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)
+record is_order_relation (C:Type) (le:C→C→Prop) : Type ≝
+ { or_reflexive: reflexive ? le;
+ or_transitive: transitive ? le;
+ or_antisimmetric: antisimmetric ? le
}.
-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
+record ordered_set: Type ≝
+ { os_carrier:> Type;
+ os_le: os_carrier → os_carrier → Prop;
+ os_order_relation_properties:> is_order_relation ? os_le
}.
+interpretation "Ordered Sets le" 'leq a b =
+ (cic:/matita/ordered_sets/os_le.con _ a b).
+
theorem antisimmetric_to_cotransitive_to_transitive:
∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
transitive ? le.
].
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:ordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
+definition is_decreasing ≝ λO:ordered_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:ordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
+definition is_lower_bound ≝ λO:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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:ordered_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).
+ ∀O:ordered_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:ordered_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).
+ ∀O:ordered_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.
-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' ])
- }.
+definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
-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
- }.
+interpretation "Ordered set lt" 'lt a b =
+ (cic:/matita/ordered_sets/lt.con _ a b).
-definition inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_below_sequence ? O → O.
+definition reverse_ordered_set: ordered_set → ordered_set.
intros;
- elim
- (dsc_inf ? O (dscos_dedekind_sigma_complete_properties ? O) b);
- [ apply a
- | apply (lower_bound ? ? b)
- | apply lower_bound_is_lower_bound
- ]
+ apply mk_ordered_set;
+ [2:apply (λx,y:o.y ≤ x)
+ | skip
+ | 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.
-
-lemma inf_is_inf:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_below_sequence ? O.
- is_inf ? ? a (inf ? ? a).
+
+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:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+ is_lower_bound O a l → is_upper_bound (reverse_ordered_set O) a l.
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));
+ unfold;
+ intro;
+ unfold;
+ unfold reverse_ordered_set;
simplify;
- assumption.
+ apply H.
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;
+lemma is_upper_bound_reverse_is_lower_bound:
+ ∀O:ordered_set.∀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;
- 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.
+ apply H.
qed.
-definition sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- bounded_above_sequence ? O → O.
+lemma reverse_is_lower_bound_is_upper_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+ is_lower_bound (reverse_ordered_set O) a l → is_upper_bound O a l.
intros;
- elim
- (dsc_sup ? O (dscos_dedekind_sigma_complete_properties ? O) b);
- [ apply a
- | apply (upper_bound ? ? b)
- | apply upper_bound_is_upper_bound
- ].
+ unfold in H;
+ unfold reverse_ordered_set in H;
+ apply H.
qed.
-lemma sup_is_sup:
- ∀C:Type.∀O:dedekind_sigma_complete_ordered_set C.
- ∀a:bounded_above_sequence ? O.
- is_sup ? ? a (sup ? ? a).
+lemma reverse_is_upper_bound_is_lower_bound:
+ ∀O:ordered_set.∀a:nat→O.∀l:O.
+ is_upper_bound (reverse_ordered_set O) a l → is_lower_bound O a l.
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.
+ unfold in H;
+ unfold reverse_ordered_set in H;
+ apply H.
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 is_inf_to_reverse_is_sup:
+ ∀O:ordered_set.∀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 (reverse_ordered_set O));
+ [ apply is_lower_bound_reverse_is_upper_bound;
+ apply inf_lower_bound;
+ assumption
+ | intros;
+ change in v with (os_carrier O);
+ change with (v ≤ l);
+ apply (inf_greatest_lower_bound ? ? ? H);
+ apply reverse_is_upper_bound_is_lower_bound;
+ 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))));
+
+lemma is_sup_to_reverse_is_inf:
+ ∀O:ordered_set.∀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 (reverse_ordered_set O));
+ [ apply is_upper_bound_reverse_is_lower_bound;
+ apply sup_upper_bound;
+ assumption
+ | intros;
+ change in v with (os_carrier O);
+ change with (l ≤ v);
+ apply (sup_least_upper_bound ? ? ? H);
+ apply reverse_is_lower_bound_is_upper_bound;
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.
+lemma reverse_is_sup_to_is_inf:
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+ is_sup (reverse_ordered_set O) a l → is_inf O a l.
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)
+ apply mk_is_inf;
+ [ apply reverse_is_upper_bound_is_lower_bound;
+ change in l with (os_carrier (reverse_ordered_set O));
+ apply sup_upper_bound;
+ assumption
+ | intros;
+ change in l with (os_carrier (reverse_ordered_set O));
+ change in v with (os_carrier (reverse_ordered_set O));
+ change with (os_le (reverse_ordered_set O) l v);
+ apply (sup_least_upper_bound ? ? ? H);
+ change in v with (os_carrier O);
+ apply is_lower_bound_reverse_is_upper_bound;
+ assumption
].
qed.
-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').
+lemma reverse_is_inf_to_is_sup:
+ ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O.
+ is_inf (reverse_ordered_set O) a l → is_sup O a l.
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
- ].
+ apply mk_is_sup;
+ [ apply reverse_is_lower_bound_is_upper_bound;
+ change in l with (os_carrier (reverse_ordered_set O));
+ apply (inf_lower_bound ? ? ? H)
+ | intros;
+ change in l with (os_carrier (reverse_ordered_set O));
+ change in v with (os_carrier (reverse_ordered_set O));
+ change with (os_le (reverse_ordered_set O) v l);
+ apply (inf_greatest_lower_bound ? ? ? H);
+ change in v with (os_carrier O);
+ apply is_upper_bound_reverse_is_lower_bound;
+ assumption
+ ].
qed.
-
-
-
-definition lt ≝ λC.λO:ordered_set C.λa,b:O.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 cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+ cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.
projects / helm.git / blobdiff