lemma le_max: ∀n,m.m ≤ max n m.
intros; unfold max; apply leb_elim; simplify; intros; [assumption] apply le_n;
-qed.
+qed.
+
+lemma max_le_l: ∀n,m,z.max n m ≤ z → n ≤ z.
+intros 3; unfold max; apply leb_elim; simplify; intros; [assumption]
+apply lt_to_le; apply (lt_to_le_to_lt ???? H1);
+apply not_le_to_lt; assumption;
+qed.
+
+lemma sym_max: ∀n,m.max n m = max m n.
+intros; apply (nat_elim2 ???? n m); simplify; intros;
+[1: elim n1; [reflexivity] rewrite < H in ⊢ (? ? ? (? %));
+ simplify; rewrite > H; reflexivity;
+|2: reflexivity
+|3: apply leb_elim; apply leb_elim; simplify;
+ [1: intros; apply le_to_le_to_eq; apply le_S_S;assumption;
+ |2,3: intros; reflexivity;
+ |4: intros; unfold max in H;
+ rewrite > (?:leb n1 m1 = false) in H; [2:
+ apply lt_to_leb_false; apply not_le_to_lt; assumption;]
+ rewrite > (?:leb m1 n1 = false) in H; [2:
+ apply lt_to_leb_false; apply not_le_to_lt; assumption;]
+ apply eq_f; assumption;]]
+qed.
+
+lemma max_le_r: ∀n,m,z.max n m ≤ z → m ≤ z.
+intros; rewrite > sym_max in H; apply (max_le_l ??? H);
+qed.
+
definition hide ≝ λT:Type.λx:T.x.
notation < "\blacksquare" non associative with precedence 50 for @{'hide}.
interpretation "hide" 'hide =
(cic:/matita/dama/property_sigma/hide.con _ _).
+interpretation "hide2" 'hide =
+ (cic:/matita/dama/property_sigma/hide.con _ _ _).
+
+definition inject ≝ λP.λa:nat.λp:P a. ex_introT ? P ? p.
+coercion cic:/matita/dama/property_sigma/inject.con 0 1.
+definition eject ≝ λP.λc: ∃n:nat.P n. match c with [ ex_introT w _ ⇒ w].
+coercion cic:/matita/dama/property_sigma/eject.con.
(* Lemma 3.6 *)
lemma sigma_cauchy:
- ∀O:ordered_uniform_space.property_sigma O →
- ∀a:sequence O.∀l:O.a ↑ l → a is_cauchy → a uniform_converges l.
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀a:sequence C.∀l:C.a ↑ l → a is_cauchy → a uniform_converges l.
intros 8; cases H1; cases H5; clear H5;
cases (H ? H3); cases H5; clear H5;
-letin m ≝ (? : sequence nat_ordered_set); [
- apply (hide (nat→nat)); intro i; elim i (i' Rec);
- [1: apply (hide nat);cases (H2 ? (H8 0)) (k _); apply k;
- |2: apply (max (hide nat ?) (S Rec)); cases (H2 ? (H8 (S i'))) (k Hk);apply k]]
-cut (m is_strictly_increasing) as Hm; [2:
- intro n; change with (S (m n) ≤ m (S n)); unfold m; whd in ⊢ (? ? %); apply (le_max ? (S (m n)));]
-lapply (selection ?? Hm a l H1) as H10;
-lapply (H9 ?? H10) as H11;
-[1: exists [apply (m 0)] intros;
- apply (ous_convex ?? H3 ? H11 (H6 (m 0)));
- simplify; repeat split;
-
-
-
\ No newline at end of file
+letin m ≝ (hide ? (let rec aux (i:nat) : nat ≝
+ match i with
+ [ O ⇒ match H2 (w i) ? with [ ex_introT k _ ⇒ k ]
+ | S i' ⇒ max (match H2 (w i) ? with [ ex_introT k _ ⇒ k ]) (S (aux i'))
+ ] in aux
+ :
+ ∀z:nat.∃k:nat.∀i,j,l.k ≤ i → k ≤ j → l ≤ z → w l 〈a i, a j〉));
+ [1,2:apply H8;
+ |3: intros 3; cases (H2 (w n) (H8 n)); simplify in ⊢ (? (? % ?) ?→?);
+ simplify in ⊢ (?→? (? % ?) ?→?);
+ intros; lapply (H10 i j) as H14;
+ [2: apply (max_le_l ??? H11);|3:apply (max_le_l ??? H12);]
+ cases (le_to_or_lt_eq ?? H13); [2: destruct H15; destruct H5; assumption]
+ generalize in match H11; generalize in match H12;
+ cases (aux n1); simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros;
+ apply H16; [3: apply le_S_S_to_le; assumption]
+ apply lt_to_le; apply (max_le_r w1); assumption;
+ |4: intros; clear H11; rewrite > H5 in H10;
+ rewrite < (le_n_O_to_eq ? H14); apply H10; assumption;]
+cut (((m : nat→nat) : sequence nat_ordered_set) is_strictly_increasing) as Hm; [2:
+ intro n; change with (S (m n) ≤ m (S n)); unfold m;
+ whd in ⊢ (? ? %); apply (le_max ? (S (m n)));]
+cut (((m : nat→nat) : sequence nat_ordered_set) is_increasing) as Hm1; [2:
+ intro n; intro L; change in L with (m (S n) < m n);
+ lapply (Hm n) as L1; change in L1 with (m n < m (S n));
+ lapply (trans_lt ??? L L1) as L3; apply (not_le_Sn_n ? L3);]
+clearbody m;
+lapply (selection ?? Hm a l H1) as H10;
+lapply (H9 ?? H10) as H11; [
+ exists [apply (m 0:nat)] intros;
+ apply (ous_convex ?? H3 ? H11 (H6 (m 0)));
+ simplify; repeat split; [intro X; cases (os_coreflexive ?? X)|2,3:apply H6;]
+ change with (a (m O) ≤ a i);
+ apply (trans_increasing ?? H4); intro; whd in H12;
+ apply (not_le_Sn_n i); apply (transitive_le ??? H12 H5)]
+clear H10; intros (p q r); change with (w p 〈a (m q),a (m r)〉);
+generalize in match (refl_eq nat (m q));
+generalize in match (m q) in ⊢ (? ? ? % → %); intro X; cases X; clear X;
+intros; simplify in H12:(? ? ? %); simplify in ⊢ (? ? (? ? ? % ?));
+generalize in match (refl_eq nat (m r));
+generalize in match (m r) in ⊢ (? ? ? % → %); intro X; cases X; clear X;
+intros; simplify in H14:(? ? ? %); simplify in ⊢ (? ? (? ? ? ? %));
+generalize in match (refl_eq nat (m p));
+generalize in match (m p) in ⊢ (? ? ? % → %); intro X; cases X; clear X;
+intros; simplify in H16:(? ? ? %);
+apply H15; [3: apply le_n] destruct H16; destruct H14; destruct H12; clear H11 H13 H15;
+[1: lapply (trans_increasing ?? Hm1 p q) as T; [apply not_lt_to_le; apply T;]
+ apply (le_to_not_lt p q H5);
+|2: lapply (trans_increasing ?? Hm1 p r) as T; [apply not_lt_to_le; apply T;]
+ apply (le_to_not_lt p r H10);]
+qed.
\ No newline at end of file
alias symbol "nleq" = "Ordered set excess".
alias symbol "leq" = "Ordered set less or equal than".
lemma trans_increasing:
+ ∀C:ordered_set.∀a:sequence C.a is_increasing → ∀n,m:nat_ordered_set. n ≤ m → a n ≤ a m.
+intros 5 (C a Hs n m); elim m; [
+ rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
+ intro X; cases (os_coreflexive ?? X);]
+cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
+[2: rewrite > H2; intro; cases (os_coreflexive ?? H1);
+|1: apply (le_transitive ???? (H ?) (Hs ?));
+ intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
+qed.
+
+lemma trans_increasing_exc:
∀C:ordered_set.∀a:sequence C.a is_increasing → ∀n,m:nat_ordered_set. m ≰ n → a n ≤ a m.
intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
intro; apply H;
∀C:ordered_set.∀m:sequence nat_ordered_set.m is_strictly_increasing →
∀a:sequence C.∀u.a ↑ u → (λx.a (m x)) ↑ u.
intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split;
-[1: intro n; simplify; apply trans_increasing; [assumption] apply (Hm n);
+[1: intro n; simplify; apply trans_increasing_exc; [assumption] apply (Hm n);
|2: intro n; simplify; apply Uu;
|3: intros (y Hy); simplify; cases (Hu ? Hy);
cases (strictly_increasing_reaches C ? Hm w);
exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [2:assumption]
- cases (trans_increasing C ? Ia ?? H1); assumption;]
+ cases (trans_increasing_exc C ? Ia ?? H1); assumption;]
qed.
(* Definition 2.7 *)
projects / helm.git / commitdiff