include "metric_space.ma".
-record premetric_lattice_ (R : ogroup) : Type ≝ {
+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 join" 'or a b =
(cic:/matita/premetric_lattice/join.con _ _ a b).
-record premetric_lattice_props (R : ogroup) (ml : premetric_lattice_ R) : Prop ≝ {
+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;
prop5: ∀a,b,c: ml. δ (a ∨ b) (a ∨ c) + δ (a ∧ b) (a ∧ c) ≤ δ b c
}.
-record pmlattice (R : ogroup) : Type ≝ {
+record pmlattice (R : todgroup) : Type ≝ {
carr :> premetric_lattice_ R;
ispremetriclattice:> premetric_lattice_props R carr
}.
include "lattice.ma".
-lemma lattice_of_pmlattice: ∀R: ogroup. pmlattice R → lattice.
+lemma lattice_of_pmlattice: ∀R: todgroup. pmlattice R → lattice.
intros (R pml); apply (mk_lattice (apart_of_metric_space ? pml));
-[apply (join ? pml)|apply (meet ? pml)]
-intros (x y z); whd; intro H; whd in H; cases H (LE AP);
-[apply (prop2a ? pml pml x y); |apply (prop2b ? pml pml x y);
+[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);]
-apply ap_symmetric; assumption;
+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.
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