(* 3.20 *)
lemma supremum_uniq:
- ∀R.∀ml:mlattice R.∀xn:sequence ml.increasing ? xn → (* BUG again the wrong coercion is chosen *)
- ∀x,y:apart_of_metric_space ? ml.supremum ?? xn x → supremum ?? xn y → x ≈ y.
+ ∀R.∀ml:mlattice R.∀xn:sequence ml.increasing ? xn →
+ ∀x,y:ml.supremum ?? xn x → supremum ?? xn y → δ x y ≈ 0.
intros (R ml xn Hxn x y Sx Sy);
elim (Sx Hxn) (_ Hx); elim (Sy Hxn) (_ Hy);
apply (tends_uniq ?? xn ?? Hx Hy);
[ O ⇒ (shift ?? xn k) O
| S n1 ⇒ (shift ?? xn k) (S n1) ∧ ank_aux n1]
in ank_aux.
+
+notation < "'a'\sup k" non associative with precedence 50 for
+ @{ 'ank $x $k }.
+
+interpretation "ank" 'ank x k =
+ (cic:/matita/infsup/ank.con _ _ x k).
+
+notation < "'a'(k \atop n)" non associative with precedence 50 for
+ @{ 'ank2 $x $k $n }.
+
+interpretation "ank2" 'ank2 x k n =
+ (cic:/matita/infsup/ank.con _ _ x k n).
definition bnk ≝
λR.λml:mlattice R.λxn:sequence ml.λk:nat.
in bnk_aux.
lemma ank_decreasing:
- ∀R.∀ml:mlattice R.∀xn:sequence ml.∀m.decreasing ? (ank ?? xn m).
-intros (R ml xn m); unfold; intro n; simplify; apply lem;
+ ∀R.∀ml:mlattice R.∀xn:sequence ml.∀k.decreasing ? (ank ?? xn k).
+intros (R ml xn k); unfold; intro n; simplify; apply lem;
qed.
(* 3.26 *)
lemma ankS:
∀R.∀ml:mlattice R.∀xn:sequence ml.∀k,n:nat.
- ((ank ?? xn k) (S n)) ≈ (xn k ∧ ank ?? xn (S k) n).
+ ((ank ?? xn k) (S n)) ≈ (xn k ∧ ank ?? xn (S k) n).
intros (R ml xn k n); elim n; simplify; [apply meet_comm]
simplify in H; apply (Eq≈ ? (feq_ml ???? (H))); clear H;
apply (Eq≈ ? (meet_assoc ????));
simplify; rewrite > sym_plus in ⊢ (? ? ? (? ? ? (? (? %))));
apply meet_comm;
qed.
+
+(* 3.27 *)
+lemma foo:
+ ∀R.∀ml:mlattice R.∀xn:sequence ml.
+ ∀alpha:sequence ml. (∀k.strong_inf ml (ank ?? xn k) (alpha k)) →
+ increasing ml alpha.
+intros (R ml xn alpha H); unfold strong_inf in H; unfold lower_bound in H; unfold;
+intro n;
+letin H2 ≝ (λk.ankS ?? xn k n); clearbody H2;
+cut (∀k.((xn k) ∧ (ank ?? xn (S k) n)) ≤ (ank ?? xn (S k) n)) as H3; [2:intro k; apply lem;]
+cut (∀k.(ank ?? xn k (S n)) ≤ (ank ?? xn (S k) n)) as H4; [2:
+ intro k; apply (le_transitive ml ???? (H3 ?));
+ apply (Le≪ ? (H2 k));
+
+elim (H (S n)) (H4 H5); intro H6; elim (H5 ? H6) (m Hm);
+lapply (H4 m) as H7;
+
+ clear H5 H6;
+
+
+
+lapply (H n) as H1; clear H; elim H1 (LB H); clear H1;
+lapply (LB (S n)) as H1; clear LB;
+lapply (ankS ?? xn n n) as H2;
+
+lapply (Le≪ (xn n∧ank R ml xn (S n) n) H2);
+
+cases H (LB H1); clear H;
+
+
+
\ No newline at end of file
(* *)
(**************************************************************************)
-
-
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 lattice_ : Type ≝ {
+ l_carr: apartness;
+ l_meet: l_carr → l_carr → l_carr;
+ l_meet_refl: ∀x.l_meet x x ≈ x;
+ l_meet_comm: ∀x,y:l_carr. l_meet x y ≈ l_meet y x;
+ l_meet_assoc: ∀x,y,z:l_carr. l_meet x (l_meet y z) ≈ l_meet (l_meet x y) z;
+ l_strong_extm: ∀x.strong_ext ? (l_meet x)
}.
-interpretation "Lattice meet" 'and a b =
- (cic:/matita/lattice/meet.con _ a b).
+definition excl ≝ λl:lattice_.λa,b:l_carr l.ap_apart (l_carr l) a (l_meet l 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;
+lemma excess_of_lattice_: lattice_ → excess.
+intro l; apply (mk_excess (l_carr l) (excl l));
+[ intro x; unfold; intro H; unfold in H; apply (ap_coreflexive (l_carr l) x);
+ apply (ap_rewr ??? (l_meet l x x) (l_meet_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 ??? (l_meet l (l_meet l x z) y) H) (H1 H2); [2:
+ left; apply ap_symmetric; apply (l_strong_extm ? y);
+ apply (ap_rewl ???? (l_meet_comm ???));
+ apply (ap_rewr ???? (l_meet_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 ??? (l_meet l x z) H1) (H2 H3); [left; assumption]
+ right; apply (l_strong_extm ? x); apply (ap_rewr ???? (l_meet_assoc ????));
assumption]
qed.
-coercion cic:/matita/lattice/excess_of_lattice.con.
+(* 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 "Lattice 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 prelattice_of_lattice_: lattice_ → prelattice.
+intro l_; apply (mk_prelattice (excess_of_lattice_ l_)); [apply (l_meet l_);]
+unfold excess_of_lattice_; try unfold apart_of_excess; simplify;
+unfold excl; simplify;
+[intro x; intro H; elim H; clear H;
+ [apply (l_meet_refl l_ x);
+ lapply (Ap≫ ? (l_meet_comm ???) t) as H; clear t;
+ lapply (l_strong_extm l_ ??? H); apply ap_symmetric; assumption
+ | lapply (Ap≪ ? (l_meet_refl ?x) t) as H; clear t;
+ lapply (l_strong_extm l_ ??? H); apply (l_meet_refl l_ x);
+ apply ap_symmetric; assumption]
+|intros 3 (x y H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (l_meet_refl ? (l_meet l_ x y)) H1) as H; clear H1;
+ lapply (l_strong_extm l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (l_meet_comm ???) H1); apply (ap_coreflexive ?? Hletin);
+ |lapply (Ap≪ ? (l_meet_refl ? (l_meet l_ y x)) H2) as H; clear H2;
+ lapply (l_strong_extm l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (l_meet_comm ???) H1);apply (ap_coreflexive ?? Hletin);]
+|intros 4 (x y z H); cases H (H1 H2); clear H;
+ [lapply (Ap≪ ? (l_meet_refl ? (l_meet l_ x (l_meet l_ y z))) H1) as H; clear H1;
+ lapply (l_strong_extm l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (eq_sym ??? (l_meet_assoc ?x y z)) H1) as H; clear H1;
+ apply (ap_coreflexive ?? H);
+ |lapply (Ap≪ ? (l_meet_refl ? (l_meet l_ (l_meet l_ x y) z)) H2) as H; clear H2;
+ lapply (l_strong_extm l_ ??? H) as H1; clear H;
+ lapply (Ap≪ ? (l_meet_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≪ ? (l_meet_refl ??) H1) as H; clear H1;
+ lapply (l_strong_extm l_ ??? H) as H1; clear H;
+ lapply (l_strong_extm l_ ??? H1) as H; clear H1;
+ cases (ap_cotransitive ??? (l_meet l_ y z) H);[left|right|right|left] try assumption;
+ [apply ap_symmetric;apply (Ap≪ ? (l_meet_comm ???));
+ |apply (Ap≫ ? (l_meet_comm ???));
+ |apply ap_symmetric;] assumption;
+|intros 4 (x y H H1); apply H; clear H; elim H1 (H H);
+ lapply (Ap≪ ? (l_meet_refl ??) H) as H1; clear H;
+ lapply (l_strong_extm l_ ??? H1) as H; clear H1;[2: apply ap_symmetric]
+ assumption
+|intros 3 (x y H);
+ cut (l_meet l_ x y ≈ l_meet l_ x (l_meet l_ y y)) as H1;[2:
+ intro; lapply (l_strong_extm ???? a); apply (l_meet_refl l_ y);
+ apply ap_symmetric; assumption;]
+ lapply (Ap≪ ? (eq_sym ??? H1) H); apply (l_meet_assoc l_ x y y);
+ assumption; ]
+qed.
+
+record lattice : Type ≝ {
+ lat_carr:> prelattice;
+ join: lat_carr → lat_carr → lat_carr;
+ join_refl: ∀x.join x x ≈ x;
+ join_comm: ∀x,y:lat_carr. join x y ≈ join y x;
+ join_assoc: ∀x,y,z:lat_carr. join x (join y z) ≈ join (join x y) z;
+ absorbjm: ∀f,g:lat_carr. join f (meet ? f g) ≈ f;
+ absorbmj: ∀f,g:lat_carr. meet ? f (join f g) ≈ f;
+ strong_extj: ∀x.strong_ext ? (join x)
+}.
+
+interpretation "Lattice join" 'or a b =
+ (cic:/matita/lattice/join.con _ a b).
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;
-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≈ (b∨(b∧a)) ? (feq_jl ???? (meet_comm ???)));
apply eq_sym; apply absorbjm;
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;
-qed.
-
-
projects / helm.git / commitdiff