+++ /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.
projects / helm.git / blobdiff