--- /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 *)
+(* *)
+(**************************************************************************)
+
+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.
+
+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.
+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 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.
+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.
+
+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.
+m \le n \land (f m)=true \land (\forall i. m < i \to i \le n \to (f i = false)).
+
+theorem max_spec_to_max: \forall f:nat \to bool.\forall n,m:nat.
+max_spec f n m \to max n f = m.
+intros 2.
+elim n
+ [elim H.elim H1.apply (le_n_O_elim ? H3).
+ apply max_O_f
+ |elim H1.
+ elim (max_S_max f n1)
+ [elim H4.
+ rewrite > H6.
+ apply le_to_le_to_eq
+ [apply not_lt_to_le.
+ unfold Not.intro.
+ apply not_eq_true_false.
+ rewrite < H5.
+ apply H3
+ [assumption|apply le_n]
+ |elim H2.assumption
+ ]
+ |elim H4.
+ rewrite > H6.
+ apply H.
+ elim H2.
+ split
+ [split
+ [elim (le_to_or_lt_eq ? ? H7)
+ [apply le_S_S_to_le.assumption
+ |apply False_ind.
+ apply not_eq_true_false.
+ rewrite < H8.
+ rewrite > H9.
+ assumption
+ ]
+ |assumption
+ ]
+ |intros.
+ apply H3
+ [assumption|apply le_S.assumption]
+ ]
+ ]
+ ]
+qed.
+
+let rec min_aux off n f \def
+ match f n with
+ [ true \Rightarrow n
+ | false \Rightarrow
+ match off with
+ [ O \Rightarrow n
+ | (S p) \Rightarrow min_aux p (S n) f]].
+
+definition min : nat \to (nat \to bool) \to nat \def
+\lambda n.\lambda f. min_aux n O f.
+
+theorem min_aux_O_f: \forall f:nat \to bool. \forall i :nat.
+min_aux O i f = i.
+intros.simplify.
+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) = true \land min_aux (S i) n f = n) \lor
+((f n) = false \land min_aux (S i) n f = min_aux i (S n) f).
+intros.simplify.elim (f n).
+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 i \land le i (off + 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.assumption.
+simplify.
+apply (bool_ind (\lambda b:bool.
+(f m = b) \to (f (match b in bool with
+[ true \Rightarrow m
+| false \Rightarrow (min_aux n (S m) f)])) = true)).
+intro; assumption.
+intro. apply H.
+elim H1.elim H3.elim H4.
+elim (le_to_or_lt_eq ? a H6).
+apply (ex_intro nat ? a).
+split.split.
+assumption.
+rewrite < plus_n_Sm; 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 \leq m \to m < (min_aux off n f) \to f m = false.
+intros 3.
+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);
+ [ unfold lt in H3;
+ apply (H (S n1));
+ [ assumption
+ | elim (min_aux_S f n n1);
+ [ elim H4;
+ elim (not_le_Sn_n n1);
+ rewrite > H6 in H2;
+ apply (lt_to_le (S n1) n1 ?).
+ apply (le_to_lt_to_lt (S n1) m n1 ? ?);
+ [apply (H3).
+ |apply (H2).
+ ]
+ | elim H4;
+ rewrite < H6;
+ assumption
+ ]
+ ]
+ | rewrite < H3 in H2 ⊢ %.
+ elim (min_aux_S f n n1);
+ [ elim H4;
+ rewrite > H6 in H2;
+ unfold lt in H2;
+ elim (not_le_Sn_n ? H2)
+ | elim H4;
+ assumption
+ ]
+ ]
+qed.
+
+
+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.
+rewrite > (min_aux_O_f f n1).apply le_n.
+elim (min_aux_S f n n1).
+elim H1.rewrite > H3.apply le_n.
+elim H1.rewrite > H3.
+apply (transitive_le ? (S n1));
+ [ apply le_n_Sn
+ | apply (H (S n1))
+ ]
+qed.
+
+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 (f n1).simplify.rewrite < plus_n_O.apply le_n.
+simplify.rewrite < plus_n_O.apply le_n.
+simplify.elim (f n1).
+simplify.
+apply (le_plus_n_r (S n) n1).
+simplify.rewrite < plus_n_Sm.
+apply (H (S n1)).
+qed.
\ No newline at end of file
projects / helm.git / blobdiff