apply ib_upper_bound_is_upper_bound.
qed.
-record is_dedekind_sigma_complete (O:ordered_set) : 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 : Type ≝
- { dscos_ordered_set:> ordered_set;
- dscos_dedekind_sigma_complete_properties:>
- is_dedekind_sigma_complete dscos_ordered_set
- }.
-
-definition inf:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_below_sequence O.
- is_inf ? a (inf ? a).
- intros;
- unfold inf;
- simplify;
- elim (dsc_inf O (dscos_dedekind_sigma_complete_properties O) a
-(lower_bound O a) (lower_bound_is_lower_bound O a));
- simplify;
- assumption.
-qed.
-
-lemma inf_proof_irrelevant:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a,a':bounded_below_sequence O.
- bbs_seq ? a = bbs_seq ? a' →
- inf ? a = inf ? a'.
- intros 3;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_inf_proof_irrelevant O O f (ib_lower_bound ? f i)
- (ib_lower_bound ? f i2) (ib_lower_bound_is_lower_bound ? f i)
- (ib_lower_bound_is_lower_bound ? f i2));
- reflexivity.
-qed.
-
-definition sup:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_above_sequence O.
- is_sup ? a (sup ? a).
- intros;
- unfold sup;
- simplify;
- elim (dsc_sup O (dscos_dedekind_sigma_complete_properties O) a
-(upper_bound O a) (upper_bound_is_upper_bound O a));
- simplify;
- assumption.
-qed.
-
-lemma sup_proof_irrelevant:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a,a':bounded_above_sequence O.
- bas_seq ? a = bas_seq ? a' →
- sup ? a = sup ? a'.
- intros 3;
- elim a 0;
- elim a';
- simplify in H;
- generalize in match i1;
- clear i1;
- rewrite > H;
- intro;
- simplify;
- rewrite < (dsc_sup_proof_irrelevant 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:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_sequence O. inf ? a ≤ sup ? a.
- intros (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))*)
- ].
-qed.
-
-lemma inf_respects_le:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀a:bounded_below_sequence O.∀m:O.
- is_upper_bound ? a m → inf ? a ≤ m.
- intros (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 O' a a
- (mk_is_bounded_above O' a m H))));
- assumption
- ].
-qed.
-
-definition is_sequentially_monotone ≝
- λO:ordered_set.λf:O→O.
- ∀a:nat→O.∀p:is_increasing ? a.
- is_increasing ? (λi.f (a i)).
-
-record is_order_continuous
- (O:dedekind_sigma_complete_ordered_set) (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:
- ∀O:dedekind_sigma_complete_ordered_set.
- ∀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;
- autobatch paramodulation library
- ]
- ].
-qed.
-
-definition is_liminf:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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 lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
-definition liminf:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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 O
- (mk_bounded_below_sequence O (\lambda j:nat.b (n+j))
- (mk_is_bounded_below O (\lambda j:nat.b (n+j)) (ib_lower_bound O b b)
- (\lambda j:nat.ib_lower_bound_is_lower_bound O b b (n+j))))
-\leq ib_upper_bound O b b);
- apply (inf_respects_le O);
- simplify;
- intro;
- apply (ib_upper_bound_is_upper_bound ? b b)
- ].
-qed.
+interpretation "Ordered set lt" 'lt a b =
+ (cic:/matita/ordered_sets/lt.con _ a b).
definition reverse_ordered_set: ordered_set → ordered_set.
intros;
].
qed.
-
-definition reverse_dedekind_sigma_complete_ordered_set:
- dedekind_sigma_complete_ordered_set → dedekind_sigma_complete_ordered_set.
- 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:
- ∀O:dedekind_sigma_complete_ordered_set.
- 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 ≝
- λO:dedekind_sigma_complete_ordered_set.
- 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:
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀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':
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀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:
- ∀O':dedekind_sigma_complete_ordered_set.
- ∀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 O' (\lambda j:nat.a (n+j))
-(mk_is_bounded_below O' (\lambda j:nat.a (n+j)) (ib_lower_bound O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound 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 O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus n j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus n j))
- (ib_lower_bound O' a a)
- (\lambda j:nat.ib_lower_bound_is_lower_bound O' a a (plus n j))))
- (ib_upper_bound O' a a)
- (\lambda n1:nat.ib_upper_bound_is_upper_bound O' a a (plus n n1)));
- intro p2;
- apply (eq_f_sup_sup_f' ? f H (mk_bounded_above_sequence O'
-(\lambda i:nat
- .inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
-(mk_is_bounded_above O'
- (\lambda i:nat
- .inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n)))))
- (ib_upper_bound O' a a) p2)));
- unfold bas_seq;
- change with
- (is_increasing ? (\lambda i:nat
-.inf O'
- (mk_bounded_below_sequence O' (\lambda j:nat.a (plus i j))
- (mk_is_bounded_below O' (\lambda j:nat.a (plus i j))
- (ib_lower_bound O' a a)
- (\lambda n:nat.ib_lower_bound_is_lower_bound O' a a (plus i n))))));
- apply tail_inf_increasing
- ].
-qed.
-
-
-
-
-definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
-
-interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ a b).
+record cotransitively_ordered_set: Type :=
+ { cos_ordered_set :> ordered_set;
+ cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
+ }.
projects / helm.git / blobdiff