X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fdama%2Flattice.ma;h=ef02134256330b43f7a4006fed38a7702a9546a6;hb=9483f7e6c85ec11f88bdb219b2cebad2039b1a74;hp=66f14b942479fb2206d93151069085fc9e82ba65;hpb=ea7c93bf4177982ff09d429a3a818e8b5f937ea1;p=helm.git

diff --git a/helm/software/matita/dama/lattice.ma b/helm/software/matita/dama/lattice.ma
index 66f14b942..ef0213425 100644
--- a/helm/software/matita/dama/lattice.ma
+++ b/helm/software/matita/dama/lattice.ma
@@ -12,88 +12,241 @@
 (*                                                                        *)
 (**************************************************************************)
 
-
-
 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;
-  strong_extj: ∀x.strong_ext ? (join x);
-  strong_extm: ∀x.strong_ext ? (meet x)
+record semi_lattice_base : Type ≝ {
+  sl_carr:> apartness;
+  sl_op: sl_carr → sl_carr → sl_carr;
+  sl_op_refl: ∀x.sl_op x x ≈ x;  
+  sl_op_comm: ∀x,y:sl_carr. sl_op x y ≈ sl_op y x;
+  sl_op_assoc: ∀x,y,z:sl_carr. sl_op x (sl_op y z) ≈ sl_op (sl_op x y) z;
+  sl_strong_extop: ∀x.strong_ext ? (sl_op x)  
 }.
 
-interpretation "Lattice meet" 'and a b =
- (cic:/matita/lattice/meet.con _ a b).
-
-interpretation "Lattice join" 'or a b =
- (cic:/matita/lattice/join.con _ a b).
-
-definition excl ≝ λl:lattice.λa,b:l.a # (a ∧ b).
-
-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]
+notation "a \cross b" left associative with precedence 50 for @{ 'op $a $b }.
+interpretation "semi lattice base operation" 'op a b = (cic:/matita/lattice/sl_op.con _ a b).
+
+lemma excess_of_semi_lattice_base: semi_lattice_base → excess.
+intro l;
+apply mk_excess;
+[1: apply mk_excess_;
+    [1: 
+    
+  apply (mk_excess_base (sl_carr l));
+    [1: apply (λa,b:sl_carr l.a # (a ✗ b));
+    |2: unfold; intros 2 (x H); simplify in H;
+        lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
+        apply (ap_coreflexive ?? H1);
+    |3: unfold; simplify; intros (x y z H1);
+        cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2:
+          lapply (Ap≪ ? (sl_op_comm ???) H2) as H21;
+          lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2;
+          lapply (sl_strong_extop ???? H22); clear H22; 
+          left; apply ap_symmetric; assumption;]
+        cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption]
+        right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31;
+        apply (sl_strong_extop ???? H31);]
+
+    |2:
+    apply apartness_of_excess_base; 
+    
+  apply (mk_excess_base (sl_carr l));
+    [1: apply (λa,b:sl_carr l.a # (a ✗ b));
+    |2: unfold; intros 2 (x H); simplify in H;
+        lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
+        apply (ap_coreflexive ?? H1);
+    |3: unfold; simplify; intros (x y z H1);
+        cases (ap_cotransitive ??? ((x ✗ z) ✗ y) H1) (H2 H2);[2:
+          lapply (Ap≪ ? (sl_op_comm ???) H2) as H21;
+          lapply (Ap≫ ? (sl_op_comm ???) H21) as H22; clear H21 H2;
+          lapply (sl_strong_extop ???? H22); clear H22; 
+          left; apply ap_symmetric; assumption;]
+        cases (ap_cotransitive ??? (x ✗ z) H2) (H3 H3);[left;assumption]
+        right; lapply (Ap≫ ? (sl_op_assoc ????) H3) as H31;
+        apply (sl_strong_extop ???? H31);]
+    
+    |3: apply refl_eq;]
+|2,3: intros (x y H); assumption;]         
 qed.    
 
-coercion cic:/matita/lattice/excess_of_lattice.con.
+record semi_lattice : Type ≝ {
+  sl_exc:> excess;
+  sl_meet: sl_exc → sl_exc → sl_exc;
+  sl_meet_refl: ∀x.sl_meet x x ≈ x;  
+  sl_meet_comm: ∀x,y. sl_meet x y ≈ sl_meet y x;
+  sl_meet_assoc: ∀x,y,z. sl_meet x (sl_meet y z) ≈ sl_meet (sl_meet x y) z;
+  sl_strong_extm: ∀x.strong_ext ? (sl_meet x);
+  sl_le_to_eqm: ∀x,y.x ≤ y → x ≈ sl_meet x y;
+  sl_lem: ∀x,y.(sl_meet x y) ≤ y 
+}.
+ 
+interpretation "semi lattice meet" 'and a b = (cic:/matita/lattice/sl_meet.con _ a b).
 
-lemma feq_ml: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b).
+lemma sl_feq_ml: ∀ml:semi_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);
+intro H1; apply H; clear H; apply (sl_strong_extm ???? H1);
 qed.
 
-lemma feq_mr: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (a ∧ c) ≈ (b ∧ c).
+lemma sl_feq_mr: ∀ml:semi_lattice.∀a,b,c:ml. a ≈ b → (a ∧ c) ≈ (b ∧ c).
 intros (l a b c H); 
-apply (Eq≈ ? (meet_comm ???)); apply (Eq≈ ?? (meet_comm ???));
-apply feq_ml; assumption;
+apply (Eq≈ ? (sl_meet_comm ???)); apply (Eq≈ ?? (sl_meet_comm ???));
+apply sl_feq_ml; assumption;
 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);
+ 
+ 
+(*
+lemma semi_lattice_of_semi_lattice_base: semi_lattice_base → semi_lattice.
+intro slb; apply (mk_semi_lattice (excess_of_semi_lattice_base slb));
+[1: apply (sl_op slb);
+|2: intro x; apply (eq_trans (excess_of_semi_lattice_base slb)); [2: 
+      apply (sl_op_refl slb);|1:skip] (sl_op slb x x)); ? (sl_op_refl slb x));
+
+ unfold excess_of_semi_lattice_base; simplify;
+    intro H; elim H;
+    [ 
+    
+    
+    lapply (ap_rewl (excess_of_semi_lattice_base slb) x ? (sl_op slb x x) 
+      (eq_sym (excess_of_semi_lattice_base slb) ?? (sl_op_refl slb x)) t);
+    change in x with (sl_carr slb);
+    apply (Ap≪ (x ✗ x)); (sl_op_refl slb x)); 
+
+whd in H; elim H; clear H;
+    [ apply (ap_coreflexive (excess_of_semi_lattice_base slb) (x ✗ x) t);
+
+prelattice (excess_of_directed l_)); [apply (sl_op l_);]
+unfold excess_of_directed; try unfold apart_of_excess; simplify;
+unfold excl; simplify;
+[intro x; intro H; elim H; clear H; 
+ [apply (sl_op_refl l_ x); 
+  lapply (Ap≫ ? (sl_op_comm ???) t) as H; clear t; 
+  lapply (sl_strong_extop l_ ??? H); apply ap_symmetric; assumption
+ | lapply (Ap≪ ? (sl_op_refl ?x) t) as H; clear t;
+   lapply (sl_strong_extop l_ ??? H); apply (sl_op_refl l_ x);
+   apply ap_symmetric; assumption]
+|intros 3 (x y H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x y)) H1) as H; clear H1;
+  lapply (sl_strong_extop l_ ??? H) as H1; clear H;
+  lapply (Ap≪ ? (sl_op_comm ???) H1); apply (ap_coreflexive ?? Hletin);
+ |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ y x)) H2) as H; clear H2;
+  lapply (sl_strong_extop l_ ??? H) as H1; clear H;
+  lapply (Ap≪ ? (sl_op_comm ???) H1);apply (ap_coreflexive ?? Hletin);]
+|intros 4 (x y z H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ x (sl_op l_ y z))) H1) as H; clear H1;
+  lapply (sl_strong_extop l_ ??? H) as H1; clear H;
+  lapply (Ap≪ ? (eq_sym ??? (sl_op_assoc ?x y z)) H1) as H; clear H1;
+  apply (ap_coreflexive ?? H);
+ |lapply (Ap≪ ? (sl_op_refl ? (sl_op l_ (sl_op l_ x y) z)) H2) as H; clear H2;
+  lapply (sl_strong_extop l_ ??? H) as H1; clear H;
+  lapply (Ap≪ ? (sl_op_assoc ?x y z) H1) as H; clear H1;
+  apply (ap_coreflexive ?? H);]
+|intros (x y z H); elim H (H1 H1); clear H;
+ lapply (Ap≪ ? (sl_op_refl ??) H1) as H; clear H1;
+ lapply (sl_strong_extop l_ ??? H) as H1; clear H;
+ lapply (sl_strong_extop l_ ??? H1) as H; clear H1;
+ cases (ap_cotransitive ??? (sl_op l_ y z) H);[left|right|right|left] try assumption;
+ [apply ap_symmetric;apply (Ap≪ ? (sl_op_comm ???));
+ |apply (Ap≫ ? (sl_op_comm ???));
+ |apply ap_symmetric;] assumption;
+|intros 4 (x y H H1); apply H; clear H; elim H1 (H H);
+ lapply (Ap≪ ? (sl_op_refl ??) H) as H1; clear H;
+ lapply (sl_strong_extop l_ ??? H1) as H; clear H1;[2: apply ap_symmetric]
+ assumption
+|intros 3 (x y H); 
+ cut (sl_op l_ x y ≈ sl_op l_ x (sl_op l_ y y)) as H1;[2:
+   intro; lapply (sl_strong_extop ???? a); apply (sl_op_refl l_ y);
+   apply ap_symmetric; assumption;]
+ lapply (Ap≪ ? (eq_sym ??? H1) H); apply (sl_op_assoc l_ x y y);
+ assumption; ]
 qed.
+*)
 
-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.
 
-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;
+record lattice_ : Type ≝ {
+  latt_mcarr:> semi_lattice;
+  latt_jcarr_: semi_lattice;
+  latt_with:  sl_exc latt_jcarr_ = dual_exc (sl_exc latt_mcarr)
+}.
+
+lemma latt_jcarr : lattice_ → semi_lattice.
+intro l;
+apply (mk_semi_lattice (dual_exc l)); 
+unfold excess_OF_lattice_;
+cases (latt_with l); simplify;
+[apply sl_meet|apply sl_meet_refl|apply sl_meet_comm|apply sl_meet_assoc|
+apply sl_strong_extm| apply sl_le_to_eqm|apply sl_lem]
+qed. 
+ 
+coercion cic:/matita/lattice/latt_jcarr.con.
+
+interpretation "Lattice meet" 'and a b =
+ (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _) a b).  
+
+interpretation "Lattice join" 'or a b =
+ (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _) a b).  
+
+record lattice : Type ≝ {
+  latt_carr:> lattice_;
+  absorbjm: ∀f,g:latt_carr. (f ∨ (f ∧ g)) ≈ f;
+  absorbmj: ∀f,g:latt_carr. (f ∧ (f ∨ g)) ≈ f
+}.
+
+notation "'meet'"        non associative with precedence 50 for @{'meet}.
+notation "'meet_refl'"   non associative with precedence 50 for @{'meet_refl}.
+notation "'meet_comm'"   non associative with precedence 50 for @{'meet_comm}.
+notation "'meet_assoc'"  non associative with precedence 50 for @{'meet_assoc}.
+notation "'strong_extm'" non associative with precedence 50 for @{'strong_extm}.
+notation "'le_to_eqm'"   non associative with precedence 50 for @{'le_to_eqm}.
+notation "'lem'"         non associative with precedence 50 for @{'lem}.
+notation "'join'"        non associative with precedence 50 for @{'join}.
+notation "'join_refl'"   non associative with precedence 50 for @{'join_refl}.
+notation "'join_comm'"   non associative with precedence 50 for @{'join_comm}.
+notation "'join_assoc'"  non associative with precedence 50 for @{'join_assoc}.
+notation "'strong_extj'" non associative with precedence 50 for @{'strong_extj}.
+notation "'le_to_eqj'"   non associative with precedence 50 for @{'le_to_eqj}.
+notation "'lej'"         non associative with precedence 50 for @{'lej}.
+
+interpretation "Lattice meet"        'meet        = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_refl"   'meet_refl   = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_comm"   'meet_comm   = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_assoc"  'meet_assoc  = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice strong_extm" 'strong_extm = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice le_to_eqm"   'le_to_eqm   = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice lem"         'lem         = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice join"        'join        = (cic:/matita/lattice/sl_meet.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_refl"   'join_refl   = (cic:/matita/lattice/sl_meet_refl.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_comm"   'join_comm   = (cic:/matita/lattice/sl_meet_comm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_assoc"  'join_assoc  = (cic:/matita/lattice/sl_meet_assoc.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice strong_extm" 'strong_extj = (cic:/matita/lattice/sl_strong_extm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice le_to_eqj"   'le_to_eqj   = (cic:/matita/lattice/sl_le_to_eqm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice lej"         'lej         = (cic:/matita/lattice/sl_lem.con (cic:/matita/lattice/latt_jcarr.con _)).
+
+notation "'feq_jl'" non associative with precedence 50 for @{'feq_jl}.
+notation "'feq_jr'" non associative with precedence 50 for @{'feq_jr}.
+notation "'feq_ml'" non associative with precedence 50 for @{'feq_ml}.
+notation "'feq_mr'" non associative with precedence 50 for @{'feq_mr}.
+interpretation "Lattice feq_jl" 'feq_jl = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice feq_jr" 'feq_jr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice feq_ml" 'feq_ml = (cic:/matita/lattice/sl_feq_ml.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice feq_mr" 'feq_mr = (cic:/matita/lattice/sl_feq_mr.con (cic:/matita/lattice/latt_mcarr.con _)).
+
+
+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).
+
+(* these coercions help unification, handmaking a bit of conversion 
+   over an open term 
+*)
+lemma le_to_ge: ∀l:lattice.∀a,b:l.a ≤ b → b ≥ a.
+intros(l a b H); apply H;
 qed.
 
-lemma lem: ∀ml:lattice.∀a,b:ml.(a ∧ b) ≤ b.
-intros; unfold le; unfold excess_of_lattice; unfold excl; simplify;
-intro H; apply (ap_coreflexive ? (a∧b));
-apply (Ap≫ (a∧(b∧b)) (feq_ml ???? (meet_refl ? b)));
-apply (Ap≫ ? (meet_assoc ????)); assumption;
+lemma ge_to_le: ∀l:lattice.∀a,b:l.b ≥ a → a ≤ b.
+intros(l a b H); apply H;
 qed.
- 
 
+coercion cic:/matita/lattice/le_to_ge.con nocomposites.
+coercion cic:/matita/lattice/ge_to_le.con nocomposites.
\ No newline at end of file