+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/orders".
-
-include "nat/nat.ma".
-include "higher_order_defs/ordering.ma".
-
-(* definitions *)
-inductive le (n:nat) : nat \to Prop \def
- | le_n : le n n
- | le_S : \forall m:nat. le n m \to le n (S m).
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'less or equal to'" 'leq x y = (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'neither less nor equal to'" 'nleq x y =
- (cic:/matita/logic/connectives/Not.con
- (cic:/matita/nat/orders/le.ind#xpointer(1/1) x y)).
-
-definition lt: nat \to nat \to Prop \def
-\lambda n,m:nat.(S n) \leq m.
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'less than'" 'lt x y = (cic:/matita/nat/orders/lt.con x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'not less than'" 'nless x y =
- (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/lt.con x y)).
-
-definition ge: nat \to nat \to Prop \def
-\lambda n,m:nat.m \leq n.
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'greater or equal to'" 'geq x y = (cic:/matita/nat/orders/ge.con x y).
-
-definition gt: nat \to nat \to Prop \def
-\lambda n,m:nat.m<n.
-
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'greater than'" 'gt x y = (cic:/matita/nat/orders/gt.con x y).
-(*CSC: the URI must disappear: there is a bug now *)
-interpretation "natural 'not greater than'" 'ngtr x y =
- (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/orders/gt.con x y)).
-
-theorem transitive_le : transitive nat le.
-unfold transitive.intros.elim H1.
-assumption.
-apply le_S.assumption.
-qed.
-
-theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
-\def transitive_le.
-
-theorem transitive_lt: transitive nat lt.
-unfold transitive.unfold lt.intros.elim H1.
-apply le_S. assumption.
-apply le_S.assumption.
-qed.
-
-theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
-\def transitive_lt.
-
-theorem le_S_S: \forall n,m:nat. n \leq m \to S n \leq S m.
-intros.elim H.
-apply le_n.
-apply le_S.assumption.
-qed.
-
-theorem le_O_n : \forall n:nat. O \leq n.
-intros.elim n.
-apply le_n.apply
-le_S. assumption.
-qed.
-
-theorem le_n_Sn : \forall n:nat. n \leq S n.
-intros. apply le_S.apply le_n.
-qed.
-
-theorem le_pred_n : \forall n:nat. pred n \leq n.
-intros.elim n.
-simplify.apply le_n.
-simplify.apply le_n_Sn.
-qed.
-
-theorem le_S_S_to_le : \forall n,m:nat. S n \leq S m \to n \leq m.
-intros.change with (pred (S n) \leq pred (S m)).
-elim H.apply le_n.apply (trans_le ? (pred n1)).assumption.
-apply le_pred_n.
-qed.
-
-theorem leS_to_not_zero : \forall n,m:nat. S n \leq m \to not_zero m.
-intros.elim H.exact I.exact I.
-qed.
-
-(* not le *)
-theorem not_le_Sn_O: \forall n:nat. S n \nleq O.
-intros.unfold Not.simplify.intros.apply (leS_to_not_zero ? ? H).
-qed.
-
-theorem not_le_Sn_n: \forall n:nat. S n \nleq n.
-intros.elim n.apply not_le_Sn_O.unfold Not.simplify.intros.cut (S n1 \leq n1).
-apply H.assumption.
-apply le_S_S_to_le.assumption.
-qed.
-
-(* le to lt or eq *)
-theorem le_to_or_lt_eq : \forall n,m:nat.
-n \leq m \to n < m \lor n = m.
-intros.elim H.
-right.reflexivity.
-left.unfold lt.apply le_S_S.assumption.
-qed.
-
-(* not eq *)
-theorem lt_to_not_eq : \forall n,m:nat. n<m \to n \neq m.
-unfold Not.intros.cut ((le (S n) m) \to False).
-apply Hcut.assumption.rewrite < H1.
-apply not_le_Sn_n.
-qed.
-
-(* le vs. lt *)
-theorem lt_to_le : \forall n,m:nat. n<m \to n \leq m.
-simplify.intros.unfold lt in H.elim H.
-apply le_S. apply le_n.
-apply le_S. assumption.
-qed.
-
-theorem lt_S_to_le : \forall n,m:nat. n < S m \to n \leq m.
-simplify.intros.
-apply le_S_S_to_le.assumption.
-qed.
-
-theorem not_le_to_lt: \forall n,m:nat. n \nleq m \to m<n.
-intros 2.
-apply (nat_elim2 (\lambda n,m.n \nleq m \to m<n)).
-intros.apply (absurd (O \leq n1)).apply le_O_n.assumption.
-unfold Not.unfold lt.intros.apply le_S_S.apply le_O_n.
-unfold Not.unfold lt.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
-assumption.
-qed.
-
-theorem lt_to_not_le: \forall n,m:nat. n<m \to m \nleq n.
-unfold Not.unfold lt.intros 3.elim H.
-apply (not_le_Sn_n n H1).
-apply H2.apply lt_to_le. apply H3.
-qed.
-
-theorem not_lt_to_le: \forall n,m:nat. Not (lt n m) \to le m n.
-simplify.intros.
-apply lt_S_to_le.
-apply not_le_to_lt.exact H.
-qed.
-
-theorem le_to_not_lt: \forall n,m:nat. le n m \to Not (lt m n).
-intros.
-change with (Not (le (S m) n)).
-apply lt_to_not_le.unfold lt.
-apply le_S_S.assumption.
-qed.
-
-(* le elimination *)
-theorem le_n_O_to_eq : \forall n:nat. n \leq O \to O=n.
-intro.elim n.reflexivity.
-apply False_ind.
-apply not_le_Sn_O.
-goal 17. apply H1.
-qed.
-
-theorem le_n_O_elim: \forall n:nat.n \leq O \to \forall P: nat \to Prop.
-P O \to P n.
-intro.elim n.
-assumption.
-apply False_ind.
-apply (not_le_Sn_O ? H1).
-qed.
-
-theorem le_n_Sm_elim : \forall n,m:nat.n \leq S m \to
-\forall P:Prop. (S n \leq S m \to P) \to (n=S m \to P) \to P.
-intros 4.elim H.
-apply H2.reflexivity.
-apply H3. apply le_S_S. assumption.
-qed.
-
-(* lt and le trans *)
-theorem lt_to_le_to_lt: \forall n,m,p:nat. lt n m \to le m p \to lt n p.
-intros.elim H1.
-assumption.unfold lt.apply le_S.assumption.
-qed.
-
-theorem le_to_lt_to_lt: \forall n,m,p:nat. le n m \to lt m p \to lt n p.
-intros 4.elim H.
-assumption.apply H2.unfold lt.
-apply lt_to_le.assumption.
-qed.
-
-theorem ltn_to_ltO: \forall n,m:nat. lt n m \to lt O m.
-intros.apply (le_to_lt_to_lt O n).
-apply le_O_n.assumption.
-qed.
-
-theorem lt_O_n_elim: \forall n:nat. lt O n \to
-\forall P:nat\to Prop. (\forall m:nat.P (S m)) \to P n.
-intro.elim n.apply False_ind.exact (not_le_Sn_O O H).
-apply H2.
-qed.
-
-(* other abstract properties *)
-theorem antisymmetric_le : antisymmetric nat le.
-unfold antisymmetric.intros 2.
-apply (nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m))).
-intros.apply le_n_O_to_eq.assumption.
-intros.apply False_ind.apply (not_le_Sn_O ? H).
-intros.apply eq_f.apply H.
-apply le_S_S_to_le.assumption.
-apply le_S_S_to_le.assumption.
-qed.
-
-theorem antisym_le: \forall n,m:nat. n \leq m \to m \leq n \to n=m
-\def antisymmetric_le.
-
-theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
-intros.
-apply (nat_elim2 (\lambda n,m.decidable (n \leq m))).
-intros.unfold decidable.left.apply le_O_n.
-intros.unfold decidable.right.exact (not_le_Sn_O n1).
-intros 2.unfold decidable.intro.elim H.
-left.apply le_S_S.assumption.
-right.unfold Not.intro.apply H1.apply le_S_S_to_le.assumption.
-qed.
-
-theorem decidable_lt: \forall n,m:nat. decidable (n < m).
-intros.exact (decidable_le (S n) m).
-qed.
-
-(* well founded induction principles *)
-
-theorem nat_elim1 : \forall n:nat.\forall P:nat \to Prop.
-(\forall m.(\forall p. (p \lt m) \to P p) \to P m) \to P n.
-intros.cut (\forall q:nat. q \le n \to P q).
-apply (Hcut n).apply le_n.
-elim n.apply (le_n_O_elim q H1).
-apply H.
-intros.apply False_ind.apply (not_le_Sn_O p H2).
-apply H.intros.apply H1.
-cut (p < S n1).
-apply lt_S_to_le.assumption.
-apply (lt_to_le_to_lt p q (S n1) H3 H2).
-qed.
-
-(* some properties of functions *)
-
-definition increasing \def \lambda f:nat \to nat.
-\forall n:nat. f n < f (S n).
-
-theorem increasing_to_monotonic: \forall f:nat \to nat.
-increasing f \to monotonic nat lt f.
-unfold monotonic.unfold lt.unfold increasing.unfold lt.intros.elim H1.apply H.
-apply (trans_le ? (f n1)).
-assumption.apply (trans_le ? (S (f n1))).
-apply le_n_Sn.
-apply H.
-qed.
-
-theorem le_n_fn: \forall f:nat \to nat. (increasing f)
-\to \forall n:nat. n \le (f n).
-intros.elim n.
-apply le_O_n.
-apply (trans_le ? (S (f n1))).
-apply le_S_S.apply H1.
-simplify in H. unfold increasing in H.unfold lt in H.apply H.
-qed.
-
-theorem increasing_to_le: \forall f:nat \to nat. (increasing f)
-\to \forall m:nat. \exists i. m \le (f i).
-intros.elim m.
-apply (ex_intro ? ? O).apply le_O_n.
-elim H1.
-apply (ex_intro ? ? (S a)).
-apply (trans_le ? (S (f a))).
-apply le_S_S.assumption.
-simplify in H.unfold increasing in H.unfold lt in H.
-apply H.
-qed.
-
-theorem increasing_to_le2: \forall f:nat \to nat. (increasing f)
-\to \forall m:nat. (f O) \le m \to
-\exists i. (f i) \le m \land m <(f (S i)).
-intros.elim H1.
-apply (ex_intro ? ? O).
-split.apply le_n.apply H.
-elim H3.elim H4.
-cut ((S n1) < (f (S a)) \lor (S n1) = (f (S a))).
-elim Hcut.
-apply (ex_intro ? ? a).
-split.apply le_S. assumption.assumption.
-apply (ex_intro ? ? (S a)).
-split.rewrite < H7.apply le_n.
-rewrite > H7.
-apply H.
-apply le_to_or_lt_eq.apply H6.
-qed.
projects / helm.git / blobdiff