1 (*********************************************************************************************************************************)
4 (* Natural Deduction proofs of the well-typedness of a Haskell term. Proofs use explicit structural rules (Gentzen-style) *)
5 (* and are in System FC extended with modal types indexed by Taha-Nielsen environment classifiers (λ^α) *)
7 (*********************************************************************************************************************************)
9 Generalizable All Variables.
10 Require Import Preamble.
11 Require Import General.
12 Require Import NaturalDeduction.
13 Require Import Coq.Strings.String.
14 Require Import Coq.Lists.List.
15 Require Import HaskGeneral.
16 Require Import HaskLiterals.
17 Require Import HaskStrongTypes.
19 (* A judgment consists of an environment shape (Γ and Δ) and a pair of trees of leveled types (the antecedent and succedent) valid
20 * in any context of that shape. Notice that the succedent contains a tree of types rather than a single type; think
21 * of [ T1 |- T2 ] as asserting that a letrec with branches having types corresponding to the leaves of T2 is well-typed
22 * in environment T1. This subtle distinction starts to matter when we get into substructural (linear, affine, ordered, etc)
27 forall Δ:CoercionEnv Γ,
28 Tree ??(LeveledHaskType Γ) ->
29 Tree ??(LeveledHaskType Γ) ->
31 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).
33 (* A Uniform Judgment (UJudg) has its environment as a type index; we'll use these to distinguish proofs that have a single,
34 * uniform context throughout the whole proof. Such proofs are important because (1) we can do left and right context
35 * expansion on them (see rules RLeft and RRight) and (2) they will form the fiber categories of our fibration later on *)
36 Inductive UJudg (Γ:TypeEnv)(Δ:CoercionEnv Γ) :=
38 Tree ??(LeveledHaskType Γ) ->
39 Tree ??(LeveledHaskType Γ) ->
41 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).
42 Notation "'ext_tree_left'" := (fun ctx j => match j with mkUJudg t s => mkUJudg _ _ (ctx,,t) s end).
43 Notation "'ext_tree_right'" := (fun ctx j => match j with mkUJudg t s => mkUJudg _ _ (t,,ctx) s end).
45 (* we can turn a UJudg into a Judg by simply internalizing the index *)
46 Definition UJudg2judg {Γ}{Δ}(ej:@UJudg Γ Δ) : Judg :=
47 match ej with mkUJudg t s => Γ > Δ > t |- s end.
48 Coercion UJudg2judg : UJudg >-> Judg.
50 (* information needed to define a case branch in a HaskProof *)
51 Record ProofCaseBranch {n}{tc:TyCon n}{Γ}{lev}{branchtype : HaskType Γ}{avars} :=
52 { pcb_scb : @StrongCaseBranch n tc Γ avars
53 ; pcb_freevars : Tree ??(LeveledHaskType Γ)
54 ; pcb_judg := scb_Γ pcb_scb > scb_Δ pcb_scb
55 > (mapOptionTree weakLT' pcb_freevars),,(unleaves (vec2list (scb_types pcb_scb)))
56 |- [weakLT' (branchtype @@ lev)]
58 Coercion pcb_scb : ProofCaseBranch >-> StrongCaseBranch.
59 Implicit Arguments ProofCaseBranch [ ].
61 (* Figure 3, production $\vdash_E$, Uniform rules *)
62 Inductive URule {Γ}{Δ} : Tree ??(UJudg Γ Δ) -> Tree ??(UJudg Γ Δ) -> Type :=
63 | RCanL : ∀ t a , URule [Γ>>Δ> [],,a |- t ] [Γ>>Δ> a |- t ]
64 | RCanR : ∀ t a , URule [Γ>>Δ> a,,[] |- t ] [Γ>>Δ> a |- t ]
65 | RuCanL : ∀ t a , URule [Γ>>Δ> a |- t ] [Γ>>Δ> [],,a |- t ]
66 | RuCanR : ∀ t a , URule [Γ>>Δ> a |- t ] [Γ>>Δ> a,,[] |- t ]
67 | RAssoc : ∀ t a b c , URule [Γ>>Δ>a,,(b,,c) |- t ] [Γ>>Δ>(a,,b),,c |- t ]
68 | RCossa : ∀ t a b c , URule [Γ>>Δ>(a,,b),,c |- t ] [Γ>>Δ> a,,(b,,c) |- t ]
69 | RLeft : ∀ h c x , URule h c -> URule (mapOptionTree (ext_tree_left x) h) (mapOptionTree (ext_tree_left x) c)
70 | RRight : ∀ h c x , URule h c -> URule (mapOptionTree (ext_tree_right x) h) (mapOptionTree (ext_tree_right x) c)
71 | RExch : ∀ t a b , URule [Γ>>Δ> (b,,a) |- t ] [Γ>>Δ> (a,,b) |- t ]
72 | RWeak : ∀ t a , URule [Γ>>Δ> [] |- t ] [Γ>>Δ> a |- t ]
73 | RCont : ∀ t a , URule [Γ>>Δ> (a,,a) |- t ] [Γ>>Δ> a |- t ].
76 (* Figure 3, production $\vdash_E$, all rules *)
77 Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
79 | RURule : ∀ Γ Δ h c, @URule Γ Δ h c -> Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c)
82 | RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t @@ (v::l) ]] [Γ > Δ > Σ |- [<[v|-t]> @@l]]
83 | REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> @@ l]] [Γ > Δ > Σ |- [t @@ (v::l) ]]
85 (* Part of GHC, but not explicitly in System FC *)
86 | RNote : ∀ h c, Note -> Rule h [ c ]
87 | RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v @@ l]]
90 | RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ @@l]]
91 | RLam : ∀ Γ Δ Σ tx te l, Γ ⊢ᴛy tx : ★ -> Rule [Γ>Δ> Σ,,[tx@@l]|- [te@@l] ] [Γ>Δ> Σ |- [tx--->te @@l]]
92 | RCast : ∀ Γ Δ Σ σ τ γ l, Δ ⊢ᴄᴏ γ : σ ∼ τ -> Rule [Γ>Δ> Σ |- [σ@@l] ] [Γ>Δ> Σ |- [τ @@l]]
93 | RBindingGroup : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ , Rule ([Γ > Δ > Σ₁ |- τ₁ ],,[Γ > Δ > Σ₂ |- τ₂ ]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ ]
94 | 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]]
95 | 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]]
96 | REmptyGroup : ∀ Γ Δ , Rule [] [Γ > Δ > [] |- [] ]
97 | 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]]
98 | RAbsT : ∀ Γ Δ Σ κ σ l,
99 Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) @@ (weakL l)]]
100 [Γ>Δ > Σ |- [HaskTAll κ σ @@ l]]
101 | RAppCo : forall Γ Δ Σ κ σ₁ σ₂ σ γ l, Δ ⊢ᴄᴏ γ : σ₁∼σ₂ ->
102 Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂:κ ⇒ σ@@l]] [Γ>Δ> Σ |- [σ @@l]]
103 | RAbsCo : ∀ Γ Δ Σ κ σ σ₁ σ₂ l,
106 Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ @@ l]]
107 [Γ > Δ > Σ |- [σ₁∼∼σ₂:κ⇒ σ @@l]]
108 | RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂, Rule [Γ > Δ > Σ₁,,τ₂ |- τ₁,,τ₂ ] [Γ > Δ > Σ₁ |- τ₁ ]
109 | RCase : forall Γ Δ lev n tc Σ avars tbranches
110 (alts:Tree ??(@ProofCaseBranch n tc Γ lev tbranches avars)),
112 ((mapOptionTree pcb_judg alts),,
113 [Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
114 [Γ > Δ > (mapOptionTreeAndFlatten pcb_freevars alts),,Σ |- [ tbranches @@ lev ] ]
116 Coercion RURule : URule >-> Rule.
119 (* A rule is considered "flat" if it is neither RBrak nor REsc *)
120 Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
121 | Flat_RURule : ∀ Γ Δ h c r , Rule_Flat (RURule Γ Δ h c r)
122 | Flat_RNote : ∀ x y z , Rule_Flat (RNote x y z)
123 | Flat_RVar : ∀ Γ Δ σ l, Rule_Flat (RVar Γ Δ σ l)
124 | Flat_RLam : ∀ Γ Δ Σ tx te q l, Rule_Flat (RLam Γ Δ Σ tx te q l)
125 | Flat_RCast : ∀ Γ Δ Σ σ τ γ q l, Rule_Flat (RCast Γ Δ Σ σ τ γ q l)
126 | Flat_RAbsT : ∀ Γ Σ κ σ a q , Rule_Flat (RAbsT Γ Σ κ σ a q )
127 | Flat_RAppT : ∀ Γ Δ Σ κ σ τ q l, Rule_Flat (RAppT Γ Δ Σ κ σ τ q l)
128 | 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)
129 | 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 )
130 | Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
131 | Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
132 | Flat_RBindingGroup : ∀ q a b c d e , Rule_Flat (RBindingGroup q a b c d e)
133 | Flat_RCase : ∀ Σ Γ T κlen κ θ l x q, Rule_Flat (RCase Σ Γ T κlen κ θ l x q).
135 (* given a proof that uses only uniform rules, we can produce a general proof *)
136 Definition UND_to_ND Γ Δ h c : ND (@URule Γ Δ) h c -> ND Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c)
137 := @nd_map' _ (@URule Γ Δ ) _ Rule (@UJudg2judg Γ Δ ) (fun h c r => nd_rule (RURule _ _ h c r)) h c.
139 Lemma no_urules_with_empty_conclusion : forall Γ Δ c h, @URule Γ Δ c h -> h=[] -> False.
142 induction 1; intros; inversion H.
143 simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
144 simpl in *; destruct c; try destruct o; simpl in *; try destruct u; inversion H; simpl in *; apply IHX; auto; inversion H1.
147 Lemma no_rules_with_empty_conclusion : forall c h, @Rule c h -> h=[] -> False.
149 destruct X; try destruct c; try destruct o; simpl in *; try inversion H.
150 apply no_urules_with_empty_conclusion in u.
155 Lemma no_urules_with_multiple_conclusions : forall Γ Δ c h,
156 @URule Γ Δ c h -> { h1:Tree ??(UJudg Γ Δ) & { h2:Tree ??(UJudg Γ Δ) & h=(h1,,h2) }} -> False.
160 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
161 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
162 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
163 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
164 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
165 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
168 destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
171 exists c1. exists c2. auto.
174 destruct X0. destruct s. destruct c; try destruct o; try destruct u; simpl in *.
177 exists c1. exists c2. auto.
179 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
180 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
181 inversion X; inversion X0; inversion H; inversion X1; destruct c; try destruct o; inversion H2; apply IHX; exists c1;exists c2; auto.
184 Lemma no_rules_with_multiple_conclusions : forall c h,
185 Rule c h -> { h1:Tree ??Judg & { h2:Tree ??Judg & h=(h1,,h2) }} -> False.
187 destruct X; try destruct c; try destruct o; simpl in *; try inversion H;
188 try apply no_urules_with_empty_conclusion in u; try apply u.
189 destruct X0; destruct s; inversion e.
191 apply (no_urules_with_multiple_conclusions _ _ h (c1,,c2)) in u. inversion u. exists c1. exists c2. auto.
192 destruct X0; destruct s; inversion e.
193 destruct X0; destruct s; inversion e.
194 destruct X0; destruct s; inversion e.
195 destruct X0; destruct s; inversion e.
196 destruct X0; destruct s; inversion e.
197 destruct X0; destruct s; inversion e.
198 destruct X0; destruct s; inversion e.
199 destruct X0; destruct s; inversion e.
200 destruct X0; destruct s; inversion e.
201 destruct X0; destruct s; inversion e.
202 destruct X0; destruct s; inversion e.
203 destruct X0; destruct s; inversion e.
204 destruct X0; destruct s; inversion e.
205 destruct X0; destruct s; inversion e.
206 destruct X0; destruct s; inversion e.
207 destruct X0; destruct s; inversion e.
208 destruct X0; destruct s; inversion e.
211 Lemma systemfc_all_rules_one_conclusion : forall h c1 c2 (r:Rule h (c1,,c2)), False.
213 eapply no_rules_with_multiple_conclusions.