From: Enrico Tassi Date: Sat, 25 Oct 2008 14:00:27 +0000 (+0000) Subject: duality is a joke X-Git-Tag: make_still_working~4640 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=c231702a57076acf0c161cdb4799bf83158175f0;p=helm.git duality is a joke --- diff --git a/helm/software/matita/contribs/dama/dama/bishop_set.ma b/helm/software/matita/contribs/dama/dama/bishop_set.ma index 3522a3bb2..d69bb2732 100644 --- a/helm/software/matita/contribs/dama/dama/bishop_set.ma +++ b/helm/software/matita/contribs/dama/dama/bishop_set.ma @@ -67,30 +67,6 @@ 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 lt ≝ λE:half_ordered_set.λa,b:E. a ≤ b ∧ a # b. - -interpretation "ordered sets less than" 'lt a b = (lt _ a b). - -lemma lt_coreflexive: ∀E.coreflexive ? (lt E). -intros 2 (E x); intro H; cases H (_ ABS); -apply (bs_coreflexive ? x ABS); -qed. - -lemma lt_transitive: ∀E.transitive ? (lt E). -intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz); -split; [apply (le_transitive E ??? Lxy Lyz)] clear H1 H2; -cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]clear Axy Ayz; -[1: cases (hos_cotransitive E ?? y H1) (X X); [cases (Lxy X)|cases (hos_coreflexive E ? X)] -|2: cases (hos_cotransitive E ?? x H2) (X X); [right;assumption|cases (Lxy X)]] -qed. - -theorem lt_to_excess: ∀E:ordered_set.∀a,b:E. (a < b) → (b ≰ a). -intros (E a b Lab); cases Lab (LEab Aab); cases Aab (H H);[cases (LEab H)] -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). diff --git a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma index d4aa1d57d..dfeaa6d35 100644 --- a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma +++ b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma @@ -44,8 +44,9 @@ cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1; qed. lemma nat_ordered_set : ordered_set. -letin hos ≝ (mk_half_ordered_set nat nat_excess ? nat_excess_cotransitive);[ - intro x; intro H; apply (not_le_Sn_n ? H);] +letin hos ≝ (mk_half_ordered_set nat (λT:Type.λf:T→T→CProp.f) ? nat_excess ? nat_excess_cotransitive); +[ intros; left; intros; reflexivity; +| intro x; intro H; apply (not_le_Sn_n ? H);] constructor 1; [ apply hos; | apply (dual_hos hos); | reflexivity] qed. @@ -63,17 +64,3 @@ lemma nat_le_to_os_le: intros 3; apply (le_to_not_lt a b);assumption; qed. -(* -lemma nat_lt_to_os_lt: - ∀a,b:nat_ordered_set.a < b → lt nat_ordered_set a b. -intros 3; split; -[1: apply nat_le_to_os_le; apply lt_to_le;assumption; -|2: right; apply H;] -qed. - -lemma os_lt_to_nat_lt: - ∀a,b:nat_ordered_set. lt nat_ordered_set a b → a < b. -intros; cases H; clear H; cases H2; -[2: apply H;| cases (H1 H)] -qed. -*) \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama/ordered_set.ma b/helm/software/matita/contribs/dama/dama/ordered_set.ma index 233df16f6..a4b40be7a 100644 --- a/helm/software/matita/contribs/dama/dama/ordered_set.ma +++ b/helm/software/matita/contribs/dama/dama/ordered_set.ma @@ -27,37 +27,48 @@ interpretation "" ' = ( (os_r _)). (* Definition 2.1 *) record half_ordered_set: Type ≝ { hos_carr:> Type; - hos_excess: hos_carr → hos_carr → CProp; - hos_coreflexive: coreflexive ? hos_excess; - hos_cotransitive: cotransitive ? hos_excess + wloss: ∀A:Type. (A → A → CProp) → A → A → CProp; + wloss_prop: (∀T,P,x,y.P x y = wloss T P x y) ∨ (∀T,P,x,y.P y x = wloss T 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). + +lemma find_leq : half_ordered_set → half_ordered_set. +intro O; constructor 1; +[1: apply (hos_carr O); +|2: apply (λT:Type.λf:T→T→CProp.f); +|3: intros; left; intros; reflexivity; +|4: apply (hos_excess_ O); +|5: intro x; lapply (hos_coreflexive O x) as H; cases (wloss_prop O); + rewrite < H1 in H; apply H; +|6: intros 4 (x y z H); cases (wloss_prop O); + rewrite > (H1 ? (hos_excess_ O)) in H ⊢ %; + rewrite > (H1 ? (hos_excess_ O)); lapply (hos_cotransitive O ?? z H); + [assumption] cases Hletin;[right|left]assumption;] +qed. + definition dual_hos : half_ordered_set → half_ordered_set. intro; constructor 1; [ apply (hos_carr h); -| apply (λx,y.hos_excess h y x); +| apply (λT,f,x,y.wloss h T 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 ?? z 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; + os_l : half_ordered_set; os_r_ : half_ordered_set; - os_with : os_r_ = dual_hos os_l_ + os_with : os_r_ = dual_hos os_l }. -definition os_l : ordered_set → half_ordered_set. -intro h; constructor 1; -[ apply (hos_carr (os_l_ h)); -| apply (λx,y.hos_excess (os_l_ h) x y); -| apply (hos_coreflexive (os_l_ h)); -| apply (hos_cotransitive (os_l_ h)); -] -qed. - definition os_r : ordered_set → half_ordered_set. -intro o; apply (dual_hos (os_l_ o)); qed. +intro o; apply (dual_hos (os_l o)); qed. lemma half2full : half_ordered_set → ordered_set. intro hos; @@ -84,26 +95,26 @@ 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 _)). +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 _)). +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 "Ordered half set less or equal than" 'leq_low a b = (le _ 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). +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 (E x H); apply (hos_coreflexive ?? H); +unfold reflexive; intros 3; apply (hos_coreflexive ? x H); qed. notation "'le_reflexive'" non associative with precedence 90 for @{'le_reflexive}. @@ -124,7 +135,7 @@ 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 ??? y H3) (H4 H4); +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. @@ -136,10 +147,10 @@ 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'. + ∀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 ??? a1 Eab) (H H); [cases (Laa1 H)] -cases (hos_cotransitive ??? b1 H) (H1 H1); [assumption] +cases (hos_cotransitive ? a b a1 Eab) (H H); [cases (Laa1 H)] +cases (hos_cotransitive ? ?? b1 H) (H1 H1); [assumption] cases (Lb1b H1); qed. @@ -149,16 +160,26 @@ notation "'exc_ge_variance'" non associative with precedence 90 for @{'exc_ge_va 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: intros (x y); apply (\fst x ≰≰ \fst y ∨ \snd x ≰≰ \snd y); -|2: intro x0; cases x0 (x y); clear x0; simplify; intro H; - cases H (X X); apply (hos_coreflexive ?? X); -|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2); - clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H; - [1: cases (hos_cotransitive ??? z1 H1); [left; left|right;left]assumption; - |2: cases (hos_cotransitive ??? z2 H1); [left;right|right;right]assumption]] +[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. diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma index 4ba3ca3af..7a52e5f06 100644 --- a/helm/software/matita/contribs/dama/dama/supremum.ma +++ b/helm/software/matita/contribs/dama/dama/supremum.ma @@ -63,16 +63,6 @@ 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). -(* se non faccio il bs_of_hos perdo dualità qui *) -lemma uniq_supremum: - ∀O:ordered_set.∀s:sequence O.∀t1,t2:O. - t1 is_supremum s → t2 is_supremum s → t1 ≈ t2. -intros (O s t1 t2 Ht1 Ht2); cases Ht1 (U1 H1); cases Ht2 (U2 H2); -apply le_le_eq; intro X; -[1: cases (H1 ? X); apply (U2 w); assumption -|2: cases (H2 ? X); apply (U1 w); assumption] -qed. - (* Fact 2.5 *) lemma h_supremum_is_upper_bound: ∀C:half_ordered_set.∀a:sequence C.∀u:C. @@ -87,14 +77,6 @@ notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'in 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 _)). -(* TEST DUALITY -lemma test_infimum_is_lower_bound_duality: - ∀C:ordered_set.∀a:sequence C.∀u:C. - u is_infimum a → ∀v.v is_lower_bound a → u ≥ v. -intros; lapply (infimum_is_lower_bound a u H v H1); assumption; -qed. -*) - (* Lemma 2.6 *) definition strictly_increasing ≝ λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n. @@ -116,16 +98,8 @@ interpretation "Ordered set strict decreasing" 'strictly_decreasing s = definition uparrow ≝ λC:half_ordered_set.λs:sequence C.λu:C. increasing ? s ∧ supremum ? s u. -(* -notation < "a \uparrow \nbsp u" non associative with precedence 45 for @{'sup_inc $a $u}. -notation > "a \uparrow u" non associative with precedence 45 for @{'sup_inc $a $u}. -*) -interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u). -(* -notation < "a \downarrow \nbsp u" non associative with precedence 45 for @{'inf_dec $a $u}. -notation > "a \downarrow u" non associative with precedence 45 for @{'inf_dec $a $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: @@ -133,9 +107,9 @@ lemma h_trans_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 (hos_coreflexive ?? X);] + 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 ?? 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. @@ -146,23 +120,24 @@ notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_de interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)). interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)). -(* TEST DUALITY -lemma test_trans_decreasing_duality: - ∀C:ordered_set.∀a:sequence C.a is_decreasing → - ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n. -intros; apply (trans_decreasing ? H ?? H1); qed. -*) +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. + ∀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 "a 'order_converges' x" non associative with precedence 45 interpretation "Order convergence" 'order_converge s u = (order_converge _ s u). (* Definition 2.8 *) -definition segment ≝ λO:half_ordered_set.λa,b:O.λx:O.(x ≤≤ b) ∧ (a ≤≤ x). +record segment (O : Type) : Type ≝ { + seg_l_ : O; + seg_u_ : O +}. + +notation > "𝕦_term 90 s p" non associative with precedence 45 for @{'upp $s $p}. +notation "𝕦 \sub term 90 s p" non associative with precedence 45 for @{'upp $s $p}. +notation > "𝕝_term 90 s p" non associative with precedence 45 for @{'low $s $p}. +notation "𝕝 \sub term 90 s p" non associative with precedence 45 for @{'low $s $p}. + +definition seg_u ≝ + λO:half_ordered_set.λs:segment O.λP: O → CProp. + wloss O ? (λl,u.P u) (seg_l_ ? s) (seg_u_ ? s). +definition seg_l ≝ + λO:half_ordered_set.λs:segment O.λP: O → CProp. + wloss O ? (λl,u.P u) (seg_u_ ? s) (seg_l_ ? s). + +interpretation "uppper" 'upp s P = (seg_u (os_l _) s P). +interpretation "lower" 'low s P = (seg_l (os_l _) s P). +interpretation "uppper dual" 'upp s P = (seg_l (os_r _) s P). +interpretation "lower dual" 'low s P = (seg_u (os_r _) s P). + +definition in_segment ≝ + λO:half_ordered_set.λs:segment O.λx:O. + wloss O ? (λp1,p2.p1 ∧ p2) (seg_u ? s (λu.u ≤≤ x)) (seg_l ? s (λl.x ≤≤ l)). -notation "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}. -interpretation "Ordered set sergment" 'segment a b = (segment (os_l _) a b). +notation "‡O" non associative with precedence 90 for @{'segment $O}. +interpretation "Ordered set sergment" 'segment x = (segment x). -notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45 - for @{'segment_in $a $b $x}. -interpretation "Ordered set sergment in" 'segment_in a b x= (segment (os_l _) a b x). +interpretation "Ordered set sergment in" 'mem x s = (in_segment _ s x). definition segment_ordered_set_carr ≝ - λO:half_ordered_set.λu,v:O.∃x.segment ? u v x. + λO:half_ordered_set.λs:‡O.∃x.x ∈ s. definition segment_ordered_set_exc ≝ - λO:half_ordered_set.λu,v:O. - λx,y:segment_ordered_set_carr ? u v.\fst x ≰≰ \fst y. + 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,u,v. coreflexive ? (segment_ordered_set_exc O u v). -intros 4; cases x; simplify; apply hos_coreflexive; qed. + ∀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,u,v. cotransitive ? (segment_ordered_set_exc O u v). -intros 6 (O u v x y z); cases x; cases y ; cases z; simplify; apply hos_cotransitive; + ∀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.∀u,v:O.half_ordered_set. -intros (O u v); apply (mk_half_ordered_set ?? (segment_ordered_set_corefl O u v) (segment_ordered_set_cotrans ???)); + ∀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.∀u,v:O.ordered_set. -intros (O u v); -apply half2full; apply (half_segment_ordered_set (os_l O) u v); + ∀O:ordered_set.∀s:‡O.ordered_set. +intros (O s); +apply half2full; apply (half_segment_ordered_set (os_l O) s); qed. -(* -notation < "hvbox({[a, break b]/})" non associative with precedence 90 - for @{'h_segment_set $a $b}. -notation > "hvbox({[a, break b]/})" non associative with precedence 90 - for @{'h_segment_set $a $b}. -interpretation "Half ordered set segment" 'h_segment_set a b = - (half_segment_ordered_set _ a b). +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; *) -notation < "hvbox({[a, break b]})" non associative with precedence 90 - for @{'segment_set $a $b}. -notation > "hvbox({[a, break b]})" non associative with precedence 90 - for @{'segment_set $a $b}. -interpretation "Ordered set segment" 'segment_set a b = - (segment_ordered_set _ a b). - definition hint_sequence: ∀C:ordered_set. sequence (hos_carr (os_l C)) → sequence (Type_of_ordered_set C). @@ -301,27 +308,56 @@ coercion hint_sequence2 nocomposites. coercion hint_sequence3 nocomposites. (* Lemma 2.9 - non easily dualizable *) -lemma segment_preserves_supremum: - ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. - ⌊n,\fst (a n)⌋ is_increasing ∧ - (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x. + +lemma x2sx: + ∀O:half_ordered_set. + ∀s:segment O.∀x,y:half_segment_ordered_set ? s. + \fst x ≰≰ \fst y → x ≰≰ y. +intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess; +whd in ⊢ (?→? (% ? ?) ? ? ? ?); simplify in ⊢ (?→%); +cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption; +qed. + +lemma sx2x: + ∀O:half_ordered_set. + ∀s:segment O.∀x,y:half_segment_ordered_set ? s. + x ≰≰ y → \fst x ≰≰ \fst y. +intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess; +whd in ⊢ (? (% ? ?) ? ? ? ? → ?); simplify in ⊢ (% → ?); +cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption; +qed. + +lemma h_segment_preserves_supremum: + ∀O:half_ordered_set.∀s:segment O. + ∀a:sequence (half_segment_ordered_set ? s). + ∀x:half_segment_ordered_set ? s. + increasing ? ⌊n,\fst (a n)⌋ ∧ + supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x. intros; split; cases H; clear H; -[1: apply H1; +[1: intro n; lapply (H1 n) as K; clear H1 H2; + intro; apply K; clear K; apply (sx2x ???? H); |2: cases H2; split; clear H2; - [1: apply H; - |2: clear H; intro y0; apply (H3 (\fst y0));]] + [1: intro n; lapply (H n) as K; intro W; apply K; + apply (sx2x ???? W); + |2: clear H1 H; intros (y0 Hy0); cases (H3 (\fst y0));[exists[apply w]] + [1: change in H with (\fst (a w) ≰≰ \fst y0); apply (x2sx ???? H); + |2: apply (sx2x ???? Hy0);]]] qed. -lemma segment_preserves_infimum: - ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. +notation "'segment_preserves_supremum'" non associative with precedence 90 for @{'segment_preserves_supremum}. +notation "'segment_preserves_infimum'" non associative with precedence 90 for @{'segment_preserves_infimum}. + +interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)). +interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)). + +(* TEST, ma quanto godo! *) +lemma segment_preserves_infimum2: + ∀O:ordered_set.∀s:‡O.∀a:sequence {[s]}.∀x:{[s]}. ⌊n,\fst (a n)⌋ is_decreasing ∧ (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x. -intros; split; cases H; clear H; -[1: apply H1; -|2: cases H2; split; clear H2; - [1: apply H; - |2: clear H; intro y0; apply (H3 (\fst y0));]] +intros; apply (segment_preserves_infimum s a x H); qed. +*) (* Definition 2.10 *) alias symbol "pi2" = "pair pi2".