRecBindings Γ'' Δ'' ξ'' lev'' tree')
: LetRecSubproofs Γ'' Δ'' ξ'' lev'' tree' branches' :=
match branches' as B in ELetRecBindings G D X L T return LetRecSubproofs G D X L T B with
| ELR_nil Γ Δ ξ lev => lrsp_nil _ _ _ _
- | ELR_leaf Γ Δ ξ l v t e => lrsp_leaf Γ Δ ξ l v t e (expr2proof _ _ _ _ e)
+ | ELR_leaf Γ Δ ξ l v t e => lrsp_leaf Γ Δ ξ l v t e (expr2proof _ _ _ _ _ e)
| ELR_branch Γ Δ ξ lev t1 t2 b1 b2 => lrsp_cons _ _ _ _ _ _ _ _ (subproofs _ _ _ _ _ b1) (subproofs _ _ _ _ _ b2)
end
) _ _ _ _ tree branches)
- | EEsc Γ Δ ξ ec t lev e => let case_EEsc := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | EBrak Γ Δ ξ ec t lev e => let case_EBrak := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ECast Γ Δ ξ γ t1 t2 lev e => let case_ECast := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ENote Γ Δ ξ t n e => let case_ENote := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ETyLam Γ Δ ξ κ σ l e => let case_ETyLam := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => let case_ECoLam := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ECoApp Γ Δ κ σ₁ σ₂ σ γ ξ l e => let case_ECoApp := tt in (fun e' => _) (expr2proof _ _ _ _ e)
- | ETyApp Γ Δ κ σ τ ξ l e => let case_ETyApp := tt in (fun e' => _) (expr2proof _ _ _ _ e)
+ | EEsc Γ Δ ξ ec t lev e => let case_EEsc := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | EBrak Γ Δ ξ ec t lev e => let case_EBrak := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ECast Γ Δ ξ γ t1 t2 lev e => let case_ECast := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ENote Γ Δ ξ t _ n e => let case_ENote := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ETyLam Γ Δ ξ κ σ l e => let case_ETyLam := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => let case_ECoLam := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ECoApp Γ Δ κ σ₁ σ₂ σ γ ξ l e => let case_ECoApp := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
+ | ETyApp Γ Δ κ σ τ ξ l e => let case_ETyApp := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
| ECase Γ Δ ξ l tc tbranches atypes e alts' =>
let dcsp :=
((fix mkdcsp (alts:
& Expr (sac_gamma sac Γ)
(sac_delta sac Γ atypes (weakCK'' Δ))
(scbwv_xi scb ξ l)
- (weakLT' (tbranches@@l)) } })
+ (weakT' tbranches) (weakL' l) } })
: ND Rule [] (mapOptionTree (fun x => pcb_judg (projT2 (mkProofCaseBranch x))) alts) :=
match alts as ALTS return ND Rule []
(mapOptionTree (fun x => pcb_judg (projT2 (mkProofCaseBranch x))) ALTS) with
| T_Leaf (Some x) =>
match x as X return ND Rule [] [pcb_judg (projT2 (mkProofCaseBranch X))] with
existT sac (existT scbx ex) =>
- (fun e' => let case_leaf := tt in _) (expr2proof _ _ _ _ ex)
+ (fun e' => let case_leaf := tt in _) (expr2proof _ _ _ _ _ ex)
end
end) alts')
- in let case_ECase := tt in (fun e' => _) (expr2proof _ _ _ _ e)
+ in let case_ECase := tt in (fun e' => _) (expr2proof _ _ _ _ _ e)
end
- ); clear exp ξ' τ' Γ' Δ' expr2proof; try clear mkdcsp.
+ ); clear exp ξ' τ' Γ' Δ' l' expr2proof; try clear mkdcsp.
destruct case_EGlobal.
apply nd_rule.
auto.
destruct case_ENote.
- destruct t.
eapply nd_comp; [ idtac | eapply nd_rule; apply RNote ].
apply e'.
auto.
| (vv,wev)::rest => update_chi (update_chi' χ rest) vv wev
end.
- Fixpoint exprToWeakExpr {Γ}{Δ}{ξ}{τ}(χ:VV->???WeakExprVar)(exp:@Expr _ eqVV Γ Δ ξ τ)
+ Fixpoint exprToWeakExpr {Γ}{Δ}{ξ}{τ}{l}(χ:VV->???WeakExprVar)(exp:@Expr _ eqVV Γ Δ ξ τ l)
: InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ
-> UniqM WeakExpr :=
- match exp as E in @Expr _ _ G D X L return InstantiatedTypeEnv (fun _ => WeakTypeVar) G -> UniqM WeakExpr with
+ match exp as E in @Expr _ _ G D X T L return InstantiatedTypeEnv (fun _ => WeakTypeVar) G -> UniqM WeakExpr with
| EVar Γ' _ ξ' ev => fun ite => match χ ev with OK v => return WEVar v | Error s => failM s end
| EGlobal Γ' _ ξ' g v lev => fun ite => bind tv' = mapM (ilist_to_list (ilmap (fun κ x => typeToWeakType x ite) v))
; return (fold_left (fun x y => WETyApp x y) tv' (WEVar g))
| EBrak Γ' _ _ ec t _ e => fun ite => bind t' = typeToWeakType t ite
; bind e' = exprToWeakExpr χ e ite
; return WEBrak hetmet_brak (ec _ ite) e' t'
- | ENote _ _ _ _ n e => fun ite => bind e' = exprToWeakExpr χ e ite
+ | ENote _ _ _ _ _ n e => fun ite => bind e' = exprToWeakExpr χ e ite
; return WENote n e'
| ETyApp Γ Δ κ σ τ ξ l e => fun ite => bind t' = typeToWeakType τ ite
; bind e' = exprToWeakExpr χ e ite
; bind tbranches' = @typeToWeakType Γ _ tbranches ite
; bind escrut' = exprToWeakExpr χ escrut ite
; bind branches' =
- ((fix caseBranches (tree:Tree ??{sac : _ & { scb : StrongCaseBranchWithVVs VV _ _ _ sac & Expr _ _ _ _ } })
+ ((fix caseBranches (tree:Tree ??{sac : _ & { scb : StrongCaseBranchWithVVs VV _ _ _ sac & Expr _ _ _ _ _ } })
: UniqM (Tree ??(WeakAltCon*list WeakTypeVar*list WeakCoerVar*list WeakExprVar*WeakExpr)) :=
match tree with
| T_Leaf None => return []
end.
- Fixpoint strongExprToWeakExpr (us:UniqSupply){Γ}{Δ}{ξ}{τ}(exp:@Expr _ eqVV Γ Δ ξ τ)
+ Fixpoint strongExprToWeakExpr (us:UniqSupply){Γ}{Δ}{ξ}{τ}{l}(exp:@Expr _ eqVV Γ Δ ξ τ l)
(ite:InstantiatedTypeEnv (fun _ => WeakTypeVar) Γ)
: ???WeakExpr :=
match exprToWeakExpr (fun v => Error ("unbound variable " +++ toString v)) exp ite with
(* un-letrec-ify multi branch letrecs *)
| WELetRec mlr e => WELetRec mlr (simplifyWeakExpr e )
end.
-*)
\ No newline at end of file
+*)
Defined.
(* attempt to "cast" an expression by simply checking if it already had the desired type, and failing otherwise *)
-Definition castExpr (we:WeakExpr)(err_msg:string) {Γ} {Δ} {ξ} {τ} τ' (e:@Expr _ CoreVarEqDecidable Γ Δ ξ τ)
- : ???(@Expr _ CoreVarEqDecidable Γ Δ ξ τ').
+Definition castExpr (we:WeakExpr)(err_msg:string) {Γ} {Δ} {ξ} {τ} {l} τ' l' (e:@Expr _ CoreVarEqDecidable Γ Δ ξ τ l)
+ : ???(@Expr _ CoreVarEqDecidable Γ Δ ξ τ' l').
apply (addErrorMessage ("castExpr " +++ err_msg)).
intros.
- destruct τ as [τ l].
- destruct τ' as [τ' l'].
destruct (eqd_dec l l'); [ idtac
| apply (Error ("level mismatch in castExpr, invoked by "+++err_msg+++eol+++
" got: " +++(fold_left (fun x y => y+++","+++y) (map (toString ○ haskTyVarToType) l) "")+++eol+++
(ig:CoreVar -> bool)
(τ:HaskType Γ ★)
(lev:HaskLevel Γ),
- WeakExpr -> ???(@Expr _ CoreVarEqDecidable Γ Δ ξ (τ @@ lev) ).
+ WeakExpr -> ???(@Expr _ CoreVarEqDecidable Γ Δ ξ τ lev ).
refine ((
fix weakExprToStrongExpr
(Γ:TypeEnv)
(ig:CoreVar -> bool)
(τ:HaskType Γ ★)
(lev:HaskLevel Γ)
- (we:WeakExpr) : ???(@Expr _ CoreVarEqDecidable Γ Δ ξ (τ @@ lev) ) :=
+ (we:WeakExpr) : ???(@Expr _ CoreVarEqDecidable Γ Δ ξ τ lev ) :=
addErrorMessage ("in weakExprToStrongExpr " +++ toString we)
match we with
| WEVar v => if ig v
- then OK ((EGlobal Γ Δ ξ (mkGlobal Γ τ v) INil lev) : Expr Γ Δ ξ (τ@@lev))
- else castExpr we ("WEVar "+++toString (v:CoreVar)) (τ @@ lev) (EVar Γ Δ ξ v)
+ then OK ((EGlobal Γ Δ ξ (mkGlobal Γ τ v) INil lev) : Expr Γ Δ ξ τ lev)
+ else castExpr we ("WEVar "+++toString (v:CoreVar)) τ lev (EVar Γ Δ ξ v)
- | WELit lit => castExpr we ("WELit "+++toString lit) (τ @@ lev) (ELit Γ Δ ξ lit lev)
+ | WELit lit => castExpr we ("WELit "+++toString lit) τ lev (ELit Γ Δ ξ lit lev)
| WELam ev ebody => weakTypeToTypeOfKind φ ev ★ >>= fun tv =>
weakTypeOfWeakExpr ebody >>= fun tbody =>
let ξ' := update_xi ξ lev (((ev:CoreVar),tv)::nil) in
let ig' := update_ig ig ((ev:CoreVar)::nil) in
weakExprToStrongExpr Γ Δ φ ψ ξ' ig' tbody' lev ebody >>= fun ebody' =>
- castExpr we "WELam" (τ@@lev) (ELam Γ Δ ξ tv tbody' lev ev ebody')
+ castExpr we "WELam" τ lev (ELam Γ Δ ξ tv tbody' lev ev ebody')
| WEBrak _ ec e tbody => φ (`ec) >>= fun ec' =>
weakTypeToTypeOfKind φ tbody ★ >>= fun tbody' =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig tbody' ((ec')::lev) e >>= fun e' =>
- castExpr we "WEBrak" (τ@@lev) (EBrak Γ Δ ξ ec' tbody' lev e')
+ castExpr we "WEBrak" τ lev (EBrak Γ Δ ξ ec' tbody' lev e')
| WEEsc _ ec e tbody => φ ec >>= fun ec'' =>
weakTypeToTypeOfKind φ tbody ★ >>= fun tbody' =>
match lev with
| nil => Error "ill-leveled escapification"
| ec'::lev' => weakExprToStrongExpr Γ Δ φ ψ ξ ig (<[ ec' |- tbody' ]>) lev' e
- >>= fun e' => castExpr we "WEEsc" (τ@@lev) (EEsc Γ Δ ξ ec' tbody' lev' e')
+ >>= fun e' => castExpr we "WEEsc" τ lev (EEsc Γ Δ ξ ec' tbody' lev' e')
end
| WECSP _ ec e tbody => Error "FIXME: CSP not supported beyond HaskWeak stage"
- | WENote n e => weakExprToStrongExpr Γ Δ φ ψ ξ ig τ lev e >>= fun e' => OK (ENote _ _ _ _ n e')
+ | WENote n e => weakExprToStrongExpr Γ Δ φ ψ ξ ig τ lev e >>= fun e' => OK (ENote _ _ _ _ _ n e')
| WELet v ve ebody => weakTypeToTypeOfKind φ v ★ >>= fun tv =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig tv lev ve >>= fun ve' =>
weakTypeToTypeOfKind φ2 te ★ >>= fun τ' =>
weakExprToStrongExpr _ (weakCE Δ) φ2
(fun x => (ψ x) >>= fun y => OK (weakCV y)) (weakLT○ξ) ig _ (weakL lev) e
- >>= fun e' => castExpr we "WETyLam2" _ (ETyLam Γ Δ ξ tv (mkTAll' τ') lev e')
+ >>= fun e' => castExpr we "WETyLam2" _ _ (ETyLam Γ Δ ξ tv (mkTAll' τ') lev e')
| WETyApp e t => weakTypeOfWeakExpr e >>= fun te =>
match te with
weakTypeToTypeOfKind φ2 te' ★ >>= fun te'' =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig (mkTAll te'') lev e >>= fun e' =>
weakTypeToTypeOfKind φ t (wtv:Kind) >>= fun t' =>
- castExpr we "WETyApp" _ (ETyApp Γ Δ wtv (mkTAll' te'') t' ξ lev e')
+ castExpr we "WETyApp" _ _ (ETyApp Γ Δ wtv (mkTAll' te'') t' ξ lev e')
| _ => Error ("weakTypeToType: WETyApp body with type "+++toString te)
end
weakTypeToTypeOfKind φ t2 κ >>= fun t2'' =>
weakTypeToTypeOfKind φ t3 ★ >>= fun t3'' =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig (t1'' ∼∼ t2'' ⇒ τ) lev e >>= fun e' =>
- castExpr we "WECoApp" _ e' >>= fun e'' =>
+ castExpr we "WECoApp" _ _ e' >>= fun e'' =>
OK (ECoApp Γ Δ κ t1'' t2''
(weakCoercionToHaskCoercion _ _ _ co) τ ξ lev e'')
end
weakTypeToTypeOfKind φ t1 cv >>= fun t1' =>
weakTypeToTypeOfKind φ t2 cv >>= fun t2' =>
weakExprToStrongExpr Γ (_ :: Δ) φ (weakPsi ψ) ξ ig te' lev e >>= fun e' =>
- castExpr we "WECoLam" _ (ECoLam Γ Δ cv te' t1' t2' ξ lev e')
+ castExpr we "WECoLam" _ _ (ECoLam Γ Δ cv te' t1' t2' ξ lev e')
| WECast e co => let (t1,t2) := weakCoercionTypes co in
weakTypeToTypeOfKind φ t1 ★ >>= fun t1' =>
weakTypeToTypeOfKind φ t2 ★ >>= fun t2' =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig t1' lev e >>= fun e' =>
- castExpr we "WECast" _
+ castExpr we "WECast" _ _
(ECast Γ Δ ξ t1' t2' (weakCoercionToHaskCoercion _ _ _ co) lev e')
| WELetRec rb e =>
weakTypeToTypeOfKind φ tbranches ★ >>= fun tbranches' =>
(fix mkTree (t:Tree ??(WeakAltCon*list WeakTypeVar*list WeakCoerVar*list WeakExprVar*WeakExpr)) : ???(Tree
??{ sac : _ & {scb : StrongCaseBranchWithVVs CoreVar CoreVarEqDecidable tc avars' sac &
- Expr (sac_gamma sac Γ) (sac_delta sac Γ avars' (weakCK'' Δ))(scbwv_xi scb ξ lev)(weakLT' (tbranches' @@ lev))}}) :=
+ Expr (sac_gamma sac Γ) (sac_delta sac Γ avars' (weakCK'' Δ))(scbwv_xi scb ξ lev)(weakT' tbranches')(weakL' lev)}}) :=
match t with
| T_Leaf None => OK []
| T_Leaf (Some (ac,extyvars,coervars,exprvars,ebranch)) =>
end) alts >>= fun tree =>
weakExprToStrongExpr Γ Δ φ ψ ξ ig (caseType tc avars') lev escrut >>= fun escrut' =>
- castExpr we "ECase" (τ@@lev) (ECase Γ Δ ξ lev tc tbranches' avars' escrut' tree)
+ castExpr we "ECase" τ lev (ECase Γ Δ ξ lev tc tbranches' avars' escrut' tree)
end)); try clear binds; try apply ConcatenableString.
destruct case_some.
--- /dev/null
+(*********************************************************************************************************************************)
+(* NaturalDeductionContext: *)
+(* *)
+(* Manipulations of a context in natural deduction proofs. *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+Require Import NaturalDeduction.
+
+Section NaturalDeductionContext.
+
+ (* Figure 3, production $\vdash_E$, Uniform rules *)
+ Inductive Arrange {T} : Tree ??T -> Tree ??T -> Type :=
+ | AId : forall a , Arrange a a
+ | ACanL : forall a , Arrange ( [],,a ) ( a )
+ | ACanR : forall a , Arrange ( a,,[] ) ( a )
+ | AuCanL : forall a , Arrange ( a ) ( [],,a )
+ | AuCanR : forall a , Arrange ( a ) ( a,,[] )
+ | AAssoc : forall a b c , Arrange (a,,(b,,c) ) ((a,,b),,c )
+ | AuAssoc : forall a b c , Arrange ((a,,b),,c ) ( a,,(b,,c) )
+ | AExch : forall a b , Arrange ( (b,,a) ) ( (a,,b) )
+ | AWeak : forall a , Arrange ( [] ) ( a )
+ | ACont : forall a , Arrange ( (a,,a) ) ( a )
+ | ALeft : forall {h}{c} x , Arrange h c -> Arrange ( x,,h ) ( x,,c)
+ | ARight : forall {h}{c} x , Arrange h c -> Arrange ( h,,x ) ( c,,x)
+ | AComp : forall {a}{b}{c}, Arrange a b -> Arrange b c -> Arrange a c
+ .
+
+ (* "Arrange" objects are parametric in the type of the leaves of the tree *)
+ Definition arrangeMap :
+ forall {T} (Σ₁ Σ₂:Tree ??T) {R} (f:T -> R),
+ Arrange Σ₁ Σ₂ ->
+ Arrange (mapOptionTree f Σ₁) (mapOptionTree f Σ₂).
+ intros.
+ induction X; simpl.
+ apply AId.
+ apply ACanL.
+ apply ACanR.
+ apply AuCanL.
+ apply AuCanR.
+ apply AAssoc.
+ apply AuAssoc.
+ apply AExch.
+ apply AWeak.
+ apply ACont.
+ apply ALeft; auto.
+ apply ARight; auto.
+ eapply AComp; [ apply IHX1 | apply IHX2 ].
+ Defined.
+
+ (* a frequently-used Arrange - swap the middle two elements of a four-element sequence *)
+ Definition arrangeSwapMiddle {T} (a b c d:Tree ??T) :
+ Arrange ((a,,b),,(c,,d)) ((a,,c),,(b,,d)).
+ eapply AComp.
+ apply AuAssoc.
+ eapply AComp.
+ eapply ALeft.
+ eapply AComp.
+ eapply AAssoc.
+ eapply ARight.
+ apply AExch.
+ eapply AComp.
+ eapply ALeft.
+ eapply AuAssoc.
+ eapply AAssoc.
+ Defined.
+
+ (* like AExch, but works on nodes which are an Assoc away from being adjacent *)
+ Definition pivotContext {T} a b c : @Arrange T ((a,,b),,c) ((a,,c),,b) :=
+ AComp (AComp (AuAssoc _ _ _) (ALeft a (AExch c b))) (AAssoc _ _ _).
+
+ (* like AExch, but works on nodes which are an Assoc away from being adjacent *)
+ Definition pivotContext' {T} a b c : @Arrange T (a,,(b,,c)) (b,,(a,,c)) :=
+ AComp (AComp (AAssoc _ _ _) (ARight c (AExch b a))) (AuAssoc _ _ _).
+
+ Definition copyAndPivotContext {T} a b c : @Arrange T ((a,,b),,(c,,b)) ((a,,c),,b).
+ eapply AComp; [ idtac | apply (ALeft (a,,c) (ACont b)) ].
+ eapply AComp; [ idtac | apply AuAssoc ].
+ eapply AComp; [ idtac | apply (ARight b (pivotContext a b c)) ].
+ apply AAssoc.
+ Defined.
+
+ (* given any set of TreeFlags on a tree, we can Arrange all of the flagged nodes into the left subtree *)
+ Definition arrangePartition :
+ 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 AuCanL.
+ apply AuCanR.
+ simpl.
+ apply AuCanL.
+ simpl in *.
+ eapply AComp; [ idtac | apply arrangeSwapMiddle ].
+ eapply AComp.
+ eapply ALeft.
+ apply IHΣ2.
+ eapply ARight.
+ apply IHΣ1.
+ Defined.
+
+ (* inverse of arrangePartition *)
+ Definition arrangeUnPartition :
+ 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 ACanL.
+ apply ACanR.
+ simpl.
+ apply ACanL.
+ simpl in *.
+ eapply AComp; [ apply arrangeSwapMiddle | idtac ].
+ eapply AComp.
+ eapply ALeft.
+ apply IHΣ2.
+ eapply ARight.
+ apply IHΣ1.
+ Defined.
+
+ (* we can decide if a tree consists exclusively of (T_Leaf None)'s *)
+ 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.
+
+ (* if a tree is empty, we can Arrange it to [] *)
+ Definition arrangeCancelEmptyTree : 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 AId.
+ simpl in *.
+ destruct t; try destruct o; inversion H.
+ set (IHq1 _ H1) as x1.
+ set (IHq2 _ H2) as x2.
+ eapply AComp.
+ eapply ARight.
+ rewrite <- H1.
+ apply x1.
+ eapply AComp.
+ apply ACanL.
+ rewrite <- H2.
+ apply x2.
+ Defined.
+
+ (* if a tree is empty, we can Arrange it from [] *)
+ Definition arrangeUnCancelEmptyTree : 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 AId.
+ simpl in *.
+ destruct t; try destruct o; inversion H.
+ set (IHq1 _ H1) as x1.
+ set (IHq2 _ H2) as x2.
+ eapply AComp.
+ apply AuCanL.
+ eapply AComp.
+ eapply ARight.
+ apply x1.
+ eapply AComp.
+ eapply ALeft.
+ apply x2.
+ rewrite H.
+ apply AId.
+ Defined.
+
+ (* given an Arrange from Σ₁ to Σ₂ and any predicate on tree nodes, we can construct an Arrange from (dropT Σ₁) to (dropT Σ₂) *)
+ Lemma arrangeDrop {T} pred
+ : forall (Σ₁ Σ₂: Tree ??T), Arrange Σ₁ Σ₂ -> Arrange (dropT (mkFlags pred Σ₁)) (dropT (mkFlags pred Σ₂)).
+
+ refine ((fix arrangeTake t1 t2 (arr:Arrange t1 t2) :=
+ match arr as R in Arrange A B return Arrange (dropT (mkFlags pred A)) (dropT (mkFlags pred B)) with
+ | AId a => let case_AId := tt in AId _
+ | ACanL a => let case_ACanL := tt in _
+ | ACanR a => let case_ACanR := tt in _
+ | AuCanL a => let case_AuCanL := tt in _
+ | AuCanR a => let case_AuCanR := tt in _
+ | 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 _
+ | ACont a => let case_ACont := tt in _
+ | ALeft a b c r' => let case_ALeft := tt in ALeft _ (arrangeTake _ _ r')
+ | ARight a b c r' => let case_ARight := tt in ARight _ (arrangeTake _ _ r')
+ | AComp a b c r1 r2 => let case_AComp := tt in AComp (arrangeTake _ _ r1) (arrangeTake _ _ r2)
+ end)); clear arrangeTake; intros.
+
+ destruct case_ACanL.
+ simpl; destruct (pred None); simpl; apply ACanL.
+
+ destruct case_ACanR.
+ simpl; destruct (pred None); simpl; apply ACanR.
+
+ destruct case_AuCanL.
+ simpl; destruct (pred None); simpl; apply AuCanL.
+
+ destruct case_AuCanR.
+ simpl; destruct (pred None); simpl; apply AuCanR.
+
+ destruct case_AWeak.
+ simpl; destruct (pred None); simpl; apply AWeak.
+
+ destruct case_ACont.
+ simpl; destruct (pred None); simpl; apply ACont.
+
+ Defined.
+
+ (* given an Arrange from Σ₁ to Σ₂ and any predicate on tree nodes, we can construct an Arrange from (takeT Σ₁) to (takeT Σ₂) *)
+ (*
+ Lemma arrangeTake {T} pred
+ : forall (Σ₁ Σ₂: Tree ??T), Arrange Σ₁ Σ₂ -> Arrange (takeT' (mkFlags pred Σ₁)) (takeT' (mkFlags pred Σ₂)).
+ unfold takeT'.
+ *)
+
+End NaturalDeductionContext.
eapply nd_prod.
apply nd_id.
apply (PCF_Arrange [h] ([],,[h]) [h0]).
- apply RuCanL.
- eapply nd_comp; [ idtac | apply (PCF_Arrange ([],,a) a [h0]); apply RCanL ].
+ apply AuCanL.
+ eapply nd_comp; [ idtac | apply (PCF_Arrange ([],,a) a [h0]); apply ACanL ].
apply nd_rule.
(*
set (@RLet Γ Δ [] (a@@@(ec::nil)) h0 h (ec::nil)) as q.
; cnd_expand_right := fun a b c => PCF_right Γ Δ lev c a b }.
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
+ exists (RArrange _ _ _ _ _ (AuAssoc _ _ _)).
apply (PCF_RArrange _ _ lev ((a,,b),,c) (a,,(b,,c)) x).
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
+ exists (RArrange _ _ _ _ _ (AAssoc _ _ _)).
apply (PCF_RArrange _ _ lev (a,,(b,,c)) ((a,,b),,c) x).
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCanL _)).
+ exists (RArrange _ _ _ _ _ (ACanL _)).
apply (PCF_RArrange _ _ lev ([],,a) _ _).
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RCanR _)).
+ exists (RArrange _ _ _ _ _ (ACanR _)).
apply (PCF_RArrange _ _ lev (a,,[]) _ _).
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RuCanL _)).
+ exists (RArrange _ _ _ _ _ (AuCanL _)).
apply (PCF_RArrange _ _ lev _ ([],,a) _).
intros; apply nd_rule. unfold PCFRule. simpl.
- exists (RArrange _ _ _ _ _ (RuCanR _)).
+ exists (RArrange _ _ _ _ _ (AuCanR _)).
apply (PCF_RArrange _ _ lev _ (a,,[]) _).
Defined.
projects / coq-hetmet.git / commitdiff