set "baseuri" "cic:/matita/nat/orders".
-include "nat/plus.ma".
+include "nat/nat.ma".
include "higher_order_defs/ordering.ma".
(* definitions *)
| 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).
-alias symbol "leq" (instance 0) = "natural 'less or equal to'".
+(*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.
-simplify.intros.elim H1.
+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.
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.
+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.
qed.
(* not le *)
-theorem not_le_Sn_O: \forall n:nat. \lnot (S n \leq O).
-intros.simplify.intros.apply leS_to_not_zero ? ? H.
+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. \lnot (S n \leq n).
-intros.elim n.apply not_le_Sn_O.simplify.intros.cut S n1 \leq n1.
+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.
-(* ??? this needs not le *)
-theorem S_pred: \forall n:nat.O<n \to n=S (pred n).
-intro.elim n.apply False_ind.exact not_le_Sn_O O H.
-apply eq_f.apply pred_Sn.
+(* 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.elim H.
+simplify.intros.unfold lt in H.elim H.
apply le_S. apply le_n.
apply le_S. assumption.
qed.
-theorem not_le_to_lt: \forall n,m:nat. \lnot (n \leq m) \to m<n.
+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.\lnot (n \leq m) \to m<n).
-intros.apply absurd (O \leq n1).apply le_O_n.assumption.
-simplify.intros.apply le_S_S.apply le_O_n.
-simplify.intros.apply le_S_S.apply H.intros.apply H1.apply le_S_S.
+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 \lnot (m \leq n).
-simplify.intros 3.elim H.
-apply not_le_Sn_n n H1.
+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.
-(* non si applica not_le_Sn_O *)
-apply (not_le_Sn_O ? H1).
+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.
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.
-simplify.intros 2.
-apply nat_elim2 (\lambda n,m.(n \leq m \to m \leq n \to n=m)).
+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 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.
theorem decidable_le: \forall n,m:nat. decidable (n \leq m).
intros.
-apply nat_elim2 (\lambda n,m.decidable (n \leq m)).
-intros.simplify.left.apply le_O_n.
-intros.simplify.right.exact not_le_Sn_O n1.
-intros 2.simplify.intro.elim H.
+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.intro.apply H1.apply le_S_S_to_le.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