Added summation formula for the power of a binomial.

[helm.git] / helm / software / matita / dama / lattice.ma

diff --git a/helm/software/matita/dama/lattice.ma b/helm/software/matita/dama/lattice.ma
index 398b1af89dc0c6c69f7e7dbc3203a0b47773483d..768c86df1c2af47785bf08b8a762524e72be9426 100644 (file)
--- a/helm/software/matita/dama/lattice.ma
+++ b/helm/software/matita/dama/lattice.ma
@@ -14,18 +14,22 @@
 
 set "baseuri" "cic:/matita/lattice/".
 
-include "excedence.ma".
+include "excess.ma".
 
 record lattice : Type ≝ {
   l_carr:> apartness;
   join: l_carr → l_carr → l_carr;
   meet: l_carr → l_carr → l_carr;
+  join_refl: ∀x.join x x ≈ x;
+  meet_refl: ∀x.meet x x ≈ x;  
   join_comm: ∀x,y:l_carr. join x y ≈ join y x;
   meet_comm: ∀x,y:l_carr. meet x y ≈ meet y x;
   join_assoc: ∀x,y,z:l_carr. join x (join y z) ≈ join (join x y) z;
   meet_assoc: ∀x,y,z:l_carr. meet x (meet y z) ≈ meet (meet x y) z;
   absorbjm: ∀f,g:l_carr. join f (meet f g) ≈ f;
-  absorbmj: ∀f,g:l_carr. meet f (join f g) ≈ f
+  absorbmj: ∀f,g:l_carr. meet f (join f g) ≈ f;
+  strong_extj: ∀x.strong_ext ? (join x);
+  strong_extm: ∀x.strong_ext ? (meet x)
 }.
 
 interpretation "Lattice meet" 'and a b =
@@ -34,60 +38,50 @@ interpretation "Lattice meet" 'and a b =
 interpretation "Lattice join" 'or a b =
  (cic:/matita/lattice/join.con _ a b).
 
-(*
-include "ordered_set.ma".
+definition excl ≝ λl:lattice.λa,b:l.a # (a ∧ b).
 
-record lattice (C:Type) (join,meet:C→C→C) : Prop \def
- { (* abelian semigroup properties *)
-   l_comm_j: symmetric ? join;
-   l_associative_j: associative ? join;
-   l_comm_m: symmetric ? meet;
-   l_associative_m: associative ? meet;
-   (* other properties *)
-   l_adsorb_j_m: ∀f,g:C. join f (meet f g) = f;
-   l_adsorb_m_j: ∀f,g:C. meet f (join f g) = f
- }.
+lemma excess_of_lattice: lattice → excess.
+intro l; apply (mk_excess l (excl l));
+[ intro x; unfold; intro H; unfold in H; apply (ap_coreflexive l x);
+  apply (ap_rewr ??? (x∧x) (meet_refl l x)); assumption;
+| intros 3 (x y z); unfold excl; intro H;
+  cases (ap_cotransitive ??? (x∧z∧y) H) (H1 H2); [2:
+    left; apply ap_symmetric; apply (strong_extm ? y); 
+    apply (ap_rewl ???? (meet_comm ???));
+    apply (ap_rewr ???? (meet_comm ???));
+    assumption]
+  cases (ap_cotransitive ??? (x∧z) H1) (H2 H3); [left; assumption]
+  right; apply (strong_extm ? x); apply (ap_rewr ???? (meet_assoc ????));
+  assumption]
+qed.    
 
-record lattice : Type \def
- { l_carrier:> Type;
-   l_join: l_carrier→l_carrier→l_carrier;
-   l_meet: l_carrier→l_carrier→l_carrier;
-   l_lattice_properties:> is_lattice ? l_join l_meet
- }.
+coercion cic:/matita/lattice/excess_of_lattice.con.
 
-definition le \def λL:lattice.λf,g:L. (f ∧ g) = f.
+lemma feq_ml: ∀ml: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 (strong_extm ???? H1);
+qed.
+
+lemma feq_jl: ∀ml: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 (strong_extj ???? H1);
+qed.
 
-definition ordered_set_of_lattice: lattice → ordered_set.
- intros (L);
- apply mk_ordered_set;
-  [2: apply (le L)
-  | skip
-  | apply mk_is_order_relation;
-     [ unfold reflexive;
-       intros;
-       unfold;
-       rewrite < (l_adsorb_j_m ? ? ? L ? x) in ⊢ (? ? (? ? ? %) ?);
-       rewrite > l_adsorb_m_j;
-        [ reflexivity
-        | apply (l_lattice_properties L)
-        ]
-     | intros;
-       unfold transitive;
-       unfold le;
-       intros;
-       rewrite < H;
-       rewrite > (l_associative_m ? ? ? L);
-       rewrite > H1;
-       reflexivity
-     | unfold antisimmetric;
-       unfold le;
-       intros;
-       rewrite < H;
-       rewrite > (l_comm_m ? ? ? L);
-       assumption
-     ]
-  ]
+lemma le_to_eqm: ∀ml:lattice.∀a,b:ml. a ≤ b → a ≈ (a ∧ b).
+intros (l a b H); 
+  unfold le in H; unfold excess_of_lattice in H;
+  unfold excl in H; simplify in H; 
+unfold eq; assumption;
 qed.
 
-coercion cic:/matita/lattices/ordered_set_of_lattice.con.
-*)
+lemma le_to_eqj: ∀ml:lattice.∀a,b:ml. a ≤ b → b ≈ (a ∨ b).
+intros (l a b H); lapply (le_to_eqm ??? H) as H1;
+lapply (feq_jl ??? b H1) as H2;
+apply (Eq≈ ?? (join_comm ???));
+apply (Eq≈ (b∨a∧b) ? H2); clear H1 H2 H;
+apply (Eq≈ (b∨(b∧a)) ? (feq_jl ???? (meet_comm ???)));
+apply eq_sym; apply absorbjm;
+qed.
+
+
+