Preuve exemple.

Lambda.v

@@ -44,7 +44,6 @@ Inductive typed : context -> term -> type -> Prop :=
 
 Notation "u @ v" := (App u v) (at level 20) : lambda_scope.
 Notation "∇ u" := (Nabla u) (at level 20) : lambda_scope.
-Notation "u , v" := (Couple u v) (at level 20) : lambda_scope.
 Notation "σ -> τ" := (Arr σ τ) : lambda_scope.
 Notation "σ /\ τ" := (And σ τ) : lambda_scope.
 Notation "σ \/ τ" := (Or σ τ) : lambda_scope.
@@ -85,4 +84,23 @@ Proof.
   - apply R_Or_e with (A := to_form τ1) (B := to_form τ2); intuition.
   - apply R_Or_i1 with (B := to_form τ). intuition.
   - apply R_Or_i2 with (A := to_form σ). intuition.
+Qed.
+
+Lemma ex : Pr Intuiti ([] ⊢ (Pred "A" [] /\ Pred "B" []) -> (Pred "A" [] \/ Pred "B" [])).
+Proof.
+  set (A := Pred "A" []). set (B := Pred "B" []).
+  set (form := (_ -> _)%form). set (typ := Atom "A" /\ Atom "B" -> Atom "A" \/ Atom "B").
+  assert (to_form typ = form).
+  { intuition. }
+  rewrite<- H.
+  assert (to_ctxt [] = []).
+  { intuition. }
+  rewrite<- H0.
+  apply CurryHoward with (u := (Abs (I1 (Pi1 (#0))))).
+  unfold typ.
+  apply Type_Abs.
+  apply Type_I1.
+  apply Type_Pi1 with (τ := Atom "B").
+  apply Type_Var.
+  intuition.
 Qed.
\ No newline at end of file

Rapport/RapportStageL3.tex

@@ -331,6 +331,7 @@ Cette partie reprend les concepts et les constructions du cours de $\lambda$-cal
 
 % TODO
 % on utilise la preuve pour redémontrer le premier résultat démontré dans le stage
+% lambda terme utilisé : lambda x . i1 pi1 x
 
 \subsubsection{Logique minimale intuitionniste}
 
@@ -524,7 +525,61 @@ u, v, \ldots ::&= x, y, \ldots & \text{variables}\\
 \end{align*}
 
 Les règles de typage sont :
-\[ \begin{array}{lr}
+%\[ \begin{array}{lr}
+%\text{\begin{prooftree}
+%\infer0[var]{\Gamma, x : \tau \vdash x : \tau}
+%\end{prooftree}} &
+%\text{\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma \Rightarrow \tau}
+%\hypo{\Gamma \vdash v : \sigma}
+%\infer2[app]{\Gamma \vdash uv : \tau}
+%\end{prooftree}} \\
+%& \\
+%\text{\begin{prooftree}
+%\hypo{\Gamma, x : \sigma \vdash u : \tau}
+%\infer1[abs]{\Gamma \vdash \lambda x \cdot u : \sigma \Rightarrow \tau}
+%\end{prooftree}} &
+%\text{\begin{prooftree}
+%\hypo{\Gamma \vdash u : \bot}
+%\infer1[$\nabla$]{\Gamma \vdash \nabla u : \tau}
+%\end{prooftree}} \\
+%& \\
+%\text{\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma \wedge \tau}
+%\infer1[$\pi_1$]{\Gamma \vdash \pi_1 u : \sigma}
+%\end{prooftree}} &
+%\text{\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma \wedge \tau}
+%\infer1[$\pi_2$]{\Gamma \vdash \pi_2 u : \tau}
+%\end{prooftree}} \\
+%& \\
+%\text{
+%\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma}
+%\hypo{\Gamma \vdash v : \tau}
+%\infer2[coup]{\Gamma \vdash \langle u, v \rangle : \sigma \wedge \tau}
+%\end{prooftree}} &
+%\text{
+%\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma \vee \tau}
+%\hypo{\Gamma, x_1 : \sigma \vdash v_1 : \mu}
+%\hypo{\Gamma, x_2 : \tau \vdash v_2 : \mu}
+%\infer3[pattern]{\Gamma \vdash \mathtt{case} \, u \, \mathtt{of} \, \iota_1 x_1 \mapsto v_1 \; | \; \iota_2 x_2 \mapsto v_2 : \mu}
+%\end{prooftree}} \\
+%& \\
+%\text{
+%\begin{prooftree}
+%\hypo{\Gamma \vdash u : \sigma}
+%\infer1[$\iota_1$]{\Gamma \vdash \iota_1 u : \sigma \vee \tau}
+%\end{prooftree}} &
+%\text{
+%\begin{prooftree}
+%\hypo{\Gamma \vdash u : \tau}
+%\infer1[$\iota_2$]{\Gamma \vdash \iota_2 u : \sigma \vee \tau}
+%\end{prooftree}}
+%\end{array} \]
+
+\[ \begin{array}{lcr}
 \text{\begin{prooftree}
 \infer0[var]{\Gamma, x : \tau \vdash x : \tau}
 \end{prooftree}} &
@@ -532,17 +587,18 @@ Les règles de typage sont :
 \hypo{\Gamma \vdash u : \sigma \Rightarrow \tau}
 \hypo{\Gamma \vdash v : \sigma}
 \infer2[app]{\Gamma \vdash uv : \tau}
-\end{prooftree}} \\
-& \\
+\end{prooftree}} &
 \text{\begin{prooftree}
 \hypo{\Gamma, x : \sigma \vdash u : \tau}
 \infer1[abs]{\Gamma \vdash \lambda x \cdot u : \sigma \Rightarrow \tau}
-\end{prooftree}} &
+\end{prooftree}} \\
+& & \\
+&
 \text{\begin{prooftree}
 \hypo{\Gamma \vdash u : \bot}
 \infer1[$\nabla$]{\Gamma \vdash \nabla u : \tau}
-\end{prooftree}} \\
-& \\
+\end{prooftree}} & \\
+& & \\
 \text{\begin{prooftree}
 \hypo{\Gamma \vdash u : \sigma \wedge \tau}
 \infer1[$\pi_1$]{\Gamma \vdash \pi_1 u : \sigma}
@@ -550,32 +606,33 @@ Les règles de typage sont :
 \text{\begin{prooftree}
 \hypo{\Gamma \vdash u : \sigma \wedge \tau}
 \infer1[$\pi_2$]{\Gamma \vdash \pi_2 u : \tau}
-\end{prooftree}} \\
-& \\
+\end{prooftree}} &
 \text{
 \begin{prooftree}
 \hypo{\Gamma \vdash u : \sigma}
 \hypo{\Gamma \vdash v : \tau}
-\infer2[coup]{\Gamma \vdash \langle u, v \rangle : \sigma \wedge \tau}
-\end{prooftree}} &
-\text{
-\begin{prooftree}
-\hypo{\Gamma \vdash u : \sigma \vee \tau}
-\hypo{\Gamma, x_1 : \sigma \vdash v_1 : \mu}
-\hypo{\Gamma, x_2 : \tau \vdash v_2 : \mu}
-\infer3[pattern]{\Gamma \vdash \mathtt{case} \, u \, \mathtt{of} \, \iota_1 x_1 \mapsto v_1 \; | \; \iota_2 x_2 \mapsto v_2 : \mu}
-\end{prooftree}} \\
-& \\
+\infer2[paire]{\Gamma \vdash \langle u, v \rangle : \sigma \wedge \tau}
+\end{prooftree}}\\
+& & \\
 \text{
 \begin{prooftree}
 \hypo{\Gamma \vdash u : \sigma}
 \infer1[$\iota_1$]{\Gamma \vdash \iota_1 u : \sigma \vee \tau}
-\end{prooftree}} &
+\end{prooftree}} & &
 \text{
 \begin{prooftree}
 \hypo{\Gamma \vdash u : \tau}
 \infer1[$\iota_2$]{\Gamma \vdash \iota_2 u : \sigma \vee \tau}
-\end{prooftree}}
+\end{prooftree}} \\
+& & \\
+&
+\text{
+\begin{prooftree}
+\hypo{\Gamma \vdash u : \sigma \vee \tau}
+\hypo{\Gamma, x_1 : \sigma \vdash v_1 : \mu}
+\hypo{\Gamma, x_2 : \tau \vdash v_2 : \mu}
+\infer3[pattern]{\Gamma \vdash \mathtt{case} \, u \, \mathtt{of} \, \iota_1 x_1 \mapsto v_1 \; | \; \iota_2 x_2 \mapsto v_2 : \mu}
+\end{prooftree}} &
 \end{array} \]
 
 \end{document}