From c077ca16ea87ba612435a47eee714b5388204d93 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Wed, 28 May 2008 10:39:13 +0000 Subject: [PATCH] the attempt of completing dama using duality frozen --- .../matita/contribs/dama/dama/lattice.ma | 56 +- .../contribs/dama/dama_duality/Makefile | 16 + .../dama_duality/Q_is_orded_divisble_group.ma | 276 ++ .../matita/contribs/dama/dama_duality/TODO | 4 + .../dama/dama_duality/attic/fields.ma | 60 + .../attic/integration_algebras.ma | 368 ++ .../dama_duality/attic/ordered_fields_ch0.ma | 151 + .../contribs/dama/dama_duality/attic/reals.ma | 172 + .../contribs/dama/dama_duality/attic/rings.ma | 103 + .../dama/dama_duality/attic/vector_spaces.ma | 151 + .../classical_pointfree/ordered_sets.ma | 424 +++ .../classical_pointfree/ordered_sets2.ma | 127 + .../dama_duality/classical_pointwise/sets.ma | 104 + .../classical_pointwise/sigma_algebra.ma | 40 + .../classical_pointwise/topology.ma | 45 + .../dama_duality/constructive_connectives.ma | 53 + .../constructive_higher_order_relations.ma | 51 + .../constructive_pointfree/lebesgue.ma | 31 + .../matita/contribs/dama/dama_duality/depends | 38 + .../dama/dama_duality/divisible_group.ma | 99 + .../dama/dama_duality/doc/DIMOSTRAZIONE | 126 + .../dama/dama_duality/doc/NotaReticoli.pdf | 3078 +++++++++++++++++ .../contribs/dama/dama_duality/excess.ma | 279 ++ .../contribs/dama/dama_duality/group.ma | 220 ++ .../contribs/dama/dama_duality/infsup.ma | 53 + .../contribs/dama/dama_duality/lattice.ma | 484 +++ .../contribs/dama/dama_duality/limit.ma | 67 + .../dama/dama_duality/metric_lattice.ma | 117 + .../dama/dama_duality/metric_space.ma | 46 + .../dama_duality/ordered_divisible_group.ma | 75 + .../dama/dama_duality/ordered_group.ma | 328 ++ .../dama/dama_duality/premetric_lattice.ma | 70 + .../dama/dama_duality/prevalued_lattice.ma | 243 ++ .../matita/contribs/dama/dama_duality/root | 1 + .../contribs/dama/dama_duality/sandwich.ma | 81 + .../dama/dama_duality/sandwich_corollary.ma | 39 + .../contribs/dama/dama_duality/sequence.ma | 21 + .../matita/contribs/dama/dama_duality/tend.ma | 28 + 38 files changed, 7716 insertions(+), 9 deletions(-) create mode 100644 helm/software/matita/contribs/dama/dama_duality/Makefile create mode 100644 helm/software/matita/contribs/dama/dama_duality/Q_is_orded_divisble_group.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/TODO create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/fields.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/integration_algebras.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/ordered_fields_ch0.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/reals.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/rings.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/attic/vector_spaces.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets2.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sets.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sigma_algebra.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/classical_pointwise/topology.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/constructive_connectives.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/constructive_higher_order_relations.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/constructive_pointfree/lebesgue.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/depends create mode 100644 helm/software/matita/contribs/dama/dama_duality/divisible_group.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/doc/DIMOSTRAZIONE create mode 100644 helm/software/matita/contribs/dama/dama_duality/doc/NotaReticoli.pdf create mode 100644 helm/software/matita/contribs/dama/dama_duality/excess.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/group.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/infsup.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/lattice.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/limit.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/metric_lattice.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/metric_space.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/ordered_divisible_group.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/ordered_group.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/premetric_lattice.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/prevalued_lattice.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/root create mode 100644 helm/software/matita/contribs/dama/dama_duality/sandwich.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/sandwich_corollary.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/sequence.ma create mode 100644 helm/software/matita/contribs/dama/dama_duality/tend.ma diff --git a/helm/software/matita/contribs/dama/dama/lattice.ma b/helm/software/matita/contribs/dama/dama/lattice.ma index 78046c688..adf751b83 100644 --- a/helm/software/matita/contribs/dama/dama/lattice.ma +++ b/helm/software/matita/contribs/dama/dama/lattice.ma @@ -185,9 +185,6 @@ definition hole ≝ λT:Type.λx:T.x. notation < "\ldots" non associative with precedence 50 for @{'hole}. interpretation "hole" 'hole = (cic:/matita/lattice/hole.con _ _). - -axiom FALSE : False. - (* SL(E,M,H2-5(#),H67(≰)) → E' → c E = c E' → H67'(≰') → SL(E,M,p2-5,H67') *) lemma subst_excess: ∀l:semi_lattice. @@ -340,15 +337,56 @@ apply (subst_excess sl); [apply (subst_excess_base_in_excess sl eb H H1 H2); |apply subst_excess_base_in_excess_preserves_apartness; |change in ⊢ (%→%→?) with ((λx.ap_carr x) (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)); simplify; - intros 3 (x y LE); - generalize in match (H3 ?? LE); - generalize in match H1 as H1;generalize in match H2 as H2; + intros 3 (x y LE); clear H3 LE; generalize in match x as x; generalize in match y as y; - cases FALSE; - (* - (reduce in H ⊢ %; cases H; simplify; intros; assumption); + generalize in match H1 as H1;generalize in match H2 as H2; + clear x y H1 H2 H4; STOP + + apply (match H return λr:Type.λm:Type_OF_semi_lattice sl=r. + ∀H2:(Πy2:exc_carr eb + .Πx2:exc_carr eb + .Or (exc_excess eb y2 x2) (exc_excess eb x2 y2) + →match H +  in eq +  return  + λright_1:Type + .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1 + .right_1→right_1→Type) +  with  + [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y2 x2) +.∀H1:Πy1:exc_carr eb + .Πx1:exc_carr eb + .match H +  in eq +  return  + λright_1:Type + .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1 + .right_1→right_1→Type) +  with  + [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y1 x1 + →Or (exc_excess eb y1 x1) (exc_excess eb x1 y1) + .∀y:ap_carr + (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) + .∀x:ap_carr + (apartness_OF_excess + (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) + .eq + (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) x + (match  + subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2 +  in eq +  return  + λright_1:apartness + .(λmatched:eq apartness (apartness_OF_semi_lattice sl) right_1 + .ap_carr right_1→ap_carr right_1→ap_carr right_1) +  with  + [refl_eq⇒sl_meet sl] x y) + + with [ refl_eq ⇒ ?]); + cases (subst_excess_base_in_excess_preserves_apartness sl eb H H1 H2); + cases H; cases (subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2); simplify; change in x:(%) with (exc_carr eb); change in y:(%) with (exc_carr eb); diff --git a/helm/software/matita/contribs/dama/dama_duality/Makefile b/helm/software/matita/contribs/dama/dama_duality/Makefile new file mode 100644 index 000000000..d40c9e674 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/Makefile @@ -0,0 +1,16 @@ +include ../../Makefile.defs + +DIR=$(shell basename $$PWD) + +$(DIR) all: + $(BIN)../matitac +$(DIR).opt opt all.opt: + $(BIN)../matitac.opt +clean: + $(BIN)../matitaclean +clean.opt: + $(BIN)../matitaclean.opt +depend: + $(BIN)../matitadep +depend.opt: + $(BIN)../matitadep.opt diff --git a/helm/software/matita/contribs/dama/dama_duality/Q_is_orded_divisble_group.ma b/helm/software/matita/contribs/dama/dama_duality/Q_is_orded_divisble_group.ma new file mode 100644 index 000000000..a0b646f33 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/Q_is_orded_divisble_group.ma @@ -0,0 +1,276 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "Q/q.ma". +include "ordered_divisible_group.ma". + +definition strong_decidable ≝ + λA:Prop.A ∨ ¬ A. + +theorem strong_decidable_to_Not_Not_eq: + ∀T:Type.∀eq: T → T → Prop.∀x,y:T. + strong_decidable (x=y) → ¬x≠y → x=y. + intros; + cases s; + [ assumption + | elim (H H1) + ] +qed. + +definition apartness_of_strong_decidable: + ∀T:Type.(∀x,y:T.strong_decidable (x=y)) → apartness. + intros; + constructor 1; + [ apply T + | apply (λx,y:T.x ≠ y); + | simplify; + intros 2; + apply (H (refl_eq ??)); + | simplify; + intros 4; + apply H; + symmetry; + assumption + | simplify; + intros; + elim (f x z); + [ elim (f z y); + [ elim H; + transitivity z; + assumption + | right; + assumption + ] + | left; + assumption + ] + ] +qed. + +theorem strong_decidable_to_strong_ext: + ∀T:Type.∀sd:∀x,y:T.strong_decidable (x=y). + ∀op:T→T. strong_ext (apartness_of_strong_decidable ? sd) op. + intros 6; + intro; + apply a; + apply eq_f; + assumption; +qed. + +theorem strong_decidable_to_transitive_to_cotransitive: + ∀T:Type.∀le:T→T→Prop.(∀x,y:T.strong_decidable (le x y)) → + transitive ? le → cotransitive ? (λx,y.¬ (le x y)). + intros; + whd; + simplify; + intros; + elim (f x z); + [ elim (f z y); + [ elim H; + apply (t ? z); + assumption + | right; + assumption + ] + | left; + assumption + ] +qed. + +theorem reflexive_to_coreflexive: + ∀T:Type.∀le:T→T→Prop.reflexive ? le → coreflexive ? (λx,y.¬(le x y)). + intros; + unfold; + simplify; + intros 2; + apply H1; + apply H; +qed. + +definition ordered_set_of_strong_decidable: + ∀T:Type.∀le:T→T→Prop.(∀x,y:T.strong_decidable (le x y)) → + transitive ? le → reflexive ? le → excess. + intros; + constructor 1; + [ constructor 1; + [ constructor 1; + [ constructor 1; + [ apply T + | apply (λx,y.¬(le x y)); + | apply reflexive_to_coreflexive; + assumption + | apply strong_decidable_to_transitive_to_cotransitive; + assumption] + no ported to duality + ] +qed. + +definition abelian_group_of_strong_decidable: + ∀T:Type.∀plus:T→T→T.∀zero:T.∀opp:T→T. + (∀x,y:T.strong_decidable (x=y)) → + associative ? plus (eq T) → + commutative ? plus (eq T) → + (∀x:T. plus zero x = x) → + (∀x:T. plus (opp x) x = zero) → + abelian_group. + intros; + constructor 1; + [apply (apartness_of_strong_decidable ? f);] + try assumption; + [ change with (associative ? plus (λx,y:T.¬x≠y)); + simplify; + intros; + intro; + apply H2; + apply a; + | intros 2; + intro; + apply a1; + apply c; + | intro; + intro; + apply a1; + apply H + | intro; + intro; + apply a1; + apply H1 + | intros; + apply strong_decidable_to_strong_ext; + assumption + ] +qed. + +definition left_neutral ≝ λC:Type.λop.λe:C. ∀x:C. op e x = x. +definition left_inverse ≝ λC:Type.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x = e. + +record nabelian_group : Type ≝ + { ncarr:> Type; + nplus: ncarr → ncarr → ncarr; + nzero: ncarr; + nopp: ncarr → ncarr; + nplus_assoc: associative ? nplus (eq ncarr); + nplus_comm: commutative ? nplus (eq ncarr); + nzero_neutral: left_neutral ? nplus nzero; + nopp_inverse: left_inverse ? nplus nzero nopp + }. + +definition abelian_group_of_nabelian_group: + ∀G:nabelian_group.(∀x,y:G.strong_decidable (x=y)) → abelian_group. + intros; + apply abelian_group_of_strong_decidable; + [2: apply (nplus G) + | skip + | apply (nzero G) + | apply (nopp G) + | assumption + | apply nplus_assoc; + | apply nplus_comm; + | apply nzero_neutral; + | apply nopp_inverse + ] +qed. + +definition Z_abelian_group: abelian_group. + apply abelian_group_of_nabelian_group; + [ constructor 1; + [ apply Z + | apply Zplus + | apply OZ + | apply Zopp + | whd; + intros; + symmetry; + apply associative_Zplus + | apply sym_Zplus + | intro; + reflexivity + | intro; + rewrite > sym_Zplus; + apply Zplus_Zopp; + ] + | simplify; + intros; + unfold; + generalize in match (eqZb_to_Prop x y); + elim (eqZb x y); + simplify in H; + [ left ; assumption + | right; assumption + ] + ] +qed. + +record nordered_set: Type ≝ + { nos_carr:> Type; + nos_le: nos_carr → nos_carr → Prop; + nos_reflexive: reflexive ? nos_le; + nos_transitive: transitive ? nos_le + }. + +definition excess_of_nordered_group: + ∀O:nordered_set.(∀x,y:O. strong_decidable (nos_le ? x y)) → excess. + intros; + constructor 1; + [ apply (nos_carr O) + | apply (λx,y.¬(nos_le ? x y)) + | apply reflexive_to_coreflexive; + apply nos_reflexive + | apply strong_decidable_to_transitive_to_cotransitive; + [ assumption + | apply nos_transitive + ] + ] +qed. + +lemma non_deve_stare_qui: reflexive ? Zle. + intro; + elim x; + [ exact I + |2,3: simplify; + apply le_n; + ] +qed. + +axiom non_deve_stare_qui3: ∀x,y:Z. x < y → x ≤ y. + +axiom non_deve_stare_qui4: ∀x,y:Z. x < y → y ≰ x. + +definition Z_excess: excess. + apply excess_of_nordered_group; + [ constructor 1; + [ apply Z + | apply Zle + | apply non_deve_stare_qui + | apply transitive_Zle + ] + | simplify; + intros; + unfold; + generalize in match (Z_compare_to_Prop x y); + cases (Z_compare x y); simplify; intro; + [ left; + apply non_deve_stare_qui3; + assumption + | left; + rewrite > H; + apply non_deve_stare_qui + | right; + apply non_deve_stare_qui4; + assumption + ] + ] +qed. \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama_duality/TODO b/helm/software/matita/contribs/dama/dama_duality/TODO new file mode 100644 index 000000000..353329bea --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/TODO @@ -0,0 +1,4 @@ +changing file resets the display-notation ref, but not the GUI tick +mettere una maction in tutti i body (ma forse non basta) +la visualizzazione dellea notazione se viene disttivata e poi se ne definisce una... la rende causa +il fatto che disabilitarla significa rimuovere quelle definite fino ad ora, non disabilitarla in senso proprio. diff --git a/helm/software/matita/contribs/dama/dama_duality/attic/fields.ma b/helm/software/matita/contribs/dama/dama_duality/attic/fields.ma new file mode 100644 index 000000000..824fdfa9e --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/attic/fields.ma @@ -0,0 +1,60 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "attic/rings.ma". + +record is_field (R:ring) (inv:∀x:R.x ≠ 0 → R) : Prop +≝ + { (* multiplicative abelian properties *) + mult_comm_: symmetric ? (mult R); + (* multiplicative group properties *) + inv_inverse_: ∀x.∀p: x ≠ 0. inv x p * x = 1 + }. + +lemma opp_opp: ∀R:ring. ∀x:R. --x=x. +intros; +apply (cancellationlaw ? (-x) ? ?); +rewrite > (opp_inverse R x); +rewrite > plus_comm; +rewrite > opp_inverse; +reflexivity. +qed. + +let rec sum (C:Type) (plus:C→C→C) (zero,one:C) (n:nat) on n ≝ + match n with + [ O ⇒ zero + | (S m) ⇒ plus one (sum C plus zero one m) + ]. + +record field : Type \def + { f_ring:> ring; + inv: ∀x:f_ring. x ≠ 0 → f_ring; + field_properties: is_field f_ring inv + }. + +theorem mult_comm: ∀F:field.symmetric ? (mult F). + intro; + apply (mult_comm_ ? ? (field_properties F)). +qed. + +theorem inv_inverse: ∀F:field.∀x:F.∀p: x ≠ 0. (inv ? x p)*x = 1. + intro; + apply (inv_inverse_ ? ? (field_properties F)). +qed. + +(*CSC: qua funzionava anche mettendo ? al posto della prima F*) +definition sum_field ≝ + λF:field. sum F (plus F) 0 1. diff --git a/helm/software/matita/contribs/dama/dama_duality/attic/integration_algebras.ma b/helm/software/matita/contribs/dama/dama_duality/attic/integration_algebras.ma new file mode 100644 index 000000000..1b775fa78 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/attic/integration_algebras.ma @@ -0,0 +1,368 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "attic/vector_spaces.ma". +include "lattice.ma". + +(**************** Riesz Spaces ********************) + +record pre_riesz_space (K:ordered_field_ch0) : Type \def + { rs_vector_space:> vector_space K; + rs_lattice_: lattice; + rs_ordered_abelian_group_: ordered_abelian_group; + rs_with1: + og_abelian_group rs_ordered_abelian_group_ = vs_abelian_group ? rs_vector_space; + rs_with2: + og_ordered_set rs_ordered_abelian_group_ = ordered_set_of_lattice rs_lattice_ + }. + +lemma rs_lattice: ∀K.pre_riesz_space K → lattice. + intros (K V); + cut (os_carrier (rs_lattice_ ? V) = V); + [ apply mk_lattice; + [ apply (carrier V) + | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut); + apply l_join + | apply (eq_rect ? ? (λC:Type.C→C→C) ? ? Hcut); + apply l_meet + | apply + (eq_rect' ? ? + (λa:Type.λH:os_carrier (rs_lattice_ ? V)=a. + is_lattice a + (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C) + (l_join (rs_lattice_ K V)) a H) + (eq_rect Type (rs_lattice_ K V) (λC:Type.C→C→C) + (l_meet (rs_lattice_ K V)) a H)) + ? ? Hcut); + simplify; + apply l_lattice_properties + ] + | transitivity (os_carrier (rs_ordered_abelian_group_ ? V)); + [ apply (eq_f ? ? os_carrier); + symmetry; + apply rs_with2 + | apply (eq_f ? ? carrier); + apply rs_with1 + ] + ]. +qed. + +coercion cic:/matita/attic/integration_algebras/rs_lattice.con. + +lemma rs_ordered_abelian_group: ∀K.pre_riesz_space K → ordered_abelian_group. + intros (K V); + apply mk_ordered_abelian_group; + [ apply mk_pre_ordered_abelian_group; + [ apply (vs_abelian_group ? (rs_vector_space ? V)) + | apply (ordered_set_of_lattice (rs_lattice ? V)) + | reflexivity + ] + | simplify; + generalize in match + (og_ordered_abelian_group_properties (rs_ordered_abelian_group_ ? V)); + intro P; + unfold in P; + elim daemon(* + apply + (eq_rect ? ? + (λO:ordered_set. + ∀f,g,h. + os_le O f g → + os_le O + (plus (abelian_group_OF_pre_riesz_space K V) f h) + (plus (abelian_group_OF_pre_riesz_space K V) g h)) + ? ? (rs_with2 ? V)); + apply + (eq_rect ? ? + (λG:abelian_group. + ∀f,g,h. + os_le (ordered_set_OF_pre_riesz_space K V) f g → + os_le (ordered_set_OF_pre_riesz_space K V) + (plus (abelian_group_OF_pre_riesz_space K V) f h) + (plus (abelian_group_OF_pre_riesz_space K V) g h)) + ? ? (rs_with1 ? V)); + simplify; + apply og_ordered_abelian_group_properties*) + ] +qed. + +coercion cic:/matita/attic/integration_algebras/rs_ordered_abelian_group.con. + +record is_riesz_space (K:ordered_field_ch0) (V:pre_riesz_space K) : Prop ≝ + { rs_compat_le_times: ∀a:K.∀f:V. 0≤a → 0≤f → 0≤a*f + }. + +record riesz_space (K:ordered_field_ch0) : Type \def + { rs_pre_riesz_space:> pre_riesz_space K; + rs_riesz_space_properties: is_riesz_space ? rs_pre_riesz_space + }. + +record is_positive_linear (K) (V:riesz_space K) (T:V→K) : Prop ≝ + { positive: ∀u:V. 0≤u → 0≤T u; + linear1: ∀u,v:V. T (u+v) = T u + T v; + linear2: ∀u:V.∀k:K. T (k*u) = k*(T u) + }. + +record sequentially_order_continuous (K) (V:riesz_space K) (T:V→K) : Prop ≝ + { soc_incr: + ∀a:nat→V.∀l:V.is_increasing ? a → is_sup V a l → + is_increasing K (λn.T (a n)) ∧ tends_to ? (λn.T (a n)) (T l) + }. + +definition absolute_value ≝ λK.λS:riesz_space K.λf:S.f ∨ -f. + +(**************** Normed Riesz spaces ****************************) + +definition is_riesz_norm ≝ + λR:real.λV:riesz_space R.λnorm:norm R V. + ∀f,g:V. absolute_value ? V f ≤ absolute_value ? V g → + n_function R V norm f ≤ n_function R V norm g. + +record riesz_norm (R:real) (V:riesz_space R) : Type ≝ + { rn_norm:> norm R V; + rn_riesz_norm_property: is_riesz_norm ? ? rn_norm + }. + +(*CSC: non fa la chiusura delle coercion verso funclass *) +definition rn_function ≝ + λR:real.λV:riesz_space R.λnorm:riesz_norm ? V. + n_function R V (rn_norm ? ? norm). + +coercion cic:/matita/attic/integration_algebras/rn_function.con 1. + +(************************** L-SPACES *************************************) +(* +record is_l_space (R:real) (V:riesz_space R) (norm:riesz_norm ? V) : Prop ≝ + { ls_banach: is_complete ? V (induced_distance ? ? norm); + ls_linear: ∀f,g:V. le ? V 0 f → le ? V 0 g → norm (f+g) = norm f + norm g + }. +*) +(******************** ARCHIMEDEAN RIESZ SPACES ***************************) + +record is_archimedean_riesz_space (K) (S:riesz_space K) : Prop +\def + { ars_archimedean: ∃u:S.∀n.∀a.∀p:n > O. + absolute_value ? S a ≤ + (inv K (sum_field K n) (not_eq_sum_field_zero K n p))* u → + a = 0 + }. + +record archimedean_riesz_space (K:ordered_field_ch0) : Type \def + { ars_riesz_space:> riesz_space K; + ars_archimedean_property: is_archimedean_riesz_space ? ars_riesz_space + }. + +definition is_weak_unit ≝ +(* This definition is by Spitters. He cites Fremlin 353P, but: + 1. that theorem holds only in f-algebras (as in Spitters, but we are + defining it on Riesz spaces) + 2. Fremlin proves |x|/\u=0 \to u=0. How do we remove the absolute value? + λR:real.λV:archimedean_riesz_space R.λunit: V. + ∀x:V. meet x unit = 0 → u = 0. + 3. Fremlin proves u > 0 implies x /\ u > 0 > 0 for Archimedean spaces + only. We pick this definition for now. +*) λR:real.λV:archimedean_riesz_space R.λe:V. + ∀v:V. 0 archimedean_riesz_space R; + irs_unit: irs_archimedean_riesz_space; + irs_weak_unit: is_weak_unit ? ? irs_unit; + integral: irs_archimedean_riesz_space → R; + irs_positive_linear: is_positive_linear ? ? integral; + irs_limit1: + ∀f:irs_archimedean_riesz_space. + tends_to ? + (λn.integral (f ∧ ((sum_field R n)*irs_unit))) + (integral f); + irs_limit2: + ∀f:irs_archimedean_riesz_space. + tends_to ? + (λn. + integral (f ∧ + ((inv ? (sum_field R (S n)) + (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n))) + ) * irs_unit))) 0; + irs_quotient_space1: + ∀f,g:irs_archimedean_riesz_space. + integral (absolute_value ? irs_archimedean_riesz_space (f - g)) = 0 → f=g + }. + +definition induced_norm_fun ≝ + λR:real.λV:integration_riesz_space R.λf:V. + integral ? V (absolute_value ? ? f). + +lemma induced_norm_is_norm: + ∀R:real.∀V:integration_riesz_space R.is_norm R V (induced_norm_fun ? V). + elim daemon.(* + intros; + apply mk_is_norm; + [ apply mk_is_semi_norm; + [ unfold induced_norm_fun; + intros; + apply positive; + [ apply (irs_positive_linear ? V) + | (* difficile *) + elim daemon + ] + | intros; + unfold induced_norm_fun; + (* facile *) + elim daemon + | intros; + unfold induced_norm_fun; + (* difficile *) + elim daemon + ] + | intros; + unfold induced_norm_fun in H; + apply irs_quotient_space1; + unfold minus; + rewrite < plus_comm; + rewrite < eq_zero_opp_zero; + rewrite > zero_neutral; + assumption + ].*) +qed. + +definition induced_norm ≝ + λR:real.λV:integration_riesz_space R. + mk_norm ? ? (induced_norm_fun ? V) (induced_norm_is_norm ? V). + +lemma is_riesz_norm_induced_norm: + ∀R:real.∀V:integration_riesz_space R. + is_riesz_norm ? ? (induced_norm ? V). + intros; + unfold is_riesz_norm; + intros; + unfold induced_norm; + simplify; + unfold induced_norm_fun; + (* difficile *) + elim daemon. +qed. + +definition induced_riesz_norm ≝ + λR:real.λV:integration_riesz_space R. + mk_riesz_norm ? ? (induced_norm ? V) (is_riesz_norm_induced_norm ? V). + +definition distance_induced_by_integral ≝ + λR:real.λV:integration_riesz_space R. + induced_distance ? ? (induced_norm R V). + +definition is_complete_integration_riesz_space ≝ + λR:real.λV:integration_riesz_space R. + is_complete ? ? (distance_induced_by_integral ? V). + +record complete_integration_riesz_space (R:real) : Type ≝ + { cirz_integration_riesz_space:> integration_riesz_space R; + cirz_complete_integration_riesz_space_property: + is_complete_integration_riesz_space ? cirz_integration_riesz_space + }. + +(* now we prove that any complete integration riesz space is an L-space *) + +(*theorem is_l_space_l_space_induced_by_integral: + ∀R:real.∀V:complete_integration_riesz_space R. + is_l_space ? ? (induced_riesz_norm ? V). + intros; + constructor 1; + [ apply cirz_complete_integration_riesz_space_property + | intros; + unfold induced_riesz_norm; + simplify; + unfold induced_norm; + simplify; + unfold induced_norm_fun; + (* difficile *) + elim daemon + ]. +qed.*) + +(**************************** f-ALGEBRAS ********************************) + +record is_algebra (K: field) (V:vector_space K) (mult:V→V→V) (one:V) : Prop +≝ + { (* ring properties *) + a_ring: is_ring V mult one; + (* algebra properties *) + a_associative_left: ∀a,f,g. a * (mult f g) = mult (a * f) g; + a_associative_right: ∀a,f,g. a * (mult f g) = mult f (a * g) + }. + +record algebra (K: field) : Type \def + { a_vector_space:> vector_space K; + a_one: a_vector_space; + a_mult: a_vector_space → a_vector_space → a_vector_space; + a_algebra_properties: is_algebra ? ? a_mult a_one + }. + +interpretation "Algebra product" 'times a b = + (cic:/matita/attic/integration_algebras/a_mult.con _ a b). + +definition ring_of_algebra ≝ + λK.λA:algebra K. + mk_ring A (a_mult ? A) (a_one ? A) + (a_ring ? ? ? ? (a_algebra_properties ? A)). + +coercion cic:/matita/attic/integration_algebras/ring_of_algebra.con. + +record pre_f_algebra (K:ordered_field_ch0) : Type ≝ + { fa_archimedean_riesz_space:> archimedean_riesz_space K; + fa_algebra_: algebra K; + fa_with: a_vector_space ? fa_algebra_ = rs_vector_space ? fa_archimedean_riesz_space + }. + +lemma fa_algebra: ∀K:ordered_field_ch0.pre_f_algebra K → algebra K. + intros (K A); + apply mk_algebra; + [ apply (rs_vector_space ? A) + | elim daemon + | elim daemon + | elim daemon + ] + qed. + +coercion cic:/matita/attic/integration_algebras/fa_algebra.con. + +record is_f_algebra (K) (A:pre_f_algebra K) : Prop ≝ +{ compat_mult_le: ∀f,g:A.0 ≤ f → 0 ≤ g → 0 ≤ f*g; + compat_mult_meet: + ∀f,g,h:A.(f ∧ g) = 0 → ((h*f) ∧ g) = 0 +}. + +record f_algebra (K:ordered_field_ch0) : Type ≝ +{ fa_pre_f_algebra:> pre_f_algebra K; + fa_f_algebra_properties: is_f_algebra ? fa_pre_f_algebra +}. + +(* to be proved; see footnote 2 in the paper by Spitters *) +axiom symmetric_a_mult: + ∀K.∀A:f_algebra K. symmetric ? (a_mult ? A). + +record integration_f_algebra (R:real) : Type \def + { ifa_integration_riesz_space:> integration_riesz_space R; + ifa_f_algebra_: f_algebra R; + ifa_with: + fa_archimedean_riesz_space ? ifa_f_algebra_ = + irs_archimedean_riesz_space ? ifa_integration_riesz_space + }. + +axiom ifa_f_algebra: ∀R:real.integration_f_algebra R → f_algebra R. + +coercion cic:/matita/attic/integration_algebras/ifa_f_algebra.con. diff --git a/helm/software/matita/contribs/dama/dama_duality/attic/ordered_fields_ch0.ma b/helm/software/matita/contribs/dama/dama_duality/attic/ordered_fields_ch0.ma new file mode 100644 index 000000000..898148d6c --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/attic/ordered_fields_ch0.ma @@ -0,0 +1,151 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "attic/fields.ma". +include "ordered_group.ma". + +(*CSC: non capisco questi alias! Una volta non servivano*) +alias id "plus" = "cic:/matita/group/plus.con". +alias symbol "plus" = "Abelian group plus". + +record pre_ordered_field_ch0: Type ≝ + { of_field:> field; + of_ordered_abelian_group_: ordered_abelian_group; + of_cotransitively_ordered_set_: cotransitively_ordered_set; + of_with1_: + cos_ordered_set of_cotransitively_ordered_set_ = + og_ordered_set of_ordered_abelian_group_; + of_with2: + og_abelian_group of_ordered_abelian_group_ = r_abelian_group of_field + }. + +lemma of_ordered_abelian_group: pre_ordered_field_ch0 → ordered_abelian_group. + intro F; + apply mk_ordered_abelian_group; + [ apply mk_pre_ordered_abelian_group; + [ apply (r_abelian_group F) + | apply (og_ordered_set (of_ordered_abelian_group_ F)) + | apply (eq_f ? ? carrier); + apply (of_with2 F) + ] + | + apply + (eq_rect' ? ? + (λG:abelian_group.λH:og_abelian_group (of_ordered_abelian_group_ F)=G. + is_ordered_abelian_group + (mk_pre_ordered_abelian_group G (of_ordered_abelian_group_ F) + (eq_f abelian_group Type carrier (of_ordered_abelian_group_ F) G + H))) + ? ? (of_with2 F)); + simplify; + apply (og_ordered_abelian_group_properties (of_ordered_abelian_group_ F)) + ] +qed. + +coercion cic:/matita/attic/ordered_fields_ch0/of_ordered_abelian_group.con. + +(*CSC: I am not able to prove this since unfold is undone by coercion composition*) +axiom of_with1: + ∀G:pre_ordered_field_ch0. + cos_ordered_set (of_cotransitively_ordered_set_ G) = + og_ordered_set (of_ordered_abelian_group G). + +lemma of_cotransitively_ordered_set : pre_ordered_field_ch0 → cotransitively_ordered_set. + intro F; + apply mk_cotransitively_ordered_set; + [ apply (og_ordered_set F) + | apply + (eq_rect ? ? (λa:ordered_set.cotransitive (os_carrier a) (os_le a)) + ? ? (of_with1 F)); + apply cos_cotransitive + ] +qed. + +coercion cic:/matita/attic/ordered_fields_ch0/of_cotransitively_ordered_set.con. + +record is_ordered_field_ch0 (F:pre_ordered_field_ch0) : Type \def + { of_mult_compat: ∀a,b:F. 0≤a → 0≤b → 0≤a*b; + of_weak_tricotomy : ∀a,b:F. a≠b → a≤b ∨ b≤a; + (* 0 characteristics *) + of_char0: ∀n. n > O → sum ? (plus F) 0 1 n ≠ 0 + }. + +record ordered_field_ch0 : Type \def + { of_pre_ordered_field_ch0:> pre_ordered_field_ch0; + of_ordered_field_properties:> is_ordered_field_ch0 of_pre_ordered_field_ch0 + }. + +(* +lemma eq_opp_x_times_opp_one_x: ∀F:ordered_field_ch0.∀x:F.-x = -1*x. + intros; + +lemma not_eq_x_zero_to_lt_zero_mult_x_x: + ∀F:ordered_field_ch0.∀x:F. x ≠ 0 → 0 < x * x. + intros; + elim (of_weak_tricotomy ? ? ? ? ? ? ? ? F ? ? H); + [ generalize in match (le_x_zero_to_le_zero_opp_x F ? H1); intro; + generalize in match (of_mult_compat ? ? ? ? ? ? ? ? F ? ? H2 H2); intro; +*) + +axiom lt_zero_to_lt_inv_zero: + ∀F:ordered_field_ch0.∀x:F.∀p:x≠0. lt F 0 x → lt F 0 (inv ? x p). + +alias symbol "lt" = "natural 'less than'". + +(* The ordering is not necessary. *) +axiom not_eq_sum_field_zero: ∀F:ordered_field_ch0.∀n. O ordered_field_ch0; + r_real_properties: is_real r_ordered_field_ch0 + }. + +definition lim: ∀R:real.∀f:nat→R.is_cauchy_seq ? f → R. + intros; + elim (r_complete ? (r_real_properties R) ? H); + exact a. +qed. + +definition max_seq: ∀R:real.∀x,y:R. nat → R. + intros (R x y); + elim (cos_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y)); + [ apply x + | apply not_eq_sum_field_zero ; + unfold; + autobatch + | apply y + | apply lt_zero_to_le_inv_zero + ]. +qed. + +axiom daemon: False. + +theorem cauchy_max_seq: ∀R:real.∀x,y:R. is_cauchy_seq ? (max_seq ? x y). +elim daemon. +(* + intros; + unfold; + intros; + exists; [ exact m | ]; (* apply (ex_intro ? ? m); *) + intros; + unfold max_seq; + elim (of_cotransitive R 0 +(inv R (sum_field R (S N)) + (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))) (x-y) +(lt_zero_to_le_inv_zero R (S N) + (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N))))); + [ simplify; + elim (of_cotransitive R 0 +(inv R (1+sum R (plus R) 0 1 m) + (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y) +(lt_zero_to_le_inv_zero R (S m) + (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m))))); + [ simplify; + rewrite > (plus_comm ? x (-x)); + rewrite > opp_inverse; + split; + [ apply (le_zero_x_to_le_opp_x_zero R ?); + apply lt_zero_to_le_inv_zero + | apply lt_zero_to_le_inv_zero + ] + | simplify; + split; + [ apply (or_transitive ? ? R ? 0); + [ apply (le_zero_x_to_le_opp_x_zero R ?) + | assumption + ] + | assumption + ] + ] + | simplify; + elim (of_cotransitive R 0 +(inv R (1+sum R (plus R) 0 1 m) + (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y) +(lt_zero_to_le_inv_zero R (S m) + (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m))))); + [ simplify; + split; + [ elim daemon + | generalize in match (le_zero_x_to_le_opp_x_zero R ? t1); + intro; + unfold minus in H1; + rewrite > eq_opp_plus_plus_opp_opp in H1; + rewrite > eq_opp_opp_x_x in H1; + rewrite > plus_comm in H1; + apply (or_transitive ? ? R ? 0); + [ assumption + | apply lt_zero_to_le_inv_zero + ] + ] + | simplify; + rewrite > (plus_comm ? y (-y)); + rewrite > opp_inverse; + split; + [ elim daemon + | apply lt_zero_to_le_inv_zero + ] + ] + ]. + elim daemon.*) +qed. + +definition max: ∀R:real.R → R → R. + intros (R x y); + apply (lim R (max_seq R x y)); + apply cauchy_max_seq. +qed. + +definition abs \def λR:real.λx:R. max R x (-x). + +lemma comparison: + ∀R:real.∀f,g:nat→R. is_cauchy_seq ? f → is_cauchy_seq ? g → + (∀n:nat.f n ≤ g n) → lim ? f ? ≤ lim ? g ?. + [ assumption + | assumption + | intros; + elim daemon + ]. +qed. + +definition to_zero ≝ + λR:real.λn. + -(inv R (sum_field R (S n)) + (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))). + +axiom is_cauchy_seq_to_zero: ∀R:real. is_cauchy_seq ? (to_zero R). + +lemma technical1: ∀R:real.lim R (to_zero R) (is_cauchy_seq_to_zero R) = 0. + intros; + unfold lim; + elim daemon. +qed. + +lemma abs_x_ge_O: ∀R:real.∀x:R. 0 ≤ abs ? x. + intros; + unfold abs; + unfold max; + rewrite < technical1; + apply comparison; + intros; + unfold to_zero; + unfold max_seq; + elim + (cos_cotransitive R 0 +(inv R (sum_field R (S n)) + (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))) (x--x) +(lt_zero_to_le_inv_zero R (S n) + (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n))))); + [ simplify; + (* facile *) + elim daemon + | simplify; + (* facile *) + elim daemon + ]. +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/attic/rings.ma b/helm/software/matita/contribs/dama/dama_duality/attic/rings.ma new file mode 100644 index 000000000..d4db003dc --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/attic/rings.ma @@ -0,0 +1,103 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "group.ma". + +record is_ring (G:abelian_group) (mult:G→G→G) (one:G) : Prop +≝ + { (* multiplicative monoid properties *) + mult_assoc_: associative ? mult; + one_neutral_left_: left_neutral ? mult one; + one_neutral_right_: right_neutral ? mult one; + (* ring properties *) + mult_plus_distr_left_: distributive_left ? mult (plus G); + mult_plus_distr_right_: distributive_right ? mult (plus G); + not_eq_zero_one_: (0 ≠ one) + }. + +record ring : Type \def + { r_abelian_group:> abelian_group; + mult: r_abelian_group → r_abelian_group → r_abelian_group; + one: r_abelian_group; + r_ring_properties: is_ring r_abelian_group mult one + }. + +theorem mult_assoc: ∀R:ring.associative ? (mult R). + intros; + apply (mult_assoc_ ? ? ? (r_ring_properties R)). +qed. + +theorem one_neutral_left: ∀R:ring.left_neutral ? (mult R) (one R). + intros; + apply (one_neutral_left_ ? ? ? (r_ring_properties R)). +qed. + +theorem one_neutral_right: ∀R:ring.right_neutral ? (mult R) (one R). + intros; + apply (one_neutral_right_ ? ? ? (r_ring_properties R)). +qed. + +theorem mult_plus_distr_left: ∀R:ring.distributive_left ? (mult R) (plus R). + intros; + apply (mult_plus_distr_left_ ? ? ? (r_ring_properties R)). +qed. + +theorem mult_plus_distr_right: ∀R:ring.distributive_right ? (mult R) (plus R). + intros; + apply (mult_plus_distr_right_ ? ? ? (r_ring_properties R)). +qed. + +theorem not_eq_zero_one: ∀R:ring.0 ≠ one R. + intros; + apply (not_eq_zero_one_ ? ? ? (r_ring_properties R)). +qed. + +interpretation "Ring mult" 'times a b = + (cic:/matita/attic/rings/mult.con _ a b). + +notation "1" with precedence 89 +for @{ 'one }. + +interpretation "Ring one" 'one = + (cic:/matita/attic/rings/one.con _). + +lemma eq_mult_zero_x_zero: ∀R:ring.∀x:R.0*x=0. + intros; + generalize in match (zero_neutral R 0); intro; + generalize in match (eq_f ? ? (λy.y*x) ? ? H); intro; clear H; + rewrite > mult_plus_distr_right in H1; + generalize in match (eq_f ? ? (λy.-(0*x)+y) ? ? H1); intro; clear H1; + rewrite < plus_assoc in H; + rewrite > opp_inverse in H; + rewrite > zero_neutral in H; + assumption. +qed. + +lemma eq_mult_x_zero_zero: ∀R:ring.∀x:R.x*0=0. +intros; +generalize in match (zero_neutral R 0); +intro; +generalize in match (eq_f ? ? (λy.x*y) ? ? H); intro; clear H; +(*CSC: qua funzionava prima della patch all'unificazione!*) +rewrite > (mult_plus_distr_left R) in H1; +generalize in match (eq_f ? ? (λy. (-(x*0)) +y) ? ? H1);intro; +clear H1; +rewrite < plus_assoc in H; +rewrite > opp_inverse in H; +rewrite > zero_neutral in H; +assumption. +qed. + diff --git a/helm/software/matita/contribs/dama/dama_duality/attic/vector_spaces.ma b/helm/software/matita/contribs/dama/dama_duality/attic/vector_spaces.ma new file mode 100644 index 000000000..5002b022c --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/attic/vector_spaces.ma @@ -0,0 +1,151 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "attic/reals.ma". + +record is_vector_space (K: field) (G:abelian_group) (emult:K→G→G) : Prop +≝ + { vs_nilpotent: ∀v. emult 0 v = 0; + vs_neutral: ∀v. emult 1 v = v; + vs_distributive: ∀a,b,v. emult (a + b) v = (emult a v) + (emult b v); + vs_associative: ∀a,b,v. emult (a * b) v = emult a (emult b v) + }. + +record vector_space (K:field): Type \def +{ vs_abelian_group :> abelian_group; + emult: K → vs_abelian_group → vs_abelian_group; + vs_vector_space_properties :> is_vector_space ? vs_abelian_group emult +}. + +interpretation "Vector space external product" 'times a b = + (cic:/matita/attic/vector_spaces/emult.con _ _ a b). + +record is_semi_norm (R:real) (V: vector_space R) (semi_norm:V→R) : Prop \def + { sn_positive: ∀x:V. zero R ≤ semi_norm x; + sn_omogeneous: ∀a:R.∀x:V. semi_norm (a*x) = (abs ? a) * semi_norm x; + sn_triangle_inequality: ∀x,y:V. semi_norm (x + y) ≤ semi_norm x + semi_norm y + }. + +theorem eq_semi_norm_zero_zero: + ∀R:real.∀V:vector_space R.∀semi_norm:V→R. + is_semi_norm ? ? semi_norm → + semi_norm 0 = 0. + intros; + (* facile *) + elim daemon. +qed. + +record is_norm (R:real) (V:vector_space R) (norm:V → R) : Prop ≝ + { n_semi_norm:> is_semi_norm ? ? norm; + n_properness: ∀x:V. norm x = 0 → x = 0 + }. + +record norm (R:real) (V:vector_space R) : Type ≝ + { n_function:1> V→R; + n_norm_properties: is_norm ? ? n_function + }. + +record is_semi_distance (R:real) (C:Type) (semi_d: C→C→R) : Prop ≝ + { sd_positive: ∀x,y:C. zero R ≤ semi_d x y; + sd_properness: ∀x:C. semi_d x x = 0; + sd_triangle_inequality: ∀x,y,z:C. semi_d x z ≤ semi_d x y + semi_d z y + }. + +record is_distance (R:real) (C:Type) (d:C→C→R) : Prop ≝ + { d_semi_distance:> is_semi_distance ? ? d; + d_properness: ∀x,y:C. d x y = 0 → x=y + }. + +record distance (R:real) (V:vector_space R) : Type ≝ + { d_function:2> V→V→R; + d_distance_properties: is_distance ? ? d_function + }. + +definition induced_distance_fun ≝ + λR:real.λV:vector_space R.λnorm:norm ? V. + λf,g:V.norm (f - g). + +theorem induced_distance_is_distance: + ∀R:real.∀V:vector_space R.∀norm:norm ? V. + is_distance ? ? (induced_distance_fun ? ? norm). +elim daemon.(* + intros; + apply mk_is_distance; + [ apply mk_is_semi_distance; + [ unfold induced_distance_fun; + intros; + apply sn_positive; + apply n_semi_norm; + apply (n_norm_properties ? ? norm) + | unfold induced_distance_fun; + intros; + unfold minus; + rewrite < plus_comm; + rewrite > opp_inverse; + apply eq_semi_norm_zero_zero; + apply n_semi_norm; + apply (n_norm_properties ? ? norm) + | unfold induced_distance_fun; + intros; + (* ??? *) + elim daemon + ] + | unfold induced_distance_fun; + intros; + generalize in match (n_properness ? ? norm ? ? H); + [ intro; + (* facile *) + elim daemon + | apply (n_norm_properties ? ? norm) + ] + ].*) +qed. + +definition induced_distance ≝ + λR:real.λV:vector_space R.λnorm:norm ? V. + mk_distance ? ? (induced_distance_fun ? ? norm) + (induced_distance_is_distance ? ? norm). + +definition tends_to : + ∀R:real.∀V:vector_space R.∀d:distance ? V.∀f:nat→V.∀l:V.Prop. +apply + (λR:real.λV:vector_space R.λd:distance ? V.λf:nat→V.λl:V. + ∀n:nat.∃m:nat.∀j:nat. m ≤ j → + d (f j) l ≤ inv R (sum_field ? (S n)) ?); + apply not_eq_sum_field_zero; + unfold; + autobatch. +qed. + +definition is_cauchy_seq : ∀R:real.\forall V:vector_space R. +\forall d:distance ? V.∀f:nat→V.Prop. + apply + (λR:real.λV: vector_space R. \lambda d:distance ? V. + \lambda f:nat→V. + ∀m:nat. + ∃n:nat.∀N. n ≤ N → + -(inv R (sum_field ? (S m)) ?) ≤ d (f N) (f n) ∧ + d (f N) (f n)≤ inv R (sum_field R (S m)) ?); + apply not_eq_sum_field_zero; + unfold; + autobatch. +qed. + +definition is_complete ≝ + λR:real.λV:vector_space R. + λd:distance ? V. + ∀f:nat→V. is_cauchy_seq ? ? d f→ + ex V (λl:V. tends_to ? ? d f l). diff --git a/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets.ma b/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets.ma new file mode 100644 index 000000000..2630da77c --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets.ma @@ -0,0 +1,424 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "excess.ma". + +record is_dedekind_sigma_complete (O:excess) : Type ≝ + { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? a m → ex ? (λs:O.is_inf O a s); + dsc_inf_proof_irrelevant: + ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? a m.∀p':is_lower_bound ? a m'. + (match dsc_inf a m p with [ ex_intro s _ ⇒ s ]) = + (match dsc_inf a m' p' with [ ex_intro s' _ ⇒ s' ]); + dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? a m → ex ? (λs:O.is_sup O a s); + dsc_sup_proof_irrelevant: + ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? a m.∀p':is_upper_bound ? a m'. + (match dsc_sup a m p with [ ex_intro s _ ⇒ s ]) = + (match dsc_sup a m' p' with [ ex_intro s' _ ⇒ s' ]) + }. + +record dedekind_sigma_complete_ordered_set : Type ≝ + { dscos_ordered_set:> excess; + dscos_dedekind_sigma_complete_properties:> + is_dedekind_sigma_complete dscos_ordered_set + }. + +definition inf: + ∀O:dedekind_sigma_complete_ordered_set. + bounded_below_sequence O → O. + intros; + elim + (dsc_inf O (dscos_dedekind_sigma_complete_properties O) b); + [ apply a + | apply (lower_bound ? b) + | apply lower_bound_is_lower_bound + ] +qed. + +lemma inf_is_inf: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a:bounded_below_sequence O. + is_inf ? a (inf ? a). + intros; + unfold inf; + simplify; + elim (dsc_inf O (dscos_dedekind_sigma_complete_properties O) a +(lower_bound O a) (lower_bound_is_lower_bound O a)); + simplify; + assumption. +qed. + +lemma inf_proof_irrelevant: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a,a':bounded_below_sequence O. + bbs_seq ? a = bbs_seq ? a' → + inf ? a = inf ? a'. + intros 3; + elim a 0; + elim a'; + simplify in H; + generalize in match i1; + clear i1; + rewrite > H; + intro; + simplify; + rewrite < (dsc_inf_proof_irrelevant O O f (ib_lower_bound ? f i) + (ib_lower_bound ? f i2) (ib_lower_bound_is_lower_bound ? f i) + (ib_lower_bound_is_lower_bound ? f i2)); + reflexivity. +qed. + +definition sup: + ∀O:dedekind_sigma_complete_ordered_set. + bounded_above_sequence O → O. + intros; + elim + (dsc_sup O (dscos_dedekind_sigma_complete_properties O) b); + [ apply a + | apply (upper_bound ? b) + | apply upper_bound_is_upper_bound + ]. +qed. + +lemma sup_is_sup: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a:bounded_above_sequence O. + is_sup ? a (sup ? a). + intros; + unfold sup; + simplify; + elim (dsc_sup O (dscos_dedekind_sigma_complete_properties O) a +(upper_bound O a) (upper_bound_is_upper_bound O a)); + simplify; + assumption. +qed. + +lemma sup_proof_irrelevant: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a,a':bounded_above_sequence O. + bas_seq ? a = bas_seq ? a' → + sup ? a = sup ? a'. + intros 3; + elim a 0; + elim a'; + simplify in H; + generalize in match i1; + clear i1; + rewrite > H; + intro; + simplify; + rewrite < (dsc_sup_proof_irrelevant O O f (ib_upper_bound ? f i2) + (ib_upper_bound ? f i) (ib_upper_bound_is_upper_bound ? f i2) + (ib_upper_bound_is_upper_bound ? f i)); + reflexivity. +qed. + +axiom daemon: False. + +theorem inf_le_sup: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a:bounded_sequence O. inf ? a ≤ sup ? a. + intros (O'); + apply (or_transitive ? ? O' ? (a O)); + [ elim daemon (*apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? a))*) + | elim daemon (*apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? a))*) + ]. +qed. + +lemma inf_respects_le: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a:bounded_below_sequence O.∀m:O. + is_upper_bound ? a m → inf ? a ≤ m. + intros (O'); + apply (or_transitive ? ? O' ? (sup ? (mk_bounded_sequence ? a ? ?))); + [ apply (bbs_is_bounded_below ? a) + | apply (mk_is_bounded_above ? ? m H) + | apply inf_le_sup + | apply + (sup_least_upper_bound ? ? ? + (sup_is_sup ? (mk_bounded_sequence O' a a + (mk_is_bounded_above O' a m H)))); + assumption + ]. +qed. + +definition is_sequentially_monotone ≝ + λO:excess.λf:O→O. + ∀a:nat→O.∀p:is_increasing ? a. + is_increasing ? (λi.f (a i)). + +record is_order_continuous + (O:dedekind_sigma_complete_ordered_set) (f:O→O) : Prop +≝ + { ioc_is_sequentially_monotone: is_sequentially_monotone ? f; + ioc_is_upper_bound_f_sup: + ∀a:bounded_above_sequence O. + is_upper_bound ? (λi.f (a i)) (f (sup ? a)); + ioc_respects_sup: + ∀a:bounded_above_sequence O. + is_increasing ? a → + f (sup ? a) = + sup ? (mk_bounded_above_sequence ? (λi.f (a i)) + (mk_is_bounded_above ? ? (f (sup ? a)) + (ioc_is_upper_bound_f_sup a))); + ioc_is_lower_bound_f_inf: + ∀a:bounded_below_sequence O. + is_lower_bound ? (λi.f (a i)) (f (inf ? a)); + ioc_respects_inf: + ∀a:bounded_below_sequence O. + is_decreasing ? a → + f (inf ? a) = + inf ? (mk_bounded_below_sequence ? (λi.f (a i)) + (mk_is_bounded_below ? ? (f (inf ? a)) + (ioc_is_lower_bound_f_inf a))) + }. + +theorem tail_inf_increasing: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a:bounded_below_sequence O. + let y ≝ λi.mk_bounded_below_sequence ? (λj.a (i+j)) ? in + let x ≝ λi.inf ? (y i) in + is_increasing ? x. + [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a)); + simplify; + intro; + apply (ib_lower_bound_is_lower_bound ? a a) + | intros; + unfold is_increasing; + intro; + unfold x in ⊢ (? ? ? %); + apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? (y (S n)))); + change with (is_lower_bound ? (y (S n)) (inf ? (y n))); + unfold is_lower_bound; + intro; + generalize in match (inf_lower_bound ? ? ? (inf_is_inf ? (y n)) (S n1)); + (*CSC: coercion per FunClass inserita a mano*) + suppose (inf ? (y n) ≤ bbs_seq ? (y n) (S n1)) (H); + cut (bbs_seq ? (y n) (S n1) = bbs_seq ? (y (S n)) n1); + [ rewrite < Hcut; + assumption + | unfold y; + simplify; + autobatch paramodulation library + ] + ]. +qed. + +definition is_liminf: + ∀O:dedekind_sigma_complete_ordered_set. + bounded_below_sequence O → O → Prop. + intros; + apply + (is_sup ? (λi.inf ? (mk_bounded_below_sequence ? (λj.b (i+j)) ?)) t); + apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b)); + simplify; + intros; + apply (ib_lower_bound_is_lower_bound ? b b). +qed. + +definition liminf: + ∀O:dedekind_sigma_complete_ordered_set. + bounded_sequence O → O. + intros; + apply + (sup ? + (mk_bounded_above_sequence ? + (λi.inf ? + (mk_bounded_below_sequence ? + (λj.b (i+j)) ?)) ?)); + [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? b b)); + simplify; + intros; + apply (ib_lower_bound_is_lower_bound ? b b) + | apply (mk_is_bounded_above ? ? (ib_upper_bound ? b b)); + unfold is_upper_bound; + intro; + change with + (inf O + (mk_bounded_below_sequence O (\lambda j:nat.b (n+j)) + (mk_is_bounded_below O (\lambda j:nat.b (n+j)) (ib_lower_bound O b b) + (\lambda j:nat.ib_lower_bound_is_lower_bound O b b (n+j)))) +\leq ib_upper_bound O b b); + apply (inf_respects_le O); + simplify; + intro; + apply (ib_upper_bound_is_upper_bound ? b b) + ]. +qed. + + +definition reverse_dedekind_sigma_complete_ordered_set: + dedekind_sigma_complete_ordered_set → dedekind_sigma_complete_ordered_set. + intros; + apply mk_dedekind_sigma_complete_ordered_set; + [ apply (reverse_ordered_set d) + | elim daemon + (*apply mk_is_dedekind_sigma_complete; + [ intros; + elim (dsc_sup ? ? d a m) 0; + [ generalize in match H; clear H; + generalize in match m; clear m; + elim d; + simplify in a1; + simplify; + change in a1 with (Type_OF_ordered_set ? (reverse_ordered_set ? o)); + apply (ex_intro ? ? a1); + simplify in H1; + change in m with (Type_OF_ordered_set ? o); + apply (is_sup_to_reverse_is_inf ? ? ? ? H1) + | generalize in match H; clear H; + generalize in match m; clear m; + elim d; + simplify; + change in t with (Type_OF_ordered_set ? o); + simplify in t; + apply reverse_is_lower_bound_is_upper_bound; + assumption + ] + | apply is_sup_reverse_is_inf; + | apply m + | + ]*) + ]. +qed. + +definition reverse_bounded_sequence: + ∀O:dedekind_sigma_complete_ordered_set. + bounded_sequence O → + bounded_sequence (reverse_dedekind_sigma_complete_ordered_set O). + intros; + apply mk_bounded_sequence; + [ apply bs_seq; + unfold reverse_dedekind_sigma_complete_ordered_set; + simplify; + elim daemon + | elim daemon + | elim daemon + ]. +qed. + +definition limsup ≝ + λO:dedekind_sigma_complete_ordered_set. + λa:bounded_sequence O. + liminf (reverse_dedekind_sigma_complete_ordered_set O) + (reverse_bounded_sequence O a). + +notation "hvbox(〈a〉)" + non associative with precedence 45 +for @{ 'hide_everything_but $a }. + +interpretation "mk_bounded_above_sequence" 'hide_everything_but a += (cic:/matita/classical_pointfree/ordered_sets/bounded_above_sequence.ind#xpointer(1/1/1) _ _ a _). + +interpretation "mk_bounded_below_sequence" 'hide_everything_but a += (cic:/matita/classical_pointfree/ordered_sets/bounded_below_sequence.ind#xpointer(1/1/1) _ _ a _). + +theorem eq_f_sup_sup_f: + ∀O':dedekind_sigma_complete_ordered_set. + ∀f:O'→O'. ∀H:is_order_continuous ? f. + ∀a:bounded_above_sequence O'. + ∀p:is_increasing ? a. + f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) ?). + [ apply (mk_is_bounded_above ? ? (f (sup ? a))); + apply ioc_is_upper_bound_f_sup; + assumption + | intros; + apply ioc_respects_sup; + assumption + ]. +qed. + +theorem eq_f_sup_sup_f': + ∀O':dedekind_sigma_complete_ordered_set. + ∀f:O'→O'. ∀H:is_order_continuous ? f. + ∀a:bounded_above_sequence O'. + ∀p:is_increasing ? a. + ∀p':is_bounded_above ? (λi.f (a i)). + f (sup ? a) = sup ? (mk_bounded_above_sequence ? (λi.f (a i)) p'). + intros; + rewrite > (eq_f_sup_sup_f ? f H a H1); + apply sup_proof_irrelevant; + reflexivity. +qed. + +theorem eq_f_liminf_sup_f_inf: + ∀O':dedekind_sigma_complete_ordered_set. + ∀f:O'→O'. ∀H:is_order_continuous ? f. + ∀a:bounded_sequence O'. + let p1 := ? in + f (liminf ? a) = + sup ? + (mk_bounded_above_sequence ? + (λi.f (inf ? + (mk_bounded_below_sequence ? + (λj.a (i+j)) + ?))) + p1). + [ apply (mk_is_bounded_below ? ? (ib_lower_bound ? a a)); + simplify; + intro; + apply (ib_lower_bound_is_lower_bound ? a a) + | apply (mk_is_bounded_above ? ? (f (sup ? a))); + unfold is_upper_bound; + intro; + apply (or_transitive ? ? O' ? (f (a n))); + [ generalize in match (ioc_is_lower_bound_f_inf ? ? H); + intro H1; + simplify in H1; + rewrite > (plus_n_O n) in ⊢ (? ? ? (? (? ? ? %))); + apply (H1 (mk_bounded_below_sequence O' (\lambda j:nat.a (n+j)) +(mk_is_bounded_below O' (\lambda j:nat.a (n+j)) (ib_lower_bound O' a a) + (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (n+j)))) O); + | elim daemon (*apply (ioc_is_upper_bound_f_sup ? ? ? H)*) + ] + | intros; + unfold liminf; + clearbody p1; + generalize in match (\lambda n:nat +.inf_respects_le O' + (mk_bounded_below_sequence O' (\lambda j:nat.a (plus n j)) + (mk_is_bounded_below O' (\lambda j:nat.a (plus n j)) + (ib_lower_bound O' a a) + (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (plus n j)))) + (ib_upper_bound O' a a) + (\lambda n1:nat.ib_upper_bound_is_upper_bound O' a a (plus n n1))); + intro p2; + apply (eq_f_sup_sup_f' ? f H (mk_bounded_above_sequence O' +(\lambda i:nat + .inf O' + (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j)) + (mk_is_bounded_below O' (\lambda j:nat.a (plus i j)) + (ib_lower_bound O' a a) + (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))) +(mk_is_bounded_above O' + (\lambda i:nat + .inf O' + (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j)) + (mk_is_bounded_below O' (\lambda j:nat.a (plus i j)) + (ib_lower_bound O' a a) + (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))) + (ib_upper_bound O' a a) p2))); + unfold bas_seq; + change with + (is_increasing ? (\lambda i:nat +.inf O' + (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j)) + (mk_is_bounded_below O' (\lambda j:nat.a (plus i j)) + (ib_lower_bound O' a a) + (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))); + apply tail_inf_increasing + ]. +qed. + diff --git a/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets2.ma b/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets2.ma new file mode 100644 index 000000000..7e74cbba2 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/classical_pointfree/ordered_sets2.ma @@ -0,0 +1,127 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "classical_pointfree/ordered_sets.ma". + +theorem le_f_inf_inf_f: + ∀O':dedekind_sigma_complete_ordered_set. + ∀f:O'→O'. ∀H:is_order_continuous ? f. + ∀a:bounded_below_sequence O'. + let p := ? in + f (inf ? a) ≤ inf ? (mk_bounded_below_sequence ? (λi. f (a i)) p). + [ apply mk_is_bounded_below; + [2: apply ioc_is_lower_bound_f_inf; + assumption + | skip + ] + | intros; + clearbody p; + apply (inf_greatest_lower_bound ? ? ? (inf_is_inf ? ?)); + simplify; + intro; + letin b := (λi.match i with [ O ⇒ inf ? a | S _ ⇒ a n]); + change with (f (b O) ≤ f (b (S O))); + apply (ioc_is_sequentially_monotone ? ? H); + simplify; + clear b; + intro; + elim n1; simplify; + [ apply (inf_lower_bound ? ? ? (inf_is_inf ? ?)); + | apply (or_reflexive O' ? (dscos_ordered_set O')) + ] + ]. +qed. + +theorem le_to_le_sup_sup: + ∀O':dedekind_sigma_complete_ordered_set. + ∀a,b:bounded_above_sequence O'. + (∀i.a i ≤ b i) → sup ? a ≤ sup ? b. + intros; + apply (sup_least_upper_bound ? ? ? (sup_is_sup ? a)); + unfold; + intro; + apply (or_transitive ? ? O'); + [2: apply H + | skip + | apply (sup_upper_bound ? ? ? (sup_is_sup ? b)) + ]. +qed. + +interpretation "mk_bounded_sequence" 'hide_everything_but a += (cic:/matita/classical_pointfree/ordered_sets/bounded_sequence.ind#xpointer(1/1/1) _ _ a _ _). + +lemma reduce_bas_seq: + ∀O:ordered_set.∀a:nat→O.∀p.∀i. + bas_seq ? (mk_bounded_above_sequence ? a p) i = a i. + intros; + reflexivity. +qed. + +(*lemma reduce_bbs_seq: + ∀C.∀O:ordered_set C.∀a:nat→O.∀p.∀i. + bbs_seq ? ? (mk_bounded_below_sequence ? ? a p) i = a i. + intros; + reflexivity. +qed.*) + +axiom inf_extensional: + ∀O:dedekind_sigma_complete_ordered_set. + ∀a,b:bounded_below_sequence O. + (∀i.a i = b i) → inf ? a = inf O b. + +lemma eq_to_le: ∀O:ordered_set.∀x,y:O.x=y → x ≤ y. + intros; + rewrite > H; + apply (or_reflexive ? ? O). +qed. + +theorem fatou: + ∀O':dedekind_sigma_complete_ordered_set. + ∀f:O'→O'. ∀H:is_order_continuous ? f. + ∀a:bounded_sequence O'. + let pb := ? in + let pa := ? in + f (liminf ? a) ≤ liminf ? (mk_bounded_sequence ? (λi. f (a i)) pb pa). + [ letin bas ≝ (bounded_above_sequence_of_bounded_sequence ? a); + apply mk_is_bounded_above; + [2: apply (ioc_is_upper_bound_f_sup ? ? H bas) + | skip + ] + | letin bbs ≝ (bounded_below_sequence_of_bounded_sequence ? a); + apply mk_is_bounded_below; + [2: apply (ioc_is_lower_bound_f_inf ? ? H bbs) + | skip + ] + | intros; + rewrite > eq_f_liminf_sup_f_inf in ⊢ (? ? % ?); + [ unfold liminf; + apply le_to_le_sup_sup; + intro; + rewrite > reduce_bas_seq; + rewrite > reduce_bas_seq; + apply (or_transitive ? ? O'); + [2: apply le_f_inf_inf_f; + assumption + | skip + | apply eq_to_le; + apply inf_extensional; + intro; + reflexivity + ] + | assumption + ] + ]. +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sets.ma b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sets.ma new file mode 100644 index 000000000..f03c3e7f2 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sets.ma @@ -0,0 +1,104 @@ +(**************************************************************************) +(* ___ *) +(* ||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/connectives.ma". + + +definition set ≝ λX:Type.X → Prop. + +definition member_of : ∀X.set X → X → Prop≝ λX.λA:set X.λx.A x. + +notation "hvbox(x break ∈ A)" with precedence 60 +for @{ 'member_of $x $A }. + +interpretation "Member of" 'member_of x A = + (cic:/matita/classical_pointwise/sets/member_of.con _ A x). + +notation "hvbox(x break ∉ A)" with precedence 60 +for @{ 'not_member_of $x $A }. + +interpretation "Not member of" 'not_member_of x A = + (cic:/matita/logic/connectives/Not.con + (cic:/matita/classical_pointwise/sets/member_of.con _ A x)). + +definition emptyset : ∀X.set X ≝ λX:Type.λx:X.False. + +notation "∅︀" with precedence 100 for @{ 'emptyset }. + +interpretation "Emptyset" 'emptyset = + (cic:/matita/classical_pointwise/sets/emptyset.con _). + +definition subset: ∀X. set X → set X → Prop≝ λX.λA,B:set X.∀x. x ∈ A → x ∈ B. + +notation "hvbox(A break ⊆ B)" with precedence 60 +for @{ 'subset $A $B }. + +interpretation "Subset" 'subset A B = + (cic:/matita/classical_pointwise/sets/subset.con _ A B). + +definition intersection: ∀X. set X → set X → set X ≝ + λX.λA,B:set X.λx. x ∈ A ∧ x ∈ B. + +notation "hvbox(A break ∩ B)" with precedence 70 +for @{ 'intersection $A $B }. + +interpretation "Intersection" 'intersection A B = + (cic:/matita/classical_pointwise/sets/intersection.con _ A B). + +definition union: ∀X. set X → set X → set X ≝ λX.λA,B:set X.λx. x ∈ A ∨ x ∈ B. + +notation "hvbox(A break ∪ B)" with precedence 65 +for @{ 'union $A $B }. + +interpretation "Union" 'union A B = + (cic:/matita/classical_pointwise/sets/union.con _ A B). + +definition seq ≝ λX:Type.nat → X. + +definition nth ≝ λX.λA:seq X.λi.A i. + +notation "hvbox(A \sub i)" with precedence 100 +for @{ 'nth $A $i }. + +interpretation "nth" 'nth A i = + (cic:/matita/classical_pointwise/sets/nth.con _ A i). + +definition countable_union: ∀X. seq (set X) → set X ≝ + λX.λA:seq (set X).λx.∃j.x ∈ A \sub j. + +notation "∪ \sub (ident i opt (: ty)) B" with precedence 65 +for @{ 'big_union ${default @{(λ${ident i}:$ty.$B)} @{(λ${ident i}.$B)}}}. + +interpretation "countable_union" 'big_union η.t = + (cic:/matita/classical_pointwise/sets/countable_union.con _ t). + +definition complement: ∀X. set X \to set X ≝ λX.λA:set X.λx. x ∉ A. + +notation "A \sup 'c'" with precedence 100 +for @{ 'complement $A }. + +interpretation "Complement" 'complement A = + (cic:/matita/classical_pointwise/sets/complement.con _ A). + +definition inverse_image: ∀X,Y.∀f: X → Y.set Y → set X ≝ + λX,Y,f,B,x. f x ∈ B. + +notation "hvbox(f \sup (-1))" with precedence 100 +for @{ 'finverse $f }. + +interpretation "Inverse image" 'finverse f = + (cic:/matita/classical_pointwise/sets/inverse_image.con _ _ f). diff --git a/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sigma_algebra.ma b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sigma_algebra.ma new file mode 100644 index 000000000..580fe9645 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/sigma_algebra.ma @@ -0,0 +1,40 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "classical_pointwise/topology.ma". + +record is_sigma_algebra (X:Type) (A: set X) (M: set (set X)) : Prop ≝ + { siga_subset: ∀B.B ∈ M → B ⊆ A; + siga_full: A ∈ M; + siga_compl: ∀B.B ∈ M → B \sup c ∈ M; + siga_enumerable_union: + ∀B:seq (set X).(∀i.(B \sub i) ∈ M) → (∪ \sub i B \sub i) ∈ M + }. + +record sigma_algebra : Type ≝ + { siga_carrier:> Type; + siga_domain:> set siga_carrier; + M: set (set siga_carrier); + siga_is_sigma_algebra:> is_sigma_algebra ? siga_domain M + }. + +(*definition is_measurable_map ≝ + λX:sigma_algebra.λY:topological_space.λf:X → Y. + ∀V. V ∈ O Y → f \sup -1 V ∈ M X.*) +definition is_measurable_map ≝ + λX:sigma_algebra.λY:topological_space.λf:X → Y. + ∀V. V ∈ O Y → inverse_image ? ? f V ∈ M X. + diff --git a/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/topology.ma b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/topology.ma new file mode 100644 index 000000000..72c9dbb4d --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/classical_pointwise/topology.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "classical_pointwise/sets.ma". + +record is_topology (X) (A: set X) (O: set (set X)) : Prop ≝ + { top_subset: ∀B. B ∈ O → B ⊆ A; + top_empty: ∅︀ ∈ O; + top_full: A ∈ O; + top_intersection: ∀B,C. B ∈ O → C ∈ O → B ∩ C ∈ O; + top_countable_union: + ∀B.(∀i.(B \sub i) ∈ O) → (∪ \sub i (B \sub i)) ∈ O + }. + +record topological_space : Type ≝ + { top_carrier:> Type; + top_domain:> set top_carrier; + O: set (set top_carrier); + top_is_topological_space:> is_topology ? top_domain O + }. + +(*definition is_continuous_map ≝ + λX,Y: topological_space.λf: X → Y. + ∀V. V ∈ O Y → (f \sup -1) V ∈ O X.*) +definition is_continuous_map ≝ + λX,Y: topological_space.λf: X → Y. + ∀V. V ∈ O Y → inverse_image ? ? f V ∈ O X. + +record continuous_map (X,Y: topological_space) : Type ≝ + { cm_f:> X → Y; + cm_is_continuous_map: is_continuous_map ? ? cm_f + }. diff --git a/helm/software/matita/contribs/dama/dama_duality/constructive_connectives.ma b/helm/software/matita/contribs/dama/dama_duality/constructive_connectives.ma new file mode 100644 index 000000000..78e2ec571 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/constructive_connectives.ma @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "logic/connectives.ma". + +inductive Or (A,B:Type) : Type ≝ + Left : A → Or A B + | Right : B → Or A B. + +interpretation "constructive or" 'or x y = + (cic:/matita/constructive_connectives/Or.ind#xpointer(1/1) x y). + +inductive And (A,B:Type) : Type ≝ + | Conj : A → B → And A B. + +interpretation "constructive and" 'and x y = + (cic:/matita/constructive_connectives/And.ind#xpointer(1/1) x y). + +inductive exT (A:Type) (P:A→Type) : Type ≝ + ex_introT: ∀w:A. P w → exT A P. + +inductive ex (A:Type) (P:A→Prop) : Type ≝ + ex_intro: ∀w:A. P w → ex A P. + +(* +notation < "hvbox(Σ ident i opt (: ty) break . p)" + right associative with precedence 20 +for @{ 'sigma ${default + @{\lambda ${ident i} : $ty. $p)} + @{\lambda ${ident i} . $p}}}. +*) + +interpretation "constructive exists" 'exists \eta.x = + (cic:/matita/constructive_connectives/ex.ind#xpointer(1/1) _ x). +interpretation "constructive exists (Type)" 'exists \eta.x = + (cic:/matita/constructive_connectives/exT.ind#xpointer(1/1) _ x). + +alias id "False" = "cic:/matita/logic/connectives/False.ind#xpointer(1/1)". +definition Not ≝ λx:Type.x → False. + +interpretation "constructive not" 'not x = + (cic:/matita/constructive_connectives/Not.con x). diff --git a/helm/software/matita/contribs/dama/dama_duality/constructive_higher_order_relations.ma b/helm/software/matita/contribs/dama/dama_duality/constructive_higher_order_relations.ma new file mode 100644 index 000000000..8d195396c --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/constructive_higher_order_relations.ma @@ -0,0 +1,51 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "constructive_connectives.ma". +include "higher_order_defs/relations.ma". + +definition cotransitive ≝ + λC:Type.λlt:C→C→Type.∀x,y,z:C. lt x y → lt x z ∨ lt z y. + +definition coreflexive ≝ λC:Type.λlt:C→C→Type. ∀x:C. ¬ (lt x x). + +definition antisymmetric ≝ + λC:Type.λle:C→C→Type.λeq:C→C→Type.∀x,y:C.le x y → le y x → eq x y. + +definition symmetric ≝ + λC:Type.λle:C→C→Type.∀x,y:C.le x y → le y x. + +definition transitive ≝ + λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z. + +definition associative ≝ + λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z). + +definition commutative ≝ + λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x). + +alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con". +theorem antisimmetric_to_cotransitive_to_transitive: + ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le. +intros (T f Af cT); unfold transitive; intros (x y z fxy fyz); +lapply (cT ??z fxy) as H; cases H; [assumption] cases (Af ? ? fyz H1); +qed. + +lemma Or_symmetric: symmetric ? Or. +unfold; intros (x y H); cases H; [right|left] assumption; +qed. + + diff --git a/helm/software/matita/contribs/dama/dama_duality/constructive_pointfree/lebesgue.ma b/helm/software/matita/contribs/dama/dama_duality/constructive_pointfree/lebesgue.ma new file mode 100644 index 000000000..c7e5d7c5d --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/constructive_pointfree/lebesgue.ma @@ -0,0 +1,31 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "metric_lattice.ma". +include "sequence.ma". +include "constructive_connectives.ma". + +(* Section 3.2 *) + +(* 3.21 *) + + +(* 3.17 *) + + +(* 3.20 *) +lemma uniq_sup: ∀O:ogroup.∀s:sequence O.∀x,y:O. is_sup ? s x → is_sup ? s y → x ≈ y. +intros; \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama_duality/depends b/helm/software/matita/contribs/dama/dama_duality/depends new file mode 100644 index 000000000..dcbfcc6f0 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/depends @@ -0,0 +1,38 @@ +metric_lattice.ma excess.ma lattice.ma metric_space.ma +metric_space.ma ordered_divisible_group.ma +sandwich.ma metric_lattice.ma nat/orders.ma nat/plus.ma tend.ma +premetric_lattice.ma lattice.ma metric_space.ma +ordered_group.ma group.ma +divisible_group.ma group.ma nat/orders.ma +ordered_divisible_group.ma divisible_group.ma nat/orders.ma nat/times.ma ordered_group.ma +sequence.ma excess.ma +constructive_connectives.ma logic/connectives.ma +group.ma excess.ma +prevalued_lattice.ma ordered_group.ma +excess.ma constructive_connectives.ma constructive_higher_order_relations.ma higher_order_defs/relations.ma nat/plus.ma +sandwich_corollary.ma sandwich.ma +Q_is_orded_divisble_group.ma Q/q.ma ordered_divisible_group.ma +limit.ma excess.ma infsup.ma metric_lattice.ma tend.ma +lattice.ma excess.ma +tend.ma metric_space.ma nat/orders.ma sequence.ma +constructive_higher_order_relations.ma constructive_connectives.ma higher_order_defs/relations.ma +infsup.ma excess.ma sequence.ma +constructive_pointfree/lebesgue.ma constructive_connectives.ma metric_lattice.ma sequence.ma +classical_pointwise/topology.ma classical_pointwise/sets.ma +classical_pointwise/sigma_algebra.ma classical_pointwise/topology.ma +classical_pointwise/sets.ma logic/connectives.ma nat/nat.ma +classical_pointfree/ordered_sets.ma excess.ma +classical_pointfree/ordered_sets2.ma classical_pointfree/ordered_sets.ma +attic/fields.ma attic/rings.ma +attic/reals.ma attic/ordered_fields_ch0.ma +attic/integration_algebras.ma attic/vector_spaces.ma lattice.ma +attic/vector_spaces.ma attic/reals.ma +attic/rings.ma group.ma +attic/ordered_fields_ch0.ma group.ma attic/fields.ma ordered_group.ma +Q/q.ma +higher_order_defs/relations.ma +logic/connectives.ma +nat/nat.ma +nat/orders.ma +nat/plus.ma +nat/times.ma diff --git a/helm/software/matita/contribs/dama/dama_duality/divisible_group.ma b/helm/software/matita/contribs/dama/dama_duality/divisible_group.ma new file mode 100644 index 000000000..3a79b11bb --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/divisible_group.ma @@ -0,0 +1,99 @@ +(**************************************************************************) +(* ___ *) +(* ||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/orders.ma". +include "group.ma". + +let rec gpow (G : abelian_group) (x : G) (n : nat) on n : G ≝ + match n with [ O ⇒ 0 | S m ⇒ x + gpow ? x m]. + +interpretation "additive abelian group pow" 'times n x = + (cic:/matita/divisible_group/gpow.con _ x n). + +record dgroup : Type ≝ { + dg_carr:> abelian_group; + dg_prop: ∀x:dg_carr.∀n:nat.∃y.S n * y ≈ x +}. + +lemma divide: ∀G:dgroup.G → nat → G. +intros (G x n); cases (dg_prop G x n); apply w; +qed. + +interpretation "divisible group divide" 'divide x n = + (cic:/matita/divisible_group/divide.con _ x n). + +lemma divide_divides: + ∀G:dgroup.∀x:G.∀n. S n * (x / n) ≈ x. +intro G; cases G; unfold divide; intros (x n); simplify; +cases (f x n); simplify; exact H; +qed. + +lemma feq_mul: ∀G:dgroup.∀x,y:G.∀n.x≈y → n * x ≈ n * y. +intros (G x y n H); elim n; [apply eq_reflexive] +simplify; apply (Eq≈ (x + (n1 * y)) H1); +apply (Eq≈ (y+n1*y) H (eq_reflexive ??)); +qed. + +lemma div1: ∀G:dgroup.∀x:G.x/O ≈ x. +intro G; cases G; unfold divide; intros; simplify; +cases (f x O); simplify; simplify in H; intro; apply H; +apply (Ap≪ ? (plus_comm ???)); +apply (Ap≪ w (zero_neutral ??)); assumption; +qed. + +lemma apmul_ap: ∀G:dgroup.∀x,y:G.∀n.S n * x # S n * y → x # y. +intros 4 (G x y n); elim n; [2: + simplify in a; + cases (applus ????? a); [assumption] + apply f; assumption;] +apply (plus_cancr_ap ??? 0); assumption; +qed. + +lemma plusmul: ∀G:dgroup.∀x,y:G.∀n.n * (x+y) ≈ n * x + n * y. +intros (G x y n); elim n; [ + simplify; apply (Eq≈ 0 ? (zero_neutral ? 0)); apply eq_reflexive] +simplify; apply eq_sym; apply (Eq≈ (x+y+(n1*x+n1*y))); [ + apply (Eq≈ (x+(n1*x+(y+(n1*y))))); [ + apply eq_sym; apply plus_assoc;] + apply (Eq≈ (x+((n1*x+y+(n1*y))))); [ + apply feq_plusl; apply plus_assoc;] + apply (Eq≈ (x+(y+n1*x+n1*y))); [ + apply feq_plusl; apply feq_plusr; apply plus_comm;] + apply (Eq≈ (x+(y+(n1*x+n1*y)))); [ + apply feq_plusl; apply eq_sym; apply plus_assoc;] + apply plus_assoc;] +apply feq_plusl; apply eq_sym; assumption; +qed. + +lemma mulzero: ∀G:dgroup.∀n.n*0 ≈ 0. [2: apply dg_carr; apply G] +intros; elim n; [simplify; apply eq_reflexive] +simplify; apply (Eq≈ ? (zero_neutral ??)); assumption; +qed. + +let rec gpowS (G : abelian_group) (x : G) (n : nat) on n : G ≝ + match n with [ O ⇒ x | S m ⇒ gpowS ? x m + x]. + +lemma gpowS_gpow: ∀G:dgroup.∀e:G.∀n. S n * e ≈ gpowS ? e n. +intros (G e n); elim n; simplify; [ + apply (Eq≈ ? (plus_comm ???));apply zero_neutral] +apply (Eq≈ ?? (plus_comm ???)); +apply (Eq≈ (e+S n1*e) ? H); clear H; simplify; apply eq_reflexive; +qed. + +lemma divpow: ∀G:dgroup.∀e:G.∀n. e ≈ gpowS ? (e/n) n. +intros (G e n); apply (Eq≈ ?? (gpowS_gpow ?(e/n) n)); +apply eq_sym; apply divide_divides; +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/doc/DIMOSTRAZIONE b/helm/software/matita/contribs/dama/dama_duality/doc/DIMOSTRAZIONE new file mode 100644 index 000000000..197c3ff97 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/doc/DIMOSTRAZIONE @@ -0,0 +1,126 @@ +############### Costruttivizzazione di Fremlin ###################### + +Prerequisiti: + 1. definizione di exceeds + 2. definizione di <= in termini di < (sui reali) + 2. definizione di sup forte (sui reali) + +======================================== + +Lemma: liminf f a_n <= limsup f a_n +Per definizione di <= dobbiamo dimostrare: + ~(limsup f a_n < liminf f a_n) +Supponiamo limsup f a_n < liminf f a_n. +Ovvero inf_n sup_{k>n} f a_k < sup_m inf_{h>m} f a_h +? Quindi esiste un l tale che + inf_n sup_{k>n} f a_k + l/2 = sup_m inf_{h>m} f a_h - l/2 +? Per definizione di inf forte abbiamo + esiste un n' tale che + sup_{k>n'} f a_k < inf_n sup_{k>n} + l + = sup_m inf_{h>m} f a_h - l/2 +? Per definizione di sup forte abbiamo + esiste un n'' tale che + sup_{k>n'} f a_k < sup_m inf_{h>m} f a_h - l/2 < inf_{h>n''} f a_k + Sia N il max tra n' e n''. Allora: + sup_{k>N} f a_k < sup_{k>n'} f a_k < inf_{h>n''} f a_k < inf_{h>N} f a_k + Assurdo per lemma precedente. +Qed. + +======================================= + +Lebesgue costruttivo: + a_n bounded da b ovvero \forall n, a_n < b + f strongly monotone ovvero f x < f y => y -<= x + liminf f a_n # limsup f a_n => liminf a_n < (o #) limsup a_n +Dimostrazione: + per ipotesi + liminf f a_n # limsup f a_n + quindi + liminf f a_n > limsup f a_n \/ liminf f a_n < limsup f a_n. + per casi: + Caso 1: + Usiamo il lemma liminf f a_n <= limsup f a_n => assurdo + Caso 2: + Usiamo Fatou e Fatou rovesciato: + f (liminf a_n) <= liminf (f a_n) (per fatou) + < limsup (f a_n) (per ipotesi) + <= f (limsup a_n) (per fatou rovesciato) + Per monotonia forte della f otteniamo: + limsup a_n -<= liminf a_n + (Da cui: + liminf a_n # limsup a_n) +Qed. + +############### Costruttivizzazione di Weber-Zoli ###################### + +Prerequisiti: + 1. does_not_approach_zero x_n = + \exists delta. \exists sottosuccessione j. + \forall n. x_(j n) > delta + 2. does_not_have_sup = ??? (vedi prerequisito ????? sotto da soddisfare) + 3. sigma_and_esaustiva su [a,b] x_n = + d(a_n,x) does_not_approach_zero => a_n does_not_have_sup x + ????? inf x_i does_not_have_sup x => liminf x_i # x + +======================================= + +Sviluppi futuri: + Spezzare sigma_and_esaustiva in sigma + esaustiva o qualcosa del + genere. Probabilmente sigma diventa + d(a,a_1) ~<= \bigsum_{i=n}^\infty d(a_n,a_{n+1}) => + a_n does_not_have_sup a + La prova del lemma 5 in versione positiva e' ancora da fare. + L'esaustivita' deve essere rimpiazzata da un concetto tipo located. + +======================================= + +Due carabinieri: + a_n <= x_n <= b_n + d(x_n,x) does_not_approach_zero => + d(a_n,x) does_not_approach_zero \/ + d(b_n,x) does_not_approach_zero +Dimostrazione: + Per ipotesi esiste un \delta e una sottosuccessione y tale che + \delta < d(y_n,x) + <= d(y_n,a_n) + d(a_n,x) + <= d(b_n,a_n) + d(a_n,x) + <= d(b_n,x) + 2d(a_n,x) + We conclude (?????? costruttivamente vero per > 0 e vero classicamente) + d(b_n,x) > \delta/4 \/ d(a_n,x) > \delta/4 + and thus + d(a_n,x) does_not_approach_zero \/ + d(b_n,x) does_not_approach_zero +Qed. + +======================================= + +Lebsegue costruttivo: + x_n in [a,b], a_n <= x_n <= b_n per ogni n + d sigma_and_esaustiva su [a,b]; + d(x_n,liminf x_n) does_not_approach_zero \/ + d(x_n,limsup x_n) does_not_approach_zero => + liminf x_n # limsup x_n (possiamo concludere che eccede? forse no) +Dimostrazione: + Fissiamo un x tale che d(x_n,x) does_not_approach_zero. + Per ipotesi d(x_n,x) does_not_approach_zero + Siano a_n := inf_{i>=n} x_i e b_n := sup_{i>=n} x_i. + Per i due carabinieri: + d(a_n,x) does_not_approach_zero \/ d(b_n,x) does_not_approach_zero + Per definizione di sigma_and_esaustiva su [a,b] + a_n does_not_have_sup x \/ b_n does_not_have_inf x + Quindi, per definizione di liminf e limsup e per ????????? + liminf x_n # x \/ limsup x_n # x + Facendo discharging di x concludiamo + \forall x t.c. d(x_n,x) does_not_approach zero, + liminf x_n # x \/ limsup x_n # x + Per ipotesi possiamo istanziare x con liminf x_n oppure con + limsup x_n. + Nel primo caso otteniamo + liminf x_n # liminf x_n \/ limsup x_n # liminf x_n + Poiche' la prima ipotesi e' falsa concludiamo + limsup x_n # liminf x_n + Nel secondo caso otteniamo + liminf x_n # limsup x_n \/ limsup x_n # limsup x_n \/ + Poiche' la seconda ipotesi e' falsa concludiamo anche in questo caso + limsup x_n # liminf x_n +Qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/doc/NotaReticoli.pdf b/helm/software/matita/contribs/dama/dama_duality/doc/NotaReticoli.pdf new file mode 100644 index 000000000..76a6842e9 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/doc/NotaReticoli.pdf @@ -0,0 +1,3078 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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b/helm/software/matita/contribs/dama/dama_duality/excess.ma @@ -0,0 +1,279 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "higher_order_defs/relations.ma". +include "nat/plus.ma". +include "constructive_higher_order_relations.ma". +include "constructive_connectives.ma". + +record excess_base : Type ≝ { + exc_carr:> Type; + exc_excess: exc_carr → exc_carr → Type; + exc_coreflexive: coreflexive ? exc_excess; + exc_cotransitive: cotransitive ? exc_excess +}. + +interpretation "Excess base excess" 'nleq a b = (cic:/matita/excess/exc_excess.con _ a b). + +(* E(#,≰) → E(#,sym(≰)) *) +lemma make_dual_exc: excess_base → excess_base. +intro E; +apply (mk_excess_base (exc_carr E)); + [ apply (λx,y:E.y≰x);|apply exc_coreflexive; + | unfold cotransitive; simplify; intros (x y z H); + cases (exc_cotransitive E ??z H);[right|left]assumption] +qed. + +record excess_dual : Type ≝ { + exc_dual_base:> excess_base; + exc_dual_dual_ : excess_base; + exc_with: exc_dual_dual_ = make_dual_exc exc_dual_base +}. + +lemma mk_excess_dual_smart: excess_base → excess_dual. +intro; apply mk_excess_dual; [apply e| apply (make_dual_exc e)|reflexivity] +qed. + +definition exc_dual_dual: excess_dual → excess_base. +intro E; apply (make_dual_exc E); +qed. + +coercion cic:/matita/excess/exc_dual_dual.con. + +record apartness : Type ≝ { + ap_carr:> Type; + ap_apart: ap_carr → ap_carr → Type; + ap_coreflexive: coreflexive ? ap_apart; + ap_symmetric: symmetric ? ap_apart; + ap_cotransitive: cotransitive ? ap_apart +}. + +notation "hvbox(a break # b)" non associative with precedence 50 for @{ 'apart $a $b}. +interpretation "apartness" 'apart x y = (cic:/matita/excess/ap_apart.con _ x y). + +definition apartness_of_excess_base: excess_base → apartness. +intros (E); apply (mk_apartness E (λa,b:E. a ≰ b ∨ b ≰ a)); +[1: unfold; cases E; simplify; clear E; intros (x); unfold; + intros (H1); apply (H x); cases H1; assumption; +|2: unfold; intros(x y H); cases H; clear H; [right|left] assumption; +|3: intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy); + cases Axy (H H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1; + [left; left|right; left|right; right|left; right] assumption] +qed. + +record excess_ : Type ≝ { + exc_exc:> excess_dual; + exc_ap_: apartness; + exc_with1: ap_carr exc_ap_ = exc_carr exc_exc +}. + +definition exc_ap: excess_ → apartness. +intro E; apply (mk_apartness E); unfold Type_OF_excess_; +cases (exc_with1 E); simplify; +[apply (ap_apart (exc_ap_ E)); +|apply ap_coreflexive;|apply ap_symmetric;|apply ap_cotransitive] +qed. + +coercion cic:/matita/excess/exc_ap.con. + +interpretation "Excess excess_" 'nleq a b = + (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess_1.con _) a b). + +record excess : Type ≝ { + excess_carr:> excess_; + ap2exc: ∀y,x:excess_carr. y # x → y ≰ x ∨ x ≰ y; + exc2ap: ∀y,x:excess_carr.y ≰ x ∨ x ≰ y → y # x +}. + +interpretation "Excess excess" 'nleq a b = + (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b). + +interpretation "Excess (dual) excess" 'ngeq a b = + (cic:/matita/excess/exc_excess.con (cic:/matita/excess/excess_base_OF_excess.con _) a b). + +definition strong_ext ≝ λA:apartness.λop:A→A.∀x,y. op x # op y → x # y. + +definition le ≝ λE:excess_base.λa,b:E. ¬ (a ≰ b). + +interpretation "Excess less or equal than" 'leq a b = + (cic:/matita/excess/le.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b). + +interpretation "Excess less or equal than" 'geq a b = + (cic:/matita/excess/le.con (cic:/matita/excess/excess_base_OF_excess.con _) a b). + +lemma le_reflexive: ∀E.reflexive ? (le E). +unfold reflexive; intros 3 (E x H); apply (exc_coreflexive ?? H); +qed. + +lemma le_transitive: ∀E.transitive ? (le E). +unfold transitive; intros 7 (E x y z H1 H2 H3); cases (exc_cotransitive ??? y H3) (H4 H4); +[cases (H1 H4)|cases (H2 H4)] +qed. + +definition eq ≝ λA:apartness.λa,b:A. ¬ (a # b). + +notation "hvbox(a break ≈ b)" non associative with precedence 50 for @{ 'napart $a $b}. +interpretation "Apartness alikeness" 'napart a b = (cic:/matita/excess/eq.con _ a b). +interpretation "Excess alikeness" 'napart a b = (cic:/matita/excess/eq.con (cic:/matita/excess/excess_base_OF_excess1.con _) a b). +interpretation "Excess (dual) alikeness" 'napart a b = (cic:/matita/excess/eq.con (cic:/matita/excess/excess_base_OF_excess.con _) a b). + +lemma eq_reflexive:∀E:apartness. reflexive ? (eq E). +intros (E); unfold; intros (x); apply ap_coreflexive; +qed. + +lemma eq_sym_:∀E:apartness.symmetric ? (eq E). +unfold symmetric; intros 5 (E x y H H1); cases (H (ap_symmetric ??? H1)); +qed. + +lemma eq_sym:∀E:apartness.∀x,y:E.x ≈ y → y ≈ x ≝ eq_sym_. + +(* SETOID REWRITE *) +coercion cic:/matita/excess/eq_sym.con. + +lemma eq_trans_: ∀E:apartness.transitive ? (eq E). +(* bug. intros k deve fare whd quanto basta *) +intros 6 (E x y z Exy Eyz); intro Axy; cases (ap_cotransitive ???y Axy); +[apply Exy|apply Eyz] assumption. +qed. + +lemma eq_trans:∀E:apartness.∀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 = (cic:/matita/excess/eq_trans.con _ _ _). + +alias id "antisymmetric" = "cic:/matita/constructive_higher_order_relations/antisymmetric.con". +lemma le_antisymmetric: + ∀E:excess.antisymmetric ? (le (excess_base_OF_excess1 E)) (eq E). +intros 5 (E x y Lxy Lyx); intro H; +cases (ap2exc ??? H); [apply Lxy;|apply Lyx] assumption; +qed. + +definition lt ≝ λE:excess.λa,b:E. a ≤ b ∧ a # b. + +interpretation "ordered sets less than" 'lt a b = (cic:/matita/excess/lt.con _ a b). + +lemma lt_coreflexive: ∀E.coreflexive ? (lt E). +intros 2 (E x); intro H; cases H (_ ABS); +apply (ap_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 ???? Lxy Lyz)] clear H1 H2; +elim (ap2exc ??? Axy) (H1 H1); elim (ap2exc ??? Ayz) (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)] +clear Axy Ayz;lapply (exc_cotransitive (exc_dual_base E)) as c; unfold cotransitive in c; +lapply (exc_coreflexive (exc_dual_base E)) as r; unfold coreflexive in r; +[1: lapply (c ?? y H1) as H3; elim H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)] +|2: lapply (c ?? x H2) as H3; elim H3 (H4 H4); [apply exc2ap; right; assumption|cases (Lxy H4)]] +qed. + +theorem lt_to_excess: ∀E:excess.∀a,b:E. (a < b) → (b ≰ a). +intros (E a b Lab); elim Lab (LEab Aab); +elim (ap2exc ??? Aab) (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *) +qed. + +lemma le_rewl: ∀E:excess.∀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; apply exc2ap; right; assumption; +qed. + +lemma le_rewr: ∀E:excess.∀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; apply exc2ap; left; assumption; +qed. + +notation > "'Le'≪" non associative with precedence 50 for @{'lerewritel}. +interpretation "le_rewl" 'lerewritel = (cic:/matita/excess/le_rewl.con _ _ _). +notation > "'Le'≫" non associative with precedence 50 for @{'lerewriter}. +interpretation "le_rewr" 'lerewriter = (cic:/matita/excess/le_rewr.con _ _ _). + +lemma ap_rewl: ∀A:apartness.∀x,z,y:A. x ≈ y → y # z → x # z. +intros (A x z y Exy Ayz); cases (ap_cotransitive ???x Ayz); [2:assumption] +cases (Exy (ap_symmetric ??? a)); +qed. + +lemma ap_rewr: ∀A:apartness.∀x,z,y:A. x ≈ y → z # y → z # x. +intros (A x z y Exy Azy); apply ap_symmetric; apply (ap_rewl ???? Exy); +apply ap_symmetric; assumption; +qed. + +notation > "'Ap'≪" non associative with precedence 50 for @{'aprewritel}. +interpretation "ap_rewl" 'aprewritel = (cic:/matita/excess/ap_rewl.con _ _ _). +notation > "'Ap'≫" non associative with precedence 50 for @{'aprewriter}. +interpretation "ap_rewr" 'aprewriter = (cic:/matita/excess/ap_rewr.con _ _ _). + +alias symbol "napart" = "Apartness alikeness". +lemma exc_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y ≰ z → x ≰ z. +intros (A x z y Exy Ayz); elim (exc_cotransitive ???x Ayz); [2:assumption] +cases Exy; apply exc2ap; right; assumption; +qed. + +lemma exc_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z ≰ y → z ≰ x. +intros (A x z y Exy Azy); elim (exc_cotransitive ???x Azy); [assumption] +elim (Exy); apply exc2ap; left; assumption; +qed. + +notation > "'Ex'≪" non associative with precedence 50 for @{'excessrewritel}. +interpretation "exc_rewl" 'excessrewritel = (cic:/matita/excess/exc_rewl.con _ _ _). +notation > "'Ex'≫" non associative with precedence 50 for @{'excessrewriter}. +interpretation "exc_rewr" 'excessrewriter = (cic:/matita/excess/exc_rewr.con _ _ _). + +lemma lt_rewr: ∀A:excess.∀x,z,y:A. x ≈ y → z < y → z < x. +intros (A x y z E H); split; elim H; +[apply (Le≫ ? (eq_sym ??? E));|apply (Ap≫ ? E)] assumption; +qed. + +lemma lt_rewl: ∀A:excess.∀x,z,y:A. x ≈ y → y < z → x < z. +intros (A x y z E H); split; elim H; +[apply (Le≪ ? (eq_sym ??? E));| apply (Ap≪ ? E);] assumption; +qed. + +notation > "'Lt'≪" non associative with precedence 50 for @{'ltrewritel}. +interpretation "lt_rewl" 'ltrewritel = (cic:/matita/excess/lt_rewl.con _ _ _). +notation > "'Lt'≫" non associative with precedence 50 for @{'ltrewriter}. +interpretation "lt_rewr" 'ltrewriter = (cic:/matita/excess/lt_rewr.con _ _ _). + +lemma lt_le_transitive: ∀A:excess.∀x,y,z:A.x < y → y ≤ z → x < z. +intros (A x y z LT LE); cases LT (LEx APx); split; [apply (le_transitive ???? LEx LE)] +apply exc2ap; cases (ap2exc ??? APx) (EXx EXx); [cases (LEx EXx)] +cases (exc_cotransitive ??? z EXx) (EXz EXz); [cases (LE EXz)] +right; assumption; +qed. + +lemma le_lt_transitive: ∀A:excess.∀x,y,z:A.x ≤ y → y < z → x < z. +intros (A x y z LE LT); cases LT (LEx APx); split; [apply (le_transitive ???? LE LEx)] +elim (ap2exc ??? APx) (EXx EXx); [cases (LEx EXx)] +elim (exc_cotransitive ??? x EXx) (EXz EXz); [apply exc2ap; right; assumption] +cases LE; assumption; +qed. + +lemma le_le_eq: ∀E:excess.∀a,b:E. a ≤ b → b ≤ a → a ≈ b. +intros (E x y L1 L2); intro H; cases (ap2exc ??? H); [apply L1|apply L2] assumption; +qed. + +lemma eq_le_le: ∀E:excess.∀a,b:E. a ≈ b → a ≤ b ∧ b ≤ a. +intros (E x y H); whd in H; +split; intro; apply H; apply exc2ap; [left|right] assumption. +qed. + +lemma ap_le_to_lt: ∀E:excess.∀a,c:E.c # a → c ≤ a → c < a. +intros; split; assumption; +qed. + +definition total_order_property : ∀E:excess. Type ≝ + λE:excess. ∀a,b:E. a ≰ b → b < a. + diff --git a/helm/software/matita/contribs/dama/dama_duality/group.ma b/helm/software/matita/contribs/dama/dama_duality/group.ma new file mode 100644 index 000000000..104dcf274 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/group.ma @@ -0,0 +1,220 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "excess.ma". + +definition left_neutral ≝ λC:apartness.λop.λe:C. ∀x:C. op e x ≈ x. +definition right_neutral ≝ λC:apartness.λop. λe:C. ∀x:C. op x e ≈ x. +definition left_inverse ≝ λC:apartness.λop.λe:C.λinv:C→C. ∀x:C. op (inv x) x ≈ e. +definition right_inverse ≝ λC:apartness.λop.λe:C.λ inv: C→ C. ∀x:C. op x (inv x) ≈ e. +(* ALLOW DEFINITION WITH SOME METAS *) + +definition distributive_left ≝ + λA:apartness.λf:A→A→A.λg:A→A→A. + ∀x,y,z. f x (g y z) ≈ g (f x y) (f x z). + +definition distributive_right ≝ + λA:apartness.λf:A→A→A.λg:A→A→A. + ∀x,y,z. f (g x y) z ≈ g (f x z) (f y z). + +record abelian_group : Type ≝ + { carr:> apartness; + plus: carr → carr → carr; + zero: carr; + opp: carr → carr; + plus_assoc_: associative ? plus (eq carr); + plus_comm_: commutative ? plus (eq carr); + zero_neutral_: left_neutral ? plus zero; + opp_inverse_: left_inverse ? plus zero opp; + plus_strong_ext: ∀z.strong_ext ? (plus z) +}. + +notation "0" with precedence 89 for @{ 'zero }. + +interpretation "Abelian group zero" 'zero = + (cic:/matita/group/zero.con _). + +interpretation "Abelian group plus" 'plus a b = + (cic:/matita/group/plus.con _ a b). + +interpretation "Abelian group opp" 'uminus a = + (cic:/matita/group/opp.con _ a). + +definition minus ≝ + λG:abelian_group.λa,b:G. a + -b. + +interpretation "Abelian group minus" 'minus a b = + (cic:/matita/group/minus.con _ a b). + +lemma plus_assoc: ∀G:abelian_group.∀x,y,z:G.x+(y+z)≈x+y+z ≝ plus_assoc_. +lemma plus_comm: ∀G:abelian_group.∀x,y:G.x+y≈y+x ≝ plus_comm_. +lemma zero_neutral: ∀G:abelian_group.∀x:G.0+x≈x ≝ zero_neutral_. +lemma opp_inverse: ∀G:abelian_group.∀x:G.-x+x≈0 ≝ opp_inverse_. + +definition ext ≝ λA:apartness.λop:A→A. ∀x,y. x ≈ y → op x ≈ op y. + +lemma strong_ext_to_ext: ∀A:apartness.∀op:A→A. strong_ext ? op → ext ? op. +intros 6 (A op SEop x y Exy); intro Axy; apply Exy; apply SEop; assumption; +qed. + +lemma feq_plusl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x+y ≈ x+z. +intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_ext ? x)); +assumption; +qed. + +coercion cic:/matita/group/feq_plusl.con nocomposites. + +lemma plus_strong_extr: ∀G:abelian_group.∀z:G.strong_ext ? (λx.x + z). +intros 5 (G z x y A); simplify in A; +lapply (plus_comm ? z x) as E1; lapply (plus_comm ? z y) as E2; +lapply (Ap≪ ? E1 A) as A1; lapply (Ap≫ ? E2 A1) as A2; +apply (plus_strong_ext ???? A2); +qed. + +lemma plus_cancl_ap: ∀G:abelian_group.∀x,y,z:G.z+x # z + y → x # y. +intros; apply plus_strong_ext; assumption; +qed. + +lemma plus_cancr_ap: ∀G:abelian_group.∀x,y,z:G.x+z # y+z → x # y. +intros; apply plus_strong_extr; assumption; +qed. + +lemma feq_plusr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → y+x ≈ z+x. +intros (G x y z Eyz); apply (strong_ext_to_ext ?? (plus_strong_extr ? x)); +assumption; +qed. + +coercion cic:/matita/group/feq_plusr.con nocomposites. + +(* generation of coercions to make *_rew[lr] easier *) +lemma feq_plusr_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → y+x ≈ z+x. +compose feq_plusr with eq_sym (H); apply H; assumption; +qed. +coercion cic:/matita/group/feq_plusr_sym_.con nocomposites. +lemma feq_plusl_sym_: ∀G:abelian_group.∀x,y,z:G.z ≈ y → x+y ≈ x+z. +compose feq_plusl with eq_sym (H); apply H; assumption; +qed. +coercion cic:/matita/group/feq_plusl_sym_.con nocomposites. + +lemma fap_plusl: ∀G:abelian_group.∀x,y,z:G. y # z → x+y # x+z. +intros (G x y z Ayz); apply (plus_strong_ext ? (-x)); +apply (Ap≪ ((-x + x) + y)); +[1: apply plus_assoc; +|2: apply (Ap≫ ((-x +x) +z)); + [1: apply plus_assoc; + |2: apply (Ap≪ (0 + y)); + [1: apply (feq_plusr ???? (opp_inverse ??)); + |2: apply (Ap≪ ? (zero_neutral ? y)); + apply (Ap≫ (0 + z) (opp_inverse ??)); + apply (Ap≫ ? (zero_neutral ??)); assumption;]]] +qed. + +lemma fap_plusr: ∀G:abelian_group.∀x,y,z:G. y # z → y+x # z+x. +intros (G x y z Ayz); apply (plus_strong_extr ? (-x)); +apply (Ap≪ (y + (x + -x))); +[1: apply (eq_sym ??? (plus_assoc ????)); +|2: apply (Ap≫ (z + (x + -x))); + [1: apply (eq_sym ??? (plus_assoc ????)); + |2: apply (Ap≪ (y + (-x+x)) (plus_comm ? x (-x))); + apply (Ap≪ (y + 0) (opp_inverse ??)); + apply (Ap≪ (0 + y) (plus_comm ???)); + apply (Ap≪ y (zero_neutral ??)); + apply (Ap≫ (z + (-x+x)) (plus_comm ? x (-x))); + apply (Ap≫ (z + 0) (opp_inverse ??)); + apply (Ap≫ (0 + z) (plus_comm ???)); + apply (Ap≫ z (zero_neutral ??)); + assumption]] +qed. + +lemma applus: ∀E:abelian_group.∀x,a,y,b:E.x + a # y + b → x # y ∨ a # b. +intros; cases (ap_cotransitive ??? (y+a) a1); [left|right] +[apply (plus_cancr_ap ??? a)|apply (plus_cancl_ap ??? y)] +assumption; +qed. + +lemma plus_cancl: ∀G:abelian_group.∀y,z,x:G. x+y ≈ x+z → y ≈ z. +intros 6 (G y z x E Ayz); apply E; apply fap_plusl; assumption; +qed. + +lemma plus_cancr: ∀G:abelian_group.∀y,z,x:G. y+x ≈ z+x → y ≈ z. +intros 6 (G y z x E Ayz); apply E; apply fap_plusr; assumption; +qed. + +theorem eq_opp_plus_plus_opp_opp: + ∀G:abelian_group.∀x,y:G. -(x+y) ≈ -x + -y. +intros (G x y); apply (plus_cancr ??? (x+y)); +apply (Eq≈ 0 (opp_inverse ??)); +apply (Eq≈ (-x + -y + x + y)); [2: apply (eq_sym ??? (plus_assoc ????))] +apply (Eq≈ (-y + -x + x + y)); [2: repeat apply feq_plusr; apply plus_comm] +apply (Eq≈ (-y + (-x + x) + y)); [2: apply feq_plusr; apply plus_assoc;] +apply (Eq≈ (-y + 0 + y)); + [2: apply feq_plusr; apply feq_plusl; apply eq_sym; apply opp_inverse] +apply (Eq≈ (-y + y)); + [2: apply feq_plusr; apply eq_sym; + apply (Eq≈ (0+-y)); [apply plus_comm|apply zero_neutral]] +apply eq_sym; apply opp_inverse. +qed. + +theorem eq_opp_opp_x_x: ∀G:abelian_group.∀x:G.--x ≈ x. +intros (G x); apply (plus_cancl ??? (-x)); +apply (Eq≈ (--x + -x) (plus_comm ???)); +apply (Eq≈ 0 (opp_inverse ??)); +apply eq_sym; apply opp_inverse; +qed. + +theorem eq_zero_opp_zero: ∀G:abelian_group.0 ≈ -0. [assumption] +intro G; apply (plus_cancr ??? 0); +apply (Eq≈ 0); [apply zero_neutral;] +apply eq_sym; apply opp_inverse; +qed. + +lemma feq_oppr: ∀G:abelian_group.∀x,y,z:G. y ≈ z → x ≈ -y → x ≈ -z. +intros (G x y z H1 H2); apply (plus_cancr ??? z); +apply (Eq≈ 0 ? (opp_inverse ??)); +apply (Eq≈ (-y + z) H2); +apply (Eq≈ (-y + y) H1); +apply (Eq≈ 0 (opp_inverse ??)); +apply eq_reflexive; +qed. + +lemma feq_oppl: ∀G:abelian_group.∀x,y,z:G. y ≈ z → -y ≈ x → -z ≈ x. +intros (G x y z H1 H2); apply eq_sym; apply (feq_oppr ??y); +[2:apply eq_sym] assumption; +qed. + +lemma feq_opp: ∀G:abelian_group.∀x,y:G. x ≈ y → -x ≈ -y. +intros (G x y H); apply (feq_oppl ??y ? H); apply eq_reflexive; +qed. + +coercion cic:/matita/group/feq_opp.con nocomposites. + +lemma eq_opp_sym: ∀G:abelian_group.∀x,y:G. y ≈ x → -x ≈ -y. +compose feq_opp with eq_sym (H); apply H; assumption; +qed. + +coercion cic:/matita/group/eq_opp_sym.con nocomposites. + +lemma eq_opp_plusr: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(x + z) ≈ -(y + z). +compose feq_plusr with feq_opp(H); apply H; assumption; +qed. + +coercion cic:/matita/group/eq_opp_plusr.con nocomposites. + +lemma eq_opp_plusl: ∀G:abelian_group.∀x,y,z:G. x ≈ y → -(z + x) ≈ -(z + y). +compose feq_plusl with feq_opp(H); apply H; assumption; +qed. + +coercion cic:/matita/group/eq_opp_plusl.con nocomposites. diff --git a/helm/software/matita/contribs/dama/dama_duality/infsup.ma b/helm/software/matita/contribs/dama/dama_duality/infsup.ma new file mode 100644 index 000000000..cc3292fd0 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/infsup.ma @@ -0,0 +1,53 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "sequence.ma". + +definition upper_bound ≝ λO:excess.λa:sequence O.λu:O.∀n:nat.a n ≤ u. + +definition weak_sup ≝ + λO:excess.λs:sequence O.λx. + upper_bound ? s x ∧ (∀y:O.upper_bound ? s y → x ≤ y). + +definition strong_sup ≝ + λO:excess.λs:sequence O.λx.upper_bound ? s x ∧ (∀y:O.x ≰ y → ∃n.s n ≰ y). + +definition increasing ≝ λO:excess.λa:sequence O.∀n:nat.a n ≤ a (S n). + +notation < "x \nbsp 'is_upper_bound' \nbsp s" non associative with precedence 50 for @{'upper_bound $_ $s $x}. +notation < "x \nbsp 'is_lower_bound' \nbsp s" non associative with precedence 50 for @{'lower_bound $_ $s $x}. +notation < "s \nbsp 'is_increasing'" non associative with precedence 50 for @{'increasing $_ $s}. +notation < "s \nbsp 'is_decreasing'" non associative with precedence 50 for @{'decreasing $_ $s}. +notation < "x \nbsp 'is_strong_sup' \nbsp s" non associative with precedence 50 for @{'strong_sup $_ $s $x}. +notation < "x \nbsp 'is_strong_inf' \nbsp s" non associative with precedence 50 for @{'strong_inf $_ $s $x}. + +notation > "x 'is_upper_bound' s 'in' e" non associative with precedence 50 for @{'upper_bound $e $s $x}. +notation > "x 'is_lower_bound' s 'in' e" non associative with precedence 50 for @{'lower_bound $e $s $x}. +notation > "s 'is_increasing' 'in' e" non associative with precedence 50 for @{'increasing $e $s}. +notation > "s 'is_decreasing' 'in' e" non associative with precedence 50 for @{'decreasing $e $s}. +notation > "x 'is_strong_sup' s 'in' e" non associative with precedence 50 for @{'strong_sup $e $s $x}. +notation > "x 'is_strong_inf' s 'in' e" non associative with precedence 50 for @{'strong_inf $e $s $x}. + +interpretation "Excess upper bound" 'upper_bound e s x = (cic:/matita/infsup/upper_bound.con e s x). +interpretation "Excess lower bound" 'lower_bound e s x = (cic:/matita/infsup/upper_bound.con (cic:/matita/excess/dual_exc.con e) s x). +interpretation "Excess increasing" 'increasing e s = (cic:/matita/infsup/increasing.con e s). +interpretation "Excess decreasing" 'decreasing e s = (cic:/matita/infsup/increasing.con (cic:/matita/excess/dual_exc.con e) s). +interpretation "Excess strong sup" 'strong_sup e s x = (cic:/matita/infsup/strong_sup.con e s x). +interpretation "Excess strong inf" 'strong_inf e s x = (cic:/matita/infsup/strong_sup.con (cic:/matita/excess/dual_exc.con e) s x). + +lemma strong_sup_is_weak: + ∀O:excess.∀s:sequence O.∀x:O.strong_sup ? s x → weak_sup ? s x. +intros (O s x Ssup); elim Ssup (Ubx M); clear Ssup; split; [assumption] +intros 3 (y Uby E); cases (M ? E) (n En); unfold in Uby; cases (Uby ? En); +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/lattice.ma b/helm/software/matita/contribs/dama/dama_duality/lattice.ma new file mode 100644 index 000000000..adf751b83 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/lattice.ma @@ -0,0 +1,484 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "excess.ma". + +record semi_lattice_base : Type ≝ { + sl_carr:> apartness; + sl_op: sl_carr → sl_carr → sl_carr; + sl_op_refl: ∀x.sl_op x x ≈ x; + sl_op_comm: ∀x,y:sl_carr. sl_op x y ≈ sl_op y x; + sl_op_assoc: ∀x,y,z:sl_carr. sl_op x (sl_op y z) ≈ sl_op (sl_op x y) z; + sl_strong_extop: ∀x.strong_ext ? (sl_op x) +}. + +notation "a \cross b" left associative with precedence 50 for @{ 'op $a $b }. +interpretation "semi lattice base operation" 'op a b = (cic:/matita/lattice/sl_op.con _ a b). + +lemma excess_of_semi_lattice_base: semi_lattice_base → excess. +intro l; +apply mk_excess; +[1: apply mk_excess_; + [1: apply mk_excess_dual_smart; + + apply (mk_excess_base (sl_carr l)); + [1: apply (λa,b:sl_carr l.a # (a ✗ b)); + |2: unfold; intros 2 (x H); simplify in H; + lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H; + apply (ap_coreflexive ?? H1); + |3: unfold; simplify; intros (x y z H1); + cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2: + lapply (Ap≪ ? (sl_op_comm ???) H2) as H21; + lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2; + lapply (sl_strong_extop ???? H22); clear H22; + left; apply ap_symmetric; assumption;] + cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption] + right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31; + apply (sl_strong_extop ???? H31);] + + |2: + apply apartness_of_excess_base; + + apply (mk_excess_base (sl_carr l)); + [1: apply (λa,b:sl_carr l.a # (a ✗ b)); + |2: unfold; intros 2 (x H); simplify in H; + lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H; + apply (ap_coreflexive ?? H1); + |3: unfold; simplify; intros (x y z H1); + cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2: + lapply (Ap≪ ? (sl_op_comm ???) H2) as H21; + lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2; + lapply (sl_strong_extop ???? H22); clear H22; + left; apply ap_symmetric; assumption;] + cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption] + right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31; + apply (sl_strong_extop ???? H31);] + + |3: apply refl_eq;] +|2,3: intros (x y H); assumption;] +qed. + +record semi_lattice : Type ≝ { + sl_exc:> excess; + sl_meet: sl_exc → sl_exc → sl_exc; + sl_meet_refl: ∀x.sl_meet x x ≈ x; + sl_meet_comm: ∀x,y. sl_meet x y ≈ sl_meet y x; + sl_meet_assoc: ∀x,y,z. sl_meet x (sl_meet y z) ≈ sl_meet (sl_meet x y) z; + sl_strong_extm: ∀x.strong_ext ? (sl_meet x); + sl_le_to_eqm: ∀x,y.x ≤ y → x ≈ sl_meet x y; + sl_lem: ∀x,y.(sl_meet x y) ≤ y +}. + +interpretation "semi lattice meet" 'and a b = (cic:/matita/lattice/sl_meet.con _ a b). + +lemma sl_feq_ml: ∀ml:semi_lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b). +intros (l a b c H); unfold eq in H ⊢ %; unfold Not in H ⊢ %; +intro H1; apply H; clear H; apply (sl_strong_extm ???? H1); +qed. + +lemma sl_feq_mr: ∀ml:semi_lattice.∀a,b,c:ml. a ≈ b → (a ∧ c) ≈ (b ∧ c). +intros (l a b c H); +apply (Eq≈ ? (sl_meet_comm ???)); apply (Eq≈ ?? (sl_meet_comm ???)); +apply sl_feq_ml; assumption; +qed. + + +(* +lemma semi_lattice_of_semi_lattice_base: semi_lattice_base → semi_lattice. +intro slb; apply (mk_semi_lattice (excess_of_semi_lattice_base slb)); +[1: apply (sl_op slb); +|2: intro x; apply (eq_trans (excess_of_semi_lattice_base slb)); [2: + apply (sl_op_refl slb);|1:skip] (sl_op slb x x)); ? (sl_op_refl slb x)); + + unfold excess_of_semi_lattice_base; simplify; + intro H; elim H; + [ + + + lapply (ap_rewl (excess_of_semi_lattice_base slb) x ? (sl_op slb x x) + (eq_sym (excess_of_semi_lattice_base slb) ?? (sl_op_refl slb x)) t); + change in x with (sl_carr slb); + apply (Ap≪ (x ✗ x)); (sl_op_refl slb x)); + +whd in H; elim H; clear H; + [ apply (ap_coreflexive (excess_of_semi_lattice_base slb) (x ✗ x) t); + +prelattice (excess_of_directed l_)); [apply (sl_op l_);] +unfold excess_of_directed; try unfold apart_of_excess; simplify; +unfold excl; simplify; +[intro x; intro H; elim H; clear H; + [apply (sl_op_refl l_ x); + lapply (Ap≫ ? (sl_op_comm ???) t) as H; clear t; + lapply (sl_strong_extop l_ ??? H); apply ap_symmetric; assumption + | lapply (Ap≪ ? (sl_op_refl ?x) t) as H; clear t; + lapply (sl_strong_extop l_ ??? H); apply (sl_op_refl l_ x); + apply ap_symmetric; assumption] +|intros 3 (x y H); cases H (H1 H2); clear H; + [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x y)) H1) as H; clear H1; + lapply (sl_strong_extop l_ ??? H) as H1; clear H; + lapply (Ap≪ ? (sl_op_comm ???) H1); apply (ap_coreflexive ?? Hletin); + |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ y x)) H2) as H; clear H2; + lapply (sl_strong_extop l_ ??? H) as H1; clear H; + lapply (Ap≪ ? (sl_op_comm ???) H1);apply (ap_coreflexive ?? Hletin);] +|intros 4 (x y z H); cases H (H1 H2); clear H; + [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x (sl_op l_ y z))) H1) as H; clear H1; + lapply (sl_strong_extop l_ ??? H) as H1; clear H; + lapply (Ap≪ ? (eq_sym ??? (sl_op_assoc ?x y z)) H1) as H; clear H1; + apply (ap_coreflexive ?? H); + |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ (sl_op l_ x y) z)) H2) as H; clear H2; + lapply (sl_strong_extop l_ ??? H) as H1; clear H; + lapply (Ap≪ ? (sl_op_assoc ?x y z) H1) as H; clear H1; + apply (ap_coreflexive ?? H);] +|intros (x y z H); elim H (H1 H1); clear H; + lapply (Ap≪ ? (sl_op_refl ??) H1) as H; clear H1; + lapply (sl_strong_extop l_ ??? H) as H1; clear H; + lapply (sl_strong_extop l_ ??? H1) as H; clear H1; + cases (ap_cotransitive ??? (sl_op l_ y z) H);[left|right|right|left] try assumption; + [apply ap_symmetric;apply (Ap≪ ? (sl_op_comm ???)); + |apply (Ap≫ ? (sl_op_comm ???)); + |apply ap_symmetric;] assumption; +|intros 4 (x y H H1); apply H; clear H; elim H1 (H H); + lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H; + lapply (sl_strong_extop l_ ??? H1) as H; clear H1;[2: apply ap_symmetric] + assumption +|intros 3 (x y H); + cut (sl_op l_ x y ≈ sl_op l_ x (sl_op l_ y y)) as H1;[2: + intro; lapply (sl_strong_extop ???? a); apply (sl_op_refl l_ y); + apply ap_symmetric; assumption;] + lapply (Ap≪ ? (eq_sym ??? H1) H); apply (sl_op_assoc l_ x y y); + assumption; ] +qed. +*) + +(* ED(≰,≱) → EB(≰') → ED(≰',≱') *) +lemma subst_excess_base: excess_dual → excess_base → excess_dual. +intros; apply (mk_excess_dual_smart e1); +qed. + +(* E_(ED(≰,≱),AP(#),c ED = c AP) → ED' → c DE' = c E_ → E_(ED',#,p) *) +lemma subst_dual_excess: ∀e:excess_.∀e1:excess_dual.exc_carr e = exc_carr e1 → excess_. +intros (e e1 p); apply (mk_excess_ e1 e); cases p; reflexivity; +qed. + +(* E(E_,H1,H2) → E_' → H1' → H2' → E(E_',H1',H2') *) +alias symbol "nleq" = "Excess excess_". +lemma subst_excess_: ∀e:excess. ∀e1:excess_. + (∀y,x:e1. y # x → y ≰ x ∨ x ≰ y) → + (∀y,x:e1.y ≰ x ∨ x ≰ y → y # x) → + excess. +intros (e e1 H1 H2); apply (mk_excess e1 H1 H2); +qed. + +definition hole ≝ λT:Type.λx:T.x. + +notation < "\ldots" non associative with precedence 50 for @{'hole}. +interpretation "hole" 'hole = (cic:/matita/lattice/hole.con _ _). + +(* SL(E,M,H2-5(#),H67(≰)) → E' → c E = c E' → H67'(≰') → SL(E,M,p2-5,H67') *) +lemma subst_excess: + ∀l:semi_lattice. + ∀e:excess. + ∀p:exc_ap l = exc_ap e. + (∀x,y:e.(le (exc_dual_base e)) x y → x ≈ (?(sl_meet l)) x y) → + (∀x,y:e.(le (exc_dual_base e)) ((?(sl_meet l)) x y) y) → + semi_lattice. +[1,2:intro M; + change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e); + cases p; apply M; +|intros (l e p H1 H2); + apply (mk_semi_lattice e); + [ change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e); + cases p; simplify; apply (sl_meet l); + |2: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_refl; + |3: change in ⊢ (% → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_comm; + |4: change in ⊢ (% → % → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_assoc; + |5: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_strong_extm; + |6: clear H2; apply hole; apply H1; + |7: clear H1; apply hole; apply H2;]] +qed. + +lemma excess_of_excess_base: excess_base → excess. +intro eb; +apply mk_excess; + [apply (mk_excess_ (mk_excess_dual_smart eb)); + [apply (apartness_of_excess_base eb); + |reflexivity] + |2,3: intros; assumption] +qed. + +lemma subst_excess_preserves_aprtness: + ∀l:semi_lattice. + ∀e:excess. + ∀p,H1,H2. + exc_ap l = exc_ap (subst_excess l e p H1 H2). +intros; +unfold subst_excess; +simplify; assumption; +qed. + + +lemma subst_excess__preserves_aprtness: + ∀l:excess. + ∀e:excess_base. + ∀p,H1,H2. + exc_ap l = apartness_OF_excess (subst_excess_ l (subst_dual_excess l (subst_excess_base l e) p) H1 H2). +intros 3; (unfold subst_excess_; unfold subst_dual_excess; unfold subst_excess_base; unfold exc_ap; unfold mk_excess_dual_smart; simplify); +(unfold subst_excess_base in p; unfold mk_excess_dual_smart in p; simplify in p); +intros; cases p; +reflexivity; +qed. + +lemma subst_excess_base_in_excess_: + ∀d:excess_. + ∀eb:excess_base. + ∀p:exc_carr d = exc_carr eb. + excess_. +intros (e_ eb); +apply (subst_dual_excess e_); + [apply (subst_excess_base e_ eb); + |assumption] +qed. + +lemma subst_excess_base_in_excess: + ∀d:excess. + ∀eb:excess_base. + ∀p:exc_carr d = exc_carr eb. + (∀y1,x1:eb. (?(ap_apart d)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) → + (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart d)) y2 x2) → + excess. +[1,3,4:apply Type|2,5:intro f; cases p; apply f] +intros (d eb p H1 H2); +apply (subst_excess_ d); + [apply (subst_excess_base_in_excess_ d eb p); + |apply hole; clear H2; + change in ⊢ (%→%→?) with (exc_carr eb); + change in ⊢ (?→?→?→? (? % ? ?) (? % ? ?)) with eb; intros (y x H3); + apply H1; generalize in match H3; + unfold ap_apart; unfold subst_excess_base_in_excess_; + unfold subst_dual_excess; simplify; + generalize in match x; + generalize in match y; + cases p; simplify; intros; assumption; + |apply hole; clear H1; + change in ⊢ (%→%→?) with (exc_carr eb); + change in ⊢ (?→?→? (? % ? ?) (? % ? ?)→?) with eb; intros (y x H3); + unfold ap_apart; unfold subst_excess_base_in_excess_; + unfold subst_dual_excess; simplify; generalize in match (H2 ?? H3); + generalize in match x; generalize in match y; cases p; + intros; assumption;] +qed. + +lemma tech1: ∀e:excess. + ∀eb:excess_base. + ∀p,H1,H2. + exc_ap e = exc_ap_ (subst_excess_base_in_excess e eb p H1 H2). +intros (e eb p H1 H2); +unfold subst_excess_base_in_excess; +unfold subst_excess_; simplify; +unfold subst_excess_base_in_excess_; +unfold subst_dual_excess; simplify; reflexivity; +qed. + +lemma tech2: + ∀e:excess_.∀eb.∀p. + exc_ap e = exc_ap (mk_excess_ (subst_excess_base e eb) (exc_ap e) p). +intros (e eb p);unfold exc_ap; simplify; cases p; simplify; reflexivity; +qed. + +(* +lemma eq_fap: + ∀a1,b1,a2,b2,a3,b3,a4,b4,a5,b5. + a1=b1 → a2=b2 → a3=b3 → a4=b4 → a5=b5 → mk_apartness a1 a2 a3 a4 a5 = mk_apartness b1 b2 b3 b4 b5. +intros; cases H; cases H1; cases H2; cases H3; cases H4; reflexivity; +qed. +*) + +lemma subst_excess_base_in_excess_preserves_apartness: + ∀e:excess. + ∀eb:excess_base. + ∀H,H1,H2. + apartness_OF_excess e = + apartness_OF_excess (subst_excess_base_in_excess e eb H H1 H2). +intros (e eb p H1 H2); +unfold subst_excess_base_in_excess; +unfold subst_excess_; unfold subst_excess_base_in_excess_; +unfold subst_dual_excess; unfold apartness_OF_excess; +simplify in ⊢ (? ? ? (? %)); +rewrite < (tech2 e eb ); +reflexivity; +qed. + + + +alias symbol "nleq" = "Excess base excess". +lemma subst_excess_base_in_semi_lattice: + ∀sl:semi_lattice. + ∀eb:excess_base. + ∀p:exc_carr sl = exc_carr eb. + (∀y1,x1:eb. (?(ap_apart sl)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) → + (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart sl)) y2 x2) → + (∀x3,y3:eb.(le eb) x3 y3 → (?(eq sl)) x3 ((?(sl_meet sl)) x3 y3)) → + (∀x4,y4:eb.(le eb) ((?(sl_meet sl)) x4 y4) y4) → + semi_lattice. +[2:apply Prop|3,7,9,10:apply Type|4:apply (exc_carr eb)|1,5,6,8,11:intro f; cases p; apply f;] +intros (sl eb H H1 H2 H3 H4); +apply (subst_excess sl); + [apply (subst_excess_base_in_excess sl eb H H1 H2); + |apply subst_excess_base_in_excess_preserves_apartness; + |change in ⊢ (%→%→?) with ((λx.ap_carr x) (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)); simplify; + intros 3 (x y LE); clear H3 LE; + generalize in match x as x; generalize in match y as y; + generalize in match H1 as H1;generalize in match H2 as H2; + clear x y H1 H2 H4; STOP + + apply (match H return λr:Type.λm:Type_OF_semi_lattice sl=r. + ∀H2:(Πy2:exc_carr eb + .Πx2:exc_carr eb + .Or (exc_excess eb y2 x2) (exc_excess eb x2 y2) + →match H +  in eq +  return  + λright_1:Type + .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1 + .right_1→right_1→Type) +  with  + [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y2 x2) +.∀H1:Πy1:exc_carr eb + .Πx1:exc_carr eb + .match H +  in eq +  return  + λright_1:Type + .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1 + .right_1→right_1→Type) +  with  + [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y1 x1 + →Or (exc_excess eb y1 x1) (exc_excess eb x1 y1) + .∀y:ap_carr + (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) + .∀x:ap_carr + (apartness_OF_excess + (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) + .eq + (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) x + (match  + subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2 +  in eq +  return  + λright_1:apartness + .(λmatched:eq apartness (apartness_OF_semi_lattice sl) right_1 + .ap_carr right_1→ap_carr right_1→ap_carr right_1) +  with  + [refl_eq⇒sl_meet sl] x y) + + with [ refl_eq ⇒ ?]); + + + cases (subst_excess_base_in_excess_preserves_apartness sl eb H H1 H2); + cases H; + cases (subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2); simplify; + change in x:(%) with (exc_carr eb); + change in y:(%) with (exc_carr eb); + generalize in match OK; generalize in match x as x; generalize in match y as y; + cases H; simplify; (* funge, ma devo fare 2 rewrite in un colpo solo *) + *) + |cases FALSE; + ] +qed. + +record lattice_ : Type ≝ { + latt_mcarr:> semi_lattice; + latt_jcarr_: semi_lattice; + W1:?; W2:?; W3:?; W4:?; W5:?; + latt_with1: latt_jcarr_ = subst_excess_base_in_semi_lattice latt_jcarr_ + (excess_base_OF_semi_lattice latt_mcarr) W1 W2 W3 W4 W5 +}. + +lemma latt_jcarr : lattice_ → semi_lattice. +intro l; apply (subst_excess_base_in_semi_lattice (latt_jcarr_ l) (excess_base_OF_semi_lattice (latt_mcarr l)) (W1 l) (W2 l) (W3 l) (W4 l) (W5 l)); +qed. + +coercion cic:/matita/lattice/latt_jcarr.con. + +interpretation "Lattice meet" 'and a b = + (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _) a b). + +interpretation "Lattice join" 'or a b = + (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _) a b). + +record lattice : Type ≝ { + latt_carr:> lattice_; + absorbjm: ∀f,g:latt_carr. (f ∨ (f ∧ g)) ≈ f; + absorbmj: ∀f,g:latt_carr. (f ∧ (f ∨ g)) ≈ f +}. + +notation "'meet'" non associative with precedence 50 for @{'meet}. +notation "'meet_refl'" non associative with precedence 50 for @{'meet_refl}. +notation "'meet_comm'" non associative with precedence 50 for @{'meet_comm}. +notation "'meet_assoc'" non associative with precedence 50 for @{'meet_assoc}. +notation "'strong_extm'" non associative with precedence 50 for @{'strong_extm}. +notation "'le_to_eqm'" non associative with precedence 50 for @{'le_to_eqm}. +notation "'lem'" non associative with precedence 50 for @{'lem}. +notation "'join'" non associative with precedence 50 for @{'join}. +notation "'join_refl'" non associative with precedence 50 for @{'join_refl}. +notation "'join_comm'" non associative with precedence 50 for @{'join_comm}. +notation "'join_assoc'" non associative with precedence 50 for @{'join_assoc}. +notation "'strong_extj'" non associative with precedence 50 for @{'strong_extj}. +notation "'le_to_eqj'" non associative with precedence 50 for @{'le_to_eqj}. +notation "'lej'" non associative with precedence 50 for @{'lej}. + +interpretation "Lattice meet" 'meet = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice meet_refl" 'meet_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice meet_comm" 'meet_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice meet_assoc" 'meet_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice strong_extm" 'strong_extm = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice le_to_eqm" 'le_to_eqm = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice lem" 'lem = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice join" 'join = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice join_refl" 'join_refl = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice join_comm" 'join_comm = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice join_assoc" 'join_assoc = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice strong_extm" 'strong_extj = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice le_to_eqj" 'le_to_eqj = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice lej" 'lej = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_jcarr.con _)). + +notation "'feq_jl'" non associative with precedence 50 for @{'feq_jl}. +notation "'feq_jr'" non associative with precedence 50 for @{'feq_jr}. +notation "'feq_ml'" non associative with precedence 50 for @{'feq_ml}. +notation "'feq_mr'" non associative with precedence 50 for @{'feq_mr}. +interpretation "Lattice feq_jl" 'feq_jl = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice feq_jr" 'feq_jr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_jcarr.con _)). +interpretation "Lattice feq_ml" 'feq_ml = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_mcarr.con _)). +interpretation "Lattice feq_mr" 'feq_mr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_mcarr.con _)). + + +interpretation "Lattive meet le" 'leq a b = + (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice1.con _) a b). + +interpretation "Lattive join le (aka ge)" 'geq a b = + (cic:/matita/excess/le.con (cic:/matita/lattice/excess_OF_lattice.con _) a b). + +(* these coercions help unification, handmaking a bit of conversion + over an open term +*) +lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a. +intros(l a b H); apply H; +qed. + +lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b. +intros(l a b H); apply H; +qed. + +coercion cic:/matita/lattice/le_to_ge.con nocomposites. +coercion cic:/matita/lattice/ge_to_le.con nocomposites. \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama_duality/limit.ma b/helm/software/matita/contribs/dama/dama_duality/limit.ma new file mode 100644 index 000000000..1250511e8 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/limit.ma @@ -0,0 +1,67 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "infsup.ma". + +definition shift ≝ λT:Type.λs:sequence T.λk:nat.λn.s (n+k). + +(* 3.28 *) +definition limsup ≝ + λE:excess.λxn:sequence E.λx:E.∃alpha:sequence E. + (∀k.(alpha k) is_strong_sup (shift ? xn k) in E) ∧ + x is_strong_inf alpha in E. + +notation < "x \nbsp 'is_limsup' \nbsp s" non associative with precedence 50 for @{'limsup $_ $s $x}. +notation < "x \nbsp 'is_liminf' \nbsp s" non associative with precedence 50 for @{'liminf $_ $s $x}. +notation > "x 'is_limsup' s 'in' e" non associative with precedence 50 for @{'limsup $e $s $x}. +notation > "x 'is_liminf' s 'in' e" non associative with precedence 50 for @{'liminf $e $s $x}. + +interpretation "Excess limsup" 'limsup e s x = (cic:/matita/limit/limsup.con e s x). +interpretation "Excess liminf" 'liminf e s x = (cic:/matita/limit/limsup.con (cic:/matita/excess/dual_exc.con e) s x). + +(* 3.29 *) +definition lim ≝ + λE:excess.λxn:sequence E.λx:E. x is_limsup xn in E ∧ x is_liminf xn in E. + +notation < "x \nbsp 'is_lim' \nbsp s" non associative with precedence 50 for @{'lim $_ $s $x}. +notation > "x 'is_lim' s 'in' e" non associative with precedence 50 for @{'lim $e $s $x}. +interpretation "Excess lim" 'lim e s x = (cic:/matita/limit/lim.con e s x). + +lemma sup_decreasing: + ∀E:excess.∀xn:sequence E. + ∀alpha:sequence E. (∀k.(alpha k) is_strong_sup xn in E) → + alpha is_decreasing in E. +intros (E xn alpha H); unfold strong_sup in H; unfold upper_bound in H; unfold; +intro r; +elim (H r) (H1r H2r); +elim (H (S r)) (H1sr H2sr); clear H H2r H1sr; +intro e; cases (H2sr (alpha r) e) (w Hw); clear e H2sr; +cases (H1r w Hw); +qed. + +include "tend.ma". +include "metric_lattice.ma". + +(* 3.30 *) +lemma lim_tends: + ∀R.∀ml:mlattice R.∀xn:sequence ml.∀x:ml. + x is_lim xn in ml → xn ⇝ x. +intros (R ml xn x Hl); unfold lim in Hl; unfold limsup in Hl; +cases Hl (e1 e2); cases e1 (an Han); cases e2 (bn Hbn); clear Hl e1 e2; +cases Han (SSan SIxan); cases Hbn (SSbn SIxbn); clear Han Hbn; +cases SIxan (LBxan Hxg); cases SIxbn (UPxbn Hxl); clear SIxbn SIxan; +change in UPxbn:(%) with (x is_lower_bound bn in ml); +unfold upper_bound in UPxbn LBxan; change +intros (e He); +(* 2.6 OC *) diff --git a/helm/software/matita/contribs/dama/dama_duality/metric_lattice.ma b/helm/software/matita/contribs/dama/dama_duality/metric_lattice.ma new file mode 100644 index 000000000..010360757 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/metric_lattice.ma @@ -0,0 +1,117 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "metric_space.ma". +include "lattice.ma". + +record mlattice_ (R : todgroup) : Type ≝ { + ml_mspace_: metric_space R; + ml_lattice:> lattice; + ml_with: ms_carr ? ml_mspace_ = Type_OF_lattice ml_lattice +}. + +lemma ml_mspace: ∀R.mlattice_ R → metric_space R. +intros (R ml); apply (mk_metric_space R (Type_OF_mlattice_ ? ml)); +unfold Type_OF_mlattice_; cases (ml_with ? ml); simplify; +[apply (metric ? (ml_mspace_ ? ml));|apply (mpositive ? (ml_mspace_ ? ml)); +|apply (mreflexive ? (ml_mspace_ ? ml));|apply (msymmetric ? (ml_mspace_ ? ml)); +|apply (mtineq ? (ml_mspace_ ? ml))] +qed. + +coercion cic:/matita/metric_lattice/ml_mspace.con. + +alias symbol "plus" = "Abelian group plus". +alias symbol "leq" = "Excess less or equal than". +record mlattice (R : todgroup) : Type ≝ { + ml_carr :> mlattice_ R; + ml_prop1: ∀a,b:ml_carr. 0 < δ a b → a # b; + ml_prop2: ∀a,b,c:ml_carr. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ (δ b c) +}. + +interpretation "Metric lattice leq" 'leq a b = + (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice1.con _ _) a b). +interpretation "Metric lattice geq" 'geq a b = + (cic:/matita/excess/le.con (cic:/matita/metric_lattice/excess_OF_mlattice.con _ _) a b). + +lemma eq_to_ndlt0: ∀R.∀ml:mlattice R.∀a,b:ml. a ≈ b → ¬ 0 < δ a b. +intros (R ml a b E); intro H; apply E; apply ml_prop1; +assumption; +qed. + +lemma eq_to_dzero: ∀R.∀ml:mlattice R.∀x,y:ml.x ≈ y → δ x y ≈ 0. +intros (R ml x y H); intro H1; apply H; clear H; +apply ml_prop1; split [apply mpositive] apply ap_symmetric; +assumption; +qed. + +lemma meq_l: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δx y ≈ δz y. +intros (R ml x y z); apply le_le_eq; +[ apply (le_transitive ???? (mtineq ???y z)); + apply (le_rewl ??? (0+δz y) (eq_to_dzero ???? H)); + apply (le_rewl ??? (δz y) (zero_neutral ??)); apply le_reflexive; +| apply (le_transitive ???? (mtineq ???y x)); + apply (le_rewl ??? (0+δx y) (eq_to_dzero ??z x H)); + apply (le_rewl ??? (δx y) (zero_neutral ??)); apply le_reflexive;] +qed. + +(* 3.3 *) +lemma meq_r: ∀R.∀ml:mlattice R.∀x,y,z:ml. x≈z → δy x ≈ δy z. +intros; apply (eq_trans ???? (msymmetric ??y x)); +apply (eq_trans ????? (msymmetric ??z y)); apply meq_l; assumption; +qed. + +lemma dap_to_lt: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → 0 < δ x y. +intros; split [apply mpositive] apply ap_symmetric; assumption; +qed. + +lemma dap_to_ap: ∀R.∀ml:mlattice R.∀x,y:ml. δ x y # 0 → x # y. +intros (R ml x y H); apply ml_prop1; split; [apply mpositive;] +apply ap_symmetric; assumption; +qed. + +(* 3.11 *) +lemma le_mtri: + ∀R.∀ml:mlattice R.∀x,y,z:ml. x ≤ y → y ≤ z → δ x z ≈ δ x y + δ y z. +intros (R ml x y z Lxy Lyz); apply le_le_eq; [apply mtineq] +apply (le_transitive ????? (ml_prop2 ?? (y) ??)); +cut ( δx y+ δy z ≈ δ(y∨x) (y∨z)+ δ(y∧x) (y∧z)); [ + apply (le_rewr ??? (δx y+ δy z)); [assumption] apply le_reflexive] +lapply (le_to_eqm y x Lxy) as Dxm; lapply (le_to_eqm z y Lyz) as Dym; +lapply (le_to_eqj x y Lxy) as Dxj; lapply (le_to_eqj y z Lyz) as Dyj; clear Lxy Lyz; + STOP +apply (Eq≈ (δ(x∧y) y + δy z) (meq_l ????? Dxm)); +apply (Eq≈ (δ(x∧y) (y∧z) + δy z) (meq_r ????? Dym)); +apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) z));[ + apply feq_plusl; apply meq_l; clear Dyj Dxm Dym; assumption] +apply (Eq≈ (δ(x∧y) (y∧z) + δ(y∨x) (z∨y))); [ + apply (feq_plusl ? (δ(x∧y) (y∧z)) ?? (meq_r ??? (y∨x) ? Dyj));] +apply (Eq≈ ? (plus_comm ???)); +apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)));[ + apply feq_plusr; apply meq_r; apply (join_comm ??);] +apply feq_plusl; +apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm ??))); +apply eq_reflexive; +qed. + + +(* 3.17 conclusione: δ x y ≈ 0 *) +(* 3.20 conclusione: δ x y ≈ 0 *) +(* 3.21 sup forte + strong_sup x ≝ ∀n. s n ≤ x ∧ ∀y x ≰ y → ∃n. s n ≰ y + strong_sup_zoli x ≝ ∀n. s n ≤ x ∧ ∄y. y#x ∧ y ≤ x +*) +(* 3.22 sup debole (più piccolo dei maggioranti) *) +(* 3.23 conclusion: δ x sup(...) ≈ 0 *) +(* 3.25 vero nel reticolo e basta (niente δ) *) +(* 3.36 conclusion: δ x y ≈ 0 *) diff --git a/helm/software/matita/contribs/dama/dama_duality/metric_space.ma b/helm/software/matita/contribs/dama/dama_duality/metric_space.ma new file mode 100644 index 000000000..2266fe9e9 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/metric_space.ma @@ -0,0 +1,46 @@ +(**************************************************************************) +(* ___ *) +(* ||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_divisible_group.ma". + +record metric_space (R: todgroup) : Type ≝ { + ms_carr :> Type; + metric: ms_carr → ms_carr → R; + mpositive: ∀a,b:ms_carr. 0 ≤ metric a b; + mreflexive: ∀a. metric a a ≈ 0; + msymmetric: ∀a,b. metric a b ≈ metric b a; + mtineq: ∀a,b,c:ms_carr. metric a b ≤ metric a c + metric c b +}. + +notation < "\nbsp \delta a \nbsp b" non associative with precedence 80 for @{ 'delta2 $a $b}. +interpretation "metric" 'delta2 a b = (cic:/matita/metric_space/metric.con _ _ a b). +notation "\delta" non associative with precedence 80 for @{ 'delta }. +interpretation "metric" 'delta = (cic:/matita/metric_space/metric.con _ _). + +lemma apart_of_metric_space: ∀R.metric_space R → apartness. +intros (R ms); apply (mk_apartness ? (λa,b:ms.0 < δ a b)); unfold; +[1: intros 2 (x H); cases H (H1 H2); clear H; + lapply (Ap≫ ? (eq_sym ??? (mreflexive ??x)) H2); + apply (ap_coreflexive R 0); assumption; +|2: intros (x y H); cases H; split; + [1: apply (Le≫ ? (msymmetric ????)); assumption + |2: apply (Ap≫ ? (msymmetric ????)); assumption] +|3: simplify; intros (x y z H); elim H (LExy Axy); + lapply (mtineq ?? x y z) as H1; elim (ap2exc ??? Axy) (H2 H2); [cases (LExy H2)] + clear LExy; lapply (lt_le_transitive ???? H H1) as LT0; + apply (lt0plus_orlt ????? LT0); apply mpositive;] +qed. + +lemma ap2delta: ∀R.∀m:metric_space R.∀x,y:m.ap_apart (apart_of_metric_space ? m) x y → 0 < δ x y. +intros 2 (R m); cases m 0; simplify; intros; assumption; qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/ordered_divisible_group.ma b/helm/software/matita/contribs/dama/dama_duality/ordered_divisible_group.ma new file mode 100644 index 000000000..a9dfc4a58 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/ordered_divisible_group.ma @@ -0,0 +1,75 @@ +(**************************************************************************) +(* ___ *) +(* ||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/orders.ma". +include "nat/times.ma". +include "ordered_group.ma". +include "divisible_group.ma". + +record todgroup : Type ≝ { + todg_order:> togroup; + todg_division_: dgroup; + todg_with_: dg_carr todg_division_ = og_abelian_group todg_order +}. + +lemma todg_division: todgroup → dgroup. +intro G; apply (mk_dgroup G); unfold abelian_group_OF_todgroup; +cases (todg_with_ G); exact (dg_prop (todg_division_ G)); +qed. + +coercion cic:/matita/ordered_divisible_group/todg_division.con. + +lemma mul_ge: ∀G:todgroup.∀x:G.∀n.0 ≤ x → 0 ≤ n * x. +intros (G x n); elim n; simplify; [apply le_reflexive] +apply (le_transitive ???? H1); +apply (Le≪ (0+(n1*x)) (zero_neutral ??)); +apply fle_plusr; assumption; +qed. + +lemma lt_ltmul: ∀G:todgroup.∀x,y:G.∀n. x < y → S n * x < S n * y. +intros; elim n; [simplify; apply flt_plusr; assumption] +simplify; apply (ltplus); [assumption] assumption; +qed. + +lemma ltmul_lt: ∀G:todgroup.∀x,y:G.∀n. S n * x < S n * y → x < y. +intros 4; elim n; [apply (plus_cancr_lt ??? 0); assumption] +simplify in l; cases (ltplus_orlt ????? l); [assumption] +apply f; assumption; +qed. + +lemma divide_preserves_lt: ∀G:todgroup.∀e:G.∀n.0 sym_plus; simplify; apply (Lt≪ (0+(y+n*y))); [ + apply eq_sym; apply zero_neutral] + apply flt_plusr; assumption;] +apply (lt_transitive ???? l); rewrite > sym_plus; simplify; +rewrite > (sym_plus n); simplify; repeat apply flt_plusl; +apply (Lt≪ (0+(n1+n)*y)); [apply eq_sym; apply zero_neutral] +apply flt_plusr; assumption; +qed. + diff --git a/helm/software/matita/contribs/dama/dama_duality/ordered_group.ma b/helm/software/matita/contribs/dama/dama_duality/ordered_group.ma new file mode 100644 index 000000000..7d25273aa --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/ordered_group.ma @@ -0,0 +1,328 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "group.ma". + +record pogroup_ : Type ≝ { + og_abelian_group_: abelian_group; + og_excess:> excess; + og_with: carr og_abelian_group_ = exc_ap og_excess +}. + +lemma og_abelian_group: pogroup_ → abelian_group. +intro G; apply (mk_abelian_group G); unfold apartness_OF_pogroup_; +cases (og_with G); simplify; +[apply (plus (og_abelian_group_ G));|apply zero;|apply opp +|apply plus_assoc|apply plus_comm|apply zero_neutral|apply opp_inverse|apply plus_strong_ext] +qed. + +coercion cic:/matita/ordered_group/og_abelian_group.con. + +record pogroup : Type ≝ { + og_carr:> pogroup_; + plus_cancr_exc: ∀f,g,h:og_carr. f+h ≰ g+h → f ≰ g +}. + +lemma fexc_plusr: + ∀G:pogroup.∀x,y,z:G. x ≰ y → x+z ≰ y + z. +intros 5 (G x y z L); apply (plus_cancr_exc ??? (-z)); +apply (Ex≪ (x + (z + -z)) (plus_assoc ????)); +apply (Ex≪ (x + (-z + z)) (plus_comm ??z)); +apply (Ex≪ (x+0) (opp_inverse ??)); +apply (Ex≪ (0+x) (plus_comm ???)); +apply (Ex≪ x (zero_neutral ??)); +apply (Ex≫ (y + (z + -z)) (plus_assoc ????)); +apply (Ex≫ (y + (-z + z)) (plus_comm ??z)); +apply (Ex≫ (y+0) (opp_inverse ??)); +apply (Ex≫ (0+y) (plus_comm ???)); +apply (Ex≫ y (zero_neutral ??) L); +qed. + +coercion cic:/matita/ordered_group/fexc_plusr.con nocomposites. + +lemma plus_cancl_exc: ∀G:pogroup.∀f,g,h:G. h+f ≰ h+g → f ≰ g. +intros 5 (G x y z L); apply (plus_cancr_exc ??? z); +apply (Ex≪ (z+x) (plus_comm ???)); +apply (Ex≫ (z+y) (plus_comm ???) L); +qed. + +lemma fexc_plusl: + ∀G:pogroup.∀x,y,z:G. x ≰ y → z+x ≰ z+y. +intros 5 (G x y z L); apply (plus_cancl_exc ??? (-z)); +apply (Ex≪? (plus_assoc ??z x)); +apply (Ex≫? (plus_assoc ??z y)); +apply (Ex≪ (0+x) (opp_inverse ??)); +apply (Ex≫ (0+y) (opp_inverse ??)); +apply (Ex≪? (zero_neutral ??)); +apply (Ex≫? (zero_neutral ??) L); +qed. + +coercion cic:/matita/ordered_group/fexc_plusl.con nocomposites. + +lemma plus_cancr_le: + ∀G:pogroup.∀x,y,z:G.x+z ≤ y + z → x ≤ y. +intros 5 (G x y z L); +apply (Le≪ (0+x) (zero_neutral ??)); +apply (Le≪ (x+0) (plus_comm ???)); +apply (Le≪ (x+(-z+z)) (opp_inverse ??)); +apply (Le≪ (x+(z+ -z)) (plus_comm ??z)); +apply (Le≪ (x+z+ -z) (plus_assoc ????)); +apply (Le≫ (0+y) (zero_neutral ??)); +apply (Le≫ (y+0) (plus_comm ???)); +apply (Le≫ (y+(-z+z)) (opp_inverse ??)); +apply (Le≫ (y+(z+ -z)) (plus_comm ??z)); +apply (Le≫ (y+z+ -z) (plus_assoc ????)); +intro H; apply L; clear L; apply (plus_cancr_exc ??? (-z) H); +qed. + +lemma fle_plusl: ∀G:pogroup. ∀f,g,h:G. f≤g → h+f≤h+g. +intros (G f g h); +apply (plus_cancr_le ??? (-h)); +apply (Le≪ (f+h+ -h) (plus_comm ? f h)); +apply (Le≪ (f+(h+ -h)) (plus_assoc ????)); +apply (Le≪ (f+(-h+h)) (plus_comm ? h (-h))); +apply (Le≪ (f+0) (opp_inverse ??)); +apply (Le≪ (0+f) (plus_comm ???)); +apply (Le≪ (f) (zero_neutral ??)); +apply (Le≫ (g+h+ -h) (plus_comm ? h ?)); +apply (Le≫ (g+(h+ -h)) (plus_assoc ????)); +apply (Le≫ (g+(-h+h)) (plus_comm ??h)); +apply (Le≫ (g+0) (opp_inverse ??)); +apply (Le≫ (0+g) (plus_comm ???)); +apply (Le≫ (g) (zero_neutral ??) H); +qed. + +lemma fle_plusr: ∀G:pogroup. ∀f,g,h:G. f≤g → f+h≤g+h. +intros (G f g h H); apply (Le≪? (plus_comm ???)); +apply (Le≫? (plus_comm ???)); apply fle_plusl; assumption; +qed. + +lemma plus_cancl_le: + ∀G:pogroup.∀x,y,z:G.z+x ≤ z+y → x ≤ y. +intros 5 (G x y z L); +apply (Le≪ (0+x) (zero_neutral ??)); +apply (Le≪ ((-z+z)+x) (opp_inverse ??)); +apply (Le≪ (-z+(z+x)) (plus_assoc ????)); +apply (Le≫ (0+y) (zero_neutral ??)); +apply (Le≫ ((-z+z)+y) (opp_inverse ??)); +apply (Le≫ (-z+(z+y)) (plus_assoc ????)); +apply (fle_plusl ??? (-z) L); +qed. + +lemma plus_cancl_lt: + ∀G:pogroup.∀x,y,z:G.z+x < z+y → x < y. +intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancl_le; assumption] +apply (plus_cancl_ap ???? LE); +qed. + +lemma plus_cancr_lt: + ∀G:pogroup.∀x,y,z:G.x+z < y+z → x < y. +intros 5 (G x y z L); elim L (A LE); split; [apply plus_cancr_le; assumption] +apply (plus_cancr_ap ???? LE); +qed. + + +lemma exc_opp_x_zero_to_exc_zero_x: + ∀G:pogroup.∀x:G.-x ≰ 0 → 0 ≰ x. +intros (G x H); apply (plus_cancr_exc ??? (-x)); +apply (Ex≫? (plus_comm ???)); +apply (Ex≫? (opp_inverse ??)); +apply (Ex≪? (zero_neutral ??) H); +qed. + +lemma le_zero_x_to_le_opp_x_zero: + ∀G:pogroup.∀x:G.0 ≤ x → -x ≤ 0. +intros (G x Px); apply (plus_cancr_le ??? x); +apply (Le≪ 0 (opp_inverse ??)); +apply (Le≫ x (zero_neutral ??) Px); +qed. + +lemma lt_zero_x_to_lt_opp_x_zero: + ∀G:pogroup.∀x:G.0 < x → -x < 0. +intros (G x Px); apply (plus_cancr_lt ??? x); +apply (Lt≪ 0 (opp_inverse ??)); +apply (Lt≫ x (zero_neutral ??) Px); +qed. + +lemma exc_zero_opp_x_to_exc_x_zero: + ∀G:pogroup.∀x:G. 0 ≰ -x → x ≰ 0. +intros (G x H); apply (plus_cancl_exc ??? (-x)); +apply (Ex≫? (plus_comm ???)); +apply (Ex≪? (opp_inverse ??)); +apply (Ex≫? (zero_neutral ??) H); +qed. + +lemma le_x_zero_to_le_zero_opp_x: + ∀G:pogroup.∀x:G. x ≤ 0 → 0 ≤ -x. +intros (G x Lx0); apply (plus_cancr_le ??? x); +apply (Le≫ 0 (opp_inverse ??)); +apply (Le≪ x (zero_neutral ??)); +assumption; +qed. + +lemma lt_x_zero_to_lt_zero_opp_x: + ∀G:pogroup.∀x:G. x < 0 → 0 < -x. +intros (G x Lx0); apply (plus_cancr_lt ??? x); +apply (Lt≫ 0 (opp_inverse ??)); +apply (Lt≪ x (zero_neutral ??)); +assumption; +qed. + +lemma lt_opp_x_zero_to_lt_zero_x: + ∀G:pogroup.∀x:G. -x < 0 → 0 < x. +intros (G x Lx0); apply (plus_cancr_lt ??? (-x)); +apply (Lt≪ (-x) (zero_neutral ??)); +apply (Lt≫ (-x+x) (plus_comm ???)); +apply (Lt≫ 0 (opp_inverse ??)); +assumption; +qed. + +lemma lt0plus_orlt: + ∀G:pogroup. ∀x,y:G. 0 ≤ x → 0 ≤ y → 0 < x + y → 0 < x ∨ 0 < y. +intros (G x y LEx LEy LT); cases LT (H1 H2); cases (ap_cotransitive ??? y H2); +[right; split; assumption|left;split;[assumption]] +apply (plus_cancr_ap ??? y); apply (Ap≪? (zero_neutral ??)); +assumption; +qed. + +lemma le0plus_le: + ∀G:pogroup.∀a,b,c:G. 0 ≤ b → a + b ≤ c → a ≤ c. +intros (G a b c L H); apply (le_transitive ????? H); +apply (plus_cancl_le ??? (-a)); +apply (Le≪ 0 (opp_inverse ??)); +apply (Le≫ (-a + a + b) (plus_assoc ????)); +apply (Le≫ (0 + b) (opp_inverse ??)); +apply (Le≫ b (zero_neutral ??)); +assumption; +qed. + +lemma le_le0plus: + ∀G:pogroup.∀a,b:G. 0 ≤ a → 0 ≤ b → 0 ≤ a + b. +intros (G a b L1 L2); apply (le_transitive ???? L1); +apply (plus_cancl_le ??? (-a)); +apply (Le≪ 0 (opp_inverse ??)); +apply (Le≫ (-a + a + b) (plus_assoc ????)); +apply (Le≫ (0 + b) (opp_inverse ??)); +apply (Le≫ b (zero_neutral ??)); +assumption; +qed. + +lemma flt_plusl: + ∀G:pogroup.∀x,y,z:G.x < y → z + x < z + y. +intros (G x y z H); cases H; split; [apply fle_plusl; assumption] +apply fap_plusl; assumption; +qed. + +lemma flt_plusr: + ∀G:pogroup.∀x,y,z:G.x < y → x + z < y + z. +intros (G x y z H); cases H; split; [apply fle_plusr; assumption] +apply fap_plusr; assumption; +qed. + + +lemma ltxy_ltyyxx: ∀G:pogroup.∀x,y:G. y < x → y+y < x+x. +intros; apply (lt_transitive ?? (y+x));[2: + apply (Lt≪? (plus_comm ???)); + apply (Lt≫? (plus_comm ???));] +apply flt_plusl;assumption; +qed. + +lemma lew_opp : ∀O:pogroup.∀a,b,c:O.0 ≤ b → a ≤ c → a + -b ≤ c. +intros (O a b c L0 L); +apply (le_transitive ????? L); +apply (plus_cancl_le ??? (-a)); +apply (Le≫ 0 (opp_inverse ??)); +apply (Le≪ (-a+a+-b) (plus_assoc ????)); +apply (Le≪ (0+-b) (opp_inverse ??)); +apply (Le≪ (-b) (zero_neutral ?(-b))); +apply le_zero_x_to_le_opp_x_zero; +assumption; +qed. + +lemma ltw_opp : ∀O:pogroup.∀a,b,c:O.0 < b → a < c → a + -b < c. +intros (O a b c P L); +apply (lt_transitive ????? L); +apply (plus_cancl_lt ??? (-a)); +apply (Lt≫ 0 (opp_inverse ??)); +apply (Lt≪ (-a+a+-b) (plus_assoc ????)); +apply (Lt≪ (0+-b) (opp_inverse ??)); +apply (Lt≪ ? (zero_neutral ??)); +apply lt_zero_x_to_lt_opp_x_zero; +assumption; +qed. + +record togroup : Type ≝ { + tog_carr:> pogroup; + tog_total: ∀x,y:tog_carr.x≰y → y < x +}. + +lemma lexxyy_lexy: ∀G:togroup. ∀x,y:G. x+x ≤ y+y → x ≤ y. +intros (G x y H); intro H1; lapply (tog_total ??? H1) as H2; +lapply (ltxy_ltyyxx ??? H2) as H3; lapply (lt_to_excess ??? H3) as H4; +cases (H H4); +qed. + +lemma eqxxyy_eqxy: ∀G:togroup.∀x,y:G. x + x ≈ y + y → x ≈ y. +intros (G x y H); cases (eq_le_le ??? H); apply le_le_eq; +apply lexxyy_lexy; assumption; +qed. + +lemma applus_orap: ∀G:abelian_group. ∀x,y:G. 0 # x + y → 0 #x ∨ 0#y. +intros; cases (ap_cotransitive ??? y a); [right; assumption] +left; apply (plus_cancr_ap ??? y); apply (Ap≪y (zero_neutral ??)); +assumption; +qed. + +lemma ltplus: ∀G:pogroup.∀a,b,c,d:G. a < b → c < d → a+c < b + d. +intros (G a b c d H1 H2); +lapply (flt_plusr ??? c H1) as H3; +apply (lt_transitive ???? H3); +apply flt_plusl; assumption; +qed. + +lemma excplus_orexc: ∀G:pogroup.∀a,b,c,d:G. a+c ≰ b + d → a ≰ b ∨ c ≰ d. +intros (G a b c d H1 H2); +cases (exc_cotransitive ??? (a + d) H1); [ + right; apply (plus_cancl_exc ??? a); assumption] +left; apply (plus_cancr_exc ??? d); assumption; +qed. + +lemma leplus: ∀G:pogroup.∀a,b,c,d:G. a ≤ b → c ≤ d → a+c ≤ b + d. +intros (G a b c d H1 H2); intro H3; cases (excplus_orexc ????? H3); +[apply H1|apply H2] assumption; +qed. + +lemma leplus_lt_le: ∀G:togroup.∀x,y:G. 0 ≤ x + y → x < 0 → 0 ≤ y. +intros; intro; apply H; lapply (lt_to_excess ??? l); +lapply (tog_total ??? e); +lapply (tog_total ??? Hletin); +lapply (ltplus ????? Hletin2 Hletin1); +apply (Ex≪ (0+0)); [apply eq_sym; apply zero_neutral] +apply lt_to_excess; assumption; +qed. + +lemma ltplus_orlt: ∀G:togroup.∀a,b,c,d:G. a+c < b + d → a < b ∨ c < d. +intros (G a b c d H1 H2); lapply (lt_to_excess ??? H1); +cases (excplus_orexc ????? Hletin); [left|right] apply tog_total; assumption; +qed. + +lemma excplus: ∀G:togroup.∀a,b,c,d:G.a ≰ b → c ≰ d → a + c ≰ b + d. +intros (G a b c d L1 L2); +lapply (fexc_plusr ??? (c) L1) as L3; +elim (exc_cotransitive ??? (b+d) L3); [assumption] +lapply (plus_cancl_exc ???? b1); lapply (tog_total ??? Hletin); +cases Hletin1; cases (H L2); +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/premetric_lattice.ma b/helm/software/matita/contribs/dama/dama_duality/premetric_lattice.ma new file mode 100644 index 000000000..6c904055c --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/premetric_lattice.ma @@ -0,0 +1,70 @@ +(**************************************************************************) +(* ___ *) +(* ||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 "metric_space.ma". + +record premetric_lattice_ (R : todgroup) : Type ≝ { + pml_carr:> metric_space R; + meet: pml_carr → pml_carr → pml_carr; + join: pml_carr → pml_carr → pml_carr +}. + +interpretation "valued lattice meet" 'and a b = + (cic:/matita/premetric_lattice/meet.con _ _ a b). + +interpretation "valued lattice join" 'or a b = + (cic:/matita/premetric_lattice/join.con _ _ a b). + +record premetric_lattice_props (R : todgroup) (ml : premetric_lattice_ R) : Prop ≝ { + prop1a: ∀a : ml.δ (a ∧ a) a ≈ 0; + prop1b: ∀a : ml.δ (a ∨ a) a ≈ 0; + prop2a: ∀a,b: ml. δ (a ∨ b) (b ∨ a) ≈ 0; + prop2b: ∀a,b: ml. δ (a ∧ b) (b ∧ a) ≈ 0; + prop3a: ∀a,b,c: ml. δ (a ∨ (b ∨ c)) ((a ∨ b) ∨ c) ≈ 0; + prop3b: ∀a,b,c: ml. δ (a ∧ (b ∧ c)) ((a ∧ b) ∧ c) ≈ 0; + prop4a: ∀a,b: ml. δ (a ∨ (a ∧ b)) a ≈ 0; + prop4b: ∀a,b: ml. δ (a ∧ (a ∨ b)) a ≈ 0; + prop5: ∀a,b,c: ml. δ (a ∨ b) (a ∨ c) + δ (a ∧ b) (a ∧ c) ≤ δ b c +}. + +record pmlattice (R : todgroup) : Type ≝ { + carr :> premetric_lattice_ R; + ispremetriclattice:> premetric_lattice_props R carr +}. + +include "lattice.ma". + +lemma lattice_of_pmlattice: ∀R: todgroup. pmlattice R → lattice. +intros (R pml); not ported to duality +apply (mk_lattice (apart_of_metric_space ? pml)); +[apply (join ? pml)|apply (meet ? pml) +|3,4,5,6,7,8,9,10: intros (x y z); whd; intro H; whd in H; cases H (LE AP);] +[apply (prop1b ? pml pml x); |apply (prop1a ? pml pml x); +|apply (prop2a ? pml pml x y); |apply (prop2b ? pml pml x y); +|apply (prop3a ? pml pml x y z);|apply (prop3b ? pml pml x y z); +|apply (prop4a ? pml pml x y); |apply (prop4b ? pml pml x y);] +try (apply ap_symmetric; assumption); intros 4 (x y z H); change with (0 < (δ y z)); +[ change in H with (0 < δ (x ∨ y) (x ∨ z)); + apply (lt_le_transitive ???? H); + apply (le0plus_le ???? (mpositive ? pml ??) (prop5 ? pml pml x y z)); +| change in H with (0 < δ (x ∧ y) (x ∧ z)); + apply (lt_le_transitive ???? H); + apply (le0plus_le ???? (mpositive ? pml (x∨y) (x∨z))); + apply (le_rewl ??? ? (plus_comm ???)); + apply (prop5 ? pml pml);] +qed. + +coercion cic:/matita/premetric_lattice/lattice_of_pmlattice.con. \ No newline at end of file diff --git a/helm/software/matita/contribs/dama/dama_duality/prevalued_lattice.ma b/helm/software/matita/contribs/dama/dama_duality/prevalued_lattice.ma new file mode 100644 index 000000000..53b2b0a1b --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/prevalued_lattice.ma @@ -0,0 +1,243 @@ +(**************************************************************************) +(* ___ *) +(* ||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_group.ma". + +record vlattice (R : togroup) : Type ≝ { + wl_carr:> Type; + value: wl_carr → R; + join: wl_carr → wl_carr → wl_carr; + meet: wl_carr → wl_carr → wl_carr; + meet_refl: ∀x. value (meet x x) ≈ value x; + join_refl: ∀x. value (join x x) ≈ value x; + meet_comm: ∀x,y. value (meet x y) ≈ value (meet y x); + join_comm: ∀x,y. value (join x y) ≈ value (join y x); + join_assoc: ∀x,y,z. value (join x (join y z)) ≈ value (join (join x y) z); + meet_assoc: ∀x,y,z. value (meet x (meet y z)) ≈ value (meet (meet x y) z); + meet_wins1: ∀x,y. value (join x (meet x y)) ≈ value x; + meet_wins2: ∀x,y. value (meet x (join x y)) ≈ value x; + modular_mjp: ∀x,y. value (join x y) + value (meet x y) ≈ value x + value y; + join_meet_le: ∀x,y,z. value (join x (meet y z)) ≤ value (join x y); + meet_join_le: ∀x,y,z. value (meet x y) ≤ value (meet x (join y z)) +}. + +interpretation "valued lattice meet" 'and a b = + (cic:/matita/prevalued_lattice/meet.con _ _ a b). + +interpretation "valued lattice join" 'or a b = + (cic:/matita/prevalued_lattice/join.con _ _ a b). + +notation < "\nbsp \mu a" non associative with precedence 80 for @{ 'value2 $a}. +interpretation "lattice value" 'value2 a = (cic:/matita/prevalued_lattice/value.con _ _ a). + +notation "\mu" non associative with precedence 80 for @{ 'value }. +interpretation "lattice value" 'value = (cic:/matita/prevalued_lattice/value.con _ _). + +lemma feq_joinr: ∀R.∀L:vlattice R.∀x,y,z:L. + μ x ≈ μ y → μ (z ∧ x) ≈ μ (z ∧ y) → μ (z ∨ x) ≈ μ (z ∨ y). +intros (R L x y z H H1); +apply (plus_cancr ??? (μ(z∧x))); +apply (Eq≈ (μz + μx) (modular_mjp ????)); +apply (Eq≈ (μz + μy) H); clear H; +apply (Eq≈ (μ(z∨y) + μ(z∧y)) (modular_mjp ??z y)); +apply (plus_cancl ??? (- μ (z ∨ y))); +apply (Eq≈ ? (plus_assoc ????)); +apply (Eq≈ (0+ μ(z∧y)) (opp_inverse ??)); +apply (Eq≈ ? (zero_neutral ??)); +apply (Eq≈ (- μ(z∨y)+ μ(z∨y)+ μ(z∧x)) ? (plus_assoc ????)); +apply (Eq≈ (0+ μ(z∧x)) ? (opp_inverse ??)); +apply (Eq≈ (μ (z ∧ x)) H1 (zero_neutral ??)); +qed. + +lemma modularj: ∀R.∀L:vlattice R.∀y,z:L. μ(y∨z) ≈ μy + μz + -μ (y ∧ z). +intros (R L y z); +lapply (modular_mjp ?? y z) as H1; +apply (plus_cancr ??? (μ(y ∧ z))); +apply (Eq≈ ? H1); clear H1; +apply (Eq≈ ?? (plus_assoc ????)); +apply (Eq≈ (μy+ μz + 0) ? (opp_inverse ??)); +apply (Eq≈ ?? (plus_comm ???)); +apply (Eq≈ (μy + μz) ? (eq_sym ??? (zero_neutral ??))); +apply eq_reflexive. +qed. + +lemma modularm: ∀R.∀L:vlattice R.∀y,z:L. μ(y∧z) ≈ μy + μz + -μ (y ∨ z). +(* CSC: questa è la causa per cui la hint per cercare i duplicati ci sta 1 mese *) +(* exact modularj; *) +intros (R L y z); +lapply (modular_mjp ?? y z) as H1; +apply (plus_cancl ??? (μ(y ∨ z))); +apply (Eq≈ ? H1); clear H1; +apply (Eq≈ ?? (plus_comm ???)); +apply (Eq≈ ?? (plus_assoc ????)); +apply (Eq≈ (μy+ μz + 0) ? (opp_inverse ??)); +apply (Eq≈ ?? (plus_comm ???)); +apply (Eq≈ (μy + μz) ? (eq_sym ??? (zero_neutral ??))); +apply eq_reflexive. +qed. + +lemma modularmj: ∀R.∀L:vlattice R.∀x,y,z:L.μ(x∧(y∨z))≈(μx + μ(y ∨ z) + - μ(x∨(y∨z))). +intros (R L x y z); +lapply (modular_mjp ?? x (y ∨ z)) as H1; +apply (Eq≈ (μ(x∨(y∨z))+ μ(x∧(y∨z)) +-μ(x∨(y∨z))) ? (feq_plusr ???? H1)); clear H1; +apply (Eq≈ ? ? (plus_comm ???)); +apply (Eq≈ (- μ(x∨(y∨z))+ μ(x∨(y∨z))+ μ(x∧(y∨z))) ? (plus_assoc ????)); +apply (Eq≈ (0+μ(x∧(y∨z))) ? (opp_inverse ??)); +apply (Eq≈ (μ(x∧(y∨z))) ? (zero_neutral ??)); +apply eq_reflexive. +qed. + +lemma modularjm: ∀R.∀L:vlattice R.∀x,y,z:L.μ(x∨(y∧z))≈(μx + μ(y ∧ z) + - μ(x∧(y∧z))). +intros (R L x y z); +lapply (modular_mjp ?? x (y ∧ z)) as H1; +apply (Eq≈ (μ(x∧(y∧z))+ μ(x∨(y∧z)) +-μ(x∧(y∧z)))); [2: apply feq_plusr; apply (eq_trans ???? (plus_comm ???)); apply H1] clear H1; +apply (Eq≈ ? ? (plus_comm ???)); +apply (Eq≈ (- μ(x∧(y∧z))+ μ(x∧(y∧z))+ μ(x∨y∧z)) ? (plus_assoc ????)); +apply (Eq≈ (0+ μ(x∨y∧z)) ? (opp_inverse ??)); +apply eq_sym; apply zero_neutral; +qed. + +lemma step1_3_57': ∀R.∀L:vlattice R.∀x,y,z:L. + μ(x ∨ (y ∧ z)) ≈ (μ x) + (μ y) + μ z + -μ (y ∨ z) + -μ (z ∧ (x ∧ y)). +intros (R L x y z); +apply (Eq≈ ? (modularjm ?? x y z)); +apply (Eq≈ ( μx+ (μy+ μz+- μ(y∨z)) +- μ(x∧(y∧z)))); [ + apply feq_plusr; apply feq_plusl; apply (modularm ?? y z);] +apply (Eq≈ (μx+ μy+ μz+- μ(y∨z)+- μ(x∧(y∧z)))); [2: + apply feq_plusl; apply feq_opp; + apply (Eq≈ ? (meet_assoc ?????)); + apply (Eq≈ ? (meet_comm ????)); + apply eq_reflexive;] +apply feq_plusr; apply (Eq≈ ? (plus_assoc ????)); +apply feq_plusr; apply plus_assoc; +qed. + +lemma step1_3_57: ∀R.∀L:vlattice R.∀x,y,z:L. + μ(x ∧ (y ∨ z)) ≈ (μ x) + (μ y) + μ z + -μ (y ∧ z) + -μ (z ∨ (x ∨ y)). +intros (R L x y z); +apply (Eq≈ ? (modularmj ?? x y z)); +apply (Eq≈ ( μx+ (μy+ μz+- μ(y∧z)) +- μ(x∨(y∨z)))); [ + apply feq_plusr; apply feq_plusl; apply (modularj ?? y z);] +apply (Eq≈ (μx+ μy+ μz+- μ(y∧z)+- μ(x∨(y∨z)))); [2: + apply feq_plusl; apply feq_opp; + apply (Eq≈ ? (join_assoc ?????)); + apply (Eq≈ ? (join_comm ????)); + apply eq_reflexive;] +apply feq_plusr; apply (Eq≈ ? (plus_assoc ????)); +apply feq_plusr; apply plus_assoc; +qed. + +(* LEMMA 3.57 *) + +lemma join_meet_le_join: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∨ (y ∧ z)) ≤ μ (x ∨ z). +intros (R L x y z); +apply (le_rewl ??? ? (eq_sym ??? (step1_3_57' ?????))); +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ -μ(z∧x∧y))); [ + apply feq_plusl; apply feq_opp; apply (eq_trans ?? ? ?? (eq_sym ??? (meet_assoc ?????))); apply eq_reflexive;] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ (- ( μ(z∧x)+ μy+- μ((z∧x)∨y))))); [ + apply feq_plusl; apply feq_opp; apply eq_sym; apply modularm] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+ (- μ(z∧x)+ -μy+-- μ((z∧x)∨y)))); [ + apply feq_plusl; apply (Eq≈ (- (μ(z∧x)+ μy) + -- μ((z∧x)∨y))); [ + apply feq_plusr; apply eq_sym; apply eq_opp_plus_plus_opp_opp;] + apply eq_sym; apply eq_opp_plus_plus_opp_opp;] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μ(z∧x)+- μy+ μ(y∨(z∧x))))); [ + repeat apply feq_plusl; apply eq_sym; apply (Eq≈ (μ((z∧x)∨y)) (eq_opp_opp_x_x ??)); + apply join_comm;] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μ(z∧x)+- μy)+ μ(y∨(z∧x)))); [ + apply eq_sym; apply plus_assoc;] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+(- μy + - μ(z∧x))+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; repeat apply feq_plusl; apply plus_comm;] +apply (le_rewl ??? (μx+ μy+ μz+- μ(y∨z)+- μy + - μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply eq_sym; apply plus_assoc;] +apply (le_rewl ??? (μx+ μy+ μz+- μy + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply (Eq≈ ( μx+ μy+ μz+(- μy+- μ(y∨z))) (eq_sym ??? (plus_assoc ????))); + apply feq_plusl; apply plus_comm;] +apply (le_rewl ??? (μx+ μy+ -μy+ μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply (Eq≈ (μx+ μy+( -μy+ μz)) (eq_sym ??? (plus_assoc ????))); + apply feq_plusl; apply plus_comm;] +apply (le_rewl ??? (μx+ 0 + μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply feq_plusl; apply eq_sym; apply (eq_trans ?? ? ? (plus_comm ???)); + apply opp_inverse; apply eq_reflexive;] +apply (le_rewl ??? (μx+ μz + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_comm ???)); + apply eq_sym; apply zero_neutral;] +apply (le_rewl ??? (μz+ μx + - μ(y∨z)+- μ(z∧x)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply plus_comm;] +apply (le_rewl ??? (μz+ μx +- μ(z∧x)+ - μ(y∨z)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply (eq_trans ?? ? ? (eq_sym ??? (plus_assoc ????))); apply feq_plusl; + apply plus_comm;] +apply (le_rewl ??? (μ(z∨x)+ - μ(y∨z)+ μ(y∨(z∧x)))); [ + repeat apply feq_plusr; apply modularj;] +apply (le_rewl ??? (μ(z∨x)+ (- μ(y∨z)+ μ(y∨(z∧x)))) (plus_assoc ????)); +apply (le_rewr ??? (μ(x∨z) + 0)); [apply (eq_trans ?? ? ? (plus_comm ???)); apply zero_neutral] +apply (le_rewr ??? (μ(x∨z) + (-μ(y∨z) + μ(y∨z)))); [ apply feq_plusl; apply opp_inverse] +apply (le_rewr ??? (μ(z∨x) + (-μ(y∨z) + μ(y∨z)))); [ apply feq_plusr; apply join_comm;] +repeat apply fle_plusl; apply join_meet_le; +qed. + +lemma meet_le_meet_join: ∀R.∀L:vlattice R.∀x,y,z:L.μ (x ∧ z) ≤ μ (x ∧ (y ∨ z)). +intros (R L x y z); +apply (le_rewr ??? ? (eq_sym ??? (step1_3_57 ?????))); +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ -μ(z∨x∨y))); [ + apply feq_plusl; apply feq_opp; apply (eq_trans ?? ? ?? (eq_sym ??? (join_assoc ?????))); apply eq_reflexive;] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ (- ( μ(z∨x)+ μy+- μ((z∨x)∧y))))); [ + apply feq_plusl; apply feq_opp; apply eq_sym; apply modularj] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+ (- μ(z∨x)+ -μy+-- μ((z∨x)∧y)))); [ + apply feq_plusl; apply (Eq≈ (- (μ(z∨x)+ μy) + -- μ((z∨x)∧y))); [ + apply feq_plusr; apply eq_sym; apply eq_opp_plus_plus_opp_opp;] + apply eq_sym; apply eq_opp_plus_plus_opp_opp;] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μ(z∨x)+- μy+ μ(y∧(z∨x))))); [ + repeat apply feq_plusl; apply eq_sym; apply (Eq≈ (μ((z∨x)∧y)) (eq_opp_opp_x_x ??)); + apply meet_comm;] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μ(z∨x)+- μy)+ μ(y∧(z∨x)))); [ + apply eq_sym; apply plus_assoc;] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+(- μy + - μ(z∨x))+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; repeat apply feq_plusl; apply plus_comm;] +apply (le_rewr ??? (μx+ μy+ μz+- μ(y∧z)+- μy + - μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply eq_sym; apply plus_assoc;] +apply (le_rewr ??? (μx+ μy+ μz+- μy + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply (Eq≈ ( μx+ μy+ μz+(- μy+- μ(y∧z))) (eq_sym ??? (plus_assoc ????))); + apply feq_plusl; apply plus_comm;] +apply (le_rewr ??? (μx+ μy+ -μy+ μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply (Eq≈ ?? (plus_assoc ????)); + apply (Eq≈ (μx+ μy+( -μy+ μz)) (eq_sym ??? (plus_assoc ????))); + apply feq_plusl; apply plus_comm;] +apply (le_rewr ??? (μx+ 0 + μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply feq_plusl; apply eq_sym; apply (eq_trans ?? ? ? (plus_comm ???)); + apply opp_inverse; apply eq_reflexive;] +apply (le_rewr ??? (μx+ μz + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_comm ???)); + apply eq_sym; apply zero_neutral;] +apply (le_rewr ??? (μz+ μx + - μ(y∧z)+- μ(z∨x)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply plus_comm;] +apply (le_rewr ??? (μz+ μx +- μ(z∨x)+ - μ(y∧z)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply (eq_trans ?? ? ?? (plus_assoc ????)); + apply (eq_trans ?? ? ? (eq_sym ??? (plus_assoc ????))); apply feq_plusl; + apply plus_comm;] +apply (le_rewr ??? (μ(z∧x)+ - μ(y∧z)+ μ(y∧(z∨x)))); [ + repeat apply feq_plusr; apply modularm;] +apply (le_rewr ??? (μ(z∧x)+ (- μ(y∧z)+ μ(y∧(z∨x)))) (plus_assoc ????)); +apply (le_rewl ??? (μ(x∧z) + 0)); [apply (eq_trans ?? ? ? (plus_comm ???)); apply zero_neutral] +apply (le_rewl ??? (μ(x∧z) + (-μ(y∧z) + μ(y∧z)))); [ apply feq_plusl; apply opp_inverse] +apply (le_rewl ??? (μ(z∧x) + (-μ(y∧z) + μ(y∧z)))); [ apply feq_plusr; apply meet_comm;] +repeat apply fle_plusl; apply meet_join_le; +qed. diff --git a/helm/software/matita/contribs/dama/dama_duality/root b/helm/software/matita/contribs/dama/dama_duality/root new file mode 100644 index 000000000..c57405b94 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/root @@ -0,0 +1 @@ +baseuri=cic:/matita/ diff --git a/helm/software/matita/contribs/dama/dama_duality/sandwich.ma b/helm/software/matita/contribs/dama/dama_duality/sandwich.ma new file mode 100644 index 000000000..aaea369f5 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama_duality/sandwich.ma @@ -0,0 +1,81 @@ +(**************************************************************************) +(* ___ *) +(* ||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/plus.ma". +include "nat/orders.ma". + +lemma ltwl: ∀a,b,c:nat. b + a < c → a < c. +intros 3 (x y z); elim y (H z IH H); [apply H] +apply IH; apply lt_S_to_lt; apply H; +qed. + +lemma ltwr: ∀a,b,c:nat. a + b < c → a < c. +intros 3 (x y z); rewrite > sym_plus; apply ltwl; +qed. + +include "tend.ma". +include "metric_lattice.ma". + +alias symbol "leq" = "ordered sets less or equal than". +alias symbol "and" = "constructive and". +theorem sandwich: +let ugo ≝ excess_OF_lattice1 in + ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.∀x:ml. + (∀n. (xn n ≤ an n) ∧ (bn n ≤ xn n)) → + an ⇝ x → bn ⇝ x → xn ⇝ x. +intros (R ml an bn xn x H Ha Hb); +unfold tends0 in Ha Hb ⊢ %; unfold d2s in Ha Hb ⊢ %; intros (e He); +alias num (instance 0) = "natural number". +cases (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha; +cases (Hb (e/2) (divide_preserves_lt ??? He)) (n2 H2); clear Hb; +apply (ex_introT ?? (n1+n2)); intros (n3 Lt_n1n2_n3); +lapply (ltwr ??? Lt_n1n2_n3) as Lt_n1n3; lapply (ltwl ??? Lt_n1n2_n3) as Lt_n2n3; +cases (H1 ? Lt_n1n3) (c daxe); cases (H2 ? Lt_n2n3) (c dbxe); +cases (H n3) (H7 H8); clear Lt_n1n3 Lt_n2n3 Lt_n1n2_n3 c H1 H2 H n1 n2; +(* the main inequality *) +cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x) as main_ineq; [2: + apply (le_transitive ???? (mtineq ???? (an n3))); + cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))) as H11; [2: + lapply (le_mtri ?? ??? H8 H7) as H9; clear H7 H8; + lapply (Eq≈ ? (msymmetric ????) H9) as H10; clear H9; + lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? H10) as H9; clear H10; + apply (Eq≈ ? H9); clear H9; + apply (Eq≈ (δ(xn n3) (an n3)+ δ(bn n3) (xn n3)+- δ(xn n3) (bn n3)) (plus_comm ??( δ(xn n3) (an n3)))); + apply (Eq≈ (δ(xn n3) (an n3)+ δ(bn n3) (xn n3)+- δ(bn n3) (xn n3)) (feq_opp ??? (msymmetric ????))); + apply (Eq≈ ( δ(xn n3) (an n3)+ (δ(bn n3) (xn n3)+- δ(bn n3) (xn n3))) (plus_assoc ????)); + apply (Eq≈ (δ(xn n3) (an n3) + (- δ(bn n3) (xn n3) + δ(bn n3) (xn n3))) (plus_comm ??(δ(bn n3) (xn n3)))); + apply (Eq≈ (δ(xn n3) (an n3) + 0) (opp_inverse ??)); + apply (Eq≈ ? (plus_comm ???)); + apply (Eq≈ ? (zero_neutral ??)); + apply (Eq≈ ? (msymmetric ????)); + apply eq_reflexive;] + apply (Le≪ (δ(an n3) (xn n3)+ δ(an n3) x) (msymmetric ??(an n3)(xn n3))); + apply (Le≪ (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x) H11); + apply (Le≪ (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x) (plus_comm ??(δ(an n3) (bn n3)))); + apply (Le≪ (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????)); + apply (Le≪ ((δ(an n3) (bn n3)+δ(an n3) x)+- δ(xn n3) (bn n3)) (plus_comm ???)); + apply lew_opp; [apply mpositive] apply fle_plusr; + apply (Le≫ ? (plus_comm ???)); + apply (Le≫ ( δ(an n3) x+ δx (bn n3)) (msymmetric ????)); + apply mtineq;] +split; [ (* first the trivial case: -e< δ(xn n3) x *) + apply (lt_le_transitive ????? (mpositive ????)); + apply lt_zero_x_to_lt_opp_x_zero; assumption;] +(* the main goal: δ(xn n3) x