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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "logic/equality.ma".
+include "nat/nat.ma".
+include "datatypes/constructors.ma".
+
+theorem stupid:
+  (S O) = O \to (\forall p:Prop. p \to Not p).
+  intros.
+  generalize in match I.
+  destruct H.
+qed.
+
+inductive bar_list (A:Set): Set \def
+  | bar_nil: bar_list A
+  | bar_cons: A \to bar_list A \to bar_list A.
+
+theorem stupid2:
+  \forall A:Set.\forall x:A.\forall l:bar_list A.
+  bar_nil A = bar_cons A x l \to False.
+  intros.
+  destruct H.
+qed.
+
+inductive dt (A:Type): Type \to Type \def
+ | k1: \forall T:Type. dt A T
+ | k2: \forall T:Type. \forall T':Type. dt A (T \to T').
+ 
+theorem stupid3:
+ k1 False (False → True) = k2 False False True → False.
+ intros;
+ destruct H.
+qed.
+
+inductive dddt (A:Type): Type \to Type \def
+ | kkk1: dddt A nat
+ | kkk2: dddt A nat.
+ 
+theorem stupid4: kkk1 False = kkk2 False \to False.
+ intros;
+ destruct H.
+qed.
+
+theorem recursive: S (S (S O)) = S (S O) \to False.
+ intros;
+ destruct H.
+qed.
+
+inductive complex (A,B : Type) : B → A → Type ≝
+| C1 : ∀x:nat.∀a:A.∀b:B. complex A B b a
+| C2 : ∀a,a1:A.∀b,b1:B.∀x:nat. complex A B b1 a1 → complex A B b a.
+
+theorem recursive1: ∀ x,y : nat. 
+  (C1 ? ? O     (Some ? x) y) = 
+  (C1 ? ? (S O) (Some ? x) y) → False.
+intros; destruct H.
+qed.
+
+theorem recursive2: ∀ x,y,z,t : nat. 
+  (C1 ? ? t (Some ? x) y) = 
+  (C1 ? ? z (Some ? x) y) → t=z.
+intros; destruct H; reflexivity.
+qed.
+
+theorem recursive3: ∀ x,y,z,t : nat. 
+  C2 ? ? (None ?) ? (S O) ? z (C1 ? ? (S O) (Some ? x) y) = 
+  C2 ? ? (None ?) ? (S O) ? t (C1 ? ? (S O) (Some ? x) y) → z=t.
+intros; destruct H; reflexivity.
+qed.
+
+theorem recursive4: ∀ x,y,z,t : nat. 
+  C2 ? ? (None ?) ? (S O) ? z (C1 ? ? (S O) (Some ? z) y) = 
+  C2 ? ? (None ?) ? (S O) ? t (C1 ? ? (S O) (Some ? x) y) → z=t.
+intros; destruct H; reflexivity.
+qed.
+
+theorem recursive2: ∀ x,y : nat. 
+  C2 ? ? (None ?) ? (S O) ? O (C1 ? ? O     (Some ? x) y) = 
+  C2 ? ? (None ?) ? (S O) ? O (C1 ? ? (S O) (Some ? x) y) → False.
+intros; destruct H.
+qed.
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