(* *)
(**************************************************************************)
-
-
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 directed : Type ≝ {
+ dir_carr: apartness;
+ dir_op: dir_carr → dir_carr → dir_carr;
+ dir_op_refl: ∀x.dir_op x x ≈ x;
+ dir_op_comm: ∀x,y:dir_carr. dir_op x y ≈ dir_op y x;
+ dir_op_assoc: ∀x,y,z:dir_carr. dir_op x (dir_op y z) ≈ dir_op (dir_op x y) z;
+ dir_strong_extop: ∀x.strong_ext ? (dir_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).
+definition excl ≝
+ λl:directed.λa,b:dir_carr l.ap_apart (dir_carr l) a (dir_op l 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;
+lemma excess_of_directed: directed → excess.
+intro l; apply (mk_excess (dir_carr l) (excl l));
+[ intro x; unfold; intro H; unfold in H; apply (ap_coreflexive (dir_carr l) x);
+ apply (ap_rewr ??? (dir_op l x x) (dir_op_refl ? 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 ???));
+ cases (ap_cotransitive ??? (dir_op l (dir_op l x z) y) H) (H1 H2); [2:
+ left; apply ap_symmetric; apply (dir_strong_extop ? y);
+ apply (ap_rewl ???? (dir_op_comm ???));
+ apply (ap_rewr ???? (dir_op_comm ???));
assumption]
- cases (ap_cotransitive ??? (x∧z) H1) (H2 H3); [left; assumption]
- right; apply (strong_extm ? x); apply (ap_rewr ???? (meet_assoc ????));
+ cases (ap_cotransitive ??? (dir_op l x z) H1) (H2 H3); [left; assumption]
+ right; apply (dir_strong_extop ? x); apply (ap_rewr ???? (dir_op_assoc ????));
assumption]
qed.
-coercion cic:/matita/lattice/excess_of_lattice.con.
+record prelattice : Type ≝ {
+ pl_carr:> excess;
+ meet: pl_carr → pl_carr → pl_carr;
+ meet_refl: ∀x.meet x x ≈ x;
+ meet_comm: ∀x,y:pl_carr. meet x y ≈ meet y x;
+ meet_assoc: ∀x,y,z:pl_carr. meet x (meet y z) ≈ meet (meet x y) z;
+ strong_extm: ∀x.strong_ext ? (meet x);
+ le_to_eqm: ∀x,y.x ≤ y → x ≈ meet x y;
+ lem: ∀x,y.(meet x y) ≤ y
+}.
+
+interpretation "prelattice meet" 'and a b =
+ (cic:/matita/lattice/meet.con _ a b).
-lemma feq_ml: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (c ∧ a) ≈ (c ∧ b).
+lemma feq_ml: ∀ml:prelattice.∀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_mr: ∀ml:lattice.∀a,b,c:ml. a ≈ b → (a ∧ c) ≈ (b ∧ c).
+lemma feq_mr: ∀ml:prelattice.∀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;
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.
-
-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;
+
+lemma prelattice_of_directed: directed → prelattice.
+intro l_; apply (mk_prelattice (excess_of_directed l_)); [apply (dir_op l_);]
+unfold excess_of_directed; try unfold apart_of_excess; simplify;
+unfold excl; simplify;
+[intro x; intro H; elim H; clear H;
+ [apply (dir_op_refl l_ x);
+ lapply (Ap≫ ? (dir_op_comm ???) t) as H; clear t;
+ lapply (dir_strong_extop l_ ??? H); apply ap_symmetric; assumption
+ | lapply (Ap≪ ? (dir_op_refl ?x) t) as H; clear t;
+ lapply (dir_strong_extop l_ ??? H); apply (dir_op_refl l_ x);
+ apply ap_symmetric; assumption]
+|intros 3 (x y H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (dir_op_refl ? (dir_op l_ x y)) H1) as H; clear H1;
+ lapply (dir_strong_extop l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (dir_op_comm ???) H1); apply (ap_coreflexive ?? Hletin);
+ |lapply (Ap≪ ? (dir_op_refl ? (dir_op l_ y x)) H2) as H; clear H2;
+ lapply (dir_strong_extop l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (dir_op_comm ???) H1);apply (ap_coreflexive ?? Hletin);]
+|intros 4 (x y z H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (dir_op_refl ? (dir_op l_ x (dir_op l_ y z))) H1) as H; clear H1;
+ lapply (dir_strong_extop l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (eq_sym ??? (dir_op_assoc ?x y z)) H1) as H; clear H1;
+ apply (ap_coreflexive ?? H);
+ |lapply (Ap≪ ? (dir_op_refl ? (dir_op l_ (dir_op l_ x y) z)) H2) as H; clear H2;
+ lapply (dir_strong_extop l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (dir_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≪ ? (dir_op_refl ??) H1) as H; clear H1;
+ lapply (dir_strong_extop l_ ??? H) as H1; clear H;
+ lapply (dir_strong_extop l_ ??? H1) as H; clear H1;
+ cases (ap_cotransitive ??? (dir_op l_ y z) H);[left|right|right|left] try assumption;
+ [apply ap_symmetric;apply (Ap≪ ? (dir_op_comm ???));
+ |apply (Ap≫ ? (dir_op_comm ???));
+ |apply ap_symmetric;] assumption;
+|intros 4 (x y H H1); apply H; clear H; elim H1 (H H);
+ lapply (Ap≪ ? (dir_op_refl ??) H) as H1; clear H;
+ lapply (dir_strong_extop l_ ??? H1) as H; clear H1;[2: apply ap_symmetric]
+ assumption
+|intros 3 (x y H);
+ cut (dir_op l_ x y ≈ dir_op l_ x (dir_op l_ y y)) as H1;[2:
+ intro; lapply (dir_strong_extop ???? a); apply (dir_op_refl l_ y);
+ apply ap_symmetric; assumption;]
+ lapply (Ap≪ ? (eq_sym ??? H1) H); apply (dir_op_assoc l_ x y y);
+ 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;
-qed.
+record lattice_ : Type ≝ {
+ latt_mcarr:> prelattice;
+ latt_jcarr_: prelattice;
+ latt_with: pl_carr latt_jcarr_ = dual_exc (pl_carr latt_mcarr)
+}.
-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;
-qed.
+lemma latt_jcarr : lattice_ → prelattice.
+intro l;
+apply (mk_prelattice (dual_exc l)); unfold excess_OF_lattice_;
+cases (latt_with l); simplify;
+[apply meet|apply meet_refl|apply meet_comm|apply meet_assoc|
+apply strong_extm| apply le_to_eqm|apply lem]
+qed.
+coercion cic:/matita/lattice/latt_jcarr.con.
+
+interpretation "Lattice meet" 'and a b =
+ (cic:/matita/lattice/meet.con (cic:/matita/lattice/latt_mcarr.con _) a b).
+
+interpretation "Lattice join" 'or a b =
+ (cic:/matita/lattice/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/meet.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_refl" 'meet_refl = (cic:/matita/lattice/meet_refl.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_comm" 'meet_comm = (cic:/matita/lattice/meet_comm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice meet_assoc" 'meet_assoc = (cic:/matita/lattice/meet_assoc.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice strong_extm" 'strong_extm = (cic:/matita/lattice/strong_extm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice le_to_eqm" 'le_to_eqm = (cic:/matita/lattice/le_to_eqm.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice lem" 'lem = (cic:/matita/lattice/lem.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice join" 'join = (cic:/matita/lattice/meet.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_refl" 'join_refl = (cic:/matita/lattice/meet_refl.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_comm" 'join_comm = (cic:/matita/lattice/meet_comm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice join_assoc" 'join_assoc = (cic:/matita/lattice/meet_assoc.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice strong_extm" 'strong_extj = (cic:/matita/lattice/strong_extm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice le_to_eqj" 'le_to_eqj = (cic:/matita/lattice/le_to_eqm.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice lej" 'lej = (cic:/matita/lattice/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}.
+interpretation "Lattice feq_jl" 'feq_jl = (cic:/matita/lattice/feq_ml.con (cic:/matita/lattice/latt_jcarr.con _)).
+interpretation "Lattice feq_jr" 'feq_jr = (cic:/matita/lattice/feq_mr.con (cic:/matita/lattice/latt_jcarr.con _)).
+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_ml" 'feq_ml = (cic:/matita/lattice/feq_ml.con (cic:/matita/lattice/latt_mcarr.con _)).
+interpretation "Lattice feq_mr" 'feq_mr = (cic:/matita/lattice/feq_mr.con (cic:/matita/lattice/latt_mcarr.con _)).
projects / helm.git / blobdiff