include "metric_space.ma".
include "lattice.ma".
-record mlattice (R : ogroup) : Type ≝ {
+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)
}.
-lemma ml_mspace: ∀R.mlattice R → metric_space R.
-intros (R ml); apply (mk_metric_space R ml); unfold Type_OF_mlattice;
+lemma ml_mspace: ∀R.mlattice_ R → metric_space R.
+intros (R ml); apply (mk_metric_space R 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));
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
+}.
+
+record mlattice (R : todgroup) : Type ≝ {
+ ml_carr :> mlattice_ R;
+ ml_props:> is_mlattice R ml_carr
+}.
+
+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);
+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;
+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 ?? ml); 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 ?? ml (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);]
+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 feq_plusl;
+apply (Eq≈ (δ(y∧x) (y∧z))); [apply (meq_l ????? (meet_comm ???));]
+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
+ 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 *)
\ No newline at end of file
projects / helm.git / blobdiff