(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/minimization".
-
include "nat/minus.ma".
let rec max i f \def
rewrite > H2.simplify.apply le_n.
qed.
+theorem max_f_g: \forall f,g,n. (\forall i. i \le n \to f i = g i) \to
+max n f = max n g.
+intros 3.
+elim n
+ [simplify.
+ rewrite > (H O)
+ [reflexivity
+ |apply le_n
+ ]
+ |simplify.
+ rewrite > H
+ [rewrite > H1
+ [reflexivity
+ |apply le_n
+ ]
+ |intros.
+ apply H1.
+ apply le_S.
+ assumption
+ ]
+ ]
+qed.
+
+theorem le_max_f_max_g: \forall f,g,n. (\forall i. i \le n \to f i = true \to g i =true) \to
+max n f \le max n g.
+intros 3.
+elim n
+ [simplify.
+ elim (f O);apply le_O_n
+ |simplify.
+ apply (bool_elim ? (f (S n1)));intro
+ [rewrite > (H1 (S n1) ? H2)
+ [apply le_n
+ |apply le_n
+ ]
+ |cases (g(S n1))
+ [simplify.
+ apply le_S.
+ apply le_max_n
+ |simplify.
+ apply H.
+ intros.
+ apply H1
+ [apply le_S.assumption
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+
+theorem max_O : \forall f:nat \to bool. \forall n:nat.
+(\forall i:nat. le i n \to f i = false) \to max n f = O.
+intros 2.elim n
+ [simplify.rewrite > H
+ [reflexivity
+ |apply le_O_n
+ ]
+ |simplify.rewrite > H1
+ [simplify.apply H.
+ intros.
+ apply H1.
+ apply le_S.
+ assumption
+ |apply le_n
+ ]
+ ]
+qed.
+
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.
reflexivity.
qed.
+theorem exists_forall_le:\forall f,n.
+(\exists i. i \le n \land f i = true) \lor
+(\forall i. i \le n \to f i = false).
+intros.
+elim n
+ [apply (bool_elim ? (f O));intro
+ [left.apply (ex_intro ? ? O).
+ split[apply le_n|assumption]
+ |right.intros.
+ apply (le_n_O_elim ? H1).
+ assumption
+ ]
+ |elim H
+ [elim H1.elim H2.
+ left.apply (ex_intro ? ? a).
+ split[apply le_S.assumption|assumption]
+ |apply (bool_elim ? (f (S n1)));intro
+ [left.apply (ex_intro ? ? (S n1)).
+ split[apply le_n|assumption]
+ |right.intros.
+ elim (le_to_or_lt_eq ? ? H3)
+ [apply H1.
+ apply le_S_S_to_le.
+ apply H4
+ |rewrite > H4.
+ assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem exists_max_forall_false:\forall f,n.
+((\exists i. i \le n \land f i = true) \land (f (max n f) = true))\lor
+((\forall i. i \le n \to f i = false) \land (max n f) = O).
+intros.
+elim (exists_forall_le f n)
+ [left.split
+ [assumption
+ |apply f_max_true.assumption
+ ]
+ |right.split
+ [assumption
+ |apply max_O.assumption
+ ]
+ ]
+qed.
+
+theorem false_to_lt_max: \forall f,n,m.O < n \to
+f n = false \to max m f \le n \to max m f < n.
+intros.
+elim (le_to_or_lt_eq ? ? H2)
+ [assumption
+ |elim (exists_max_forall_false f m)
+ [elim H4.
+ apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H6.
+ rewrite > H3.
+ assumption
+ |elim H4.
+ rewrite > H6.
+ assumption
+ ]
+ ]
+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.
+elim (max_S_max f n1) in H1 ⊢ %.
+elim H1.
absurd (m \le S n1).assumption.
-apply lt_to_not_le.rewrite < H6.assumption.
-elim H3.
+apply lt_to_not_le.rewrite < H5.assumption.
+elim H1.
apply (le_n_Sm_elim m n1 H2).
intro.
-apply H.rewrite < H6.assumption.
+apply H.rewrite < H5.assumption.
apply le_S_S_to_le.assumption.
-intro.rewrite > H7.assumption.
+intro.rewrite > H6.assumption.
+qed.
+
+theorem f_false_to_le_max: \forall f,n,p. (∃i:nat.i≤n∧f i=true) \to
+(\forall m. p < m \to f m = false)
+\to max n f \le p.
+intros.
+apply not_lt_to_le.intro.
+apply not_eq_true_false.
+rewrite < (H1 ? H2).
+apply sym_eq.
+apply f_max_true.
+assumption.
qed.
definition max_spec \def \lambda f:nat \to bool.\lambda n,m: nat.
reflexivity.
qed.
+theorem f_min_true: \forall f:nat \to bool. \forall m:nat.
+(\exists i. le i m \land f i = true) \to
+f (min m f) = true.
+intros.unfold min.
+apply f_min_aux_true.
+elim H.clear H.elim H1.clear H1.
+apply (ex_intro ? ? a).
+split
+ [split
+ [apply le_O_n
+ |rewrite < plus_n_O.assumption
+ ]
+ |assumption
+ ]
+qed.
+
theorem lt_min_aux_to_false : \forall f:nat \to bool.
\forall n,off,m:nat. n \leq m \to m < (min_aux off n f) \to f m = false.
intros 3.
-generalize in match n; clear n.
+generalize in match n; clear n;
elim off.absurd (le n1 m).assumption.
apply lt_to_not_le.rewrite < (min_aux_O_f f n1).assumption.
elim (le_to_or_lt_eq ? ? H1);
lemma le_min_aux : \forall f:nat \to bool.
\forall n,off:nat. n \leq (min_aux off n f).
intros 3.
-generalize in match n. clear n.
-elim off.
+elim off in n ⊢ %.
rewrite > (min_aux_O_f f n1).apply le_n.
elim (min_aux_S f n n1).
elim H1.rewrite > H3.apply le_n.
theorem le_min_aux_r : \forall f:nat \to bool.
\forall n,off:nat. (min_aux off n f) \le n+off.
intros.
-generalize in match n. clear n.
-elim off.simplify.
+elim off in n ⊢ %.simplify.
elim (f n1).simplify.rewrite < plus_n_O.apply le_n.
simplify.rewrite < plus_n_O.apply le_n.
simplify.elim (f n1).