-(**************************************************************************)
-(* ___ *)
-(* ||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 "excess.ma".
-
-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)
-}.
-
-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_dual_smart;
-
- 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.
-
-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 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 (sl_strong_extm ???? H1);
-qed.
-
-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≈ ? (sl_meet_comm ???)); apply (Eq≈ ?? (sl_meet_comm ???));
-apply sl_feq_ml; assumption;
-qed.
-
-
-(*
-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.
-*)
-
-(* ED(≰,≱) → EB(≰') → ED(≰',≱') *)
-lemma subst_excess_base: excess_dual → excess_base → excess_dual.
-intros; apply (mk_excess_dual_smart e1);
-qed.
-
-(* E_(ED(≰,≱),AP(#),c ED = c AP) → ED' → c DE' = c E_ → E_(ED',#,p) *)
-lemma subst_dual_excess: ∀e:excess_.∀e1:excess_dual.exc_carr e = exc_carr e1 → excess_.
-intros (e e1 p); apply (mk_excess_ e1 e); cases p; reflexivity;
-qed.
-
-(* E(E_,H1,H2) → E_' → H1' → H2' → E(E_',H1',H2') *)
-alias symbol "nleq" = "Excess excess_".
-lemma subst_excess_: ∀e:excess. ∀e1:excess_.
- (∀y,x:e1. y # x → y ≰ x ∨ x ≰ y) →
- (∀y,x:e1.y ≰ x ∨ x ≰ y → y # x) →
- excess.
-intros (e e1 H1 H2); apply (mk_excess e1 H1 H2);
-qed.
-
-definition hole ≝ λT:Type.λx:T.x.
-
-notation < "\ldots" non associative with precedence 50 for @{'hole}.
-interpretation "hole" 'hole = (cic:/matita/lattice/hole.con _ _).
-
-(* SL(E,M,H2-5(#),H67(≰)) → E' → c E = c E' → H67'(≰') → SL(E,M,p2-5,H67') *)
-lemma subst_excess:
- ∀l:semi_lattice.
- ∀e:excess.
- ∀p:exc_ap l = exc_ap e.
- (∀x,y:e.(le (exc_dual_base e)) x y → x ≈ (?(sl_meet l)) x y) →
- (∀x,y:e.(le (exc_dual_base e)) ((?(sl_meet l)) x y) y) →
- semi_lattice.
-[1,2:intro M;
- change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
- cases p; apply M;
-|intros (l e p H1 H2);
- apply (mk_semi_lattice e);
- [ change with ((λx.ap_carr x) e -> (λx.ap_carr x) e -> (λx.ap_carr x) e);
- cases p; simplify; apply (sl_meet l);
- |2: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_refl;
- |3: change in ⊢ (% → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_comm;
- |4: change in ⊢ (% → % → % → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_meet_assoc;
- |5: change in ⊢ (% → ?) with ((λx.ap_carr x) e); cases p; simplify; apply sl_strong_extm;
- |6: clear H2; apply hole; apply H1;
- |7: clear H1; apply hole; apply H2;]]
-qed.
-
-lemma excess_of_excess_base: excess_base → excess.
-intro eb;
-apply mk_excess;
- [apply (mk_excess_ (mk_excess_dual_smart eb));
- [apply (apartness_of_excess_base eb);
- |reflexivity]
- |2,3: intros; assumption]
-qed.
-
-lemma subst_excess_preserves_aprtness:
- ∀l:semi_lattice.
- ∀e:excess.
- ∀p,H1,H2.
- exc_ap l = exc_ap (subst_excess l e p H1 H2).
-intros;
-unfold subst_excess;
-simplify; assumption;
-qed.
-
-
-lemma subst_excess__preserves_aprtness:
- ∀l:excess.
- ∀e:excess_base.
- ∀p,H1,H2.
- exc_ap l = apartness_OF_excess (subst_excess_ l (subst_dual_excess l (subst_excess_base l e) p) H1 H2).
-intros 3; (unfold subst_excess_; unfold subst_dual_excess; unfold subst_excess_base; unfold exc_ap; unfold mk_excess_dual_smart; simplify);
-(unfold subst_excess_base in p; unfold mk_excess_dual_smart in p; simplify in p);
-intros; cases p;
-reflexivity;
-qed.
-
-lemma subst_excess_base_in_excess_:
- ∀d:excess_.
- ∀eb:excess_base.
- ∀p:exc_carr d = exc_carr eb.
- excess_.
-intros (e_ eb);
-apply (subst_dual_excess e_);
- [apply (subst_excess_base e_ eb);
- |assumption]
-qed.
-
-lemma subst_excess_base_in_excess:
- ∀d:excess.
- ∀eb:excess_base.
- ∀p:exc_carr d = exc_carr eb.
- (∀y1,x1:eb. (?(ap_apart d)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) →
- (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart d)) y2 x2) →
- excess.
-[1,3,4:apply Type|2,5:intro f; cases p; apply f]
-intros (d eb p H1 H2);
-apply (subst_excess_ d);
- [apply (subst_excess_base_in_excess_ d eb p);
- |apply hole; clear H2;
- change in ⊢ (%→%→?) with (exc_carr eb);
- change in ⊢ (?→?→?→? (? % ? ?) (? % ? ?)) with eb; intros (y x H3);
- apply H1; generalize in match H3;
- unfold ap_apart; unfold subst_excess_base_in_excess_;
- unfold subst_dual_excess; simplify;
- generalize in match x;
- generalize in match y;
- cases p; simplify; intros; assumption;
- |apply hole; clear H1;
- change in ⊢ (%→%→?) with (exc_carr eb);
- change in ⊢ (?→?→? (? % ? ?) (? % ? ?)→?) with eb; intros (y x H3);
- unfold ap_apart; unfold subst_excess_base_in_excess_;
- unfold subst_dual_excess; simplify; generalize in match (H2 ?? H3);
- generalize in match x; generalize in match y; cases p;
- intros; assumption;]
-qed.
-
-lemma tech1: ∀e:excess.
- ∀eb:excess_base.
- ∀p,H1,H2.
- exc_ap e = exc_ap_ (subst_excess_base_in_excess e eb p H1 H2).
-intros (e eb p H1 H2);
-unfold subst_excess_base_in_excess;
-unfold subst_excess_; simplify;
-unfold subst_excess_base_in_excess_;
-unfold subst_dual_excess; simplify; reflexivity;
-qed.
-
-lemma tech2:
- ∀e:excess_.∀eb.∀p.
- exc_ap e = exc_ap (mk_excess_ (subst_excess_base e eb) (exc_ap e) p).
-intros (e eb p);unfold exc_ap; simplify; cases p; simplify; reflexivity;
-qed.
-
-(*
-lemma eq_fap:
- ∀a1,b1,a2,b2,a3,b3,a4,b4,a5,b5.
- a1=b1 → a2=b2 → a3=b3 → a4=b4 → a5=b5 → mk_apartness a1 a2 a3 a4 a5 = mk_apartness b1 b2 b3 b4 b5.
-intros; cases H; cases H1; cases H2; cases H3; cases H4; reflexivity;
-qed.
-*)
-
-lemma subst_excess_base_in_excess_preserves_apartness:
- ∀e:excess.
- ∀eb:excess_base.
- ∀H,H1,H2.
- apartness_OF_excess e =
- apartness_OF_excess (subst_excess_base_in_excess e eb H H1 H2).
-intros (e eb p H1 H2);
-unfold subst_excess_base_in_excess;
-unfold subst_excess_; unfold subst_excess_base_in_excess_;
-unfold subst_dual_excess; unfold apartness_OF_excess;
-simplify in ⊢ (? ? ? (? %));
-rewrite < (tech2 e eb );
-reflexivity;
-qed.
-
-
-
-alias symbol "nleq" = "Excess base excess".
-lemma subst_excess_base_in_semi_lattice:
- ∀sl:semi_lattice.
- ∀eb:excess_base.
- ∀p:exc_carr sl = exc_carr eb.
- (∀y1,x1:eb. (?(ap_apart sl)) y1 x1 → y1 ≰ x1 ∨ x1 ≰ y1) →
- (∀y2,x2:eb.y2 ≰ x2 ∨ x2 ≰ y2 → (?(ap_apart sl)) y2 x2) →
- (∀x3,y3:eb.(le eb) x3 y3 → (?(eq sl)) x3 ((?(sl_meet sl)) x3 y3)) →
- (∀x4,y4:eb.(le eb) ((?(sl_meet sl)) x4 y4) y4) →
- semi_lattice.
-[2:apply Prop|3,7,9,10:apply Type|4:apply (exc_carr eb)|1,5,6,8,11:intro f; cases p; apply f;]
-intros (sl eb H H1 H2 H3 H4);
-apply (subst_excess sl);
- [apply (subst_excess_base_in_excess sl eb H H1 H2);
- |apply subst_excess_base_in_excess_preserves_apartness;
- |change in ⊢ (%→%→?) with ((λx.ap_carr x) (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)); simplify;
- intros 3 (x y LE); clear H3 LE;
- generalize in match x as x; generalize in match y as y;
- generalize in match H1 as H1;generalize in match H2 as H2;
- clear x y H1 H2 H4; STOP
-
- apply (match H return λr:Type.λm:Type_OF_semi_lattice sl=r.
- ∀H2:(Πy2:exc_carr eb
- .Πx2:exc_carr eb
- .Or (exc_excess eb y2 x2) (exc_excess eb x2 y2)
- →match H
- in eq
- return
- λright_1:Type
- .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1
- .right_1→right_1→Type)
- with
- [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y2 x2)
-.∀H1:Πy1:exc_carr eb
- .Πx1:exc_carr eb
- .match H
- in eq
- return
- λright_1:Type
- .(λmatched:eq Type (Type_OF_semi_lattice sl) right_1
- .right_1→right_1→Type)
- with
- [refl_eq⇒ap_apart (apartness_OF_semi_lattice sl)] y1 x1
- →Or (exc_excess eb y1 x1) (exc_excess eb x1 y1)
- .∀y:ap_carr
- (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2))
- .∀x:ap_carr
- (apartness_OF_excess
- (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2))
- .eq
- (apartness_OF_excess (subst_excess_base_in_excess (sl_exc sl) eb H H1 H2)) x
- (match
- subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2
- in eq
- return
- λright_1:apartness
- .(λmatched:eq apartness (apartness_OF_semi_lattice sl) right_1
- .ap_carr right_1→ap_carr right_1→ap_carr right_1)
- with
- [refl_eq⇒sl_meet sl] x y)
-
- with [ refl_eq ⇒ ?]);
-
-
- cases (subst_excess_base_in_excess_preserves_apartness sl eb H H1 H2);
- cases H;
- cases (subst_excess_base_in_excess_preserves_apartness (sl_exc sl) eb H H1 H2); simplify;
- change in x:(%) with (exc_carr eb);
- change in y:(%) with (exc_carr eb);
- generalize in match OK; generalize in match x as x; generalize in match y as y;
- cases H; simplify; (* funge, ma devo fare 2 rewrite in un colpo solo *)
- *)
- |cases FALSE;
- ]
-qed.
-
-record lattice_ : Type ≝ {
- latt_mcarr:> semi_lattice;
- latt_jcarr_: semi_lattice;
- W1:?; W2:?; W3:?; W4:?; W5:?;
- latt_with1: latt_jcarr_ = subst_excess_base_in_semi_lattice latt_jcarr_
- (excess_base_OF_semi_lattice latt_mcarr) W1 W2 W3 W4 W5
-}.
-
-lemma latt_jcarr : lattice_ → semi_lattice.
-intro l; apply (subst_excess_base_in_semi_lattice (latt_jcarr_ l) (excess_base_OF_semi_lattice (latt_mcarr l)) (W1 l) (W2 l) (W3 l) (W4 l) (W5 l));
-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 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.
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