From 179574c117d34a39cebeaa66673cda83974e135a Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 20 Nov 2008 16:04:25 +0000 Subject: [PATCH] ... --- .../dama/dama/models/discrete_uniformity.ma | 48 ------ .../models/increasing_supremum_stabilizes.ma | 138 ------------------ .../contribs/dama/dama/models/nat_lebesgue.ma | 26 ---- .../dama/dama/models/nat_order_continuous.ma | 39 ----- .../dama/dama/models/nat_ordered_uniform.ma | 32 ---- .../contribs/dama/dama/models/nat_uniform.ma | 22 --- 6 files changed, 305 deletions(-) delete mode 100644 helm/software/matita/contribs/dama/dama/models/discrete_uniformity.ma delete mode 100644 helm/software/matita/contribs/dama/dama/models/increasing_supremum_stabilizes.ma delete mode 100644 helm/software/matita/contribs/dama/dama/models/nat_lebesgue.ma delete mode 100644 helm/software/matita/contribs/dama/dama/models/nat_order_continuous.ma delete mode 100644 helm/software/matita/contribs/dama/dama/models/nat_ordered_uniform.ma delete mode 100644 helm/software/matita/contribs/dama/dama/models/nat_uniform.ma diff --git a/helm/software/matita/contribs/dama/dama/models/discrete_uniformity.ma b/helm/software/matita/contribs/dama/dama/models/discrete_uniformity.ma deleted file mode 100644 index 76461f3f4..000000000 --- a/helm/software/matita/contribs/dama/dama/models/discrete_uniformity.ma +++ /dev/null @@ -1,48 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "uniform.ma". -include "bishop_set_rewrite.ma". - -definition discrete_uniform_space_of_bishop_set:bishop_set → uniform_space. -intro C; apply (mk_uniform_space C); -[1: intro; - alias symbol "pi2" = "pair pi2". - alias symbol "pi1" = "pair pi1". - alias symbol "and" = "logical and". - apply (∃d:unit.∀n:C squareB.(\fst n ≈ \snd n → P n) ∧ (P n → \fst n ≈ \snd n)); -|2: intros 4 (U H x Hx); simplify in Hx; - cases H (_ H1); cases (H1 x); apply H2; apply Hx; -|3: intros; cases H (_ PU); cases H1 (_ PV); - exists[apply (λx.U x ∧ V x)] split; - [1: exists[apply something] intro; cases (PU n); cases (PV n); - split; intros; try split;[apply H2;|apply H4] try assumption; - apply H3; cases H6; assumption; - |2: simplify; intros; assumption;] -|4: intros; cases H (_ PU); exists [apply U] split; - [1: exists [apply something] intro n; cases (PU n); - split; intros; try split;[apply H1;|apply H2] assumption; - |2: intros 2 (x Hx); cases Hx; cases H1; clear H1; - cases (PU 〈(\fst x),x1〉); lapply (H4 H2) as H5; simplify in H5; - cases (PU 〈x1,(\snd x)〉); lapply (H7 H3) as H8; simplify in H8; - generalize in match H5; generalize in match H8; - generalize in match (PU x); clear H6 H7 H1 H4 H2 H3 H5 H8; - cases x; simplify; intros; cases H1; clear H1; apply H4; - apply (eq_trans ???? H3 H2);] -|5: intros; cases H (_ H1); split; cases x; - [2: cases (H1 〈b,b1〉); intro; apply H2; cases (H1 〈b1,b〉); - lapply (H6 H4) as H7; apply eq_sym; apply H7; - |1: cases (H1 〈b1,b〉); intro; apply H2; cases (H1 〈b,b1〉); - lapply (H6 H4) as H7; apply eq_sym; apply H7;]] -qed. diff --git a/helm/software/matita/contribs/dama/dama/models/increasing_supremum_stabilizes.ma b/helm/software/matita/contribs/dama/dama/models/increasing_supremum_stabilizes.ma deleted file mode 100644 index 58626ff15..000000000 --- a/helm/software/matita/contribs/dama/dama/models/increasing_supremum_stabilizes.ma +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "models/nat_uniform.ma". -include "supremum.ma". -include "nat/le_arith.ma". -include "russell_support.ma". - -lemma hint1: - ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s)) - → sequence (hos_carr (os_l (segment_ordered_set nat_ordered_set s))). -intros; assumption; -qed. - -coercion hint1 nocomposites. - -alias symbol "pi1" = "exT \fst". -alias symbol "N" = "ordered set N". -lemma increasing_supremum_stabilizes: - ∀sg:‡ℕ.∀a:sequence {[sg]}. - a is_increasing → - ∀X.X is_supremum a → ∃i.∀j.i ≤ j → \fst X = \fst (a j). -intros 4; cases X (x Hx); clear X; letin X ≝ ≪x,Hx≫; -fold normalize X; intros; cases H1; -alias symbol "N" = "Natural numbers". -letin spec ≝ (λi,j:ℕ.(𝕦_sg ≤ i ∧ x = \fst (a j)) ∨ (i < 𝕦_sg ∧ x + i ≤ 𝕦_sg + \fst (a j))); -(* x - aj <= max 0 (u - i) *) -letin m ≝ (hide ? ( - let rec aux i ≝ - match i with - [ O ⇒ O - | S m ⇒ - let pred ≝ aux m in - let apred ≝ a pred in - match cmp_nat x (\fst apred) with - [ cmp_le _ ⇒ pred - | cmp_gt nP ⇒ \fst (H3 apred ?)]] - in aux - : - ∀i:nat.∃j:nat.spec i j));[whd; apply nP;] unfold spec in aux ⊢ %; -[3: unfold X in H2; clear H4 n aux spec H3 H1 H X; - cases (cases_in_segment ??? Hx); - elim 𝕦_sg in H1 ⊢ %; intros (a Hs H); - [1: left; split; [apply le_n] - generalize in match H; - generalize in match Hx; - rewrite > (?:x = O); - [2: cases Hx; lapply (os_le_to_nat_le ?? H1); - apply (symmetric_eq nat O x ?).apply (le_n_O_to_eq x ?).apply (Hletin). - |1: intros; unfold Type_of_ordered_set in sg; - lapply (H2 O) as K; lapply (sl2l_ ?? (a O) ≪x,Hx≫ K) as P; - simplify in P:(???%); lapply (le_transitive ??? P H1) as W; - lapply (os_le_to_nat_le ?? W) as R; apply (le_n_O_to_eq (\fst (a O)) R);] - |2: right; cases Hx; rewrite > (sym_plus x O); split; [apply le_S_S; apply le_O_n]; - apply (trans_le ??? (os_le_to_nat_le ?? H3)); - apply le_plus_n_r;] -|2: clear H6; cut (x = \fst (a (aux n1))); [2: - cases (le_to_or_lt_eq ?? H5); [2: assumption] - cases (?:False); apply (H2 (aux n1) H6);] clear H5; - generalize in match Hcut; clear Hcut; intro H5; -|1: clear H6] -[2,1: - cases (aux n1) in H5 ⊢ %; intros; - change in match (a ≪w,H5≫) in H6 ⊢ % with (a w); - cases H5; clear H5; cases H7; clear H7; - [1: left; split; [ apply (le_S ?? H5); | assumption] - |3: cases (?:False); rewrite < H8 in H6; apply (not_le_Sn_n ? H6); - |*: cases (cmp_nat 𝕦_sg (S n1)); - [1,3: left; split; [1,3: assumption |2: assumption] - cut (𝕦_sg = S n1); [2: apply le_to_le_to_eq; assumption ] - clear H7 H5 H4;rewrite > Hcut in H8:(? ? (? % ?)); clear Hcut; - cut (x = S (\fst (a w))); - [2: apply le_to_le_to_eq; [2: assumption] - change in H8 with (x + n1 ≤ S (n1 + \fst (a w))); - rewrite > plus_n_Sm in H8; rewrite > sym_plus in H8; - apply (le_plus_to_le ??? H8);] - cases (H3 (a w) H6); - change with (x = \fst (a w1)); - change in H4 with (\fst (a w) < \fst (a w1)); - apply le_to_le_to_eq; [ rewrite > Hcut; assumption ] - apply (os_le_to_nat_le (\fst (a w1)) x (H2 w1)); - |*: right; split; try assumption; - [1: rewrite > sym_plus in ⊢ (? ? %); - rewrite < H6; apply le_plus_r; assumption; - |2: cases (H3 (a w) H6); - change with (x + S n1 ≤ 𝕦_sg + \fst (a w1));rewrite < plus_n_Sm; - apply (trans_le ??? (le_S_S ?? H8)); rewrite > plus_n_Sm; - apply (le_plus ???? (le_n ?) H9);]]]] -clearbody m; unfold spec in m; clear spec; -letin find ≝ ( - let rec find i u on u : nat ≝ - match u with - [ O ⇒ (m i:nat) - | S w ⇒ match eqb (\fst (a (m i))) x with - [ true ⇒ (m i:nat) - | false ⇒ find (S i) w]] - in find - : - ∀i,bound.∃j.i + bound = 𝕦_sg → x = \fst (a j)); -[1: cases (find (S n) n2); intro; change with (x = \fst (a w)); - apply H6; rewrite < H7; simplify; apply plus_n_Sm; -|2: intros; rewrite > (eqb_true_to_eq ?? H5); reflexivity -|3: intros; rewrite > sym_plus in H5; rewrite > H5; clear H5 H4 n n1; - cases (m 𝕦_sg); cases H4; clear H4; cases H5; clear H5; [assumption] - cases (not_le_Sn_n ? H4)] -clearbody find; cases (find O 𝕦_sg); -exists [apply w]; intros; change with (x = \fst (a j)); -rewrite > (H4 ?); [2: reflexivity] -apply le_to_le_to_eq; -[1: apply os_le_to_nat_le; - apply (trans_increasing ? H ? ? (nat_le_to_os_le ?? H5)); -|2: apply (trans_le ? x ?);[apply os_le_to_nat_le; apply (H2 j);] - rewrite < (H4 ?); [2: reflexivity] apply le_n;] -qed. - -lemma hint2: - ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s)) - → sequence (hos_carr (os_r (segment_ordered_set nat_ordered_set s))). -intros; assumption; -qed. - -coercion hint2 nocomposites. - -alias symbol "N" = "ordered set N". -axiom increasing_supremum_stabilizes_r: - ∀s:‡ℕ.∀a:sequence {[s]}.a is_decreasing → - ∀x.x is_infimum a → ∃i.∀j.i ≤ j → \fst x = \fst (a j). diff --git a/helm/software/matita/contribs/dama/dama/models/nat_lebesgue.ma b/helm/software/matita/contribs/dama/dama/models/nat_lebesgue.ma deleted file mode 100644 index d5e65215e..000000000 --- a/helm/software/matita/contribs/dama/dama/models/nat_lebesgue.ma +++ /dev/null @@ -1,26 +0,0 @@ - (**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "lebesgue.ma". -include "models/nat_order_continuous.ma". - -theorem nat_lebesgue_oc: - ∀a:sequence ℕ.∀s:‡ℕ.∀H:∀i:nat.a i ∈ s. - ∀x:ℕ.a order_converges x → - x ∈ s ∧ - ∀h:x ∈ s. - uniform_converge {[s]} ⌊n,≪a n,H n≫⌋ ≪x,h≫. -intros; apply lebesgue_oc; [apply nat_us_is_oc] assumption; -qed. - diff --git a/helm/software/matita/contribs/dama/dama/models/nat_order_continuous.ma b/helm/software/matita/contribs/dama/dama/models/nat_order_continuous.ma deleted file mode 100644 index 5f00dc338..000000000 --- a/helm/software/matita/contribs/dama/dama/models/nat_order_continuous.ma +++ /dev/null @@ -1,39 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "models/increasing_supremum_stabilizes.ma". -include "models/nat_ordered_uniform.ma". - -lemma nat_us_is_oc: ∀s:‡ℕ.order_continuity (segment_ordered_uniform_space ℕ s). -intros 3; split; intros 3; cases H1; cases H2; clear H2 H1; cases H; clear H; -[1: cases (increasing_supremum_stabilizes s a H1 ? H2); - exists [apply w1]; intros; - apply (restrict nat_ordered_uniform_space s ??? H4); - generalize in match (H ? H5); - cases x; intro; - generalize in match (refl_eq ? (a i):a i = a i); - generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X; - intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify; - apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space)); -|2: cases (increasing_supremum_stabilizes_r s a H1 ? H2); - exists [apply w1]; intros; - apply (restrict nat_ordered_uniform_space s ??? H4); - generalize in match (H ? H5); - cases x; intro; - generalize in match (refl_eq ? (a i):a i = a i); - generalize in match (a i) in ⊢ (?? % ? → %); intro X; cases X; clear X; - intro; rewrite < H9 in H7; simplify; rewrite < H7; simplify; - apply (us_phi1 nat_uniform_space ? H3); simplify; apply (eq_reflexive (us_carr nat_uniform_space));] -qed. - diff --git a/helm/software/matita/contribs/dama/dama/models/nat_ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/models/nat_ordered_uniform.ma deleted file mode 100644 index becbab2fb..000000000 --- a/helm/software/matita/contribs/dama/dama/models/nat_ordered_uniform.ma +++ /dev/null @@ -1,32 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "models/nat_uniform.ma". -include "bishop_set_rewrite.ma". -include "ordered_uniform.ma". - -definition nat_ordered_uniform_space:ordered_uniform_space. - apply (mk_ordered_uniform_space (mk_ordered_uniform_space_ ℕ ℕ (refl_eq ? ℕ))); -simplify; intros 10; cases H (_); cases (H7 y); apply H8; cases (H7 s); -lapply (H11 H1) as H13; apply (le_le_eq); -[2: apply (le_transitive ??? H5); apply (Le≪ ? H13); assumption; -|1: assumption; -|3: change with (le (os_r ℕ) (\snd y) (\fst y)); - apply (ge_transitive ??? H5);apply (ge_transitive ???? H4); - change with (le (os_l ℕ) (\fst s) (\snd s)); - apply (Le≫ ? H13); apply le_reflexive; -|4: assumption;] -qed. - -interpretation "Ordered uniform space N" 'N = nat_ordered_uniform_space. diff --git a/helm/software/matita/contribs/dama/dama/models/nat_uniform.ma b/helm/software/matita/contribs/dama/dama/models/nat_uniform.ma deleted file mode 100644 index 0b2d43563..000000000 --- a/helm/software/matita/contribs/dama/dama/models/nat_uniform.ma +++ /dev/null @@ -1,22 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "nat_ordered_set.ma". -include "models/discrete_uniformity.ma". - -definition nat_uniform_space: uniform_space. -apply (discrete_uniform_space_of_bishop_set ℕ); -qed. - -interpretation "Uniform space N" 'N = nat_uniform_space. \ No newline at end of file -- 2.39.2