(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/metric_lattice/".
-
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_ = ap_carr (l_carr ml_lattice)
+ ml_with_: ms_carr ? ml_mspace_ = apart_of_excess (pl_carr ml_lattice)
}.
lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice_;
+intros (R ml); apply (mk_metric_space R (apart_of_excess ml));
+unfold apartness_OF_mlattice_;
+[rewrite < (ml_with_ ? ml); apply (metric ? (ml_mspace_ ? ml))]
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))]
+[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.
-record is_mlattice (R : todgroup) (ml: mlattice_ R) : Type ≝ {
- ml_prop1: ∀a,b:ml. 0 < δ a b → a # b;
- ml_prop2: ∀a,b,c:ml. δ (a∨b) (a∨c) + δ (a∧b) (a∧c) ≤ δ b c
-}.
-
+alias symbol "plus" = "Abelian group plus".
record mlattice (R : todgroup) : Type ≝ {
ml_carr :> mlattice_ R;
- ml_props:> is_mlattice R ml_carr
+ 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
}.
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 ?? ml);
+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 ?? ml); split [apply mpositive] apply ap_symmetric;
+apply ml_prop1; split [apply mpositive] apply ap_symmetric;
assumption;
qed.
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 ?? ml); split; [apply mpositive;]
+intros (R ml x y H); apply ml_prop1; split; [apply mpositive;]
apply ap_symmetric; assumption;
qed.
+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).
+
+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.
+
+lemma eq_to_eq:∀l:lattice.∀a,b:l.
+ (eq (apart_of_excess (pl_carr (latt_jcarr l))) a b) →
+ (eq (apart_of_excess (pl_carr (latt_mcarr l))) a b).
+intros 3; unfold eq; unfold apartness_OF_lattice;
+unfold apartness_OF_lattice_1; unfold latt_jcarr; simplify;
+unfold dual_exc; simplify; intros 2 (H H1); apply H;
+cases H1;[right|left]assumption;
+qed.
+
+coercion cic:/matita/metric_lattice/eq_to_eq.con nocomposites.
+
(* 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 ?? ml (y) ??));
+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 ??? Lxy) as Dxm; lapply (le_to_eqm ??? Lyz) as Dym;
-lapply (le_to_eqj ??? Lxy) as Dxj; lapply (le_to_eqj ??? Lyz) as Dyj; clear Lxy Lyz;
-apply (Eq≈ (δ(x∧y) y + δy z)); [apply feq_plusr; apply (meq_l ????? Dxm);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δy z)); [apply feq_plusr; apply (meq_r ????? Dym);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) z)); [apply feq_plusl; apply (meq_l ????? Dxj);]
-apply (Eq≈ (δ(x∧y) (y∧z) + δ(x∨y) (y∨z))); [apply feq_plusl; apply (meq_r ????? Dyj);]
+lapply (le_to_eqm ?? Lxy) as Dxm; lapply (le_to_eqm ?? Lyz) as Dym;
+lapply (le_to_eqj ?? (le_to_ge ??? Lxy)) as Dxj; lapply (le_to_eqj ?? (le_to_ge ??? Lyz)) as Dyj; clear Lxy Lyz;
+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;
+ unfold apartness_OF_mlattice1;
+ exact (eq_to_eq ??? Dxj);]
+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_l ????? (join_comm ???));]
+apply (Eq≈ (δ(y∨x) (y∨z)+ δ(x∧y) (y∧z)));[
+ apply feq_plusr;
+ apply meq_r;
+ apply (join_comm y z);]
apply feq_plusl;
-apply (Eq≈ (δ(y∧x) (y∧z))); [apply (meq_l ????? (meet_comm ???));]
+apply (Eq≈ (δ(y∧x) (y∧z)) (meq_l ????? (meet_comm x y)));
apply eq_reflexive;
qed.
-include "sequence.ma".
-include "nat/plus.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.
-
-
-definition d2s ≝
- λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
-(*
-notation "s ⇝ 'Zero'" non associative with precedence 50 for @{'tends0 $s }.
-
-interpretation "tends to" 'tends s x =
- (cic:/matita/sequence/tends0.con _ s).
-*)
-
-axiom core1: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
-
-alias symbol "leq" = "ordered sets less or equal than".
-alias symbol "and" = "constructive and".
-alias symbol "exists" = "constructive exists (Type)".
-theorem carabinieri:
- ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.
- (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) →
- ∀x:ml. tends0 ? (d2s ? ml an x) → tends0 ? (d2s ? ml bn x) →
- tends0 ? (d2s ? ml xn x).
-intros (R ml an bn xn H x Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
-intros (e He);
-alias num (instance 0) = "natural number".
-elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha;
-elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
-constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
-lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
-elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
-elim (H n3) (H7 H8); clear H H1 H2;
-lapply (le_to_eqm ??? H7) as Dxm; lapply (le_to_eqj ??? H7) as Dym;
-(* the main step *)
-cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x); [2:
- apply (le_transitive ???? (mtineq ???? (an n3)));
- lapply (le_mtri ????? H7 H8);
- lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? Hletin);
- cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))); [2:
- apply (Eq≈ (0 + δ(an n3) (xn n3)) ? (zero_neutral ??));
- apply (Eq≈ (δ(an n3) (xn n3) + 0) ? (plus_comm ???));
- apply (Eq≈ (δ(an n3) (xn n3) + (-δ(xn n3) (bn n3) + δ(xn n3) (bn n3))) ? (opp_inverse ??));
- apply (Eq≈ (δ(an n3) (xn n3) + (δ(xn n3) (bn n3) + -δ(xn n3) (bn n3))) ? (plus_comm ?? (δ(xn n3) (bn n3))));
- apply (Eq≈ ? ? (eq_sym ??? (plus_assoc ????))); assumption;] clear Hletin1;
- apply (le_rewl ??? ( δ(an n3) (xn n3)+ δ(an n3) x));[
- apply feq_plusr; apply msymmetric;]
- apply (le_rewl ??? (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x));[
- apply feq_plusr; assumption;]
- clear Hcut Hletin Dym Dxm H8 H7 H6 H5 H4 H3;
- apply (le_rewl ??? (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x));[
- apply feq_plusr; apply plus_comm;]
- apply (le_rewl ??? (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????));
- apply (le_rewl ??? ((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_rewr ???? (plus_comm ???));
- apply (le_rewr ??? ( δ(an n3) x+ δx (bn n3)) (msymmetric ????));
- apply mtineq;]
-split; [
- apply (lt_le_transitive ????? (mpositive ????));
- split; elim He; [apply le_zero_x_to_le_opp_x_zero; assumption;]
- cases t1; [
- left; apply exc_zero_opp_x_to_exc_x_zero;
- apply (Ex≫ ? (eq_opp_opp_x_x ??));assumption;]
- right; apply exc_opp_x_zero_to_exc_zero_x;
- apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
-clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
-apply (le_lt_transitive ???? ? (core1 ?? He));
-apply (le_transitive ???? Hcut);
-apply (le_transitive ?? (e/3+ δ(an n3) x+ δ(an n3) x)); [
- repeat apply fle_plusr; cases H6; assumption;]
-apply (le_transitive ?? (e/3+ e/2 + δ(an n3) x)); [
- apply fle_plusr; apply fle_plusl; cases H4; assumption;]
-apply (le_transitive ?? (e/3+ e/2 + e/2)); [
- apply fle_plusl; cases H4; assumption;]
-apply le_reflexive;
-qed.
-
(* 3.17 conclusione: δ x y ≈ 0 *)
(* 3.20 conclusione: δ x y ≈ 0 *)
(* 3.21 sup forte
(* 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 *)
\ No newline at end of file
+(* 3.36 conclusion: δ x y ≈ 0 *)
projects / helm.git / blobdiff