-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 ordered_set (C:Type): Type ≝
- { os_pre_ordered_set:> pre_ordered_set C;
- os_order_relation_properties:> is_order_relation ? os_pre_ordered_set
- }.
-
-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
- ].
-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_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.
-
-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_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_bounded_below (C:Type) (O:ordered_set C) (a:nat→O) : Type ≝
- { ib_lower_bound: O;
- 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 ≝
- { ib_upper_bound: O;
- 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 bounded_below_sequence (C:Type) (O:ordered_set C) : Type ≝
- { bbs_seq:1> nat→O;
- bbs_is_bounded_below:> is_bounded_below ? ? bbs_seq
- }.
-
-record bounded_above_sequence (C:Type) (O:ordered_set C) : Type ≝
- { bas_seq:1> nat→O;
- bas_is_bounded_above:> is_bounded_above ? ? bas_seq
- }.
-
-record bounded_sequence (C:Type) (O:ordered_set C) : Type ≝
- { bs_seq:1> nat → O;
- 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).
-
-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).
-
-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).
-
-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.
-qed.
-
-definition upper_bound ≝
- λC.λO:ordered_set C.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.
-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))*)
- ].
-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
- ].
-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
- ]
- ].