record pre_ordered_abelian_group : Type ≝
{ og_abelian_group:> abelian_group;
- og_ordered_set_: ordered_set;
- og_with: os_carrier og_ordered_set_ = og_abelian_group
+ og_tordered_set_: tordered_set;
+ og_with: exc_carr og_tordered_set_ = og_abelian_group
}.
-lemma og_ordered_set: pre_ordered_abelian_group → ordered_set.
- intro G;
- apply mk_ordered_set;
- [ apply (carrier (og_abelian_group G))
- | apply (eq_rect ? ? (λC:Type.C→C→Prop) ? ? (og_with G));
- apply os_le
- | apply
- (eq_rect' ? ?
- (λa:Type.λH:os_carrier (og_ordered_set_ G) = a.
- is_order_relation a
- (eq_rect Type (og_ordered_set_ G) (λC:Type.C→C→Prop)
- (os_le (og_ordered_set_ G)) a H))
- ? ? (og_with G));
- simplify;
- apply (os_order_relation_properties (og_ordered_set_ G))
- ]
+lemma og_tordered_set: pre_ordered_abelian_group → tordered_set.
+intro G; apply mk_tordered_set;
+[1: apply mk_pordered_set;
+ [1: apply (mk_excedence G);
+ [1: cases G; clear G; simplify; rewrite < H; clear H;
+ cases og_tordered_set_; clear og_tordered_set_; simplify;
+ cases tos_poset; simplify; cases pos_carr; simplify; assumption;
+ |2: cases G; simplify; cases H; simplify; clear H;
+ cases og_tordered_set_; simplify; clear og_tordered_set_;
+ cases tos_poset; simplify; cases pos_carr; simplify;
+ intros; apply H;
+ |3: cases G; simplify; cases H; simplify; cases og_tordered_set_; simplify;
+ cases tos_poset; simplify; cases pos_carr; simplify;
+ intros; apply c; assumption]
+ |2: cases G; simplify;
+ cases H; simplify; clear H; cases og_tordered_set_; simplify;
+ cases tos_poset; simplify; assumption;]
+|2: simplify; (* SLOW, senza la simplify il widget muore *)
+ cases G; simplify;
+ generalize in match (tos_totality og_tordered_set_);
+ unfold total_order_property;
+ cases H; simplify; cases og_tordered_set_; simplify;
+ cases tos_poset; simplify; cases pos_carr; simplify;
+ intros; apply f; assumption;]
qed.
-coercion cic:/matita/ordered_groups/og_ordered_set.con.
+coercion cic:/matita/ordered_groups/og_tordered_set.con.
definition is_ordered_abelian_group ≝
λG:pre_ordered_abelian_group. ∀f,g,h:G. f≤g → f+h≤g+h.
is_ordered_abelian_group og_pre_ordered_abelian_group
}.
-lemma le_zero_x_to_le_opp_x_zero: ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
- intros;
- generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) H); intro;
+lemma le_zero_x_to_le_opp_x_zero:
+ ∀G:ordered_abelian_group.∀x:G.0 ≤ x → -x ≤ 0.
+intros (G x Px);
+generalize in match (og_ordered_abelian_group_properties ? ? ? (-x) Px); intro;
+(* ma cazzo, qui bisogna rifare anche i gruppi con ≈ ? *)
rewrite > zero_neutral in H1;
rewrite > plus_comm in H1;
rewrite > opp_inverse in H1;
[apply le_reflexive|apply le_transitive|apply le_antisymmetric]
qed.
-definition total_order : ∀E:excedence. Type ≝
- λE:excedence. ∀a,b:E. a ≰ b → a < b.
-
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:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.
intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con.
definition lower_bound ≝
- λO:ordered_set.λb:bounded_below_sequence O.
+ λO:pordered_set.λb:bounded_below_sequence O.
ib_lower_bound ? b (bbs_is_bounded_below ? b).
lemma lower_bound_is_lower_bound:
- ∀O:ordered_set.∀b:bounded_below_sequence O.
+ ∀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.
+intros; unfold lower_bound; apply ib_lower_bound_is_lower_bound.
qed.
definition upper_bound ≝
- λO:ordered_set.λb:bounded_above_sequence O.
+ λO:pordered_set.λb:bounded_above_sequence O.
ib_upper_bound ? b (bas_is_bounded_above ? b).
lemma upper_bound_is_upper_bound:
- ∀O:ordered_set.∀b:bounded_above_sequence O.
+ ∀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.
+intros; unfold upper_bound; apply ib_upper_bound_is_upper_bound.
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).
-
-definition reverse_ordered_set: ordered_set → ordered_set.
- intros;
- 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
- ]
- ].
+lemma Or_symmetric: symmetric ? Or.
+unfold; intros (x y H); cases H; [right|left] assumption;
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).
+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 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.
+
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;
- 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:
- ∀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;
- 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:
- ∀O:ordered_set.∀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:
- ∀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 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:
- ∀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
- ].
+ ∀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:
- ∀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
- ].
+ ∀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:
- ∀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 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
- ].
+ ∀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:
- ∀O:ordered_set.∀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 (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
- ].
+ ∀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 total_order_property : ∀E:excedence. Type ≝
+ λE:excedence. ∀a,b:E. a ≰ b → a < b.
+
+record tordered_set : Type ≝ {
+ tos_poset:> pordered_set;
+ tos_totality: total_order_property tos_poset
+}.
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