Ajout de Lambda.v au dépôt (c'est plus facile pour voir les modif).

Lambda.v

+
+(* A link with Lambda Calculus. *)
+
+Require Import Defs Mix List.
+Import ListNotations.
+
+Open Scope list.
+
+Inductive term :=
+  | Var : nat -> term
+  | App : term -> term -> term
+  | Abs : term -> term.
+
+Inductive type :=
+  | Arr : type -> type -> type
+  | Atom : name -> type.
+
+Definition context := list type.
+
+Inductive typed : context -> term -> type -> Prop :=
+  | Type_Ax : forall Γ τ n, nth_error Γ n = Some τ -> typed Γ (Var n) τ
+  | Type_App : forall Γ u v σ μ, typed Γ u (Arr σ μ) -> typed Γ v σ -> typed Γ (App u v) μ
+  | Type_Abs : forall Γ u σ μ,  typed (σ :: Γ) u μ -> typed Γ (Abs u) (Arr σ μ).
+
+Notation "u @ v" := (App u v) (at level 20) : lambda_scope.
+Notation "σ -> μ" := (Arr σ μ) : lambda_scope.
+Notation "# n" := (Var n) (at level 20, format "# n") : lambda_scope.
+
+Delimit Scope lambda_scope with lam.
+Bind Scope lambda_scope with term.
+
+Open Scope lambda_scope.
+
+Coercion Var : nat >-> term.
+Check Abs 0 @ 0.
+
+Fixpoint to_form (t : type) : formula :=
+  match t with
+    | Arr u v => ((to_form u) -> (to_form v))%form
+    | Atom a => Pred a []
+  end.
+
+Definition to_ctxt (Γ : context) := List.map to_form Γ.
+
+Lemma In_in τ Γ :
+  In τ Γ -> In (to_form τ) (to_ctxt Γ).
+Proof.
+  induction Γ.
+  - intros. inversion H.
+  - intros.
+    assert (a :: Γ = [a] ++ Γ). { easy. }
+    rewrite H0 in H. clear H0.
+    apply in_app_or in H.
+    destruct H.
+    + inversion H; [ | inversion H0 ].
+      unfold to_ctxt. rewrite List.map_cons.
+      rewrite H0.
+      apply in_eq.
+    + unfold to_ctxt. rewrite List.map_cons.
+      apply in_cons.
+      apply IHΓ. assumption.
+Qed.
+
+Theorem CurryHoward Γ u τ :
+  typed Γ u τ -> Pr Intuiti (to_ctxt Γ ⊢ to_form τ).
+Proof.
+  revert Γ. revert τ.
+  induction u.
+  - intros.
+    inversion H. clear Γ0 τ0 n0 H1 H0 H3.
+    apply R_Ax.
+    apply In_in.
+    apply nth_error_In with (n := n); auto.
+  - intros.
+    inversion H. clear H2 H0 H1 H4.
+    apply R_Imp_e with (A := to_form σ).
+    + specialize IHu1 with (τ := σ -> τ) (Γ := Γ).
+      cbn in IHu1.
+      intuition.
+    + specialize IHu2 with (τ := σ) (Γ := Γ).
+      intuition.
+  - intros.
+    inversion H. clear Γ0 u0 H1 H0.
+    rewrite<- H3 in H. clear H3.
+    specialize IHu with (τ := μ) (Γ := σ :: Γ).
+    cbn. apply R_Imp_i.
+    intuition.
+Qed.
\ No newline at end of file