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+(**************************************************************************)
+(*       ___	                                                          *)
+(*      ||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         *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "datatypes/bool.ma".
+include "datatypes/compare.ma".
+include "nat/orders.ma".
+
+let rec eqb n m \def 
+match n with 
+  [ O \Rightarrow 
+     match m with 
+     [ O \Rightarrow true
+	   | (S q) \Rightarrow false] 
+  | (S p) \Rightarrow
+	   match m with 
+     [ O \Rightarrow false
+	   | (S q) \Rightarrow eqb p q]].
+	   
+theorem eqb_to_Prop: \forall n,m:nat. 
+match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+intros.
+apply (nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m])).
+intro.elim n1.
+simplify.reflexivity.
+simplify.apply not_eq_O_S.
+intro.
+simplify.unfold Not.
+intro. apply (not_eq_O_S n1).apply sym_eq.assumption.
+intros.simplify.
+generalize in match H.
+elim ((eqb n1 m1)).
+simplify.apply eq_f.apply H1.
+simplify.unfold Not.intro.apply H1.apply inj_S.assumption.
+qed.
+
+theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
+(n=m \to (P true)) \to (n \neq m \to (P false)) \to (P (eqb n m)). 
+intros.
+cut 
+(match (eqb n m) with
+[ true  \Rightarrow n = m
+| false \Rightarrow n \neq m] \to (P (eqb n m))).
+apply Hcut.apply eqb_to_Prop.
+elim (eqb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
+
+theorem eqb_n_n: \forall n. eqb n n = true.
+intro.elim n.simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem eqb_true_to_eq: \forall n,m:nat.
+eqb n m = true \to n = m.
+intros.
+change with 
+match true with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop. 
+qed.
+
+theorem eqb_false_to_not_eq: \forall n,m:nat.
+eqb n m = false \to n \neq m.
+intros.
+change with 
+match false with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq m].
+rewrite < H.
+apply eqb_to_Prop. 
+qed.
+
+theorem eq_to_eqb_true: \forall n,m:nat.
+n = m \to eqb n m = true.
+intros.apply (eqb_elim n m).
+intros. reflexivity.
+intros.apply False_ind.apply (H1 H).
+qed.
+
+theorem not_eq_to_eqb_false: \forall n,m:nat.
+\lnot (n = m) \to eqb n m = false.
+intros.apply (eqb_elim n m).
+intros. apply False_ind.apply (H H1).
+intros.reflexivity.
+qed.
+
+let rec leb n m \def 
+match n with 
+    [ O \Rightarrow true
+    | (S p) \Rightarrow
+	match m with 
+        [ O \Rightarrow false
+	| (S q) \Rightarrow leb p q]].
+
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
+P (leb n m).
+apply nat_elim2; intros; simplify
+  [apply H.apply le_O_n
+  |apply H1.apply not_le_Sn_O.
+  |apply H;intros
+    [apply H1.apply le_S_S.assumption.
+    |apply H2.unfold Not.intros.apply H3.apply le_S_S_to_le.assumption
+    ]
+  ]
+qed.
+
+theorem leb_true_to_le:\forall n,m.
+leb n m = true \to (n \le m).
+intros 2.
+apply leb_elim
+  [intros.assumption
+  |intros.destruct H1.
+  ]
+qed.
+
+theorem leb_false_to_not_le:\forall n,m.
+leb n m = false \to \lnot (n \le m).
+intros 2.
+apply leb_elim
+  [intros.destruct H1
+  |intros.assumption
+  ]
+qed.
+(*
+theorem decidable_le: \forall n,m. n \leq m \lor n \nleq m. 
+intros.
+apply (leb_elim n m)
+  [intro.left.assumption
+  |intro.right.assumption
+  ]
+qed.
+*)
+
+theorem le_to_leb_true: \forall n,m. n \leq m \to leb n m = true.
+intros.apply leb_elim;intros
+  [reflexivity
+  |apply False_ind.apply H1.apply H.
+  ]
+qed.
+
+theorem lt_to_leb_false: \forall n,m. m < n \to leb n m = false.
+intros.apply leb_elim;intros
+  [apply False_ind.apply (le_to_not_lt ? ? H1). assumption
+  |reflexivity
+  ]
+qed.
+
+theorem leb_to_Prop: \forall n,m:nat. 
+match (leb n m) with
+[ true  \Rightarrow n \leq m 
+| false \Rightarrow n \nleq m].
+apply nat_elim2;simplify
+  [exact le_O_n
+  |exact not_le_Sn_O
+  |intros 2.simplify.
+   elim ((leb n m));simplify
+    [apply le_S_S.apply H
+    |unfold Not.intros.apply H.apply le_S_S_to_le.assumption
+    ]
+  ]
+qed.
+
+(*
+theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
+(n \leq m \to (P true)) \to (n \nleq m \to (P false)) \to
+P (leb n m).
+intros.
+cut 
+(match (leb n m) with
+[ true  \Rightarrow n \leq m
+| false \Rightarrow n \nleq m] \to (P (leb n m))).
+apply Hcut.apply leb_to_Prop.
+elim (leb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
+*)
+
+definition ltb ≝λn,m. leb n m ∧ notb (eqb n m).
+
+theorem ltb_to_Prop :
+ ∀n,m.
+  match ltb n m with
+  [ true ⇒ n < m
+  | false ⇒ n ≮ m
+  ].
+intros;
+unfold ltb;
+apply leb_elim;
+apply eqb_elim;
+intros;
+simplify;
+[ rewrite < H;
+  apply le_to_not_lt;
+  constructor 1
+| apply (not_eq_to_le_to_lt ? ? H H1)
+| rewrite < H;
+  apply le_to_not_lt;
+  constructor 1
+| apply le_to_not_lt;
+  generalize in match (not_le_to_lt ? ? H1);
+  clear H1;
+  intro;
+  apply lt_to_le;
+  assumption
+].
+qed.
+
+theorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop.
+(n < m → (P true)) → (n ≮ m → (P false)) →
+P (ltb n m).
+intros.
+cut
+(match (ltb n m) with
+[ true  ⇒ n < m
+| false ⇒ n ≮ m] → (P (ltb n m))).
+apply Hcut.apply ltb_to_Prop.
+elim (ltb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
+
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
+
+theorem nat_compare_n_n: \forall n:nat. nat_compare n n = EQ.
+intro.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem nat_compare_S_S: \forall n,m:nat. 
+nat_compare n m = nat_compare (S n) (S m).
+intros.simplify.reflexivity.
+qed.
+
+theorem nat_compare_pred_pred: 
+\forall n,m:nat.lt O n \to lt O m \to 
+eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
+intros.
+apply (lt_O_n_elim n H).
+apply (lt_O_n_elim m H1).
+intros.
+simplify.reflexivity.
+qed.
+
+theorem nat_compare_to_Prop: \forall n,m:nat. 
+match (nat_compare n m) with
+  [ LT \Rightarrow n < m
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ].
+intros.
+apply (nat_elim2 (\lambda n,m.match (nat_compare n m) with
+  [ LT \Rightarrow n < m
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ])).
+intro.elim n1.simplify.reflexivity.
+simplify.unfold lt.apply le_S_S.apply le_O_n.
+intro.simplify.unfold lt.apply le_S_S. apply le_O_n.
+intros 2.simplify.elim ((nat_compare n1 m1)).
+simplify. unfold lt. apply le_S_S.apply H.
+simplify. apply eq_f. apply H.
+simplify. unfold lt.apply le_S_S.apply H.
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat. 
+nat_compare n m = compare_invert (nat_compare m n).
+intros. 
+apply (nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n))).
+intros.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intro.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intros.simplify.elim H.reflexivity.
+qed.
+     
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to 
+(P (nat_compare n m)).
+intros.
+cut (match (nat_compare n m) with
+[ LT \Rightarrow n < m
+| EQ \Rightarrow n=m
+| GT \Rightarrow m < n] \to
+(P (nat_compare n m))).
+apply Hcut.apply nat_compare_to_Prop.
+elim ((nat_compare n m)).
+apply ((H H3)).
+apply ((H1 H3)).
+apply ((H2 H3)).
+qed.