From 9bcbaf15316cfb7fbe831af1efb89da954f40b72 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 20 Nov 2008 16:06:24 +0000 Subject: [PATCH] ... --- .../matita/contribs/dama/dama/bishop_set.ma | 89 ---- .../contribs/dama/dama/bishop_set_rewrite.ma | 87 ---- .../matita/contribs/dama/dama/depends | 30 +- .../matita/contribs/dama/dama/depends.png | Bin 235258 -> 69994 bytes .../matita/contribs/dama/dama/lebesgue.ma | 125 ----- .../contribs/dama/dama/models/q_bars.ma | 2 +- .../contribs/dama/dama/models/q_rebase.ma | 2 +- .../contribs/dama/dama/nat_ordered_set.ma | 65 --- .../matita/contribs/dama/dama/ordered_set.ma | 181 ------- .../contribs/dama/dama/ordered_uniform.ma | 243 ---------- .../dama/dama/property_exhaustivity.ma | 171 ------- .../contribs/dama/dama/property_sigma.ma | 121 ----- .../contribs/dama/dama/russell_support.ma | 27 -- .../matita/contribs/dama/dama/sandwich.ma | 48 -- .../matita/contribs/dama/dama/sequence.ma | 38 -- .../matita/contribs/dama/dama/supremum.ma | 442 ------------------ .../matita/contribs/dama/dama/uniform.ma | 92 ---- 17 files changed, 6 insertions(+), 1757 deletions(-) delete mode 100644 helm/software/matita/contribs/dama/dama/bishop_set.ma delete mode 100644 helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma delete mode 100644 helm/software/matita/contribs/dama/dama/lebesgue.ma delete mode 100644 helm/software/matita/contribs/dama/dama/nat_ordered_set.ma delete mode 100644 helm/software/matita/contribs/dama/dama/ordered_set.ma delete mode 100644 helm/software/matita/contribs/dama/dama/ordered_uniform.ma delete mode 100644 helm/software/matita/contribs/dama/dama/property_exhaustivity.ma delete mode 100644 helm/software/matita/contribs/dama/dama/property_sigma.ma delete mode 100644 helm/software/matita/contribs/dama/dama/russell_support.ma delete mode 100644 helm/software/matita/contribs/dama/dama/sandwich.ma delete mode 100644 helm/software/matita/contribs/dama/dama/sequence.ma delete mode 100644 helm/software/matita/contribs/dama/dama/supremum.ma delete mode 100644 helm/software/matita/contribs/dama/dama/uniform.ma diff --git a/helm/software/matita/contribs/dama/dama/bishop_set.ma b/helm/software/matita/contribs/dama/dama/bishop_set.ma deleted file mode 100644 index d69bb2732..000000000 --- a/helm/software/matita/contribs/dama/dama/bishop_set.ma +++ /dev/null @@ -1,89 +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 "ordered_set.ma". - -(* Definition 2.2, setoid *) -record bishop_set: Type ≝ { - bs_carr:> Type; - bs_apart: bs_carr → bs_carr → CProp; - bs_coreflexive: coreflexive ? bs_apart; - bs_symmetric: symmetric ? bs_apart; - bs_cotransitive: cotransitive ? bs_apart -}. - -interpretation "bishop set apartness" 'apart x y = (bs_apart _ x y). - -definition bishop_set_of_ordered_set: ordered_set → bishop_set. -intros (E); apply (mk_bishop_set E (λa,b:E. a ≰ b ∨ b ≰ a)); -[1: intro x; simplify; intro H; cases H; clear H; - apply (exc_coreflexive x H1); -|2: intros 3 (x y H); simplify in H ⊢ %; cases H; [right|left]assumption; -|3: intros 4 (x y z H); simplify in H ⊢ %; cases H; clear H; - [ cases (exc_cotransitive x y z H1); [left;left|right;left] assumption; - | cases (exc_cotransitive y x z H1); [right;right|left;right] assumption;]] -qed. - -(* Definition 2.2 (2) *) -definition eq ≝ λA:bishop_set.λa,b:A. ¬ (a # b). - -interpretation "Bishop set alikeness" 'napart a b = (eq _ a b). - -lemma eq_reflexive:∀E:bishop_set. reflexive ? (eq E). -intros (E); unfold; intros (x); apply bs_coreflexive; -qed. - -lemma eq_sym_:∀E:bishop_set.symmetric ? (eq E). -intros 4 (E x y H); intro T; cases (H (bs_symmetric ??? T)); -qed. - -lemma eq_sym:∀E:bishop_set.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_. - -lemma eq_trans_: ∀E:bishop_set.transitive ? (eq E). -intros 6 (E x y z Exy Eyz); intro Axy; cases (bs_cotransitive ???y Axy); -[apply Exy|apply Eyz] assumption. -qed. - -coercion bishop_set_of_ordered_set. - -lemma le_antisymmetric: - ∀E:ordered_set.antisymmetric E (le (os_l E)) (eq E). -intros 5 (E x y Lxy Lyx); intro H; -cases H; [apply Lxy;|apply Lyx] assumption; -qed. - -lemma le_le_eq: ∀E:ordered_set.∀a,b:E. a ≤ b → b ≤ a → a ≈ b. -intros (E x y L1 L2); intro H; cases H; [apply L1|apply L2] assumption; -qed. - -definition bs_subset ≝ λO:bishop_set.λP,Q:O→Prop.∀x:O.P x → Q x. - -interpretation "bishop set subset" 'subseteq a b = (bs_subset _ a b). - -definition square_bishop_set : bishop_set → bishop_set. -intro S; apply (mk_bishop_set (S × S)); -[1: intros (x y); apply ((\fst x # \fst y) ∨ (\snd x # \snd y)); -|2: intro x; simplify; intro; cases H (X X); clear H; apply (bs_coreflexive ?? X); -|3: intros 2 (x y); simplify; intro H; cases H (X X); clear H; [left|right] apply (bs_symmetric ??? X); -|4: intros 3 (x y z); simplify; intro H; cases H (X X); clear H; - [1: cases (bs_cotransitive ??? (\fst z) X); [left;left|right;left]assumption; - |2: cases (bs_cotransitive ??? (\snd z) X); [left;right|right;right]assumption;]] -qed. - -notation "s 2 \atop \neq" non associative with precedence 90 - for @{ 'square_bs $s }. -notation > "s 'squareB'" non associative with precedence 90 - for @{ 'squareB $s }. -interpretation "bishop set square" 'squareB x = (square_bishop_set x). -interpretation "bishop set square" 'square_bs x = (square_bishop_set x). \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma b/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma deleted file mode 100644 index ff063e29a..000000000 --- a/helm/software/matita/contribs/dama/dama/bishop_set_rewrite.ma +++ /dev/null @@ -1,87 +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 "bishop_set.ma". - -coercion eq_sym. - -lemma eq_trans:∀E:bishop_set.∀x,z,y:E.x ≈ y → y ≈ z → x ≈ z ≝ - λE,x,y,z.eq_trans_ E x z y. - -notation > "'Eq'≈" non associative with precedence 50 - for @{'eqrewrite}. - -interpretation "eq_rew" 'eqrewrite = (eq_trans _ _ _). - -lemma le_rewl: ∀E:ordered_set.∀z,y,x:E. x ≈ y → x ≤ z → y ≤ z. -intros (E z y x Exy Lxz); apply (le_transitive ??? ? Lxz); -intro Xyz; apply Exy; right; assumption; -qed. - -lemma le_rewr: ∀E:ordered_set.∀z,y,x:E. x ≈ y → z ≤ x → z ≤ y. -intros (E z y x Exy Lxz); apply (le_transitive ??? Lxz); -intro Xyz; apply Exy; left; assumption; -qed. - -notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}. -interpretation "le_rewl" 'lerewritel = (le_rewl _ _ _). -notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}. -interpretation "le_rewr" 'lerewriter = (le_rewr _ _ _). - -lemma ap_rewl: ∀A:bishop_set.∀x,z,y:A. x ≈ y → y # z → x # z. -intros (A x z y Exy Ayz); cases (bs_cotransitive ???x Ayz); [2:assumption] -cases (eq_sym ??? Exy H); -qed. - -lemma ap_rewr: ∀A:bishop_set.∀x,z,y:A. x ≈ y → z # y → z # x. -intros (A x z y Exy Azy); apply bs_symmetric; apply (ap_rewl ???? Exy); -apply bs_symmetric; assumption; -qed. - -notation > "'Ap'≪" non associative with precedence 50 for @{'aprewritel}. -interpretation "ap_rewl" 'aprewritel = (ap_rewl _ _ _). -notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}. -interpretation "ap_rewr" 'aprewriter = (ap_rewr _ _ _). - -lemma exc_rewl: ∀A:ordered_set.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z. -intros (A x z y Exy Ayz); cases (exc_cotransitive ?? x Ayz); [2:assumption] -cases Exy; right; assumption; -qed. - -lemma exc_rewr: ∀A:ordered_set.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x. -intros (A x z y Exy Azy); cases (exc_cotransitive ??x Azy); [assumption] -cases (Exy); left; assumption; -qed. - -notation > "'Ex'≪" non associative with precedence 50 for @{'ordered_setrewritel}. -interpretation "exc_rewl" 'ordered_setrewritel = (exc_rewl _ _ _). -notation > "'Ex'≫" non associative with precedence 50 for @{'ordered_setrewriter}. -interpretation "exc_rewr" 'ordered_setrewriter = (exc_rewr _ _ _). - -(* -lemma lt_rewr: ∀A:ordered_set.∀x,z,y:A. x ≈ y → z < y → z < x. -intros (A x y z E H); split; cases H; -[apply (Le≫ ? (eq_sym ??? E));|apply (Ap≫ ? E)] assumption; -qed. - -lemma lt_rewl: ∀A:ordered_set.∀x,z,y:A. x ≈ y → y < z → x < z. -intros (A x y z E H); split; cases H; -[apply (Le≪ ? (eq_sym ??? E));| apply (Ap≪ ? E);] assumption; -qed. - -notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}. -interpretation "lt_rewl" 'ltrewritel = (lt_rewl _ _ _). -notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}. -interpretation "lt_rewr" 'ltrewriter = (lt_rewr _ _ _). -*) diff --git a/helm/software/matita/contribs/dama/dama/depends b/helm/software/matita/contribs/dama/dama/depends index 7e0d8968e..68f26f5b4 100644 --- a/helm/software/matita/contribs/dama/dama/depends +++ b/helm/software/matita/contribs/dama/dama/depends @@ -1,35 +1,13 @@ -property_exhaustivity.ma ordered_uniform.ma property_sigma.ma -sequence.ma nat/nat.ma -supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma -bishop_set_rewrite.ma bishop_set.ma -property_sigma.ma ordered_uniform.ma russell_support.ma models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma logic/cprop_connectives.ma -models/increasing_supremum_stabilizes.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma -ordered_set.ma datatypes/constructors.ma logic/cprop_connectives.ma -sandwich.ma ordered_uniform.ma -nat_ordered_set.ma bishop_set.ma nat/compare.ma -models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma +models/q_bars.ma dama/nat_ordered_set.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma models/list_support.ma list/list.ma logic/cprop_connectives.ma nat/minus.ma -models/q_bars.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma .unnamed.ma -lebesgue.ma ordered_set.ma property_exhaustivity.ma sandwich.ma -bishop_set.ma ordered_set.ma -models/nat_order_continuous.ma models/increasing_supremum_stabilizes.ma models/nat_ordered_uniform.ma -models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma -russell_support.ma logic/cprop_connectives.ma nat/nat.ma models/q_copy.ma models/q_bars.ma -models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma 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z5AG3;Z;MTLDY~gU;9L%;SHg;QXB`|3Y|_E*ucXApt00`jbJJzb8#Hy+Mzo4|;yvggh_rpWe8ZAzhRU4t$|4|l=$I)?zCBJ$!-bmqzIG)I4@@ccMI5VfbDTh>6zM`QjO%y3|Va}1@C|Ta*kVj=|j{1 z`VIc8^Kg|8=f8f_EU0$1|JQFh5C7S-iIc=t{MWCrkws}g)D9-CTM+u)d$_`{N#mv( K$BePw@_zuaA0Mg! diff --git a/helm/software/matita/contribs/dama/dama/lebesgue.ma b/helm/software/matita/contribs/dama/dama/lebesgue.ma deleted file mode 100644 index f81c5ce46..000000000 --- a/helm/software/matita/contribs/dama/dama/lebesgue.ma +++ /dev/null @@ -1,125 +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 *) -(* *) -(**************************************************************************) - -(* manca un pezzo del pullback, se inverto poi non tipa *) -include "sandwich.ma". -include "property_exhaustivity.ma". - -(* NOT DUALIZED *) -alias symbol "low" = "lower". -alias id "le" = "cic:/matita/dama/ordered_set/le.con". -lemma order_converges_bigger_lowsegment: - ∀C:ordered_set. - ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s. - ∀x:C.∀p:order_converge C a x. - ∀j. 𝕝_s ≤ (pi1exT23 ???? p j). -intros; cases p (xi yi Ux Dy Hxy); clear p; simplify; -cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy; -cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa; -intro H2; cases (SSa 𝕝_s H2) (w Hw); simplify in Hw; -lapply (H (w+j)) as K; cases (cases_in_segment ? s ? K); apply H3; apply Hw; -qed. - -alias symbol "upp" = "uppper". -alias symbol "leq" = "Ordered set less or equal than". -lemma order_converges_smaller_upsegment: - ∀C:ordered_set. - ∀a:sequence (os_l C).∀s:segment C.∀H:∀i:nat.a i ∈ s. - ∀x:C.∀p:order_converge C a x. - ∀j. (pi2exT23 ???? p j) ≤ 𝕦_s. -intros; cases p (xi yi Ux Dy Hxy); clear p; simplify; -cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy; -cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa; -intro H2; cases (SIa 𝕦_s H2) (w Hw); lapply (H (w+j)) as K; -cases (cases_in_segment ? s ? K); apply H1; apply Hw; -qed. - -(* Theorem 3.10 *) -theorem lebesgue_oc: - ∀C:ordered_uniform_space. - (∀s:‡C.order_continuity {[s]}) → - ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s. - ∀x:C.a order_converges x → - x ∈ s ∧ - ∀h:x ∈ s. - uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫. -intros; -generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2); -generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2); -cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros; -cut (∀i.xi i ∈ s) as Hxi; [2: - intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _); - lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); - simplify in K:(? ? % ?); apply (hle_transitive (os_l C) (xi i) (a i) 𝕦_s K Pu);] clear H3; -cut (∀i.yi i ∈ s) as Hyi; [2: - intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _); - lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K; - apply (le_transitive 𝕝_s ? ? ? K);apply Pl;] clear H2; -split; -[1: apply (uparrow_to_in_segment s ? Hxi ? Hx); -|2: intros 3 (h); - letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋); - letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋); - letin Ai ≝ (⌊n,≪a n, H1 n≫⌋); - apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;] - [1: intro j; cases (Hxy j); cases H3; cases H4; split; clear H3 H4; simplify in H5 H7; - [apply (l2sl_ ? s (Xi j) (Ai j) (H5 0));|apply (l2sl_ ? s (Ai j) (Yi j) (H7 0))] - |2: cases (H s Xi ≪?,h≫) (Ux Uy); apply Ux; cases Hx; split; [intro i; apply (l2sl_ ? s (Xi i) (Xi (S i)) (H3 i));] - cases H4; split; [intro i; apply (l2sl_ ? s (Xi i) ≪x,h≫ (H5 i))] - intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x_ ? s ? y Hy)] - exists [apply w] apply (x2sx_ ? s (Xi w) y H7); - |3: cases (H s Yi ≪?,h≫) (Ux Uy); apply Uy; cases Hy; split; [intro i; apply (l2sl_ ? s (Yi (S i)) (Yi i) (H3 i));] - cases H4; split; [intro i; apply (l2sl_ ? s ≪x,h≫ (Yi i) (H5 i))] - intros (y Hy);cases (H6 (\fst y));[2:apply (sx2x_ ? s y ≪x,h≫ Hy)] - exists [apply w] apply (x2sx_ ? s y (Yi w) H7);]] -qed. - - -(* Theorem 3.9 *) -theorem lebesgue_se: - ∀C:ordered_uniform_space.property_sigma C → - (∀s:‡C.exhaustive {[s]}) → - ∀a:sequence C.∀s:‡C.∀H:∀i:nat.a i ∈ s. - ∀x:C.a order_converges x → - x ∈ s ∧ - ∀h:x ∈ s. - uniform_converge {[s]} (⌊n,≪a n,H n≫⌋) ≪x,h≫. -intros (C S); -generalize in match (order_converges_bigger_lowsegment ? a s H1 ? H2); -generalize in match (order_converges_smaller_upsegment ? a s H1 ? H2); -cases H2 (xi yi Hx Hy Hxy); clear H2; simplify in ⊢ ((?→???%) → (?→???%) → ?); intros; -cut (∀i.xi i ∈ s) as Hxi; [2: - intros; apply (prove_in_segment (os_l C)); [apply (H3 i)] cases (Hxy i) (H5 _); cases H5 (H7 _); - lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); - simplify in K:(? ? % ?); apply (hle_transitive (os_l C) (xi i) (a i) 𝕦_s K Pu);] clear H3; -cut (∀i.yi i ∈ s) as Hyi; [2: - intros; apply (prove_in_segment (os_l C)); [2:apply (H2 i)] cases (Hxy i) (_ H5); cases H5 (H7 _); - lapply (H7 0) as K; cases (cases_in_segment ? s ? (H1 i)) (Pl Pu); simplify in K; - apply (le_transitive 𝕝_s ? ? ? K);apply Pl;] clear H2; -letin Xi ≝ (⌊n,≪xi n, Hxi n≫⌋); -letin Yi ≝ (⌊n,≪yi n, Hyi n≫⌋); -cases (restrict_uniform_convergence_uparrow ? S ? (H s) Xi x Hx); -cases (restrict_uniform_convergence_downarrow ? S ? (H s) Yi x Hy); -split; [1: assumption] -intros 3; -lapply (uparrow_upperlocated xi x Hx)as Ux; -lapply (downarrow_lowerlocated yi x Hy)as Uy; -letin Ai ≝ (⌊n,≪a n, H1 n≫⌋); -apply (sandwich {[s]} ≪x, h≫ Xi Yi Ai); [4: assumption;|2:apply H3;|3:apply H5] -intro j; cases (Hxy j); cases H7; cases H8; split; -[apply (l2sl_ ? s (Xi j) (Ai j) (H9 0));|apply (l2sl_ ? s (Ai j) (Yi j) (H11 0))] -qed. - - - diff --git a/helm/software/matita/contribs/dama/dama/models/q_bars.ma b/helm/software/matita/contribs/dama/dama/models/q_bars.ma index 1d2107b7c..f3224e627 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -include "nat_ordered_set.ma". +include "dama/nat_ordered_set.ma". include "models/q_support.ma". include "models/list_support.ma". include "logic/cprop_connectives.ma". diff --git a/helm/software/matita/contribs/dama/dama/models/q_rebase.ma b/helm/software/matita/contribs/dama/dama/models/q_rebase.ma index f8243d6d1..b14f0bd17 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_rebase.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_rebase.ma @@ -12,7 +12,7 @@ (* *) (**************************************************************************) -include "russell_support.ma". +include "dama/russell_support.ma". include "models/q_copy.ma". (* definition rebase_spec ≝ diff --git a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma deleted file mode 100644 index 26e2f0d29..000000000 --- a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma +++ /dev/null @@ -1,65 +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/compare.ma". -include "bishop_set.ma". - -definition nat_excess : nat → nat → CProp ≝ λn,m. m H2; reflexivity; - |2: right;apply le_S_S; assumption]] -qed. - -lemma nat_excess_cotransitive: cotransitive ? nat_excess. -intros 3 (x y z); unfold nat_excess; simplify; intros; -cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1; -[1: right; apply (trans_lt ??? H H2); -|2: right; rewrite < H2; assumption;] -qed. - -lemma nat_ordered_set : ordered_set. -letin hos ≝ (mk_half_ordered_set nat (λT,R:Type.λf:T→T→R.f) ? nat_excess ? nat_excess_cotransitive); -[ intros; left; intros; reflexivity; -| intro x; intro H; apply (not_le_Sn_n ? H);] -constructor 1; apply hos; -qed. - -interpretation "ordered set N" 'N = nat_ordered_set. - -alias id "le" = "cic:/matita/nat/orders/le.ind#xpointer(1/1)". -lemma os_le_to_nat_le: - ∀a,b:nat_ordered_set.a ≤ b → le a b. -intros; normalize in H; apply (not_lt_to_le b a H); -qed. - -lemma nat_le_to_os_le: - ∀a,b:nat_ordered_set.le a b → a ≤ b. -intros 3; apply (le_to_not_lt a b);assumption; -qed. - diff --git a/helm/software/matita/contribs/dama/dama/ordered_set.ma b/helm/software/matita/contribs/dama/dama/ordered_set.ma deleted file mode 100644 index 890164ea9..000000000 --- a/helm/software/matita/contribs/dama/dama/ordered_set.ma +++ /dev/null @@ -1,181 +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 "datatypes/constructors.ma". -include "logic/cprop_connectives.ma". - - -(* TEMPLATES -notation "''" non associative with precedence 90 for @{'}. -notation "''" non associative with precedence 90 for @{'}. - -interpretation "" ' = ( (os_l _)). -interpretation "" ' = ( (os_r _)). -*) - -(* Definition 2.1 *) -record half_ordered_set: Type ≝ { - hos_carr:> Type; - wloss: ∀A,B:Type. (A → A → B) → A → A → B; - wloss_prop: (∀T,R,P,x,y.P x y = wloss T R P x y) ∨ (∀T,R,P,x,y.P y x = wloss T R P x y); - hos_excess_: hos_carr → hos_carr → CProp; - hos_coreflexive: coreflexive ? (wloss ?? hos_excess_); - hos_cotransitive: cotransitive ? (wloss ?? hos_excess_) -}. - -definition hos_excess ≝ λO:half_ordered_set.wloss O ?? (hos_excess_ O). - -definition dual_hos : half_ordered_set → half_ordered_set. -intro; constructor 1; -[ apply (hos_carr h); -| apply (λT,R,f,x,y.wloss h T R f y x); -| intros; cases (wloss_prop h);[right|left]intros;apply H; -| apply (hos_excess_ h); -| apply (hos_coreflexive h); -| intros 4 (x y z H); simplify in H ⊢ %; cases (hos_cotransitive h y x z H); - [right|left] assumption;] -qed. - -record ordered_set : Type ≝ { - os_l : half_ordered_set -}. - -definition os_r : ordered_set → half_ordered_set. -intro o; apply (dual_hos (os_l o)); qed. - -lemma half2full : half_ordered_set → ordered_set. -intro hos; -constructor 1; apply hos; -qed. - -definition Type_of_ordered_set : ordered_set → Type. -intro o; apply (hos_carr (os_l o)); qed. - -definition Type_of_ordered_set_dual : ordered_set → Type. -intro o; apply (hos_carr (os_r o)); qed. - -coercion Type_of_ordered_set_dual. -coercion Type_of_ordered_set. - -notation "a ≰≰ b" non associative with precedence 45 for @{'nleq_low $a $b}. -interpretation "Ordered half set excess" 'nleq_low a b = (hos_excess _ a b). - -interpretation "Ordered set excess (dual)" 'ngeq a b = (hos_excess (os_r _) a b). -interpretation "Ordered set excess" 'nleq a b = (hos_excess (os_l _) a b). - -notation "'exc_coreflexive'" non associative with precedence 90 for @{'exc_coreflexive}. -notation "'cxe_coreflexive'" non associative with precedence 90 for @{'cxe_coreflexive}. - -interpretation "exc_coreflexive" 'exc_coreflexive = ((hos_coreflexive (os_l _))). -interpretation "cxe_coreflexive" 'cxe_coreflexive = ((hos_coreflexive (os_r _))). - -notation "'exc_cotransitive'" non associative with precedence 90 for @{'exc_cotransitive}. -notation "'cxe_cotransitive'" non associative with precedence 90 for @{'cxe_cotransitive}. - -interpretation "exc_cotransitive" 'exc_cotransitive = ((hos_cotransitive (os_l _))). -interpretation "cxe_cotransitive" 'cxe_cotransitive = ((hos_cotransitive (os_r _))). - -(* Definition 2.2 (3) *) -definition le ≝ λE:half_ordered_set.λa,b:E. ¬ (a ≰≰ b). - -notation "hvbox(a break ≤≤ b)" non associative with precedence 45 for @{ 'leq_low $a $b }. -interpretation "Half ordered set greater or equal than" 'leq_low a b = ((le _ a b)). - -interpretation "Ordered set greater or equal than" 'geq a b = ((le (os_r _) a b)). -interpretation "Ordered set less or equal than" 'leq a b = ((le (os_l _) a b)). - -lemma hle_reflexive: ∀E.reflexive ? (le E). -unfold reflexive; intros 3; apply (hos_coreflexive ? x H); -qed. - -notation "'le_reflexive'" non associative with precedence 90 for @{'le_reflexive}. -notation "'ge_reflexive'" non associative with precedence 90 for @{'ge_reflexive}. - -interpretation "le reflexive" 'le_reflexive = (hle_reflexive (os_l _)). -interpretation "ge reflexive" 'ge_reflexive = (hle_reflexive (os_r _)). - -(* DUALITY TESTS -lemma test_le_ge_convertible :∀o:ordered_set.∀x,y:o. x ≤ y → y ≥ x. -intros; assumption; qed. - -lemma test_ge_reflexive :∀o:ordered_set.∀x:o. x ≥ x. -intros; apply ge_reflexive. qed. - -lemma test_le_reflexive :∀o:ordered_set.∀x:o. x ≤ x. -intros; apply le_reflexive. qed. -*) - -lemma hle_transitive: ∀E.transitive ? (le E). -unfold transitive; intros 7 (E x y z H1 H2 H3); cases (hos_cotransitive E x z y H3) (H4 H4); -[cases (H1 H4)|cases (H2 H4)] -qed. - -notation "'le_transitive'" non associative with precedence 90 for @{'le_transitive}. -notation "'ge_transitive'" non associative with precedence 90 for @{'ge_transitive}. - -interpretation "le transitive" 'le_transitive = (hle_transitive (os_l _)). -interpretation "ge transitive" 'ge_transitive = (hle_transitive (os_r _)). - -(* Lemma 2.3 *) -lemma exc_hle_variance: - ∀O:half_ordered_set.∀a,b,a',b':O.a ≰≰ b → a ≤≤ a' → b' ≤≤ b → a' ≰≰ b'. -intros (O a b a1 b1 Eab Laa1 Lb1b); -cases (hos_cotransitive ? a b a1 Eab) (H H); [cases (Laa1 H)] -cases (hos_cotransitive ? ?? b1 H) (H1 H1); [assumption] -cases (Lb1b H1); -qed. - -notation "'exc_le_variance'" non associative with precedence 90 for @{'exc_le_variance}. -notation "'exc_ge_variance'" non associative with precedence 90 for @{'exc_ge_variance}. - -interpretation "exc_le_variance" 'exc_le_variance = (exc_hle_variance (os_l _)). -interpretation "exc_ge_variance" 'exc_ge_variance = (exc_hle_variance (os_r _)). - -definition square_exc ≝ - λO:half_ordered_set.λx,y:O×O.\fst x ≰≰ \fst y ∨ \snd x ≰≰ \snd y. - -lemma square_half_ordered_set: half_ordered_set → half_ordered_set. -intro O; -apply (mk_half_ordered_set (O × O)); -[1: apply (wloss O); -|2: intros; cases (wloss_prop O); [left|right] intros; apply H; -|3: apply (square_exc O); -|4: intro x; cases (wloss_prop O); rewrite < (H ?? (square_exc O) x x); clear H; - cases x; clear x; unfold square_exc; intro H; cases H; clear H; simplify in H1; - [1,3: apply (hos_coreflexive O h H1); - |*: apply (hos_coreflexive O h1 H1);] -|5: intros 3 (x0 y0 z0); cases (wloss_prop O); - do 3 rewrite < (H ?? (square_exc O)); clear H; cases x0; cases y0; cases z0; clear x0 y0 z0; - simplify; intro H; cases H; clear H; - [1: cases (hos_cotransitive ? h h2 h4 H1); [left;left|right;left] assumption; - |2: cases (hos_cotransitive ? h1 h3 h5 H1); [left;right|right;right] assumption; - |3: cases (hos_cotransitive ? h2 h h4 H1); [right;left|left;left] assumption; - |4: cases (hos_cotransitive ? h3 h1 h5 H1); [right;right|left;right] assumption;]] -qed. - -lemma square_ordered_set: ordered_set → ordered_set. -intro O; constructor 1; apply (square_half_ordered_set (os_l O)); -qed. - -notation "s 2 \atop \nleq" non associative with precedence 90 - for @{ 'square_os $s }. -notation > "s 'squareO'" non associative with precedence 90 - for @{ 'squareO $s }. -interpretation "ordered set square" 'squareO s = (square_ordered_set s). -interpretation "ordered set square" 'square_os s = (square_ordered_set s). - -definition os_subset ≝ λO:ordered_set.λP,Q:O→Prop.∀x:O.P x → Q x. - -interpretation "ordered set subset" 'subseteq a b = (os_subset _ a b). - diff --git a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma b/helm/software/matita/contribs/dama/dama/ordered_uniform.ma deleted file mode 100644 index 5a712f127..000000000 --- a/helm/software/matita/contribs/dama/dama/ordered_uniform.ma +++ /dev/null @@ -1,243 +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". - -record ordered_uniform_space_ : Type ≝ { - ous_os:> ordered_set; - ous_us_: uniform_space; - with_ : us_carr ous_us_ = bishop_set_of_ordered_set ous_os -}. - -lemma ous_unifspace: ordered_uniform_space_ → uniform_space. -intro X; apply (mk_uniform_space (bishop_set_of_ordered_set X)); -unfold bishop_set_OF_ordered_uniform_space_; -[1: rewrite < (with_ X); simplify; apply (us_unifbase (ous_us_ X)); -|2: cases (with_ X); simplify; apply (us_phi1 (ous_us_ X)); -|3: cases (with_ X); simplify; apply (us_phi2 (ous_us_ X)); -|4: cases (with_ X); simplify; apply (us_phi3 (ous_us_ X)); -|5: cases (with_ X); simplify; apply (us_phi4 (ous_us_ X))] -qed. - -coercion ous_unifspace. - -record ordered_uniform_space : Type ≝ { - ous_stuff :> ordered_uniform_space_; - ous_convex_l: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U; - ous_convex_r: ∀U.us_unifbase ous_stuff U → convex (os_r ous_stuff) U -}. - -definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set. -intro; compose ordered_set_OF_ordered_uniform_space with os_l. apply (f o); -qed. - -definition invert_os_relation ≝ - λC:ordered_set.λU:C squareO → Prop. - λx:C squareO. U 〈\snd x,\fst x〉. - -interpretation "relation invertion" 'invert a = (invert_os_relation _ a). -interpretation "relation invertion" 'invert_symbol = (invert_os_relation _). -interpretation "relation invertion" 'invert_appl a x = (invert_os_relation _ a x). - -lemma hint_segment: ∀O. - segment (Type_of_ordered_set O) → - segment (hos_carr (os_l O)). -intros; assumption; -qed. - -coercion hint_segment nocomposites. - -lemma segment_square_of_ordered_set_square: - ∀O:ordered_set.∀s:‡O.∀x:O squareO. - \fst x ∈ s → \snd x ∈ s → {[s]} squareO. -intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption; -qed. - -coercion segment_square_of_ordered_set_square with 0 2 nocomposites. - -alias symbol "pi1" (instance 4) = "exT \fst". -alias symbol "pi1" (instance 2) = "exT \fst". -lemma ordered_set_square_of_segment_square : - ∀O:ordered_set.∀s:‡O.{[s]} squareO → O squareO ≝ - λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉. - -coercion ordered_set_square_of_segment_square nocomposites. - -lemma restriction_agreement : - ∀O:ordered_uniform_space.∀s:‡O.∀P:{[s]} squareO → Prop.∀OP:O squareO → Prop.Prop. -apply(λO:ordered_uniform_space.λs:‡O. - λP:{[s]} squareO → Prop. λOP:O squareO → Prop. - ∀b:{[s]} squareO.(P b → OP b) ∧ (OP b → P b)); -qed. - -lemma unrestrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO. - restriction_agreement ? s U u → U x → u x. -intros 6; cases (H x); assumption; -qed. - -lemma restrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO. - restriction_agreement ? s U u → u x → U x. -intros 6; cases (H x); assumption; -qed. - -lemma invert_restriction_agreement: - ∀O:ordered_uniform_space.∀s:‡O. - ∀U:{[s]} squareO → Prop.∀u:O squareO → Prop. - restriction_agreement ? s U u → - restriction_agreement ? s (\inv U) (\inv u). -intros 6; cases b; -generalize in match (H 〈h1,h〉); cases h; cases h1; simplify; -intro X; cases X; split; assumption; -qed. - -lemma bs2_of_bss2: - ∀O:ordered_set.∀s:‡O.(bishop_set_of_ordered_set {[s]}) squareB → (bishop_set_of_ordered_set O) squareB ≝ - λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉. - -coercion bs2_of_bss2 nocomposites. - - -lemma a2sa : - ∀O:ordered_uniform_space.∀s:‡(ordered_set_OF_ordered_uniform_space O). - ∀x: - bs_carr - (square_bishop_set - (bishop_set_of_ordered_set - (segment_ordered_set - (ordered_set_OF_ordered_uniform_space O) s))). - (\fst x) ≈ (\snd x) → - (\fst (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)) - ≈ - (\snd (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)). -intros 3; cases x (a b); clear x; simplify in match (\fst ?); -simplify in match (\snd ?); intros 2 (K H); apply K; clear K; -cases H; -[ left; change in H1:(? ? % ?) with (\fst a); - change in H1:(? ? ? %) with (\fst b); - change in a with (hos_carr (half_segment_ordered_set ? s)); - change in b with (hos_carr (half_segment_ordered_set ? s)); - apply rule H1; -| right; change in H1:(? ? % ?) with (\fst b); - change in H1:(? ? ? %) with (\fst a); - change in a with (hos_carr (half_segment_ordered_set ? s)); - change in b with (hos_carr (half_segment_ordered_set ? s)); - apply rule H1;] -qed. - - -lemma segment_ordered_uniform_space: - ∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space. -intros (O s); apply mk_ordered_uniform_space; -[1: apply (mk_ordered_uniform_space_ {[s]}); - [1: alias symbol "and" = "constructive and". - letin f ≝ (λP:{[s]} squareO → Prop. ∃OP:O squareO → Prop. - (us_unifbase O OP) ∧ restriction_agreement ?? P OP); - apply (mk_uniform_space (bishop_set_of_ordered_set {[s]}) f); - [1: intros (U H); intro x; simplify; - cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm; - lapply (us_phi1 O w Gw x (a2sa ??? Hm)) as IH; - apply (restrict ? s ??? Hwp IH); - |2: intros (U V HU HV); cases HU (u Hu); cases HV (v Hv); clear HU HV; - cases Hu (Gu HuU); cases Hv (Gv HvV); clear Hu Hv; - cases (us_phi2 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW; - exists; [apply (λb:{[s]} squareB.w b)] split; - [1: unfold f; simplify; clearbody f; - exists; [apply w]; split; [assumption] intro b; simplify; - unfold segment_square_of_ordered_set_square; - cases b; intros; split; intros; assumption; - |2: intros 2 (x Hx); cases (Hw ? Hx); split; - [apply (restrict O s ??? HuU H)|apply (restrict O s ??? HvV H1);]] - |3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU; - cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW; - exists; [apply (λx:{[s]} squareB.w x)] split; - [1: exists;[apply w];split;[assumption] intros; simplify; intro; - unfold segment_square_of_ordered_set_square; - cases b; intros; split; intro; assumption; - |2: intros 2 (x Hx); apply (restrict O s ??? HuU); apply Hwu; - cases Hx (m Hm); exists[apply (\fst m)] apply Hm;] - |4: intros (U HU x); cases HU (u Hu); cases Hu (Gu HuU); clear HU Hu; - cases (us_phi4 O u Gu x) (Hul Hur); - split; intros; - [1: lapply (invert_restriction_agreement O s ?? HuU) as Ra; - apply (restrict O s ?? x Ra); - apply Hul; apply (unrestrict O s ??? HuU H); - |2: apply (restrict O s ??? HuU); apply Hur; - apply (unrestrict O s ??? (invert_restriction_agreement O s ?? HuU) H);]] - |2: simplify; reflexivity;] -|2: simplify; unfold convex; intros 3; cases s1; intros; - (* TODO: x2sx is for ≰, we need one for ≤ *) - cases H (u HU); cases HU (Gu HuU); clear HU H; - lapply depth=0 (ous_convex_l ?? Gu 〈\fst h,\fst h1〉 ???????) as K3; - [2: intro; apply H2; apply (x2sx_ (os_l O) s h h1 H); - |3: apply 〈\fst (\fst y),\fst (\snd y)〉; - |4: intro; change in H with (\fst (\fst y) ≰ \fst h1); apply H3; - apply (x2sx_ (os_l O) s (\fst y) h1 H); - |5: change with (\fst h ≤ \fst (\fst y)); intro; apply H4; - apply (x2sx_ (os_l O) s h (\fst y) H); - |6: change with (\fst (\snd y) ≤ \fst h1); intro; apply H5; - apply (x2sx_ (os_l O) s (\snd y) h1 H); - |7: change with (\fst (\fst y) ≤ \fst (\snd y)); intro; apply H6; - apply (x2sx_ (os_l O) s (\fst y) (\snd y) H); - |8: apply (restrict O s U u y HuU K3); - |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);] -|3: simplify; unfold convex; intros 3; cases s1; intros; - cases H (u HU); cases HU (Gu HuU); clear HU H; - lapply depth=0 (ous_convex_r ?? Gu 〈\fst h,\fst h1〉 ???????) as K3; - [2: intro; apply H2; apply (x2sx_ (os_r O) s h h1 H); - |3: apply 〈\fst (\fst y),\fst (\snd y)〉; - |4: intro; apply H3; - apply (x2sx_ (os_r O) s (\fst y) h1 H); - |5: intro; apply H4; - apply (x2sx_ (os_r O) s h (\fst y) H); - |6: intro; apply H5; - apply (x2sx_ (os_r O) s (\snd y) h1 H); - |7: intro; apply H6; - apply (x2sx_ (os_r O) s (\fst y) (\snd y) H); - |8: apply (restrict O s U u y HuU K3); - |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);] -] -qed. - -interpretation "Ordered uniform space segment" 'segset a = - (segment_ordered_uniform_space _ a). - -(* Lemma 3.2 *) -alias symbol "pi1" = "exT \fst". -lemma restric_uniform_convergence: - ∀O:ordered_uniform_space.∀s:‡O. - ∀x:{[s]}. - ∀a:sequence {[s]}. - (⌊n, \fst (a n)⌋ : sequence O) uniform_converges (\fst x) → - a uniform_converges x. -intros 7; cases H1; cases H2; clear H2 H1; -cases (H ? H3) (m Hm); exists [apply m]; intros; -apply (restrict ? s ??? H4); apply (Hm ? H1); -qed. - -definition order_continuity ≝ - λC:ordered_uniform_space.∀a:sequence C.∀x:C. - (a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x). - -lemma hint_boh1: ∀O. Type_OF_ordered_uniform_space O → hos_carr (os_l O). -intros; assumption; -qed. - -coercion hint_boh1 nocomposites. - -lemma hint_boh2: ∀O:ordered_uniform_space. hos_carr (os_l O) → Type_OF_ordered_uniform_space O. -intros; assumption; -qed. - -coercion hint_boh2 nocomposites. - diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma deleted file mode 100644 index deddf8804..000000000 --- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma +++ /dev/null @@ -1,171 +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 "ordered_uniform.ma". -include "property_sigma.ma". - -lemma h_segment_upperbound: - ∀C:half_ordered_set. - ∀s:segment C. - ∀a:sequence (half_segment_ordered_set C s). - upper_bound ? ⌊n,\fst (a n)⌋ (seg_u C s). -intros 4; simplify; cases (a n); simplify; unfold in H; -cases (wloss_prop C); rewrite < H1 in H; simplify; cases H; -assumption; -qed. - -notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}. -notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}. - -interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)). -interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)). - -lemma h_segment_preserves_uparrow: - ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s). - ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫. -intros; cases H (Ha Hx); split; -[ intro n; intro H; apply (Ha n); apply rule H; -| cases Hx; split; - [ intro n; intro H; apply (H1 n);apply rule H; - | intros; cases (H2 (\fst y)); [2: apply rule H3;] - exists [apply w] apply (x2sx_ ?? (a w) y H4);]] -qed. - -notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}. -notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}. - -interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)). -interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)). - -(* Fact 2.18 *) -lemma segment_cauchy: - ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}. - a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy. -intros 6; -alias symbol "pi1" (instance 3) = "pair pi1". -alias symbol "pi2" = "pair pi2". -apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉)); -(unfold segment_ordered_uniform_space; simplify); -exists [apply U] split; [assumption;] -intro; cases b; intros; simplify; split; intros; assumption; -qed. - -(* Definition 3.7 *) -definition exhaustive ≝ - λC:ordered_uniform_space. - ∀a,b:sequence C. - (a is_increasing → a is_upper_located → a is_cauchy) ∧ - (b is_decreasing → b is_lower_located → b is_cauchy). - -lemma h_uparrow_to_in_segment: - ∀C:half_ordered_set. - ∀s:segment C. - ∀a:sequence C. - (∀i.a i ∈ s) → - ∀x:C. uparrow C a x → - in_segment C s x. -intros (C H a H1 x H2); unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2; -cases (wloss_prop C) (W W); apply prove_in_segment; unfold; -[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; 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. - -(* Lemma 3.6 *) -lemma sigma_cauchy: - ∀C:ordered_uniform_space.property_sigma C → - ∀a:sequence C.∀l:C.(a ↑ l ∨ a ↓ l) → a is_cauchy → a uniform_converges l. -intros 8; cases (H ? H3) (w H5); cases H5 (H8 H9); clear H5; -letin spec ≝ (λz,k:nat.∀i,j,l:nat.k ≤ i → k ≤ j → l ≤ z → w l 〈a i,a j〉); -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.∃k. spec z k)); unfold spec in aux ⊢ %; - [1,2:apply H8; - |3: intros 3; cases (H2 (w n) (H8 n)); simplify in ⊢ (? (? % ?) ?→?); - simplify in ⊢ (?→? (? % ?) ?→?); - intros; lapply (H5 i j) as H14; - [2: apply (max_le_l ??? H6);|3:apply (max_le_l ??? H7);] - cases (le_to_or_lt_eq ?? H10); [2: destruct H11; destruct H4; assumption] - cases (aux n1) in H6 H7 ⊢ %; simplify in ⊢ (? (? ? %) ?→? (? ? %) ?→?); intros; - apply H6; [3: apply le_S_S_to_le; assumption] - apply lt_to_le; apply (max_le_r w1); assumption; - |4: intros; clear H6; rewrite > H4 in H5; - rewrite < (le_n_O_to_eq ? H11); apply H5; assumption;] -cut ((⌊x,(m x: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 ((⌊x,(m x: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; unfold spec in m Hm Hm1; clear spec; -cut (⌊x,a (m x)⌋ ↑ l ∨ ⌊x,a (m x)⌋ ↓ l) as H10; [2: - cases H1; - [ left; apply (selection_uparrow ? Hm a l H4); - | right; apply (selection_downarrow ? Hm a l H4);]] -lapply (H9 ?? H10) as H11; [ - exists [apply (m 0:nat)] intros; - cases H1; cases H5; cases H7; cases (us_phi4 ?? H3 〈l,a i〉); - apply H15; change with (U 〈a i,l〉); - [apply (ous_convex_l C ? H3 ? H11 (H12 (m O))); - |apply (ous_convex_r C ? H3 ? H11 (H12 (m O)));] - [1:apply (H12 i); - |3: apply (le_reflexive l); - |4: apply (H12 i); - |2:change with (a (m O) ≤ a i); - apply (trans_increasing a H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H16); - |5:apply (H12 i); - |7:apply (ge_reflexive (l : hos_carr (os_r C))); - |8:apply (H12 i); - |6:change with (a i ≤ a (m O)); - apply (trans_decreasing ? H6 (\fst (m 0)) i); intro; apply (le_to_not_lt ?? H4 H16);]] -clear H10; intros (p q r); change with (w p 〈a (m q),a (m r)〉); -generalize in match (refl_eq nat (m p)); -generalize in match (m p) in ⊢ (? ? ? % → %); intro X; cases X (w1 H15); clear X; -intros (H16); simplify in H16:(? ? ? %); destruct H16; -apply H15; [3: apply le_n] -[1: lapply (trans_increasing ? Hm1 p q) as T; [apply not_lt_to_le; apply T;] - apply (le_to_not_lt p q H4); -|2: lapply (trans_increasing ? Hm1 p r) as T; [apply not_lt_to_le; apply T;] - apply (le_to_not_lt p r H5);] -qed. diff --git a/helm/software/matita/contribs/dama/dama/russell_support.ma b/helm/software/matita/contribs/dama/dama/russell_support.ma deleted file mode 100644 index deb5fc950..000000000 --- a/helm/software/matita/contribs/dama/dama/russell_support.ma +++ /dev/null @@ -1,27 +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/nat.ma". -include "logic/cprop_connectives.ma". - -definition hide ≝ λT:Type.λx:T.x. - -notation < "\blacksquare" non associative with precedence 50 for @{'hide}. -interpretation "hide" 'hide = (hide _ _). -interpretation "hide2" 'hide = (hide _ _ _). - -definition inject ≝ λP.λa:nat.λp:P a. ex_introT ? P ? p. -coercion cic:/matita/dama/russell_support/inject.con 0 1. -definition eject ≝ λP.λc: ∃n:nat.P n. match c with [ ex_introT w _ ⇒ w]. -coercion cic:/matita/dama/russell_support/eject.con. diff --git a/helm/software/matita/contribs/dama/dama/sandwich.ma b/helm/software/matita/contribs/dama/dama/sandwich.ma deleted file mode 100644 index 68bb4453c..000000000 --- a/helm/software/matita/contribs/dama/dama/sandwich.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 "ordered_uniform.ma". - -lemma le_w_plus: ∀n,m,o.n+m≤o → m ≤ o. -intro; elim n; simplify; [assumption] -lapply (le_S ?? H1); apply H; apply le_S_S_to_le; assumption; -qed. - -alias symbol "leq" = "Ordered set less or equal than". -alias symbol "and" = "logical and". -theorem sandwich: - ∀O:ordered_uniform_space.∀l:O.∀a,b,x:sequence O. - (∀i:nat.a i ≤ x i ∧ x i ≤ b i) → - a uniform_converges l → - b uniform_converges l → - x uniform_converges l. -intros 10; -cases (us_phi3 ? ? H3) (V GV); cases GV (Gv HV); clear GV; -cases (us_phi3 ? ? Gv) (W GW); cases GW (Gw HW); clear GW; -cases (H1 ? Gw) (ma Hma); cases (H2 ? Gw) (mb Hmb); clear H1 H2; -exists [apply (ma + mb)] intros; apply (HV 〈l,(x i)〉); -unfold; simplify; exists [apply (a i)] split; -[2: apply (ous_convex_l ?? Gv 〈a i,b i〉); cases (H i) (Lax Lxb); clear H; - [1: apply HW; exists [apply l]simplify; split; - [1: cases (us_phi4 ?? Gw 〈(a i),l〉); apply H2; clear H2 H; - apply (Hma i); rewrite > sym_plus in H1; apply (le_w_plus mb); assumption; - |2: apply Hmb; apply (le_w_plus ma); assumption] - |2,3: simplify; apply (le_transitive (a i) ?? Lax Lxb); - |4: apply (le_reflexive); - |5,6: assumption;] -|1: apply HW; exists[apply l] simplify; split; - [1: apply (us_phi1 ?? Gw); unfold; apply eq_reflexive; - |2: apply Hma; rewrite > sym_plus in H1; apply (le_w_plus mb); assumption;]] -qed. - diff --git a/helm/software/matita/contribs/dama/dama/sequence.ma b/helm/software/matita/contribs/dama/dama/sequence.ma deleted file mode 100644 index 948d14f67..000000000 --- a/helm/software/matita/contribs/dama/dama/sequence.ma +++ /dev/null @@ -1,38 +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/nat.ma". - -inductive sequence (O:Type) : Type ≝ - | mk_seq : (nat → O) → sequence O. - -definition fun_of_seq: ∀O:Type.sequence O → nat → O ≝ - λO.λx:sequence O.match x with [ mk_seq f ⇒ f ]. - -coercion cic:/matita/dama/sequence/fun_of_seq.con 1. - -notation < "hvbox((\lfloor term 19 p \rfloor) \sub ident i)" with precedence 90 -for @{ 'sequence (\lambda ${ident i} : $t . $p)}. - -notation > "hvbox((\lfloor term 19 p \rfloor) \sub ident i)" with precedence 90 -for @{ 'sequence (\lambda ${ident i} . $p)}. - -notation > "hvbox(\lfloor ident i, term 19 p \rfloor)" with precedence 90 -for @{ 'sequence (\lambda ${ident i} . $p)}. - -notation "a \sub i" left associative with precedence 90 - for @{ 'sequence_appl $a $i }. - -interpretation "sequence" 'sequence \eta.x = (mk_seq _ x). -interpretation "sequence element" 'sequence_appl s i = (fun_of_seq _ s i). diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma deleted file mode 100644 index 0de61b292..000000000 --- a/helm/software/matita/contribs/dama/dama/supremum.ma +++ /dev/null @@ -1,442 +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 "datatypes/constructors.ma". -include "nat/plus.ma". -include "nat_ordered_set.ma". -include "sequence.ma". - -(* Definition 2.4 *) -definition upper_bound ≝ - λO:half_ordered_set.λa:sequence O.λu:O.∀n:nat.a n ≤≤ u. - -definition supremum ≝ - λO:half_ordered_set.λs:sequence O.λx. - upper_bound ? s x ∧ (∀y:O.x ≰≰ y → ∃n.s n ≰≰ y). - -definition increasing ≝ - λO:half_ordered_set.λa:sequence O.∀n:nat.a n ≤≤ a (S n). - -notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 45 - for @{'upper_bound $s $x}. -notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 45 - for @{'lower_bound $s $x}. -notation < "s \nbsp 'is_increasing'" non associative with precedence 45 - for @{'increasing $s}. -notation < "s \nbsp 'is_decreasing'" non associative with precedence 45 - for @{'decreasing $s}. -notation < "x \nbsp 'is_supremum' \nbsp s" non associative with precedence 45 - for @{'supremum $s $x}. -notation < "x \nbsp 'is_infimum' \nbsp s" non associative with precedence 45 - for @{'infimum $s $x}. -notation > "x 'is_upper_bound' s" non associative with precedence 45 - for @{'upper_bound $s $x}. -notation > "x 'is_lower_bound' s" non associative with precedence 45 - for @{'lower_bound $s $x}. -notation > "s 'is_increasing'" non associative with precedence 45 - for @{'increasing $s}. -notation > "s 'is_decreasing'" non associative with precedence 45 - for @{'decreasing $s}. -notation > "x 'is_supremum' s" non associative with precedence 45 - for @{'supremum $s $x}. -notation > "x 'is_infimum' s" non associative with precedence 45 - for @{'infimum $s $x}. - -interpretation "Ordered set upper bound" 'upper_bound s x = (upper_bound (os_l _) s x). -interpretation "Ordered set lower bound" 'lower_bound s x = (upper_bound (os_r _) s x). - -interpretation "Ordered set increasing" 'increasing s = (increasing (os_l _) s). -interpretation "Ordered set decreasing" 'decreasing s = (increasing (os_r _) s). - -interpretation "Ordered set strong sup" 'supremum s x = (supremum (os_l _) s x). -interpretation "Ordered set strong inf" 'infimum s x = (supremum (os_r _) s x). - -(* Fact 2.5 *) -lemma h_supremum_is_upper_bound: - ∀C:half_ordered_set.∀a:sequence C.∀u:C. - supremum ? a u → ∀v.upper_bound ? a v → u ≤≤ v. -intros 7 (C s u Hu v Hv H); cases Hu (_ H1); clear Hu; -cases (H1 ? H) (w Hw); apply Hv; [apply w] assumption; -qed. - -notation "'supremum_is_upper_bound'" non associative with precedence 90 for @{'supremum_is_upper_bound}. -notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'infimum_is_lower_bound}. - -interpretation "supremum_is_upper_bound" 'supremum_is_upper_bound = (h_supremum_is_upper_bound (os_l _)). -interpretation "infimum_is_lower_bound" 'infimum_is_lower_bound = (h_supremum_is_upper_bound (os_r _)). - -(* Lemma 2.6 *) -definition strictly_increasing ≝ - λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n. - -notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 45 - for @{'strictly_increasing $s}. -notation > "s 'is_strictly_increasing'" non associative with precedence 45 - for @{'strictly_increasing $s}. -interpretation "Ordered set strict increasing" 'strictly_increasing s = - (strictly_increasing (os_l _) s). - -notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 45 - for @{'strictly_decreasing $s}. -notation > "s 'is_strictly_decreasing'" non associative with precedence 45 - for @{'strictly_decreasing $s}. -interpretation "Ordered set strict decreasing" 'strictly_decreasing s = - (strictly_increasing (os_r _) s). - -definition uparrow ≝ - λC:half_ordered_set.λs:sequence C.λu:C. - increasing ? s ∧ supremum ? s u. - -interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u). -interpretation "Ordered set downarrow" 'fintersects s u = (uparrow (os_r _) s u). - -lemma h_trans_increasing: - ∀C:half_ordered_set.∀a:sequence C.increasing ? a → - ∀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 (hos_coreflexive ? (a n) X);] -cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1; -[2: rewrite > H2; intro; cases (hos_coreflexive ? (a (S n1)) H1); -|1: apply (hle_transitive ???? (H ?) (Hs ?)); - intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);] -qed. - -notation "'trans_increasing'" non associative with precedence 90 for @{'trans_increasing}. -notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_decreasing}. - -interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)). -interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)). - -lemma hint_nat : - Type_of_ordered_set nat_ordered_set → - hos_carr (os_l (nat_ordered_set)). -intros; assumption; -qed. - -coercion hint_nat nocomposites. - -lemma h_trans_increasing_exc: - ∀C:half_ordered_set.∀a:sequence C.increasing ? a → - ∀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; -[1: change in n1 with (hos_carr (os_l nat_ordered_set)); - change with (n H3; apply H;] - |2: cases (?:False); change in Hp with (n "a 'order_converges' x" non associative with precedence 45 - for @{'order_converge $a $x}. -interpretation "Order convergence" 'order_converge s u = (order_converge _ s u). - -(* Definition 2.8 *) -record segment (O : Type) : Type ≝ { - seg_l_ : O; - seg_u_ : O -}. - -notation > "𝕦_term 90 s" non associative with precedence 90 for @{'upp $s}. -notation "𝕦 \sub term 90 s" non associative with precedence 90 for @{'upp $s}. -notation > "𝕝_term 90 s" non associative with precedence 90 for @{'low $s}. -notation "𝕝 \sub term 90 s" non associative with precedence 90 for @{'low $s}. - -definition seg_u ≝ - λO:half_ordered_set.λs:segment O. - wloss O ?? (λl,u.l) (seg_u_ ? s) (seg_l_ ? s). -definition seg_l ≝ - λO:half_ordered_set.λs:segment O. - wloss O ?? (λl,u.l) (seg_l_ ? s) (seg_u_ ? s). - -interpretation "uppper" 'upp s = (seg_u (os_l _) s). -interpretation "lower" 'low s = (seg_l (os_l _) s). -interpretation "uppper dual" 'upp s = (seg_l (os_r _) s). -interpretation "lower dual" 'low s = (seg_u (os_r _) s). - -definition in_segment ≝ - λO:half_ordered_set.λs:segment O.λx:O. - wloss O ?? (λp1,p2.p1 ∧ p2) (seg_l ? s ≤≤ x) (x ≤≤ seg_u ? s). - -notation "‡O" non associative with precedence 90 for @{'segment $O}. -interpretation "Ordered set sergment" 'segment x = (segment x). - -interpretation "Ordered set sergment in" 'mem x s = (in_segment _ s x). - -definition segment_ordered_set_carr ≝ - λO:half_ordered_set.λs:‡O.∃x.x ∈ s. -definition segment_ordered_set_exc ≝ - λO:half_ordered_set.λs:‡O. - λx,y:segment_ordered_set_carr O s.hos_excess_ O (\fst x) (\fst y). -lemma segment_ordered_set_corefl: - ∀O,s. coreflexive ? (wloss O ?? (segment_ordered_set_exc O s)). -intros 3; cases x; cases (wloss_prop O); -generalize in match (hos_coreflexive O w); -rewrite < (H1 ?? (segment_ordered_set_exc O s)); -rewrite < (H1 ?? (hos_excess_ O)); intros; assumption; -qed. -lemma segment_ordered_set_cotrans : - ∀O,s. cotransitive ? (wloss O ?? (segment_ordered_set_exc O s)). -intros 5 (O s x y z); cases x; cases y ; cases z; clear x y z; -generalize in match (hos_cotransitive O w w1 w2); -cases (wloss_prop O); -do 3 rewrite < (H3 ?? (segment_ordered_set_exc O s)); -do 3 rewrite < (H3 ?? (hos_excess_ O)); intros; apply H4; assumption; -qed. - -lemma half_segment_ordered_set: - ∀O:half_ordered_set.∀s:segment O.half_ordered_set. -intros (O a); constructor 1; -[ apply (segment_ordered_set_carr O a); -| apply (wloss O); -| apply (wloss_prop O); -| apply (segment_ordered_set_exc O a); -| apply (segment_ordered_set_corefl O a); -| apply (segment_ordered_set_cotrans ??); -] -qed. - -lemma segment_ordered_set: - ∀O:ordered_set.∀s:‡O.ordered_set. -intros (O s); -apply half2full; apply (half_segment_ordered_set (os_l O) s); -qed. - -notation "{[ term 19 s ]}" non associative with precedence 90 for @{'segset $s}. -interpretation "Ordered set segment" 'segset s = (segment_ordered_set _ s). - -(* test : - ∀O:ordered_set.∀s: segment (os_l O).∀x:O. - in_segment (os_l O) s x - = - in_segment (os_r O) s x. -intros; try reflexivity; -*) - -lemma prove_in_segment: - ∀O:half_ordered_set.∀s:segment O.∀x:O. - (seg_l O s) ≤≤ x → x ≤≤ (seg_u O s) → x ∈ s. -intros; unfold; cases (wloss_prop O); rewrite < H2; -split; assumption; -qed. - -lemma cases_in_segment: - ∀C:half_ordered_set.∀s:segment C.∀x. x ∈ s → (seg_l C s) ≤≤ x ∧ x ≤≤ (seg_u C s). -intros; unfold in H; cases (wloss_prop C) (W W); rewrite "s 'is_upper_located'" non associative with precedence 45 - for @{'upper_located $s}. -interpretation "Ordered set upper locatedness" 'upper_located s = - (upper_located (os_l _) s). - -notation < "s \nbsp 'is_lower_located'" non associative with precedence 45 - for @{'lower_located $s}. -notation > "s 'is_lower_located'" non associative with precedence 45 - for @{'lower_located $s}. -interpretation "Ordered set lower locatedness" 'lower_located s = - (upper_located (os_r _) s). - -(* Lemma 2.12 *) -lemma h_uparrow_upperlocated: - ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a. -intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy); -cases H3 (H4 H5); clear H3; cases (hos_cotransitive C y x u Hxy) (W W); -[2: cases (H5 x W) (w Hw); left; exists [apply w] assumption; -|1: right; exists [apply u]; split; [apply W|apply H4]] -qed. - -notation "'uparrow_upperlocated'" non associative with precedence 90 for @{'uparrow_upperlocated}. -notation "'downarrow_lowerlocated'" non associative with precedence 90 for @{'downarrow_lowerlocated}. - -interpretation "uparrow_upperlocated" 'uparrow_upperlocated = (h_uparrow_upperlocated (os_l _)). -interpretation "downarrow_lowerlocated" 'downarrow_lowerlocated = (h_uparrow_upperlocated (os_r _)). diff --git a/helm/software/matita/contribs/dama/dama/uniform.ma b/helm/software/matita/contribs/dama/dama/uniform.ma deleted file mode 100644 index 759037124..000000000 --- a/helm/software/matita/contribs/dama/dama/uniform.ma +++ /dev/null @@ -1,92 +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 "supremum.ma". - -(* Definition 2.13 *) -alias symbol "pair" = "Pair construction". -alias symbol "exists" = "exists". -alias symbol "and" = "logical and". -definition compose_bs_relations ≝ - λC:bishop_set.λU,V:C squareB → Prop. - λx:C squareB.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉. - -definition compose_os_relations ≝ - λC:ordered_set.λU,V:C squareB → Prop. - λx:C squareB.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉. - -interpretation "bishop set relations composition" 'compose a b = (compose_bs_relations _ a b). -interpretation "ordered set relations composition" 'compose a b = (compose_os_relations _ a b). - -definition invert_bs_relation ≝ - λC:bishop_set.λU:C squareB → Prop. - λx:C squareB. U 〈\snd x,\fst x〉. - -notation > "\inv" with precedence 60 for @{ 'invert_symbol }. -interpretation "relation invertion" 'invert a = (invert_bs_relation _ a). -interpretation "relation invertion" 'invert_symbol = (invert_bs_relation _). -interpretation "relation invertion" 'invert_appl a x = (invert_bs_relation _ a x). - -alias symbol "exists" = "CProp exists". -alias symbol "compose" = "bishop set relations composition". -alias symbol "and" (instance 21) = "constructive and". -alias symbol "and" (instance 16) = "constructive and". -alias symbol "and" (instance 9) = "constructive and". -record uniform_space : Type ≝ { - us_carr:> bishop_set; - us_unifbase: (us_carr squareB → Prop) → CProp; - us_phi1: ∀U:us_carr squareB → Prop. us_unifbase U → - (λx:us_carr squareB.\fst x ≈ \snd x) ⊆ U; - us_phi2: ∀U,V:us_carr squareB → Prop. us_unifbase U → us_unifbase V → - ∃W:us_carr squareB → Prop.us_unifbase W ∧ (W ⊆ (λx.U x ∧ V x)); - us_phi3: ∀U:us_carr squareB → Prop. us_unifbase U → - ∃W:us_carr squareB → Prop.us_unifbase W ∧ (W ∘ W) ⊆ U; - us_phi4: ∀U:us_carr squareB → Prop. us_unifbase U → ∀x.(U x → (\inv U) x) ∧ ((\inv U) x → U x) -}. - -(* Definition 2.14 *) -alias symbol "leq" = "natural 'less or equal to'". -definition cauchy ≝ - λC:uniform_space.λa:sequence C.∀U.us_unifbase C U → - ∃n. ∀i,j. n ≤ i → n ≤ j → U 〈a i,a j〉. - -notation < "a \nbsp 'is_cauchy'" non associative with precedence 45 - for @{'cauchy $a}. -notation > "a 'is_cauchy'" non associative with precedence 45 - for @{'cauchy $a}. -interpretation "Cauchy sequence" 'cauchy s = (cauchy _ s). - -(* Definition 2.15 *) -definition uniform_converge ≝ - λC:uniform_space.λa:sequence C.λu:C. - ∀U.us_unifbase C U → ∃n. ∀i. n ≤ i → U 〈u,a i〉. - -notation < "a \nbsp (\u \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 45 - for @{'uniform_converge $a $x}. -notation > "a 'uniform_converges' x" non associative with precedence 45 - for @{'uniform_converge $a $x}. -interpretation "Uniform convergence" 'uniform_converge s u = - (uniform_converge _ s u). - -(* Lemma 2.16 *) -lemma uniform_converge_is_cauchy : - ∀C:uniform_space.∀a:sequence C.∀x:C. - a uniform_converges x → a is_cauchy. -intros (C a x Ha); intros 2 (u Hu); -cases (us_phi3 ?? Hu) (v Hv0); cases Hv0 (Hv H); clear Hv0; -cases (Ha ? Hv) (n Hn); exists [apply n]; intros; -apply H; unfold; exists [apply x]; split [2: apply (Hn ? H2)] -cases (us_phi4 ?? Hv 〈a i,x〉) (P1 P2); apply P2; -apply (Hn ? H1); -qed. -- 2.39.2