+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+
inductive True: Prop \def
I : True.
\lambda A:Prop. (A \to False).
theorem absurd : \forall A,C:Prop. A \to Not A \to C.
-cut False.elim Hcut.apply H1.assumption.
+intros.cut False.elim Hcut.apply H1.assumption.
qed.
inductive And (A,B:Prop) : Prop \def
conj : A \to B \to (And A B).
theorem proj1: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
qed.
theorem proj2: \forall A,B:Prop. (And A B) \to A.
-intro. elim H. assumption.
+intros. elim H. assumption.
qed.
inductive Or (A,B:Prop) : Prop \def
inductive eq (A:Type) (x:A) : A \to Prop \def
refl_equal : eq A x x.
+
+theorem sym_eq : \forall A:Type.\forall x,y:A. eq A x y \to eq A y x.
+intros. elim H. apply refl_equal.
+qed.
+
+theorem trans_eq : \forall A:Type.
+\forall x,y,z:A. eq A x y \to eq A y z \to eq A x z.
+intros.elim H1.assumption.
+qed.
+
+theorem f_equal: \forall A,B:Type.\forall f:A\to B.
+\forall x,y:A. eq A x y \to eq B (f x) (f y).
+intros.elim H.apply refl_equal.
+qed.
+
+theorem f_equal2: \forall A,B,C:Type.\forall f:A\to B \to C.
+\forall x1,x2:A. \forall y1,y2:B.
+eq A x1 x2\to eq B y1 y2\to eq C (f x1 y1) (f x2 y2).
+intros.elim H1.elim H.apply refl_equal.
+qed.
+
+
+inductive nat : Set \def
+ | O : nat
+ | S : nat \to nat.
+
+definition pred: nat \to nat \def
+\lambda n:nat. match n with
+[ O \Rightarrow O
+| (S u) \Rightarrow u ].
+
+theorem pred_Sn : \forall n:nat.
+(eq nat n (pred (S n))).
+intros.apply refl_equal.
+qed.
+
+theorem injective_S : \forall n,m:nat.
+(eq nat (S n) (S m)) \to (eq nat n m).
+intros.(elim (sym_eq ? ? ? (pred_Sn n))).
+(elim (sym_eq ? ? ? (pred_Sn m))).
+apply f_equal. assumption.
+qed.
+
+theorem not_eq_S : \forall n,m:nat.
+Not (eq nat n m) \to Not (eq nat (S n) (S m)).
+intros. simplify.intros.
+apply H.apply injective_S.assumption.
+qed.
+
+definition not_zero : nat \to Prop \def
+\lambda n: nat.
+ match n with
+ [ O \Rightarrow False
+ | (S p) \Rightarrow True ].
+
+theorem O_S : \forall n:nat. Not (eq nat O (S n)).
+intros.simplify.intros.
+cut (not_zero O).exact Hcut.elim (sym_eq ? ? ? H).
+exact I.
+qed.
+
+theorem n_Sn : \forall n:nat. Not (eq nat n (S n)).
+intros.elim n.apply O_S.apply not_eq_S.assumption.
+qed.
+
+definition plus : nat \to nat \to nat \def
+let rec plus (n,m:nat) \def
+ match n with
+ [ O \Rightarrow m
+ | (S p) \Rightarrow S (plus p m) ]
+in
+plus.
+
+theorem plus_n_O: \forall n:nat. eq nat n (plus n O).
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+qed.
+
+theorem plus_n_Sm : \forall n,m:nat. eq nat (S (plus n m)) (plus n (S m)).
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+qed.
+
+theorem sym_plus: \forall n,m:nat. eq nat (plus n m) (plus m n).
+intros.elim n.simplify.apply plus_n_O.
+simplify.elim (sym_eq ? ? ? H).apply plus_n_Sm.
+qed.
+
+theorem assoc_plus:
+\forall n,m,p:nat. eq nat (plus (plus n m) p) (plus n (plus m p)).
+intros.elim n.simplify.apply refl_equal.simplify.apply f_equal.assumption.
+qed.
+
+definition times : nat \to nat \to nat \def
+let rec times (n,m:nat) \def
+ match n with
+ [ O \Rightarrow O
+ | (S p) \Rightarrow (plus m (times p m)) ]
+in
+times.
+
+theorem times_n_O: \forall n:nat. eq nat O (times n O).
+intros.elim n.simplify.apply refl_equal.simplify.assumption.
+qed.
+
+theorem times_n_Sm :
+\forall n,m:nat. eq nat (plus n (times n m)) (times n (S m)).
+intros.elim n.simplify.apply refl_equal.
+simplify.apply f_equal.elim H.
+apply trans_eq ? ? (plus (plus e m) (times e m)).apply sym_eq.
+apply assoc_plus.apply trans_eq ? ? (plus (plus m e) (times e m)).
+apply f_equal2.
+apply sym_plus.apply refl_equal.apply assoc_plus.
+qed.
+
+theorem sym_times :
+\forall n,m:nat. eq nat (times n m) (times m n).
+intros.elim n.simplify.apply times_n_O.
+simplify.elim (sym_eq ? ? ? H).apply times_n_Sm.
+qed.
+
+definition minus : nat \to nat \to nat \def
+let rec minus (n,m:nat) \def
+ [\lambda n:nat.nat] match n with
+ [ O \Rightarrow O
+ | (S p) \Rightarrow
+ [\lambda n:nat.nat] match m with
+ [O \Rightarrow (S p)
+ | (S q) \Rightarrow minus p q ]]
+in
+minus.
+
+theorem nat_case :
+\forall n:nat.\forall P:nat \to Prop.
+P O \to (\forall m:nat. P (S m)) \to P n.
+intros.elim n.assumption.apply H1.
+qed.
+
+theorem nat_double_ind :
+\forall R:nat \to nat \to Prop.
+(\forall n:nat. R O n) \to
+(\forall n:nat. R (S n) O) \to
+(\forall n,m:nat. R n m \to R (S n) (S m)) \to \forall n,m:nat. R n m.
+intros.cut \forall m:nat.R n m.apply Hcut.elim n.apply H.
+apply nat_case m1.apply H1.intros.apply H2. apply H3.
+qed.
+
+inductive bool : Set \def
+ | true : bool
+ | false : bool.
+
+definition notn : bool \to bool \def
+\lambda b:bool.
+ match b with
+ [ true \Rightarrow false
+ | false \Rightarrow true ].
+
+definition andb : bool \to bool \to bool\def
+\lambda b1,b2:bool.
+ match b1 with
+ [ true \Rightarrow
+ match b2 with [true \Rightarrow true | false \Rightarrow false]
+ | false \Rightarrow false ].
+
+definition orb : bool \to bool \to bool\def
+\lambda b1,b2:bool.
+ match b1 with
+ [ true \Rightarrow
+ match b2 with [true \Rightarrow true | false \Rightarrow false]
+ | false \Rightarrow false ].
+
+definition if_then_else : bool \to Prop \to Prop \to Prop \def
+\lambda b:bool.\lambda P,Q:Prop.
+match b with
+[ true \Rightarrow P
+| false \Rightarrow Q].
+
+inductive le (n:nat) : nat \to Prop \def
+ | le_n : le n n
+ | le_S : \forall m:nat. le n m \to le n (S m).
+
+theorem trans_le: \forall n,m,p:nat. le n m \to le m p \to le n p.
+intros.
+elim H1.assumption.
+apply le_S.assumption.
+qed.
+
+theorem le_n_S: \forall n,m:nat. le n m \to le (S n) (S m).
+intros.elim H.
+apply le_n.apply le_S.assumption.
+qed.
+
+theorem le_O_n : \forall n:nat. le O n.
+intros.elim n.apply le_n.apply le_S. assumption.
+qed.
+
+theorem le_n_Sn : \forall n:nat. le n (S n).
+intros. apply le_S.apply le_n.
+qed.
+
+theorem le_pred_n : \forall n:nat. le (pred n) n.
+intros.elim n.simplify.apply le_n.simplify.
+apply le_n_Sn.
+qed.
+
+theorem not_zero_le : \forall n,m:nat. (le (S n) m ) \to not_zero m.
+intros.elim H.exact I.exact I.
+qed.
+
+theorem le_Sn_O: \forall n:nat. Not (le (S n) O).
+intros.simplify.intros.apply not_zero_le ? O H.
+qed.
+
+theorem le_n_O_eq : \forall n:nat. (le n O) \to (eq nat O n).
+intros.cut (le n O) \to (eq nat O n).apply Hcut. assumption.
+elim n.apply refl_equal.apply False_ind.apply (le_Sn_O ? H2).
+qed.
+
+theorem le_S_n : \forall n,m:nat. le (S n) (S m) \to le n m.
+intros.cut le (pred (S n)) (pred (S m)).exact Hcut.
+elim H.apply le_n.apply trans_le ? (pred x).assumption.
+apply le_pred_n.
+qed.
+
+theorem le_Sn_n : \forall n:nat. Not (le (S n) n).
+intros.elim n.apply le_Sn_O.simplify.intro.
+cut le (S e) e.apply H.assumption.apply le_S_n.assumption.
+qed.
+
+theorem le_antisym : \forall n,m:nat. (le n m) \to (le m n) \to (eq nat n m).
+intros.cut (le n m) \to (le m n) \to (eq nat n m).exact Hcut H H1.
+apply nat_double_ind (\lambda n,m.((le n m) \to (le m n) \to eq nat n m)).
+intros.whd.intros.
+apply le_n_O_eq.assumption.
+intros.whd.intro.apply sym_eq.apply le_n_O_eq.assumption.
+intros.whd.intro.apply f_equal.apply H2.
+apply le_S_n.assumption.
+apply le_S_n.assumption.
+qed.
+
+
+definition leb : nat \to nat \to bool \def
+let rec leb (n,m:nat) \def
+ [\lambda n:nat.bool] match n with
+ [ O \Rightarrow true
+ | (S p) \Rightarrow
+ [\lambda n:nat.bool] match m with
+ [ O \Rightarrow false
+ | (S q) \Rightarrow leb p q]]
+in leb.
+
+theorem le_dec: \forall n,m:nat. if_then_else (leb n m) (le n m) (Not (le n m)).
+intros.
+apply (nat_double_ind
+(\lambda n,m:nat.if_then_else (leb n m) (le n m) (Not (le n m))) ? ? ? n m).
+simplify.intros.apply le_O_n.
+simplify.exact le_Sn_O.
+intros 2.simplify.elim (leb n1 m1).
+simplify.apply le_n_S.apply H.
+simplify.intro.apply H.apply le_S_n.assumption.
+
+
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