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 HaskKinds.
16 Require Import HaskCoreTypes.
17 Require Import HaskLiteralsAndTyCons.
18 Require Import HaskStrongTypes.
19 Require Import HaskWeakVars.
21 (* A judgment consists of an environment shape (Γ and Δ) and a pair of trees of leveled types (the antecedent and succedent) valid
22 * in any context of that shape. Notice that the succedent contains a tree of types rather than a single type; think
23 * of [ T1 |- T2 ] as asserting that a letrec with branches having types corresponding to the leaves of T2 is well-typed
24 * in environment T1. This subtle distinction starts to matter when we get into substructural (linear, affine, ordered, etc)
29 forall Δ:CoercionEnv Γ,
30 Tree ??(LeveledHaskType Γ ★) ->
31 Tree ??(LeveledHaskType Γ ★) ->
33 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).
35 (* information needed to define a case branch in a HaskProof *)
36 Record ProofCaseBranch {tc:TyCon}{Γ}{Δ}{lev}{branchtype : HaskType Γ ★}{avars}{sac:@StrongAltCon tc} :=
37 { pcb_freevars : Tree ??(LeveledHaskType Γ ★)
38 ; pcb_judg := sac_Γ sac Γ > sac_Δ sac Γ avars (map weakCK' Δ)
39 > (mapOptionTree weakLT' pcb_freevars),,(unleaves (map (fun t => t@@weakL' lev)
40 (vec2list (sac_types sac Γ avars))))
41 |- [weakLT' (branchtype @@ lev)]
43 (*Coercion pcb_scb : ProofCaseBranch >-> StrongAltCon.*)
44 Implicit Arguments ProofCaseBranch [ ].
46 (* Figure 3, production $\vdash_E$, Uniform rules *)
47 Inductive Arrange {T} : Tree ??T -> Tree ??T -> Type :=
48 | RCanL : forall a , Arrange ( [],,a ) ( a )
49 | RCanR : forall a , Arrange ( a,,[] ) ( a )
50 | RuCanL : forall a , Arrange ( a ) ( [],,a )
51 | RuCanR : forall a , Arrange ( a ) ( a,,[] )
52 | RAssoc : forall a b c , Arrange (a,,(b,,c) ) ((a,,b),,c )
53 | RCossa : forall a b c , Arrange ((a,,b),,c ) ( a,,(b,,c) )
54 | RExch : forall a b , Arrange ( (b,,a) ) ( (a,,b) )
55 | RWeak : forall a , Arrange ( [] ) ( a )
56 | RCont : forall a , Arrange ( (a,,a) ) ( a )
57 | RLeft : forall {h}{c} x , Arrange h c -> Arrange ( x,,h ) ( x,,c)
58 | RRight : forall {h}{c} x , Arrange h c -> Arrange ( h,,x ) ( c,,x)
59 | RComp : forall {a}{b}{c}, Arrange a b -> Arrange b c -> Arrange a c
62 (* Figure 3, production $\vdash_E$, all rules *)
63 Inductive Rule : Tree ??Judg -> Tree ??Judg -> Type :=
65 | RArrange : ∀ Γ Δ Σ₁ Σ₂ Σ, Arrange Σ₁ Σ₂ -> Rule [Γ > Δ > Σ₁ |- Σ ] [Γ > Δ > Σ₂ |- Σ ]
68 | RBrak : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [t @@ (v::l) ]] [Γ > Δ > Σ |- [<[v|-t]> @@l]]
69 | REsc : ∀ Γ Δ t v Σ l, Rule [Γ > Δ > Σ |- [<[v|-t]> @@ l]] [Γ > Δ > Σ |- [t @@ (v::l) ]]
71 (* Part of GHC, but not explicitly in System FC *)
72 | RNote : ∀ Γ Δ Σ τ l, Note -> Rule [Γ > Δ > Σ |- [τ @@ l]] [Γ > Δ > Σ |- [τ @@ l]]
73 | RLit : ∀ Γ Δ v l, Rule [ ] [Γ > Δ > []|- [literalType v @@ l]]
76 | RVar : ∀ Γ Δ σ l, Rule [ ] [Γ>Δ> [σ@@l] |- [σ @@l]]
77 | RGlobal : ∀ Γ Δ τ l, WeakExprVar -> Rule [ ] [Γ>Δ> [] |- [τ @@l]]
78 | RLam : forall Γ Δ Σ (tx:HaskType Γ ★) te l, Rule [Γ>Δ> Σ,,[tx@@l]|- [te@@l] ] [Γ>Δ> Σ |- [tx--->te @@l]]
79 | RCast : forall Γ Δ Σ (σ₁ σ₂:HaskType Γ ★) l,
80 HaskCoercion Γ Δ (σ₁∼∼∼σ₂) -> Rule [Γ>Δ> Σ |- [σ₁@@l] ] [Γ>Δ> Σ |- [σ₂ @@l]]
81 | RBindingGroup : ∀ Γ Δ Σ₁ Σ₂ τ₁ τ₂ , Rule ([Γ > Δ > Σ₁ |- τ₁ ],,[Γ > Δ > Σ₂ |- τ₂ ]) [Γ>Δ> Σ₁,,Σ₂ |- τ₁,,τ₂ ]
82 | 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]]
83 | 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]]
84 | REmptyGroup : ∀ Γ Δ , Rule [] [Γ > Δ > [] |- [] ]
85 | 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]]
86 | RAbsT : ∀ Γ Δ Σ κ σ l,
87 Rule [(κ::Γ)> (weakCE Δ) > mapOptionTree weakLT Σ |- [ HaskTApp (weakF σ) (FreshHaskTyVar _) @@ (weakL l)]]
88 [Γ>Δ > Σ |- [HaskTAll κ σ @@ l]]
89 | RAppCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) (γ:HaskCoercion Γ Δ (σ₁∼∼∼σ₂)) σ l,
90 Rule [Γ>Δ> Σ |- [σ₁∼∼σ₂ ⇒ σ@@l]] [Γ>Δ> Σ |- [σ @@l]]
91 | RAbsCo : forall Γ Δ Σ κ (σ₁ σ₂:HaskType Γ κ) σ l,
92 Rule [Γ > ((σ₁∼∼∼σ₂)::Δ) > Σ |- [σ @@ l]]
93 [Γ > Δ > Σ |- [σ₁∼∼σ₂⇒ σ @@l]]
94 | RLetRec : forall Γ Δ Σ₁ τ₁ τ₂ lev, Rule [Γ > Δ > Σ₁,,(τ₂@@@lev) |- ([τ₁],,τ₂)@@@lev ] [Γ > Δ > Σ₁ |- [τ₁@@lev] ]
95 | RCase : forall Γ Δ lev tc Σ avars tbranches
96 (alts:Tree ??{ sac : @StrongAltCon tc & @ProofCaseBranch tc Γ Δ lev tbranches avars sac }),
98 ((mapOptionTree (fun x => pcb_judg (projT2 x)) alts),,
99 [Γ > Δ > Σ |- [ caseType tc avars @@ lev ] ])
100 [Γ > Δ > (mapOptionTreeAndFlatten (fun x => pcb_freevars (projT2 x)) alts),,Σ |- [ tbranches @@ lev ] ]
104 (* A rule is considered "flat" if it is neither RBrak nor REsc *)
105 Inductive Rule_Flat : forall {h}{c}, Rule h c -> Prop :=
106 | Flat_RArrange : ∀ Γ Δ h c r a , Rule_Flat (RArrange Γ Δ h c r a)
107 | Flat_RNote : ∀ Γ Δ Σ τ l n , Rule_Flat (RNote Γ Δ Σ τ l n)
108 | Flat_RLit : ∀ Γ Δ Σ τ , Rule_Flat (RLit Γ Δ Σ τ )
109 | Flat_RVar : ∀ Γ Δ σ l, Rule_Flat (RVar Γ Δ σ l)
110 | Flat_RLam : ∀ Γ Δ Σ tx te q , Rule_Flat (RLam Γ Δ Σ tx te q )
111 | Flat_RCast : ∀ Γ Δ Σ σ τ γ q , Rule_Flat (RCast Γ Δ Σ σ τ γ q )
112 | Flat_RAbsT : ∀ Γ Σ κ σ a q , Rule_Flat (RAbsT Γ Σ κ σ a q )
113 | Flat_RAppT : ∀ Γ Δ Σ κ σ τ q , Rule_Flat (RAppT Γ Δ Σ κ σ τ q )
114 | Flat_RAppCo : ∀ Γ Δ Σ σ₁ σ₂ σ γ q l, Rule_Flat (RAppCo Γ Δ Σ σ₁ σ₂ σ γ q l)
115 | Flat_RAbsCo : ∀ Γ Σ κ σ σ₁ σ₂ q1 q2 , Rule_Flat (RAbsCo Γ Σ κ σ σ₁ σ₂ q1 q2 )
116 | Flat_RApp : ∀ Γ Δ Σ tx te p l, Rule_Flat (RApp Γ Δ Σ tx te p l)
117 | Flat_RLet : ∀ Γ Δ Σ σ₁ σ₂ p l, Rule_Flat (RLet Γ Δ Σ σ₁ σ₂ p l)
118 | Flat_RBindingGroup : ∀ q a b c d e , Rule_Flat (RBindingGroup q a b c d e)
119 | Flat_REmptyGroup : ∀ q a , Rule_Flat (REmptyGroup q a)
120 | Flat_RCase : ∀ Σ Γ T κlen κ θ l x , Rule_Flat (RCase Σ Γ T κlen κ θ l x)
121 | Flat_RLetRec : ∀ Γ Δ Σ₁ τ₁ τ₂ lev, Rule_Flat (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
123 Lemma no_rules_with_empty_conclusion : forall c h, @Rule c h -> h=[] -> False.
125 destruct X; try destruct c; try destruct o; simpl in *; try inversion H.
128 Lemma no_rules_with_multiple_conclusions : forall c h,
129 Rule c h -> { h1:Tree ??Judg & { h2:Tree ??Judg & h=(h1,,h2) }} -> False.
131 destruct X; try destruct c; try destruct o; simpl in *; try inversion H;
132 try apply no_urules_with_empty_conclusion in u; try apply u.
133 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.