+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 *)
projects / helm.git / blobdiff