Require Import NaturalDeduction.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
-Require Import HaskGeneral.
+Require Import HaskKinds.
+Require Import HaskCoreTypes.
Require Import HaskLiterals.
+Require Import HaskTyCons.
Require Import HaskStrongTypes.
+Require Import HaskWeakVars.
(* A judgment consists of an environment shape (Γ and Δ) and a pair of trees of leveled types (the antecedent and succedent) valid
* in any context of that shape. Notice that the succedent contains a tree of types rather than a single type; think
mkJudg :
forall Γ:TypeEnv,
forall Δ:CoercionEnv Γ,
- Tree ??(LeveledHaskType Γ) ->
- Tree ??(LeveledHaskType Γ) ->
+ Tree ??(LeveledHaskType Γ ★) ->
+ Tree ??(LeveledHaskType Γ ★) ->
Judg.
Notation "Γ > Δ > a '|-' s" := (mkJudg Γ Δ a s) (at level 52, Δ at level 50, a at level 52, s at level 50).
-(* A Uniform Judgment (UJudg) has its environment as a type index; we'll use these to distinguish proofs that have a single,
- * uniform context throughout the whole proof. Such proofs are important because (1) we can do left and right context
- * expansion on them (see rules RLeft and RRight) and (2) they will form the fiber categories of our fibration later on *)
-Inductive UJudg (Γ:TypeEnv)(Δ:CoercionEnv Γ) :=
- mkUJudg :
- Tree ??(LeveledHaskType Γ) ->
- Tree ??(LeveledHaskType Γ) ->
- UJudg Γ Δ.
- Notation "Γ >> Δ > a '|-' s" := (mkUJudg Γ Δ a s) (at level 52, Δ at level 50, a at level 52, s at level 50).
- Notation "'ext_tree_left'" := (fun ctx j => match j with mkUJudg t s => mkUJudg _ _ (ctx,,t) s end).
- Notation "'ext_tree_right'" := (fun ctx j => match j with mkUJudg t s => mkUJudg _ _ (t,,ctx) s end).
-
-(* we can turn a UJudg into a Judg by simply internalizing the index *)
-Definition UJudg2judg {Γ}{Δ}(ej:@UJudg Γ Δ) : Judg :=
- match ej with mkUJudg t s => Γ > Δ > t |- s end.
- Coercion UJudg2judg : UJudg >-> Judg.
-
(* information needed to define a case branch in a HaskProof *)
-Record ProofCaseBranch {n}{tc:TyCon n}{Γ}{lev}{branchtype : HaskType Γ}{avars} :=
-{ pcb_scb : @StrongCaseBranch n tc Γ avars
-; pcb_freevars : Tree ??(LeveledHaskType Γ)
-; pcb_judg := scb_Γ pcb_scb > scb_Δ pcb_scb
- > (mapOptionTree weakLT' pcb_freevars),,(unleaves (vec2list (scb_types pcb_scb)))
- |- [weakLT' (branchtype @@ lev)]
+Record ProofCaseBranch {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc} :=
+{ pcb_freevars : Tree ??(LeveledHaskType Γ ★)
+; pcb_judg := sac_Γ sac Γ > sac_Δ sac Γ avars (map weakCK' Δ)
+ > (mapOptionTree weakLT' pcb_freevars),,(unleaves (map (fun t => t@@weakL' lev)
+ (vec2list (sac_types sac Γ avars))))
+ |- [weakLT' (branchtype @@ lev)]
}.
-Coercion pcb_scb : ProofCaseBranch >-> StrongCaseBranch.
Implicit Arguments ProofCaseBranch [ ].
(* Figure 3, production $\vdash_E$, Uniform rules *)
-Inductive URule {Γ}{Δ} : Tree ??(UJudg Γ Δ) -> Tree ??(UJudg Γ Δ) -> Type :=
-| RCanL : ∀ t a , URule [Γ>>Δ> [],,a |- t ] [Γ>>Δ> a |- t ]
-| RCanR : ∀ t a , URule [Γ>>Δ> a,,[] |- t ] [Γ>>Δ> a |- t ]
-| RuCanL : ∀ t a , URule [Γ>>Δ> a |- t ] [Γ>>Δ> [],,a |- t ]
-| RuCanR : ∀ t a , URule [Γ>>Δ> a |- t ] [Γ>>Δ> a,,[] |- t ]
-| RAssoc : ∀ t a b c , URule [Γ>>Δ>a,,(b,,c) |- t ] [Γ>>Δ>(a,,b),,c |- t ]
-| RCossa : ∀ t a b c , URule [Γ>>Δ>(a,,b),,c |- t ] [Γ>>Δ> a,,(b,,c) |- t ]
-| RLeft : ∀ h c x , URule h c -> URule (mapOptionTree (ext_tree_left x) h) (mapOptionTree (ext_tree_left x) c)
-| RRight : ∀ h c x , URule h c -> URule (mapOptionTree (ext_tree_right x) h) (mapOptionTree (ext_tree_right x) c)
-| RExch : ∀ t a b , URule [Γ>>Δ> (b,,a) |- t ] [Γ>>Δ> (a,,b) |- t ]
-| RWeak : ∀ t a , URule [Γ>>Δ> [] |- t ] [Γ>>Δ> a |- t ]
-| RCont : ∀ t a , URule [Γ>>Δ> (a,,a) |- t ] [Γ>>Δ> a |- t ].
-
+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 :=
-| RURule : ∀ Γ Δ h c, @URule Γ Δ h c -> Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c)
+| RArrange : ∀ Γ Δ Σ₁ Σ₂ Σ, Arrange Σ₁ Σ₂ -> Rule [Γ > Δ > Σ₁ |- Σ ] [Γ > Δ > Σ₂ |- Σ ]
(* λ^α rules *)
| RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t @@ (v::l) ]] [Γ > Δ > Σ |- [<[v|-t]> @@l]]
-| REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> @@ l]] [Γ > Δ > Σ |- [t @@ (v::l) ]]
+| REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> @@ l]] [Γ > Δ > Σ |- [t @@ (v::l)]]
(* Part of GHC, but not explicitly in System FC *)
-| RNote : ∀ h c, Note -> Rule h [ c ]
-| RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v @@ l]]
+| RNote : ∀ Γ Δ Σ τ l, Note -> Rule [Γ > Δ > Σ |- [τ @@ l]] [Γ > Δ > Σ |- [τ @@l]]
+| RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v @@l]]
(* SystemFC rules *)
| RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ @@l]]
-| RLam : ∀ Γ Δ Σ tx te l, Γ ⊢ᴛy tx : ★ -> Rule [Γ>Δ> Σ,,[tx@@l]|- [te@@l] ] [Γ>Δ> Σ |- [tx--->te @@l]]
-| RCast : ∀ Γ Δ Σ σ τ γ l, Δ ⊢ᴄᴏ γ : σ ∼ τ -> Rule [Γ>Δ> Σ |- [σ@@l] ] [Γ>Δ> Σ |- [τ @@l]]
-| RBindingGroup : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ , Rule ([Γ > Δ > Σ₁ |- τ₁ ],,[Γ > Δ > Σ₂ |- τ₂ ]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ ]
-| 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,,[σ₂@@l] |- [σ₁@@l] ],,[Γ>Δ> Σ₂ |- [σ₂@@l]]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₁ @@l]]
-| REmptyGroup : ∀ Γ Δ , Rule [] [Γ > Δ > [] |- [] ]
-| RAppT : ∀ Γ Δ Σ κ σ τ l, Γ ⊢ᴛy τ : κ -> Rule [Γ>Δ> Σ |- [HaskTAll κ σ @@l]] [Γ>Δ> Σ |- [substT σ τ @@l]]
+| RGlobal : forall Γ Δ l (g:Global Γ) v, Rule [ ] [Γ>Δ> [] |- [g v @@l]]
+| RLam : forall Γ Δ Σ (tx:HaskType Γ ★) te l, Rule [Γ>Δ> Σ,,[tx@@l]|- [te@@l] ] [Γ>Δ> Σ |- [tx--->te @@l]]
+| RCast : forall Γ Δ Σ (σ₁ σ₂:HaskType Γ ★) l,
+ HaskCoercion Γ Δ (σ₁∼∼∼σ₂) -> Rule [Γ>Δ> Σ |- [σ₁@@l] ] [Γ>Δ> Σ |- [σ₂ @@l]]
+
+| RJoin : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ , Rule ([Γ > Δ > Σ₁ |- τ₁ ],,[Γ > Δ > Σ₂ |- τ₂ ]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ ]
+
+| RApp : ∀ Γ Δ Σ₁ Σ₂ tx te l, Rule ([Γ>Δ> Σ₁ |- [tx@@l]],,[Γ>Δ> Σ₂ |- [tx--->te @@l]]) [Γ>Δ> Σ₁,,Σ₂ |- [te @@l]]
+
+| RLet : ∀ Γ Δ Σ₁ Σ₂ σ₁ σ₂ l, Rule ([Γ>Δ> Σ₂ |- [σ₂@@l]],,[Γ>Δ> Σ₁,,[σ₂@@l] |- [σ₁@@l] ]) [Γ>Δ> Σ₁,,Σ₂ |- [σ₁ @@l]]
+
+| RVoid : ∀ Γ Δ , Rule [] [Γ > Δ > [] |- [] ]
+
+| RAppT : forall Γ Δ Σ κ σ (τ:HaskType Γ κ) l, Rule [Γ>Δ> Σ |- [HaskTAll κ σ @@l]] [Γ>Δ> Σ |- [substT σ τ @@l]]
| RAbsT : ∀ Γ Δ Σ κ σ l,
Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) @@ (weakL l)]]
[Γ>Δ > Σ |- [HaskTAll κ σ @@ l]]
-| RAppCo : forall Γ Δ Σ κ σ₁ σ₂ σ γ l, Δ ⊢ᴄᴏ γ : σ₁∼σ₂ ->
- Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂:κ ⇒ σ@@l]] [Γ>Δ> Σ |- [σ @@l]]
-| RAbsCo : ∀ Γ Δ Σ κ σ σ₁ σ₂ l,
- Γ ⊢ᴛy σ₁:κ ->
- Γ ⊢ᴛy σ₂:κ ->
+
+| RAppCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ l,
+ Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂ ⇒ σ@@l]] [Γ>Δ> Σ |- [σ @@l]]
+| RAbsCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) σ l,
Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ @@ l]]
- [Γ > Δ > Σ |- [σ₁∼∼σ₂:κ⇒ σ @@l]]
-| RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂, Rule [Γ > Δ > Σ₁,,τ₂ |- τ₁,,τ₂ ] [Γ > Δ > Σ₁ |- τ₁ ]
-| RCase : forall Γ Δ lev n tc Σ avars tbranches
- (alts:Tree ??(@ProofCaseBranch n tc Γ lev tbranches avars)),
+ [Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ @@l]]
+
+| RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- ([τ₁],,τ₂)@@@lev ] [Γ > Δ > Σ₁ |- [τ₁@@lev] ]
+| RCase : forall Γ Δ lev tc Σ avars tbranches
+ (alts:Tree ??{ sac : @StrongAltCon tc & @ProofCaseBranch tc Γ Δ lev tbranches avars sac }),
Rule
- ((mapOptionTree pcb_judg alts),,
- [Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
- [Γ > Δ > (mapOptionTreeAndFlatten pcb_freevars alts),,Σ |- [ tbranches @@ lev ] ]
+ ((mapOptionTree (fun x => pcb_judg (projT2 x)) alts),,
+ [Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
+ [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches @@ lev ] ]
.
-Coercion RURule : URule >-> Rule.
(* A rule is considered "flat" if it is neither RBrak nor REsc *)
+(* TODO: change this to (if RBrak/REsc -> False) *)
Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
-| Flat_RURule : ∀ Γ Δ h c r , Rule_Flat (RURule Γ Δ h c r)
-| Flat_RNote : ∀ x y z , Rule_Flat (RNote x y z)
+| Flat_RArrange : ∀ Γ Δ h c r a , Rule_Flat (RArrange Γ Δ h c r a)
+| Flat_RNote : ∀ Γ Δ Σ τ l n , Rule_Flat (RNote Γ Δ Σ τ l n)
+| Flat_RLit : ∀ Γ Δ Σ τ , Rule_Flat (RLit Γ Δ Σ τ )
| Flat_RVar : ∀ Γ Δ σ l, Rule_Flat (RVar Γ Δ σ l)
-| Flat_RLam : ∀ Γ Δ Σ tx te q l, Rule_Flat (RLam Γ Δ Σ tx te q l)
-| Flat_RCast : ∀ Γ Δ Σ σ τ γ q l, Rule_Flat (RCast Γ Δ Σ σ τ γ q l)
+| Flat_RLam : ∀ Γ Δ Σ tx te q , Rule_Flat (RLam Γ Δ Σ tx te q )
+| Flat_RCast : ∀ Γ Δ Σ σ τ γ q , Rule_Flat (RCast Γ Δ Σ σ τ γ q )
| Flat_RAbsT : ∀ Γ Σ κ σ a q , Rule_Flat (RAbsT Γ Σ κ σ a q )
-| Flat_RAppT : ∀ Γ Δ Σ κ σ τ q l, Rule_Flat (RAppT Γ Δ Σ κ σ τ q l)
-| Flat_RAppCo : ∀ Γ Δ Σ κ σ₁ σ₂ σ γ q l, Rule_Flat (RAppCo Γ Δ Σ κ σ₁ σ₂ σ γ q l)
-| Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 q3 x , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 q3 x )
+| Flat_RAppT : ∀ Γ Δ Σ κ σ τ q , Rule_Flat (RAppT Γ Δ Σ κ σ τ q )
+| Flat_RAppCo : ∀ Γ Δ Σ σ₁ σ₂ σ γ q l, Rule_Flat (RAppCo Γ Δ Σ σ₁ σ₂ σ γ q l)
+| Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 )
| Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
| Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
-| Flat_RBindingGroup : ∀ q a b c d e , Rule_Flat (RBindingGroup q a b c d e)
-| Flat_RCase : ∀ Σ Γ T κlen κ θ l x q, Rule_Flat (RCase Σ Γ T κlen κ θ l x q).
-
-(* given a proof that uses only uniform rules, we can produce a general proof *)
-Definition UND_to_ND Γ Δ h c : ND (@URule Γ Δ) h c -> ND Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c)
- := @nd_map' _ (@URule Γ Δ ) _ Rule (@UJudg2judg Γ Δ ) (fun h c r => nd_rule (RURule _ _ h c r)) h c.
-
-Lemma no_urules_with_empty_conclusion : forall Γ Δ c h, @URule Γ Δ c h -> h=[] -> False.
- intro.
- intro.
- induction 1; intros; inversion H.
- simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
- simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
- Qed.
+| Flat_RJoin : ∀ q a b c d e , Rule_Flat (RJoin q a b c d e)
+| Flat_RVoid : ∀ q a , Rule_Flat (RVoid q a)
+| Flat_RCase : ∀ Σ Γ T κlen κ θ l x , Rule_Flat (RCase Σ Γ T κlen κ θ l x)
+| Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev, Rule_Flat (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
Lemma no_rules_with_empty_conclusion : forall c h, @Rule c h -> h=[] -> False.
intros.
destruct X; try destruct c; try destruct o; simpl in *; try inversion H.
- apply no_urules_with_empty_conclusion in u.
- apply u.
- auto.
- Qed.
-
-Lemma no_urules_with_multiple_conclusions : forall Γ Δ c h,
- @URule Γ Δ c h -> { h1:Tree ??(UJudg Γ Δ) & { h2:Tree ??(UJudg Γ Δ) & h=(h1,,h2) }} -> False.
- intro.
- intro.
- induction 1; intros.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
-
- apply IHX.
- destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
- inversion e.
- inversion e.
- exists c1. exists c2. auto.
-
- apply IHX.
- destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
- inversion e.
- inversion e.
- exists c1. exists c2. auto.
-
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
- inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
Qed.
Lemma no_rules_with_multiple_conclusions : forall c h,
try apply no_urules_with_empty_conclusion in u; try apply u.
destruct X0; destruct s; inversion e.
auto.
- apply (no_urules_with_multiple_conclusions _ _ h (c1,,c2)) in u. inversion u. exists c1. exists c2. auto.
+ destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
destruct X0; destruct s; inversion e.
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.
projects / coq-hetmet.git / blobdiff