--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/nat/minimization".
+
+include "nat/minus.ma".
+
+let rec max i f \def
+ match (f i) with
+ [ true \Rightarrow i
+ | false \Rightarrow
+ match i with
+ [ O \Rightarrow O
+ | (S j) \Rightarrow max j f ]].
+
+theorem max_O_f : \forall f: nat \to bool. max O f = O.
+intro. simplify.
+elim (f O).
+simplify.reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem max_S_max : \forall f: nat \to bool. \forall n:nat.
+(f (S n) = true \land max (S n) f = (S n)) \lor
+(f (S n) = false \land max (S n) f = max n f).
+intros.simplify.elim (f (S n)).
+simplify.left.split.reflexivity.reflexivity.
+simplify.right.split.reflexivity.reflexivity.
+qed.
+
+theorem le_max_n : \forall f: nat \to bool. \forall n:nat.
+max n f \le n.
+intros.elim n.rewrite > max_O_f.apply le_n.
+simplify.elim (f (S n1)).simplify.apply le_n.
+simplify.apply le_S.assumption.
+qed.
+
+theorem le_to_le_max : \forall f: nat \to bool. \forall n,m:nat.
+n\le m \to max n f \le max m f.
+intros.elim H.
+apply le_n.
+apply (trans_le ? (max n1 f)).apply H2.
+cut ((f (S n1) = true \land max (S n1) f = (S n1)) \lor
+(f (S n1) = false \land max (S n1) f = max n1 f)).
+elim Hcut.elim H3.
+rewrite > H5.
+apply le_S.apply le_max_n.
+elim H3.rewrite > H5.apply le_n.
+apply max_S_max.
+qed.
+
+theorem f_m_to_le_max: \forall f: nat \to bool. \forall n,m:nat.
+m\le n \to f m = true \to m \le max n f.
+intros 3.elim n.apply (le_n_O_elim m H).
+apply le_O_n.
+apply (le_n_Sm_elim m n1 H1).
+intro.apply (trans_le ? (max n1 f)).
+apply H.apply le_S_S_to_le.assumption.assumption.
+apply le_to_le_max.apply le_n_Sn.
+intro.simplify.rewrite < H3.
+rewrite > H2.simplify.apply le_n.
+qed.
+
+
+definition max_spec \def \lambda f:nat \to bool.\lambda n: nat.
+\exists i. (le i n) \land (f i = true) \to
+(f n) = true \land (\forall i. i < n \to (f i = false)).
+
+theorem f_max_true : \forall f:nat \to bool. \forall n:nat.
+(\exists i:nat. le i n \land f i = true) \to f (max n f) = true.
+intros 2.
+elim n.elim H.elim H1.generalize in match H3.
+apply (le_n_O_elim a H2).intro.simplify.rewrite > H4.
+simplify.assumption.
+simplify.
+apply (bool_ind (\lambda b:bool.
+(f (S n1) = b) \to (f (match b in bool with
+[ true \Rightarrow (S n1)
+| false \Rightarrow (max n1 f)])) = true)).
+simplify.intro.assumption.
+simplify.intro.apply H.
+elim H1.elim H3.generalize in match H5.
+apply (le_n_Sm_elim a n1 H4).
+intros.
+apply (ex_intro nat ? a).
+split.apply le_S_S_to_le.assumption.assumption.
+intros.apply False_ind.apply not_eq_true_false.
+rewrite < H2.rewrite < H7.rewrite > H6. reflexivity.
+reflexivity.
+qed.
+
+theorem lt_max_to_false : \forall f:nat \to bool.
+\forall n,m:nat. (max n f) < m \to m \leq n \to f m = false.
+intros 2.
+elim n.absurd (le m O).assumption.
+cut (O < m).apply (lt_O_n_elim m Hcut).exact not_le_Sn_O.
+rewrite < (max_O_f f).assumption.
+generalize in match H1.
+elim (max_S_max f n1).
+elim H3.
+absurd (m \le S n1).assumption.
+apply lt_to_not_le.rewrite < H6.assumption.
+elim H3.
+apply (le_n_Sm_elim m n1 H2).
+intro.
+apply H.rewrite < H6.assumption.
+apply le_S_S_to_le.assumption.
+intro.rewrite > H7.assumption.
+qed.
+
+let rec min_aux off n f \def
+ match f (n-off) with
+ [ true \Rightarrow (n-off)
+ | false \Rightarrow
+ match off with
+ [ O \Rightarrow n
+ | (S p) \Rightarrow min_aux p n f]].
+
+definition min : nat \to (nat \to bool) \to nat \def
+\lambda n.\lambda f. min_aux n n f.
+
+theorem min_aux_O_f: \forall f:nat \to bool. \forall i :nat.
+min_aux O i f = i.
+intros.simplify.rewrite < minus_n_O.
+elim (f i).reflexivity.
+simplify.reflexivity.
+qed.
+
+theorem min_O_f : \forall f:nat \to bool.
+min O f = O.
+intro.apply (min_aux_O_f f O).
+qed.
+
+theorem min_aux_S : \forall f: nat \to bool. \forall i,n:nat.
+(f (n -(S i)) = true \land min_aux (S i) n f = (n - (S i))) \lor
+(f (n -(S i)) = false \land min_aux (S i) n f = min_aux i n f).
+intros.simplify.elim (f (n - (S i))).
+simplify.left.split.reflexivity.reflexivity.
+simplify.right.split.reflexivity.reflexivity.
+qed.
+
+theorem f_min_aux_true: \forall f:nat \to bool. \forall off,m:nat.
+(\exists i. le (m-off) i \land le i m \land f i = true) \to
+f (min_aux off m f) = true.
+intros 2.
+elim off.elim H.elim H1.elim H2.
+cut (a = m).
+rewrite > (min_aux_O_f f).rewrite < Hcut.assumption.
+apply (antisym_le a m).assumption.rewrite > (minus_n_O m).assumption.
+simplify.
+apply (bool_ind (\lambda b:bool.
+(f (m-(S n)) = b) \to (f (match b in bool with
+[ true \Rightarrow m-(S n)
+| false \Rightarrow (min_aux n m f)])) = true)).
+simplify.intro.assumption.
+simplify.intro.apply H.
+elim H1.elim H3.elim H4.
+elim (le_to_or_lt_eq (m-(S n)) a H6).
+apply (ex_intro nat ? a).
+split.split.
+apply lt_minus_S_n_to_le_minus_n.assumption.
+assumption.assumption.
+absurd (f a = false).rewrite < H8.assumption.
+rewrite > H5.
+apply not_eq_true_false.
+reflexivity.
+qed.
+
+theorem lt_min_aux_to_false : \forall f:nat \to bool.
+\forall n,off,m:nat. (n-off) \leq m \to m < (min_aux off n f) \to f m = false.
+intros 3.
+elim off.absurd (le n m).rewrite > minus_n_O.assumption.
+apply lt_to_not_le.rewrite < (min_aux_O_f f n).assumption.
+generalize in match H1.
+elim (min_aux_S f n1 n).
+elim H3.
+absurd (n - S n1 \le m).assumption.
+apply lt_to_not_le.rewrite < H6.assumption.
+elim H3.
+elim (le_to_or_lt_eq (n -(S n1)) m).
+apply H.apply lt_minus_S_n_to_le_minus_n.assumption.
+rewrite < H6.assumption.
+rewrite < H7.assumption.
+assumption.
+qed.
+
+theorem le_min_aux : \forall f:nat \to bool.
+\forall n,off:nat. (n-off) \leq (min_aux off n f).
+intros 3.
+elim off.rewrite < minus_n_O.
+rewrite > (min_aux_O_f f n).apply le_n.
+elim (min_aux_S f n1 n).
+elim H1.rewrite > H3.apply le_n.
+elim H1.rewrite > H3.
+apply (trans_le (n-(S n1)) (n-n1)).
+apply monotonic_le_minus_r.
+apply le_n_Sn.
+assumption.
+qed.
+
+theorem le_min_aux_r : \forall f:nat \to bool.
+\forall n,off:nat. (min_aux off n f) \le n.
+intros.
+elim off.simplify.rewrite < minus_n_O.
+elim (f n).simplify.apply le_n.
+simplify.apply le_n.
+simplify.elim (f (n -(S n1))).
+simplify.apply le_plus_to_minus.
+rewrite < sym_plus.apply le_plus_n.
+simplify.assumption.
+qed.
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