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 NaturalDeductionContext.
14 Require Import Coq.Strings.String.
15 Require Import Coq.Lists.List.
16 Require Import HaskKinds.
17 Require Import HaskCoreTypes.
18 Require Import HaskLiterals.
19 Require Import HaskTyCons.
20 Require Import HaskStrongTypes.
21 Require Import HaskWeakVars.
23 (* A judgment consists of an environment shape (Γ and Δ) and a pair of trees of leveled types (the antecedent and succedent) valid
24 * in any context of that shape. Notice that the succedent contains a tree of types rather than a single type; think
25 * of [ T1 |- T2 ] as asserting that a letrec with branches having types corresponding to the leaves of T2 is well-typed
26 * in environment T1. This subtle distinction starts to matter when we get into substructural (linear, affine, ordered, etc)
31 forall Δ:CoercionEnv Γ,
32 Tree ??(LeveledHaskType Γ ★) ->
33 Tree ??(HaskType Γ ★) ->
36 Notation "Γ > Δ > a '|-' s '@' l" := (mkJudg Γ Δ a s l) (at level 52, Δ at level 50, a at level 52, s at level 50, l at level 50).
38 (* information needed to define a case branch in a HaskProof *)
39 Record ProofCaseBranch {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc} :=
40 { pcb_freevars : Tree ??(LeveledHaskType Γ ★)
41 ; pcb_judg := sac_gamma sac Γ > sac_delta sac Γ avars (map weakCK' Δ)
42 > (mapOptionTree weakLT' pcb_freevars),,(unleaves (map (fun t => t@@weakL' lev)
43 (vec2list (sac_types sac Γ avars))))
44 |- [weakT' branchtype ] @ weakL' lev
46 Implicit Arguments ProofCaseBranch [ ].
48 (* Figure 3, production $\vdash_E$, all rules *)
49 Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
51 | RArrange : ∀ Γ Δ Σ₁ Σ₂ Σ l, Arrange Σ₁ Σ₂ -> Rule [Γ > Δ > Σ₁ |- Σ @l] [Γ > Δ > Σ₂ |- Σ @l]
54 | RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t]@(v::l) ] [Γ > Δ > Σ |- [<[v|-t]> ] @l]
55 | REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> ] @l] [Γ > Δ > Σ |- [t]@(v::l) ]
57 (* Part of GHC, but not explicitly in System FC *)
58 | RNote : ∀ Γ Δ Σ τ l, Note -> Rule [Γ > Δ > Σ |- [τ ] @l] [Γ > Δ > Σ |- [τ ] @l]
59 | RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v ] @l]
62 | RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ ] @l]
63 | RGlobal : forall Γ Δ l (g:Global Γ) v, Rule [ ] [Γ>Δ> [] |- [g v ] @l]
64 | RLam : forall Γ Δ Σ (tx:HaskType Γ ★) te l, Rule [Γ>Δ> Σ,,[tx@@l]|- [te] @l] [Γ>Δ> Σ |- [tx--->te ] @l]
65 | RCast : forall Γ Δ Σ (σ₁ σ₂:HaskType Γ ★) l,
66 HaskCoercion Γ Δ (σ₁∼∼∼σ₂) -> Rule [Γ>Δ> Σ |- [σ₁] @l] [Γ>Δ> Σ |- [σ₂ ] @l]
68 (* order is important here; we want to be able to skolemize without introducing new AExch'es *)
69 | 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]
71 | 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]
72 | 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]
74 | 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]
75 | RLeft : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> (Σ@@@l),,Σ₁ |- Σ,,Σ₂@l]
76 | RRight : ∀ Γ Δ Σ₁ Σ₂ Σ l, Rule [Γ>Δ> Σ₁ |- Σ₂ @l] [Γ>Δ> Σ₁,,(Σ@@@l) |- Σ₂,,Σ@l]
78 | RVoid : ∀ Γ Δ l, Rule [] [Γ > Δ > [] |- [] @l ]
80 | 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]
81 | RAbsT : ∀ Γ Δ Σ κ σ l,
82 Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) ]@(weakL l)]
83 [Γ>Δ > Σ |- [HaskTAll κ σ ]@l]
85 | RAppCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ l,
86 Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂ ⇒ σ]@l] [Γ>Δ> Σ |- [σ ]@l]
87 | RAbsCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) σ l,
88 Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ ]@l]
89 [Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ ]@l]
91 | RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- (τ₂,,[τ₁]) @lev ] [Γ > Δ > Σ₁ |- [τ₁] @lev]
92 | RCase : forall Γ Δ lev tc Σ avars tbranches
93 (alts:Tree ??{ sac : @StrongAltCon tc & @ProofCaseBranch tc Γ Δ lev tbranches avars sac }),
95 ((mapOptionTree (fun x => pcb_judg (projT2 x)) alts),,
96 [Γ > Δ > Σ |- [ caseType tc avars ] @lev])
97 [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches ] @ lev]
101 (* A rule is considered "flat" if it is neither RBrak nor REsc *)
102 (* TODO: change this to (if RBrak/REsc -> False) *)
103 Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
104 | Flat_RArrange : ∀ Γ Δ h c r a l , Rule_Flat (RArrange Γ Δ h c r a l)
105 | Flat_RNote : ∀ Γ Δ Σ τ l n , Rule_Flat (RNote Γ Δ Σ τ l n)
106 | Flat_RLit : ∀ Γ Δ Σ τ , Rule_Flat (RLit Γ Δ Σ τ )
107 | Flat_RVar : ∀ Γ Δ σ l, Rule_Flat (RVar Γ Δ σ l)
108 | Flat_RLam : ∀ Γ Δ Σ tx te q , Rule_Flat (RLam Γ Δ Σ tx te q )
109 | Flat_RCast : ∀ Γ Δ Σ σ τ γ q , Rule_Flat (RCast Γ Δ Σ σ τ γ q )
110 | Flat_RAbsT : ∀ Γ Σ κ σ a q , Rule_Flat (RAbsT Γ Σ κ σ a q )
111 | Flat_RAppT : ∀ Γ Δ Σ κ σ τ q , Rule_Flat (RAppT Γ Δ Σ κ σ τ q )
112 | Flat_RAppCo : ∀ Γ Δ Σ σ₁ σ₂ σ γ q l, Rule_Flat (RAppCo Γ Δ Σ σ₁ σ₂ σ γ q l)
113 | Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 )
114 | Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
115 | Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
116 | Flat_RVoid : ∀ q a l, Rule_Flat (RVoid q a l)
117 | Flat_RCase : ∀ Σ Γ T κlen κ θ l x , Rule_Flat (RCase Σ Γ T κlen κ θ l x)
118 | Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev, Rule_Flat (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
120 Lemma no_rules_with_empty_conclusion : forall c h, @Rule c h -> h=[] -> False.
122 destruct X; try destruct c; try destruct o; simpl in *; try inversion H.
125 Lemma no_rules_with_multiple_conclusions : forall c h,
126 Rule c h -> { h1:Tree ??Judg & { h2:Tree ??Judg & h=(h1,,h2) }} -> False.
128 destruct X; try destruct c; try destruct o; simpl in *; try inversion H;
129 try apply no_urules_with_empty_conclusion in u; try apply u.
130 destruct X0; destruct s; inversion e.
132 destruct X0; destruct s; inversion e.
133 destruct X0; destruct s; inversion e.
134 destruct X0; destruct s; inversion e.
135 destruct X0; destruct s; inversion e.
136 destruct X0; destruct s; inversion e.
137 destruct X0; destruct s; inversion e.
138 destruct X0; destruct s; inversion e.
139 destruct X0; destruct s; inversion e.
140 destruct X0; destruct s; inversion e.
141 destruct X0; destruct s; inversion e.
142 destruct X0; destruct s; inversion e.
143 destruct X0; destruct s; inversion e.
144 destruct X0; destruct s; inversion e.
145 destruct X0; destruct s; inversion e.
146 destruct X0; destruct s; inversion e.
147 destruct X0; destruct s; inversion e.
148 destruct X0; destruct s; inversion e.
149 destruct X0; destruct s; inversion e.
150 destruct X0; destruct s; inversion e.
151 destruct X0; destruct s; inversion e.
152 destruct X0; destruct s; inversion e.
155 Lemma systemfc_all_rules_one_conclusion : forall h c1 c2 (r:Rule h (c1,,c2)), False.
157 eapply no_rules_with_multiple_conclusions.