set "baseuri" "cic:/matita/ordered_sets/".
-include "higher_order_defs/relations.ma".
-include "nat/plus.ma".
-include "constructive_connectives.ma".
-
-record pre_ordered_set (C:Type) : Type ≝
- { le_:C→C→Prop }.
-
-definition carrier_of_pre_ordered_set ≝ λC:Type.λO:pre_ordered_set C.C.
-
-coercion cic:/matita/ordered_sets/carrier_of_pre_ordered_set.con.
-
-definition os_le: ∀C.∀O:pre_ordered_set C.O → O → Prop ≝ le_.
-
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ _ a b).
-
-definition cotransitive ≝
- λC:Type.λle:C→C→Prop.∀x,y,z:C. le x y → le x z ∨ le z y.
-
-definition antisimmetric ≝
- λC:Type.λle:C→C→Prop.∀x,y:C. le x y → le y x → x=y.
-
-record is_order_relation (C) (O:pre_ordered_set C) : Type ≝
- { or_reflexive: reflexive ? (os_le ? O);
- or_transitive: transitive ? (os_le ? O);
- or_antisimmetric: antisimmetric ? (os_le ? O)
- }.
+include "excedence.ma".
+
+record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ {
+ por_reflexive: reflexive ? le;
+ por_transitive: transitive ? le;
+ por_antisimmetric: antisymmetric ? le eq
+}.
+
+record pordered_set: Type ≝ {
+ pos_carr:> excedence;
+ pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
+}.
+
+lemma pordered_set_of_excedence: excedence → pordered_set.
+intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation);
+[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
+qed.
-record ordered_set (C:Type): Type ≝
- { os_pre_ordered_set:> pre_ordered_set C;
- os_order_relation_properties:> is_order_relation ? os_pre_ordered_set
- }.
+alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con".
+alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con".
+alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
theorem antisimmetric_to_cotransitive_to_transitive:
- ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
- transitive ? le.
- intros;
- unfold transitive;
- intros;
- elim (c ? ? z H1);
- [ assumption
- | rewrite < (H ? ? H2 t);
- assumption
- ].
+ ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.
+intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
+lapply (cT ? ? fxy z) as H; cases H; [assumption] cases (Af ? ? fyz H1);
qed.
-definition is_increasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a n ≤ a (S n).
-definition is_decreasing ≝ λC.λO:ordered_set C.λa:nat→O.∀n:nat.a (S n) ≤ a n.
+definition is_increasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
+definition is_decreasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n.
-definition is_upper_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λC.λO:ordered_set C.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
+definition is_upper_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
-record is_sup (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { sup_upper_bound: is_upper_bound ? O a u;
- sup_least_upper_bound: ∀v:O. is_upper_bound ? O a v → u≤v
+record is_sup (O:pordered_set) (a:nat→O) (u:O) : Prop ≝
+ { sup_upper_bound: is_upper_bound O a u;
+ sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v
}.
-record is_inf (C:Type) (O:ordered_set C) (a:nat→O) (u:O) : Prop ≝
- { inf_lower_bound: is_lower_bound ? O a u;
- inf_greatest_lower_bound: ∀v:O. is_lower_bound ? O a v → v≤u
+record is_inf (O:pordered_set) (a:nat→O) (u:O) : Prop ≝
+ { inf_lower_bound: is_lower_bound O a u;
+ inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u
}.
-record is_bounded_below (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_below (O:pordered_set) (a:nat→O) : Type ≝
{ ib_lower_bound: O;
- ib_lower_bound_is_lower_bound: is_lower_bound ? ? a ib_lower_bound
+ ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound
}.
-record is_bounded_above (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
+record is_bounded_above (O:pordered_set) (a:nat→O) : Type ≝
{ ib_upper_bound: O;
- ib_upper_bound_is_upper_bound: is_upper_bound ? ? a ib_upper_bound
+ ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound
}.
-record is_bounded (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
- { ib_bounded_below:> is_bounded_below ? ? a;
- ib_bounded_above:> is_bounded_above ? ? a
+record is_bounded (O:pordered_set) (a:nat→O) : Type ≝
+ { ib_bounded_below:> is_bounded_below ? a;
+ ib_bounded_above:> is_bounded_above ? a
}.
-record bounded_below_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_below_sequence (O:pordered_set) : Type ≝
{ bbs_seq:1> nat→O;
- bbs_is_bounded_below:> is_bounded_below ? ? bbs_seq
+ bbs_is_bounded_below:> is_bounded_below ? bbs_seq
}.
-record bounded_above_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_above_sequence (O:pordered_set) : Type ≝
{ bas_seq:1> nat→O;
- bas_is_bounded_above:> is_bounded_above ? ? bas_seq
+ bas_is_bounded_above:> is_bounded_above ? bas_seq
}.
-record bounded_sequence (C:Type) (O:ordered_set C) : Type ≝
+record bounded_sequence (O:pordered_set) : Type ≝
{ bs_seq:1> nat → O;
- bs_is_bounded_below: is_bounded_below ? ? bs_seq;
- bs_is_bounded_above: is_bounded_above ? ? bs_seq
+ bs_is_bounded_below: is_bounded_below ? bs_seq;
+ bs_is_bounded_above: is_bounded_above ? bs_seq
}.
definition bounded_below_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
- mk_bounded_below_sequence ? ? b (bs_is_bounded_below ? ? b).
+ λO:pordered_set.λb:bounded_sequence O.
+ mk_bounded_below_sequence ? b (bs_is_bounded_below ? b).
coercion cic:/matita/ordered_sets/bounded_below_sequence_of_bounded_sequence.con.
definition bounded_above_sequence_of_bounded_sequence ≝
- λC.λO:ordered_set C.λb:bounded_sequence ? O.
- mk_bounded_above_sequence ? ? b (bs_is_bounded_above ? ? b).
+ λO:pordered_set.λb:bounded_sequence O.
+ mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con.
definition lower_bound ≝
- λC.λO:ordered_set C.λb:bounded_below_sequence ? O.
- ib_lower_bound ? ? b (bbs_is_bounded_below ? ? b).
+ λO:pordered_set.λb:bounded_below_sequence O.
+ ib_lower_bound ? b (bbs_is_bounded_below ? b).
lemma lower_bound_is_lower_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_below_sequence ? O.
- is_lower_bound ? ? b (lower_bound ? ? b).
- intros;
- unfold lower_bound;
- apply ib_lower_bound_is_lower_bound.
+ ∀O:pordered_set.∀b:bounded_below_sequence O.
+ is_lower_bound ? b (lower_bound ? b).
+intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
qed.
definition upper_bound ≝
- λC.λO:ordered_set C.λb:bounded_above_sequence ? O.
- ib_upper_bound ? ? b (bas_is_bounded_above ? ? b).
+ λO:pordered_set.λb:bounded_above_sequence O.
+ ib_upper_bound ? b (bas_is_bounded_above ? b).
lemma upper_bound_is_upper_bound:
- ∀C.∀O:ordered_set C.∀b:bounded_above_sequence ? O.
- is_upper_bound ? ? b (upper_bound ? ? b).
- intros;
- unfold upper_bound;
- apply ib_upper_bound_is_upper_bound.
-qed.
-
-record is_dedekind_sigma_complete (C:Type) (O:ordered_set C) : Type ≝
- { dsc_inf: ∀a:nat→O.∀m:O. is_lower_bound ? ? a m → ex ? (λs:O.is_inf ? O a s);
- dsc_inf_proof_irrelevant:
- ∀a:nat→O.∀m,m':O.∀p:is_lower_bound ? ? a m.∀p':is_lower_bound ? ? a m'.
- (match dsc_inf a m p with [ ex_intro s _ ⇒ s ]) =
- (match dsc_inf a m' p' with [ ex_intro s' _ ⇒ s' ]);
- dsc_sup: ∀a:nat→O.∀m:O. is_upper_bound ? ? a m → ex ? (λs:O.is_sup ? O a s);
- dsc_sup_proof_irrelevant:
- ∀a:nat→O.∀m,m':O.∀p:is_upper_bound ? ? a m.∀p':is_upper_bound ? ? a m'.
- (match dsc_sup a m p with [ ex_intro s _ ⇒ s ]) =
- (match dsc_sup a m' p' with [ ex_intro s' _ ⇒ s' ])
- }.
-
-record dedekind_sigma_complete_ordered_set (C:Type) : Type ≝
- { dscos_ordered_set:> ordered_set C;
- dscos_dedekind_sigma_complete_properties:>
- is_dedekind_sigma_complete ? dscos_ordered_set
- }.
-
-definition inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_below_sequence ? O → O.
- intros;
- elim
- (dsc_inf ? O (dscos_dedekind_sigma_complete_properties ? O) b);
- [ apply a
- | apply (lower_bound ? ? b)
- | apply lower_bound_is_lower_bound
- ]
-qed.
-
-lemma inf_is_inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.
- is_inf ? ? a (inf ? ? a).
- intros;
- unfold inf;
- simplify;
- elim (dsc_inf C O (dscos_dedekind_sigma_complete_properties C O) a
-(lower_bound C O a) (lower_bound_is_lower_bound C O a));
- simplify;
- assumption.
-qed.
-
-lemma inf_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a,a':bounded_below_sequence ? O.
- bbs_seq ? ? a = bbs_seq ? ? a' →
- inf ? ? a = inf ? ? a'.
- intros 4;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_inf_proof_irrelevant C O O f (ib_lower_bound ? ? f i2)
- (ib_lower_bound ? ? f i) (ib_lower_bound_is_lower_bound ? ? f i2)
- (ib_lower_bound_is_lower_bound ? ? f i));
- reflexivity.
+ ∀O:pordered_set.∀b:bounded_above_sequence O.
+ is_upper_bound ? b (upper_bound ? b).
+intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
qed.
-definition sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_above_sequence ? O → O.
- intros;
- elim
- (dsc_sup ? O (dscos_dedekind_sigma_complete_properties ? O) b);
- [ apply a
- | apply (upper_bound ? ? b)
- | apply upper_bound_is_upper_bound
- ].
-qed.
-
-lemma sup_is_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_above_sequence ? O.
- is_sup ? ? a (sup ? ? a).
- intros;
- unfold sup;
- simplify;
- elim (dsc_sup C O (dscos_dedekind_sigma_complete_properties C O) a
-(upper_bound C O a) (upper_bound_is_upper_bound C O a));
- simplify;
- assumption.
-qed.
-
-lemma sup_proof_irrelevant:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a,a':bounded_above_sequence ? O.
- bas_seq ? ? a = bas_seq ? ? a' →
- sup ? ? a = sup ? ? a'.
- intros 4;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_sup_proof_irrelevant C O O f (ib_upper_bound ? ? f i2)
- (ib_upper_bound ? ? f i) (ib_upper_bound_is_upper_bound ? ? f i2)
- (ib_upper_bound_is_upper_bound ? ? f i));
- reflexivity.
-qed.
-
-axiom daemon: False.
-
-theorem inf_le_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_sequence ? O. inf ? ? a ≤ sup ? ? a.
- intros (C O');
- apply (or_transitive ? ? O' ? (a O));
- [ elim daemon (*apply (inf_lower_bound ? ? ? ? (inf_is_inf ? ? a))*)
- | elim daemon (*apply (sup_upper_bound ? ? ? ? (sup_is_sup ? ? a))*)
- ].
+lemma Or_symmetric: symmetric ? Or.
+unfold; intros (x y H); cases H; [right|left] assumption;
qed.
-lemma inf_respects_le:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.∀m:O.
- is_upper_bound ? ? a m → inf ? ? a ≤ m.
- intros (C O');
- apply (or_transitive ? ? O' ? (sup ? ? (mk_bounded_sequence ? ? a ? ?)));
- [ apply (bbs_is_bounded_below ? ? a)
- | apply (mk_is_bounded_above ? ? ? m H)
- | apply inf_le_sup
- | apply
- (sup_least_upper_bound ? ? ? ?
- (sup_is_sup ? ? (mk_bounded_sequence C O' a a
- (mk_is_bounded_above C O' a m H))));
- assumption
- ].
+definition reverse_excedence: excedence → excedence.
+intros (E); apply (mk_excedence E); [apply (λx,y.exc_relation E y x)]
+cases E (T f cRf cTf); simplify;
+[1: unfold Not; intros (x H); apply (cRf x); assumption
+|2: intros (x y z); apply Or_symmetric; apply cTf; assumption;]
qed.
-definition is_sequentially_monotone ≝
- λC.λO:ordered_set C.λf:O→O.
- ∀a:nat→O.∀p:is_increasing ? ? a.
- is_increasing ? ? (λi.f (a i)).
-
-record is_order_continuous (C)
- (O:dedekind_sigma_complete_ordered_set C) (f:O→O) : Prop
-≝
- { ioc_is_sequentially_monotone: is_sequentially_monotone ? ? f;
- ioc_is_upper_bound_f_sup:
- ∀a:bounded_above_sequence ? O.
- is_upper_bound ? ? (λi.f (a i)) (f (sup ? ? a));
- ioc_respects_sup:
- ∀a:bounded_above_sequence ? O.
- is_increasing ? ? a →
- f (sup ? ? a) =
- sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i))
- (mk_is_bounded_above ? ? ? (f (sup ? ? a))
- (ioc_is_upper_bound_f_sup a)));
- ioc_is_lower_bound_f_inf:
- ∀a:bounded_below_sequence ? O.
- is_lower_bound ? ? (λi.f (a i)) (f (inf ? ? a));
- ioc_respects_inf:
- ∀a:bounded_below_sequence ? O.
- is_decreasing ? ? a →
- f (inf ? ? a) =
- inf ? ? (mk_bounded_below_sequence ? ? (λi.f (a i))
- (mk_is_bounded_below ? ? ? (f (inf ? ? a))
- (ioc_is_lower_bound_f_inf a)))
- }.
-
-theorem tail_inf_increasing:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.
- let y ≝ λi.mk_bounded_below_sequence ? ? (λj.a (i+j)) ? in
- let x ≝ λi.inf ? ? (y i) in
- is_increasing ? ? x.
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
- simplify;
- intro;
- apply (ib_lower_bound_is_lower_bound ? ? a a)
- | intros;
- unfold is_increasing;
- intro;
- unfold x in ⊢ (? ? ? ? %);
- apply (inf_greatest_lower_bound ? ? ? ? (inf_is_inf ? ? (y (S n))));
- change with (is_lower_bound ? ? (y (S n)) (inf ? ? (y n)));
- unfold is_lower_bound;
- intro;
- generalize in match (inf_lower_bound ? ? ? ? (inf_is_inf ? ? (y n)) (S n1));
- (*CSC: coercion per FunClass inserita a mano*)
- suppose (inf ? ? (y n) ≤ bbs_seq ? ? (y n) (S n1)) (H);
- cut (bbs_seq ? ? (y n) (S n1) = bbs_seq ? ? (y (S n)) n1);
- [ rewrite < Hcut;
- assumption
- | unfold y;
- simplify;
- auto paramodulation library
- ]
- ].
-qed.
-
-definition is_liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_below_sequence ? O → O → Prop.
- intros;
- apply
- (is_sup ? ? (λi.inf ? ? (mk_bounded_below_sequence ? ? (λj.b (i+j)) ?)) t);
- apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
- simplify;
- intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b).
-qed.
-
-definition liminf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_sequence ? O → O.
- intros;
- apply
- (sup ? ?
- (mk_bounded_above_sequence ? ?
- (λi.inf ? ?
- (mk_bounded_below_sequence ? ?
- (λj.b (i+j)) ?)) ?));
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? b b));
- simplify;
- intros;
- apply (ib_lower_bound_is_lower_bound ? ? b b)
- | apply (mk_is_bounded_above ? ? ? (ib_upper_bound ? ? b b));
- unfold is_upper_bound;
- intro;
- change with
- (inf C O
- (mk_bounded_below_sequence C O (\lambda j:nat.b (n+j))
- (mk_is_bounded_below C O (\lambda j:nat.b (n+j)) (ib_lower_bound C O b b)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O b b (n+j))))
-\leq ib_upper_bound C O b b);
- apply (inf_respects_le ? O);
- simplify;
- intro;
- apply (ib_upper_bound_is_upper_bound ? ? b b)
- ].
-qed.
-
-definition reverse_ordered_set: ∀C.ordered_set C → ordered_set C.
- intros;
- apply mk_ordered_set;
- [ apply mk_pre_ordered_set;
- apply (λx,y:o.y ≤ x)
- | apply mk_is_order_relation;
- [ simplify;
- intros;
- apply (or_reflexive ? ? o)
- | simplify;
- intros;
- apply (or_transitive ? ? o);
- [2: apply H1
- | skip
- | assumption
- ]
- | simplify;
- intros;
- apply (or_antisimmetric ? ? o);
- assumption
- ]
- ].
-qed.
+definition reverse_pordered_set: pordered_set → pordered_set.
+intros (p); apply (mk_pordered_set (reverse_excedence p));
+generalize in match (reverse_excedence p); intros (E); cases E (T f cRf cTf);
+simplify; apply mk_is_porder_relation; unfold; intros;
+[apply le_reflexive|apply (le_transitive ???? H H1);|apply (le_antisymmetric ??? H H1)]
+qed.
-interpretation "Ordered set ge" 'geq a b =
- (cic:/matita/ordered_sets/os_le.con _
- (cic:/matita/ordered_sets/os_pre_ordered_set.con _
- (cic:/matita/ordered_sets/reverse_ordered_set.con _ _)) a b).
-
lemma is_lower_bound_reverse_is_upper_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
- is_lower_bound ? O a l → is_upper_bound ? (reverse_ordered_set ? O) a l.
- intros;
- unfold;
- intro;
- unfold;
- unfold reverse_ordered_set;
- simplify;
- apply H.
+ ∀O:pordered_set.∀a:nat→O.∀l:O.
+ is_lower_bound O a l → is_upper_bound (reverse_pordered_set O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+unfold reverse_excedence; simplify; fold unfold le (le ? l (a n)); apply H;
qed.
lemma is_upper_bound_reverse_is_lower_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
- is_upper_bound ? O a l → is_lower_bound ? (reverse_ordered_set ? O) a l.
- intros;
- unfold;
- intro;
- unfold;
- unfold reverse_ordered_set;
- simplify;
- apply H.
+ ∀O:pordered_set.∀a:nat→O.∀l:O.
+ is_upper_bound O a l → is_lower_bound (reverse_pordered_set O) a l.
+intros (O a l H); unfold; intros (n); unfold reverse_pordered_set;
+unfold reverse_excedence; simplify; fold unfold le (le ? (a n) l); apply H;
qed.
lemma reverse_is_lower_bound_is_upper_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
- is_lower_bound ? (reverse_ordered_set ? O) a l → is_upper_bound ? O a l.
- intros;
- unfold in H;
- unfold reverse_ordered_set in H;
- apply H.
+ ∀O:pordered_set.∀a:nat→O.∀l:O.
+ is_lower_bound (reverse_pordered_set O) a l → is_upper_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+unfold reverse_excedence in H; simplify in H; apply H;
qed.
lemma reverse_is_upper_bound_is_lower_bound:
- ∀C.∀O:ordered_set C.∀a:nat→O.∀l:O.
- is_upper_bound ? (reverse_ordered_set ? O) a l → is_lower_bound ? O a l.
- intros;
- unfold in H;
- unfold reverse_ordered_set in H;
- apply H.
+ ∀O:pordered_set.∀a:nat→O.∀l:O.
+ is_upper_bound (reverse_pordered_set O) a l → is_lower_bound O a l.
+intros (O a l H); unfold; intros (n); unfold reverse_pordered_set in H;
+unfold reverse_excedence in H; simplify in H; apply H;
qed.
-
lemma is_inf_to_reverse_is_sup:
- ∀C.∀O:ordered_set C.∀a:bounded_below_sequence ? O.∀l:O.
- is_inf ? O a l → is_sup ? (reverse_ordered_set ? O) a l.
- intros;
- apply (mk_is_sup C (reverse_ordered_set ? ?));
- [ apply is_lower_bound_reverse_is_upper_bound;
- apply inf_lower_bound;
- assumption
- | intros;
- change in v with (Type_OF_ordered_set ? O);
- change with (v ≤ l);
- apply (inf_greatest_lower_bound ? ? ? ? H);
- apply reverse_is_upper_bound_is_lower_bound;
- assumption
- ].
+ ∀O:pordered_set.∀a:bounded_below_sequence O.∀l:O.
+ is_inf O a l → is_sup (reverse_pordered_set O) a l.
+intros (O a l H); apply (mk_is_sup (reverse_pordered_set O));
+[1: apply is_lower_bound_reverse_is_upper_bound; apply inf_lower_bound; assumption
+|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+ intros (m H1); apply (inf_greatest_lower_bound ? ? ? H); apply H1;]
qed.
-
+
lemma is_sup_to_reverse_is_inf:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
- is_sup ? O a l → is_inf ? (reverse_ordered_set ? O) a l.
- intros;
- apply (mk_is_inf C (reverse_ordered_set ? ?));
- [ apply is_upper_bound_reverse_is_lower_bound;
- apply sup_upper_bound;
- assumption
- | intros;
- change in v with (Type_OF_ordered_set ? O);
- change with (l ≤ v);
- apply (sup_least_upper_bound ? ? ? ? H);
- apply reverse_is_lower_bound_is_upper_bound;
- assumption
- ].
+ ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
+ is_sup O a l → is_inf (reverse_pordered_set O) a l.
+intros (O a l H); apply (mk_is_inf (reverse_pordered_set O));
+[1: apply is_upper_bound_reverse_is_lower_bound; apply sup_upper_bound; assumption
+|2: unfold reverse_pordered_set; simplify; unfold reverse_excedence; simplify;
+ intros (m H1); apply (sup_least_upper_bound ? ? ? H); apply H1;]
qed.
lemma reverse_is_sup_to_is_inf:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
- is_sup ? (reverse_ordered_set ? O) a l → is_inf ? O a l.
- intros;
- apply mk_is_inf;
- [ apply reverse_is_upper_bound_is_lower_bound;
- change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- apply sup_upper_bound;
- assumption
- | intros;
- change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- change in v with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- change with (os_le ? (reverse_ordered_set ? O) l v);
- apply (sup_least_upper_bound ? ? ? ? H);
- change in v with (Type_OF_ordered_set ? O);
- apply is_lower_bound_reverse_is_upper_bound;
- assumption
- ].
+ ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
+ is_sup (reverse_pordered_set O) a l → is_inf O a l.
+intros (O a l H); apply mk_is_inf;
+[1: apply reverse_is_upper_bound_is_lower_bound;
+ apply (sup_upper_bound (reverse_pordered_set O)); assumption
+|2: intros (v H1); apply (sup_least_upper_bound (reverse_pordered_set O) a l H v);
+ apply is_lower_bound_reverse_is_upper_bound; assumption;]
qed.
lemma reverse_is_inf_to_is_sup:
- ∀C.∀O:ordered_set C.∀a:bounded_above_sequence ? O.∀l:O.
- is_inf ? (reverse_ordered_set ? O) a l → is_sup ? O a l.
- intros;
- apply mk_is_sup;
- [ apply reverse_is_lower_bound_is_upper_bound;
- change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- apply (inf_lower_bound ? ? ? ? H)
- | intros;
- change in l with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- change in v with (Type_OF_ordered_set ? (reverse_ordered_set ? O));
- change with (os_le ? (reverse_ordered_set ? O) v l);
- apply (inf_greatest_lower_bound ? ? ? ? H);
- change in v with (Type_OF_ordered_set ? O);
- apply is_upper_bound_reverse_is_lower_bound;
- assumption
- ].
-qed.
-
-
-definition reverse_dedekind_sigma_complete_ordered_set:
- ∀C.
- dedekind_sigma_complete_ordered_set C → dedekind_sigma_complete_ordered_set C.
- intros;
- apply mk_dedekind_sigma_complete_ordered_set;
- [ apply (reverse_ordered_set ? d)
- | elim daemon
- (*apply mk_is_dedekind_sigma_complete;
- [ intros;
- elim (dsc_sup ? ? d a m) 0;
- [ generalize in match H; clear H;
- generalize in match m; clear m;
- elim d;
- simplify in a1;
- simplify;
- change in a1 with (Type_OF_ordered_set ? (reverse_ordered_set ? o));
- apply (ex_intro ? ? a1);
- simplify in H1;
- change in m with (Type_OF_ordered_set ? o);
- apply (is_sup_to_reverse_is_inf ? ? ? ? H1)
- | generalize in match H; clear H;
- generalize in match m; clear m;
- elim d;
- simplify;
- change in t with (Type_OF_ordered_set ? o);
- simplify in t;
- apply reverse_is_lower_bound_is_upper_bound;
- assumption
- ]
- | apply is_sup_reverse_is_inf;
- | apply m
- |
- ]*)
- ].
-qed.
-
-definition reverse_bounded_sequence:
- ∀C.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_sequence ? O →
- bounded_sequence ? (reverse_dedekind_sigma_complete_ordered_set ? O).
- intros;
- apply mk_bounded_sequence;
- [ apply bs_seq;
- unfold reverse_dedekind_sigma_complete_ordered_set;
- simplify;
- elim daemon
- | elim daemon
- | elim daemon
- ].
-qed.
-
-definition limsup ≝
- λC:Type.λO:dedekind_sigma_complete_ordered_set C.
- λa:bounded_sequence ? O.
- liminf ? (reverse_dedekind_sigma_complete_ordered_set ? O)
- (reverse_bounded_sequence ? O a).
-
-notation "hvbox(〈a〉)"
- non associative with precedence 45
-for @{ 'hide_everything_but $a }.
-
-interpretation "mk_bounded_above_sequence" 'hide_everything_but a
-= (cic:/matita/ordered_sets/bounded_above_sequence.ind#xpointer(1/1/1) _ _ a _).
-
-interpretation "mk_bounded_below_sequence" 'hide_everything_but a
-= (cic:/matita/ordered_sets/bounded_below_sequence.ind#xpointer(1/1/1) _ _ a _).
-
-theorem eq_f_sup_sup_f:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
- ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
- ∀a:bounded_above_sequence ? O'.
- ∀p:is_increasing ? ? a.
- f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) ?).
- [ apply (mk_is_bounded_above ? ? ? (f (sup ? ? a)));
- apply ioc_is_upper_bound_f_sup;
- assumption
- | intros;
- apply ioc_respects_sup;
- assumption
- ].
-qed.
-
-theorem eq_f_sup_sup_f':
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
- ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
- ∀a:bounded_above_sequence ? O'.
- ∀p:is_increasing ? ? a.
- ∀p':is_bounded_above ? ? (λi.f (a i)).
- f (sup ? ? a) = sup ? ? (mk_bounded_above_sequence ? ? (λi.f (a i)) p').
- intros;
- rewrite > (eq_f_sup_sup_f ? ? f H a H1);
- apply sup_proof_irrelevant;
- reflexivity.
-qed.
-
-theorem eq_f_liminf_sup_f_inf:
- ∀C.∀O':dedekind_sigma_complete_ordered_set C.
- ∀f:O'→O'. ∀H:is_order_continuous ? ? f.
- ∀a:bounded_sequence ? O'.
- let p1 := ? in
- f (liminf ? ? a) =
- sup ? ?
- (mk_bounded_above_sequence ? ?
- (λi.f (inf ? ?
- (mk_bounded_below_sequence ? ?
- (λj.a (i+j))
- ?)))
- p1).
- [ apply (mk_is_bounded_below ? ? ? (ib_lower_bound ? ? a a));
- simplify;
- intro;
- apply (ib_lower_bound_is_lower_bound ? ? a a)
- | apply (mk_is_bounded_above ? ? ? (f (sup ? ? a)));
- unfold is_upper_bound;
- intro;
- apply (or_transitive ? ? O' ? (f (a n)));
- [ generalize in match (ioc_is_lower_bound_f_inf ? ? ? H);
- intro H1;
- simplify in H1;
- rewrite > (plus_n_O n) in ⊢ (? ? ? ? (? (? ? ? ? %)));
- apply (H1 (mk_bounded_below_sequence C O' (\lambda j:nat.a (n+j))
-(mk_is_bounded_below C O' (\lambda j:nat.a (n+j)) (ib_lower_bound C O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (n+j)))) O);
- | elim daemon (*apply (ioc_is_upper_bound_f_sup ? ? ? H)*)
- ]
- | intros;
- unfold liminf;
- clearbody p1;
- generalize in match (\lambda n:nat
-.inf_respects_le C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus n j))
- (mk_is_bounded_below C O' (\lambda j:nat.a (plus n j))
- (ib_lower_bound C O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound C O' a a (plus n j))))
- (ib_upper_bound C O' a a)
- (\lambda n1:nat.ib_upper_bound_is_upper_bound C O' a a (plus n n1)));
- intro p2;
- apply (eq_f_sup_sup_f' ? ? f H (mk_bounded_above_sequence C O'
-(\lambda i:nat
- .inf C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound C O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
-(mk_is_bounded_above C O'
- (\lambda i:nat
- .inf C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound C O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n)))))
- (ib_upper_bound C O' a a) p2)));
- unfold bas_seq;
- change with
- (is_increasing ? ? (\lambda i:nat
-.inf C O'
- (mk_bounded_below_sequence C O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below C O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound C O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound C O' a a (plus i n))))));
- apply tail_inf_increasing
- ].
-qed.
-
-
-
+ ∀O:pordered_set.∀a:bounded_above_sequence O.∀l:O.
+ is_inf (reverse_pordered_set O) a l → is_sup O a l.
+intros (O a l H); apply mk_is_sup;
+[1: apply reverse_is_lower_bound_is_upper_bound;
+ apply (inf_lower_bound (reverse_pordered_set O)); assumption
+|2: intros (v H1); apply (inf_greatest_lower_bound (reverse_pordered_set O) a l H v);
+ apply is_upper_bound_reverse_is_lower_bound; assumption;]
+qed.
+
+(*
+record cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+ cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.
+*)
-definition lt ≝ λC.λO:ordered_set C.λa,b:O.a ≤ b ∧ a ≠ b.
+definition total_order_property : ∀E:excedence. Type ≝
+ λE:excedence. ∀a,b:E. a ≰ b → a < b.
-interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ _ a b).
\ No newline at end of file
+record tordered_set : Type ≝ {
+ tos_poset:> pordered_set;
+ tos_totality: total_order_property tos_poset
+}.
projects / helm.git / blobdiff