(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/lattice/".
+include "excess.ma".
-include "excedence.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)
+}.
-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)
+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.
+
+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.
+*)
+
+
+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/meet.con _ 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/join.con _ a b).
-
-definition excl ≝ λl:lattice.λa,b:l.a # (a ∧ b).
-
-lemma excedence_of_lattice: lattice → excedence.
-intro l; apply (mk_excedence 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.
+ (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/excedence_of_lattice.con.
\ No newline at end of file
+coercion cic:/matita/lattice/le_to_ge.con nocomposites.
+coercion cic:/matita/lattice/ge_to_le.con nocomposites.
\ No newline at end of file
projects / helm.git / blobdiff