--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/algebra/finite_groups/".
+
+include "algebra/groups.ma".
+
+record finite_enumerable (T:Type) : Type ≝
+ { order: nat;
+ repr: nat → T;
+ index_of: T → nat;
+ index_of_sur: ∀x.index_of x ≤ order;
+ index_of_repr: ∀n. n≤order → index_of (repr n) = n;
+ repr_index_of: ∀x. repr (index_of x) = x
+ }.
+
+notation "hvbox(C \sub i)" with precedence 89
+for @{ 'repr $C $i }.
+
+(* CSC: multiple interpretations in the same file are not considered in the
+ right order
+interpretation "Finite_enumerable representation" 'repr C i =
+ (cic:/matita/algebra/finite_groups/repr.con C _ i).*)
+
+notation < "hvbox(|C|)" with precedence 89
+for @{ 'card $C }.
+
+interpretation "Finite_enumerable order" 'card C =
+ (cic:/matita/algebra/finite_groups/order.con C _).
+
+record finite_enumerable_SemiGroup : Type ≝
+ { semigroup:> SemiGroup;
+ is_finite_enumerable:> finite_enumerable semigroup
+ }.
+
+notation < "S"
+for @{ 'semigroup_of_finite_enumerable_semigroup $S }.
+
+interpretation "Semigroup_of_finite_enumerable_semigroup"
+ 'semigroup_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/finite_groups/semigroup.con S).
+
+notation < "S"
+for @{ 'magma_of_finite_enumerable_semigroup $S }.
+
+interpretation "Magma_of_finite_enumerable_semigroup"
+ 'magma_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/finite_groups/Magma_of_finite_enumerable_SemiGroup.con S).
+
+notation < "S"
+for @{ 'type_of_finite_enumerable_semigroup $S }.
+
+interpretation "Type_of_finite_enumerable_semigroup"
+ 'type_of_finite_enumerable_semigroup S
+=
+ (cic:/matita/algebra/finite_groups/Type_of_finite_enumerable_SemiGroup.con S).
+
+interpretation "Finite_enumerable representation" 'repr S i =
+ (cic:/matita/algebra/finite_groups/repr.con S
+ (cic:/matita/algebra/finite_groups/is_finite_enumerable.con S) i).
+
+notation "hvbox(ι e)" with precedence 60
+for @{ 'index_of_finite_enumerable_semigroup $e }.
+
+interpretation "Index_of_finite_enumerable representation"
+ 'index_of_finite_enumerable_semigroup e
+=
+ (cic:/matita/algebra/finite_groups/index_of.con _
+ (cic:/matita/algebra/finite_groups/is_finite_enumerable.con _) e).
+
+
+(* several definitions/theorems to be moved somewhere else *)
+
+definition ltb ≝ λn,m. leb n m ∧ notb (eqb n m).
+
+theorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
+intros;
+elim (le_to_or_lt_eq ? ? H1);
+[ assumption
+| elim (H H2)
+].
+qed.
+
+theorem ltb_to_Prop :
+ ∀n,m.
+ match ltb n m with
+ [ true ⇒ n < m
+ | false ⇒ n ≮ m
+ ].
+intros;
+unfold ltb;
+apply leb_elim;
+apply eqb_elim;
+intros;
+simplify;
+[ rewrite < H;
+ apply le_to_not_lt;
+ constructor 1
+| apply (not_eq_to_le_to_lt ? ? H H1)
+| rewrite < H;
+ apply le_to_not_lt;
+ constructor 1
+| apply le_to_not_lt;
+ generalize in match (not_le_to_lt ? ? H1);
+ clear H1;
+ intro;
+ apply lt_to_le;
+ assumption
+].
+qed.
+
+theorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop.
+(n < m → (P true)) → (n ≮ m → (P false)) →
+P (ltb n m).
+intros.
+cut
+(match (ltb n m) with
+[ true ⇒ n < m
+| false ⇒ n ≮ m] → (P (ltb n m))).
+apply Hcut.apply ltb_to_Prop.
+elim (ltb n m).
+apply ((H H2)).
+apply ((H1 H2)).
+qed.
+
+theorem Not_lt_n_n: ∀n. n ≮ n.
+intro;
+unfold Not;
+intro;
+unfold lt in H;
+apply (not_le_Sn_n ? H).
+qed.
+
+theorem eq_pred_to_eq:
+ ∀n,m. O < n → O < m → pred n = pred m → n = m.
+intros;
+generalize in match (eq_f ? ? S ? ? H2);
+intro;
+rewrite < S_pred in H3;
+rewrite < S_pred in H3;
+assumption.
+qed.
+
+theorem le_pred_to_le:
+ ∀n,m. O < m → pred n ≤ pred m → n ≤ m.
+intros 2;
+elim n;
+[ apply le_O_n
+| simplify in H2;
+ rewrite > (S_pred m);
+ [ apply le_S_S;
+ assumption
+ | assumption
+ ]
+].
+qed.
+
+theorem le_to_le_pred:
+ ∀n,m. n ≤ m → pred n ≤ pred m.
+intros 2;
+elim n;
+[ simplify;
+ apply le_O_n
+| simplify;
+ generalize in match H1;
+ clear H1;
+ elim m;
+ [ elim (not_le_Sn_O ? H1)
+ | simplify;
+ apply le_S_S_to_le;
+ assumption
+ ]
+].
+qed.
+
+theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
+intros;
+unfold Not;
+intro;
+unfold lt in H;
+unfold lt in H1;
+generalize in match (le_S_S ? ? H);
+intro;
+generalize in match (transitive_le ? ? ? H2 H1);
+intro;
+apply (not_le_Sn_n ? H3).
+qed.
+
+theorem lt_S_S: ∀n,m. n < m → S n < S m.
+intros;
+unfold lt in H;
+apply (le_S_S ? ? H).
+qed.
+
+theorem lt_O_S: ∀n. O < S n.
+intro;
+unfold lt;
+apply le_S_S;
+apply le_O_n.
+qed.
+
+theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
+intros;
+unfold lt in H1;
+generalize in match (le_S_S_to_le ? ? H1);
+intro;
+apply cic:/matita/nat/orders/antisym_le.con;
+assumption.
+qed.
+
+theorem pigeonhole:
+ ∀n:nat.∀f:nat→nat.
+ (∀x,y.x≤n → y≤n → f x = f y → x=y) →
+ (∀m. m ≤ n → f m ≤ n) →
+ ∀x. x≤n → ∃y.f y = x ∧ y ≤ n.
+intro;
+elim n;
+[ apply (ex_intro ? ? O);
+ split;
+ [ rewrite < (le_n_O_to_eq ? H2);
+ rewrite < (le_n_O_to_eq ? (H1 O ?));
+ [ reflexivity
+ | apply le_n
+ ]
+ | apply le_n
+ ]
+| clear n;
+ letin f' ≝
+ (λx.
+ let fSn1 ≝ f (S n1) in
+ let fx ≝ f x in
+ match ltb fSn1 fx with
+ [ true ⇒ pred fx
+ | false ⇒ fx
+ ]);
+ cut (∀x,y. x ≤ n1 → y ≤ n1 → f' x = f' y → x=y);
+ [ cut (∀x. x ≤ n1 → f' x ≤ n1);
+ [ apply (nat_compare_elim (f (S n1)) x);
+ [ intro;
+ elim (H f' ? ? (pred x));
+ [ simplify in H5;
+ clear Hcut;
+ clear Hcut1;
+ clear f';
+ elim H5;
+ clear H5;
+ apply (ex_intro ? ? a);
+ split;
+ [ generalize in match (eq_f ? ? S ? ? H6);
+ clear H6;
+ intro;
+ rewrite < S_pred in H5;
+ [ generalize in match H4;
+ clear H4;
+ rewrite < H5;
+ clear H5;
+ apply (ltb_elim (f (S n1)) (f a));
+ [ simplify;
+ intros;
+ rewrite < S_pred;
+ [ reflexivity
+ | apply (ltn_to_ltO ? ? H4)
+ ]
+ | simplify;
+ intros;
+ generalize in match (not_lt_to_le ? ? H4);
+ clear H4;
+ intro;
+ generalize in match (le_n_m_to_lt_m_Sn_to_eq_n_m ? ? H6 H5);
+ intro;
+ generalize in match (H1 ? ? ? ? H4);
+ [ intro;
+ generalize in match (le_n_m_to_lt_m_Sn_to_eq_n_m ? ? H6 H5);
+ intro;
+ generalize in match (H1 ? ? ? ? H9);
+ [ intro;
+ rewrite > H10 in H7;
+ elim (not_le_Sn_n ? H7)
+ | rewrite > H8;
+ apply le_n
+ | apply le_n
+ ]
+ | apply le_S;
+ assumption
+ | apply le_n
+ ]
+ ]
+ | apply (ltn_to_ltO ? ? H4)
+ ]
+ | apply le_S;
+ assumption
+ ]
+ | apply Hcut
+ | apply Hcut1
+ | apply le_S_S_to_le;
+ rewrite < S_pred;
+ [ assumption
+ | apply (ltn_to_ltO ? ? H4)
+ ]
+ ]
+ | intros;
+ apply (ex_intro ? ? (S n1));
+ split;
+ [ assumption
+ | constructor 1
+ ]
+ | intro;
+ elim (H f' ? ? x);
+ [ simplify in H5;
+ clear Hcut;
+ clear Hcut1;
+ clear f';
+ elim H5;
+ clear H5;
+ apply (ex_intro ? ? a);
+ split;
+ [ generalize in match H4;
+ clear H4;
+ rewrite < H6;
+ clear H6;
+ apply (ltb_elim (f (S n1)) (f a));
+ [ simplify;
+ intros;
+ generalize in match (lt_S_S ? ? H5);
+ intro;
+ rewrite < S_pred in H6;
+ [ elim (lt_n_m_to_not_lt_m_Sn ? ? H4 H6)
+ | apply (ltn_to_ltO ? ? H4)
+ ]
+ | simplify;
+ intros;
+ reflexivity
+ ]
+ | apply le_S;
+ assumption
+ ]
+ | apply Hcut
+ | apply Hcut1
+ | rewrite > (pred_Sn n1);
+ simplify;
+ generalize in match (H2 (S n1));
+ intro;
+ generalize in match (lt_to_le_to_lt ? ? ? H4 (H5 (le_n ?)));
+ intro;
+ unfold lt in H6;
+ apply le_S_S_to_le;
+ assumption
+ ]
+ ]
+ | unfold f';
+ simplify;
+ intro;
+ apply (ltb_elim (f (S n1)) (f x1));
+ simplify;
+ intros;
+ [ generalize in match (H2 x1);
+ intro;
+ change in match n1 with (pred (S n1));
+ apply le_to_le_pred;
+ apply H6;
+ apply le_S;
+ assumption
+ | generalize in match (H2 (S n1) (le_n ?));
+ intro;
+ generalize in match (not_lt_to_le ? ? H4);
+ intro;
+ generalize in match (transitive_le ? ? ? H7 H6);
+ intro;
+ cut (f x1 ≠ f (S n1));
+ [ generalize in match (not_eq_to_le_to_lt ? ? Hcut1 H7);
+ intro;
+ unfold lt in H9;
+ generalize in match (transitive_le ? ? ? H9 H6);
+ intro;
+ apply le_S_S_to_le;
+ assumption
+ | unfold Not;
+ intro;
+ generalize in match (H1 ? ? ? ? H9);
+ [ intro;
+ rewrite > H10 in H5;
+ apply (not_le_Sn_n ? H5)
+ | apply le_S;
+ assumption
+ | apply le_n
+ ]
+ ]
+ ]
+ ]
+ | intros 4;
+ unfold f';
+ simplify;
+ apply (ltb_elim (f (S n1)) (f x1));
+ simplify;
+ apply (ltb_elim (f (S n1)) (f y));
+ simplify;
+ intros;
+ [ cut (f x1 = f y);
+ [ apply (H1 ? ? ? ? Hcut);
+ apply le_S;
+ assumption
+ | apply eq_pred_to_eq;
+ [ apply (ltn_to_ltO ? ? H7)
+ | apply (ltn_to_ltO ? ? H6)
+ | assumption
+ ]
+ ]
+ | (* pred (f x1) = f y absurd since y ≠ S n1 and thus f y ≠ f (S n1)
+ so that f y < f (S n1) < f x1; hence pred (f x1) = f y is absurd *)
+ cut (y < S n1);
+ [ generalize in match (lt_to_not_eq ? ? Hcut);
+ intro;
+ cut (f y ≠ f (S n1));
+ [ cut (f y < f (S n1));
+ [ rewrite < H8 in Hcut2;
+ unfold lt in Hcut2;
+ unfold lt in H7;
+ generalize in match (le_S_S ? ? Hcut2);
+ intro;
+ generalize in match (transitive_le ? ? ? H10 H7);
+ intros;
+ rewrite < (S_pred (f x1)) in H11;
+ [ elim (not_le_Sn_n ? H11)
+ | fold simplify ((f (S n1)) < (f x1)) in H7;
+ apply (ltn_to_ltO ? ? H7)
+ ]
+ | apply not_eq_to_le_to_lt;
+ [ assumption
+ | apply not_lt_to_le;
+ assumption
+ ]
+ ]
+ | unfold Not;
+ intro;
+ apply H9;
+ apply (H1 ? ? ? ? H10);
+ [ apply lt_to_le;
+ assumption
+ | constructor 1
+ ]
+ ]
+ | unfold lt;
+ apply le_S_S;
+ assumption
+ ]
+ | (* f x1 = pred (f y) absurd since it implies S (f x1) = f y and
+ f x1 ≤ f (S n1) < f y = S (f x1) so that f x1 = f (S n1); by
+ injectivity x1 = S n1 that is absurd since x1 ≤ n1 *)
+ generalize in match (eq_f ? ? S ? ? H8);
+ intro;
+ rewrite < S_pred in H9;
+ [ rewrite < H9 in H6;
+ generalize in match (not_lt_to_le ? ? H7);
+ intro;
+ unfold lt in H6;
+ generalize in match (le_S_S ? ? H10);
+ intro;
+ generalize in match (antisym_le ? ? H11 H6);
+ intro;
+ generalize in match (inj_S ? ? H12);
+ intro;
+ generalize in match (H1 ? ? ? ? H13);
+ [ intro;
+ rewrite > H14 in H4;
+ elim (not_le_Sn_n ? H4)
+ | apply le_S;
+ assumption
+ | apply le_n
+ ]
+ | apply (ltn_to_ltO ? ? H6)
+ ]
+ | apply (H1 ? ? ? ? H8);
+ apply le_S;
+ assumption
+ ]
+ ]
+].
+qed.
+
+theorem finite_enumerable_SemiGroup_to_left_cancellable_to_right_cancellable_to_isMonoid:
+ ∀G:finite_enumerable_SemiGroup.
+ left_cancellable ? (op G) →
+ right_cancellable ? (op G) →
+ ∃e:G. isMonoid (mk_PreMonoid G e).
+intros;
+letin f ≝ (λn.ι(G \sub O · G \sub n));
+cut (∀n.n ≤ order ? (is_finite_enumerable G) → ∃m.f m = n);
+[ letin EX ≝ (Hcut O ?);
+ [ apply le_O_n
+ | clearbody EX;
+ clear Hcut;
+ unfold f in EX;
+ elim EX;
+ clear EX;
+ letin HH ≝ (eq_f ? ? (repr ? (is_finite_enumerable G)) ? ? H2);
+ clearbody HH;
+ rewrite > (repr_index_of ? (is_finite_enumerable G)) in HH;
+ apply (ex_intro ? ? (G \sub a));
+ letin GOGO ≝ (refl_eq ? (repr ? (is_finite_enumerable G) O));
+ clearbody GOGO;
+ rewrite < HH in GOGO;
+ rewrite < HH in GOGO:(? ? % ?);
+ rewrite > (associative ? G) in GOGO;
+ letin GaGa ≝ (H ? ? ? GOGO);
+ clearbody GaGa;
+ clear GOGO;
+ constructor 1;
+ [ simplify;
+ apply (semigroup_properties G)
+ | unfold is_left_unit; intro;
+ letin GaxGax ≝ (refl_eq ? (G \sub a ·x));
+ clearbody GaxGax;
+ rewrite < GaGa in GaxGax:(? ? % ?);
+ rewrite > (associative ? (semigroup_properties G)) in GaxGax;
+ apply (H ? ? ? GaxGax)
+ | unfold is_right_unit; intro;
+ letin GaxGax ≝ (refl_eq ? (x·G \sub a));
+ clearbody GaxGax;
+ rewrite < GaGa in GaxGax:(? ? % ?);
+ rewrite < (associative ? (semigroup_properties G)) in GaxGax;
+ apply (H1 ? ? ? GaxGax)
+ ]
+ ]
+| intros;
+ elim (pigeonhole (order ? G) f ? ? ? H2);
+ [ apply (ex_intro ? ? a);
+ elim H3;
+ assumption
+ | intros;
+ change in H5 with (ι(G \sub O · G \sub x) = ι(G \sub O · G \sub y));
+ cut (G \sub (ι(G \sub O · G \sub x)) = G \sub (ι(G \sub O · G \sub y)));
+ [ rewrite > (repr_index_of ? ? (G \sub O · G \sub x)) in Hcut;
+ rewrite > (repr_index_of ? ? (G \sub O · G \sub y)) in Hcut;
+ generalize in match (H ? ? ? Hcut);
+ intro;
+ generalize in match (eq_f ? ? (index_of ? G) ? ? H6);
+ intro;
+ rewrite > index_of_repr in H7;
+ rewrite > index_of_repr in H7;
+ assumption
+ | apply eq_f;
+ assumption
+ ]
+ | intros;
+ apply index_of_sur
+ ]
+].
+qed.
record PreGroup : Type ≝
{ premonoid:> PreMonoid;
- opp: premonoid -> premonoid
+ inv: premonoid -> premonoid
}.
record isGroup (G:PreGroup) : Prop ≝
{ is_monoid: isMonoid G;
- opp_is_left_inverse: is_left_inverse (mk_Monoid ? is_monoid) (opp G);
- opp_is_right_inverse: is_right_inverse (mk_Monoid ? is_monoid) (opp G)
+ inv_is_left_inverse: is_left_inverse (mk_Monoid ? is_monoid) (inv G);
+ inv_is_right_inverse: is_right_inverse (mk_Monoid ? is_monoid) (inv G)
}.
record Group : Type ≝
(cic:/matita/algebra/groups/Magma_of_Group.con G).
notation "hvbox(x \sup (-1))" with precedence 89
-for @{ 'gopp $x }.
+for @{ 'ginv $x }.
-interpretation "Group inverse" 'gopp x =
- (cic:/matita/algebra/groups/opp.con _ x).
+interpretation "Group inverse" 'ginv x =
+ (cic:/matita/algebra/groups/inv.con _ x).
definition left_cancellable ≝
λT:Type. λop: T -> T -> T.
unfold left_cancellable;
unfold injective;
intros (x y z);
-rewrite < (e_is_left_unit ? (is_monoid ? (group_properties G)));
-rewrite < (e_is_left_unit ? (is_monoid ? (group_properties G)) z);
-rewrite < (opp_is_left_inverse ? (group_properties G) x);
-rewrite > (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
-rewrite > (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
+rewrite < (e_is_left_unit ? (is_monoid ? G));
+rewrite < (e_is_left_unit ? (is_monoid ? G) z);
+rewrite < (inv_is_left_inverse ? G x);
+rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
apply eq_f;
assumption.
qed.
unfold injective;
simplify;fold simplify (op G);
intros (x y z);
-rewrite < (e_is_right_unit ? (is_monoid ? (group_properties G)));
-rewrite < (e_is_right_unit ? (is_monoid ? (group_properties G)) z);
-rewrite < (opp_is_right_inverse ? (group_properties G) x);
-rewrite < (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
-rewrite < (associative ? (is_semi_group ? (is_monoid ? (group_properties G))));
+rewrite < (e_is_right_unit ? (is_monoid ? G));
+rewrite < (e_is_right_unit ? (is_monoid ? G) z);
+rewrite < (inv_is_right_inverse ? G x);
+rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
rewrite > H;
reflexivity.
qed.
-
-record finite_enumerable (T:Type) : Type ≝
- { order: nat;
- repr: nat → T;
- index_of: T → nat;
- index_of_sur: ∀x.index_of x ≤ order;
- index_of_repr: ∀n. n≤order → index_of (repr n) = n;
- repr_index_of: ∀x. repr (index_of x) = x
- }.
-
-notation "hvbox(C \sub i)" with precedence 89
-for @{ 'repr $C $i }.
-
-(* CSC: multiple interpretations in the same file are not considered in the
- right order
-interpretation "Finite_enumerable representation" 'repr C i =
- (cic:/matita/algebra/groups/repr.con C _ i).*)
-
-notation < "hvbox(|C|)" with precedence 89
-for @{ 'card $C }.
-
-interpretation "Finite_enumerable order" 'card C =
- (cic:/matita/algebra/groups/order.con C _).
-
-record finite_enumerable_SemiGroup : Type ≝
- { semigroup:> SemiGroup;
- is_finite_enumerable:> finite_enumerable semigroup
- }.
-
-notation < "S"
-for @{ 'semigroup_of_finite_enumerable_semigroup $S }.
-
-interpretation "Semigroup_of_finite_enumerable_semigroup"
- 'semigroup_of_finite_enumerable_semigroup S
-=
- (cic:/matita/algebra/groups/semigroup.con S).
-
-notation < "S"
-for @{ 'magma_of_finite_enumerable_semigroup $S }.
-
-interpretation "Magma_of_finite_enumerable_semigroup"
- 'magma_of_finite_enumerable_semigroup S
-=
- (cic:/matita/algebra/groups/Magma_of_finite_enumerable_SemiGroup.con S).
-
-notation < "S"
-for @{ 'type_of_finite_enumerable_semigroup $S }.
-
-interpretation "Type_of_finite_enumerable_semigroup"
- 'type_of_finite_enumerable_semigroup S
-=
- (cic:/matita/algebra/groups/Type_of_finite_enumerable_SemiGroup.con S).
-
-interpretation "Finite_enumerable representation" 'repr S i =
- (cic:/matita/algebra/groups/repr.con S
- (cic:/matita/algebra/groups/is_finite_enumerable.con S) i).
-
-notation "hvbox(ι e)" with precedence 60
-for @{ 'index_of_finite_enumerable_semigroup $e }.
-
-interpretation "Index_of_finite_enumerable representation"
- 'index_of_finite_enumerable_semigroup e
-=
- (cic:/matita/algebra/groups/index_of.con _
- (cic:/matita/algebra/groups/is_finite_enumerable.con _) e).
-
-
-(* several definitions/theorems to be moved somewhere else *)
-
-definition ltb ≝ λn,m. leb n m ∧ notb (eqb n m).
-
-theorem not_eq_to_le_to_lt: ∀n,m. n≠m → n≤m → n<m.
+theorem inv_inv: ∀G:Group. ∀x:G. x \sup -1 \sup -1 = x.
intros;
-elim (le_to_or_lt_eq ? ? H1);
-[ assumption
-| elim (H H2)
-].
-qed.
-
-theorem ltb_to_Prop :
- ∀n,m.
- match ltb n m with
- [ true ⇒ n < m
- | false ⇒ n ≮ m
- ].
-intros;
-unfold ltb;
-apply leb_elim;
-apply eqb_elim;
-intros;
-simplify;
-[ rewrite < H;
- apply le_to_not_lt;
- constructor 1
-| apply (not_eq_to_le_to_lt ? ? H H1)
-| rewrite < H;
- apply le_to_not_lt;
- constructor 1
-| apply le_to_not_lt;
- generalize in match (not_le_to_lt ? ? H1);
- clear H1;
- intro;
- apply lt_to_le;
- assumption
-].
-qed.
-
-theorem ltb_elim: ∀n,m:nat. ∀P:bool → Prop.
-(n < m → (P true)) → (n ≮ m → (P false)) →
-P (ltb n m).
-intros.
-cut
-(match (ltb n m) with
-[ true ⇒ n < m
-| false ⇒ n ≮ m] → (P (ltb n m))).
-apply Hcut.apply ltb_to_Prop.
-elim (ltb n m).
-apply ((H H2)).
-apply ((H1 H2)).
-qed.
-
-theorem Not_lt_n_n: ∀n. n ≮ n.
-intro;
-unfold Not;
-intro;
-unfold lt in H;
-apply (not_le_Sn_n ? H).
+apply (eq_op_x_y_op_z_y_to_eq ? (x \sup -1));
+rewrite > (inv_is_right_inverse ? G);
+rewrite > (inv_is_left_inverse ? G);
+reflexivity.
qed.
-theorem eq_pred_to_eq:
- ∀n,m. O < n → O < m → pred n = pred m → n = m.
+theorem eq_opxy_e_to_eq_x_invy:
+ ∀G:Group. ∀x,y:G. x·y=1 → x=y \sup -1.
intros;
-generalize in match (eq_f ? ? S ? ? H2);
-intro;
-rewrite < S_pred in H3;
-rewrite < S_pred in H3;
+apply (eq_op_x_y_op_z_y_to_eq ? y);
+rewrite > (inv_is_left_inverse ? G);
assumption.
qed.
-theorem le_pred_to_le:
- ∀n,m. O < m → pred n ≤ pred m → n ≤ m.
-intros 2;
-elim n;
-[ apply le_O_n
-| simplify in H2;
- rewrite > (S_pred m);
- [ apply le_S_S;
- assumption
- | assumption
- ]
-].
-qed.
-
-theorem le_to_le_pred:
- ∀n,m. n ≤ m → pred n ≤ pred m.
-intros 2;
-elim n;
-[ simplify;
- apply le_O_n
-| simplify;
- generalize in match H1;
- clear H1;
- elim m;
- [ elim (not_le_Sn_O ? H1)
- | simplify;
- apply le_S_S_to_le;
- assumption
- ]
-].
-qed.
-
-theorem lt_n_m_to_not_lt_m_Sn: ∀n,m. n < m → m ≮ S n.
+theorem eq_opxy_e_to_eq_invx_y:
+ ∀G:Group. ∀x,y:G. x·y=1 → x \sup -1=y.
intros;
-unfold Not;
-intro;
-unfold lt in H;
-unfold lt in H1;
-generalize in match (le_S_S ? ? H);
-intro;
-generalize in match (transitive_le ? ? ? H2 H1);
-intro;
-apply (not_le_Sn_n ? H3).
-qed.
-
-theorem lt_S_S: ∀n,m. n < m → S n < S m.
-intros;
-unfold lt in H;
-apply (le_S_S ? ? H).
-qed.
-
-theorem lt_O_S: ∀n. O < S n.
-intro;
-unfold lt;
-apply le_S_S;
-apply le_O_n.
+apply (eq_op_x_y_op_x_z_to_eq ? x);
+rewrite > (inv_is_right_inverse ? G);
+symmetry;
+assumption.
qed.
-theorem le_n_m_to_lt_m_Sn_to_eq_n_m: ∀n,m. n ≤ m → m < S n → n=m.
+theorem eq_opxy_z_to_eq_x_opzinvy:
+ ∀G:Group. ∀x,y,z:G. x·y=z → x = z·y \sup -1.
intros;
-unfold lt in H1;
-generalize in match (le_S_S_to_le ? ? H1);
-intro;
-apply cic:/matita/nat/orders/antisym_le.con;
+apply (eq_op_x_y_op_z_y_to_eq ? y);
+rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (inv_is_left_inverse ? G);
+rewrite > (e_is_right_unit ? (is_monoid ? G));
assumption.
qed.
-theorem pigeonhole:
- ∀n:nat.∀f:nat→nat.
- (∀x,y.x≤n → y≤n → f x = f y → x=y) →
- (∀m. m ≤ n → f m ≤ n) →
- ∀x. x≤n → ∃y.f y = x ∧ y ≤ n.
-intro;
-elim n;
-[ apply (ex_intro ? ? O);
- split;
- [ rewrite < (le_n_O_to_eq ? H2);
- rewrite < (le_n_O_to_eq ? (H1 O ?));
- [ reflexivity
- | apply le_n
- ]
- | apply le_n
- ]
-| clear n;
- letin f' ≝
- (λx.
- let fSn1 ≝ f (S n1) in
- let fx ≝ f x in
- match ltb fSn1 fx with
- [ true ⇒ pred fx
- | false ⇒ fx
- ]);
- cut (∀x,y. x ≤ n1 → y ≤ n1 → f' x = f' y → x=y);
- [ cut (∀x. x ≤ n1 → f' x ≤ n1);
- [ apply (nat_compare_elim (f (S n1)) x);
- [ intro;
- elim (H f' ? ? (pred x));
- [ simplify in H5;
- clear Hcut;
- clear Hcut1;
- clear f';
- elim H5;
- clear H5;
- apply (ex_intro ? ? a);
- split;
- [ generalize in match (eq_f ? ? S ? ? H6);
- clear H6;
- intro;
- rewrite < S_pred in H5;
- [ generalize in match H4;
- clear H4;
- rewrite < H5;
- clear H5;
- apply (ltb_elim (f (S n1)) (f a));
- [ simplify;
- intros;
- rewrite < S_pred;
- [ reflexivity
- | apply (ltn_to_ltO ? ? H4)
- ]
- | simplify;
- intros;
- generalize in match (not_lt_to_le ? ? H4);
- clear H4;
- intro;
- generalize in match (le_n_m_to_lt_m_Sn_to_eq_n_m ? ? H6 H5);
- intro;
- generalize in match (H1 ? ? ? ? H4);
- [ intro;
- generalize in match (le_n_m_to_lt_m_Sn_to_eq_n_m ? ? H6 H5);
- intro;
- generalize in match (H1 ? ? ? ? H9);
- [ intro;
- rewrite > H10 in H7;
- elim (not_le_Sn_n ? H7)
- | rewrite > H8;
- apply le_n
- | apply le_n
- ]
- | apply le_S;
- assumption
- | apply le_n
- ]
- ]
- | apply (ltn_to_ltO ? ? H4)
- ]
- | apply le_S;
- assumption
- ]
- | apply Hcut
- | apply Hcut1
- | apply le_S_S_to_le;
- rewrite < S_pred;
- [ assumption
- | apply (ltn_to_ltO ? ? H4)
- ]
- ]
- | intros;
- apply (ex_intro ? ? (S n1));
- split;
- [ assumption
- | constructor 1
- ]
- | intro;
- elim (H f' ? ? x);
- [ simplify in H5;
- clear Hcut;
- clear Hcut1;
- clear f';
- elim H5;
- clear H5;
- apply (ex_intro ? ? a);
- split;
- [ generalize in match H4;
- clear H4;
- rewrite < H6;
- clear H6;
- apply (ltb_elim (f (S n1)) (f a));
- [ simplify;
- intros;
- generalize in match (lt_S_S ? ? H5);
- intro;
- rewrite < S_pred in H6;
- [ elim (lt_n_m_to_not_lt_m_Sn ? ? H4 H6)
- | apply (ltn_to_ltO ? ? H4)
- ]
- | simplify;
- intros;
- reflexivity
- ]
- | apply le_S;
- assumption
- ]
- | apply Hcut
- | apply Hcut1
- | rewrite > (pred_Sn n1);
- simplify;
- generalize in match (H2 (S n1));
- intro;
- generalize in match (lt_to_le_to_lt ? ? ? H4 (H5 (le_n ?)));
- intro;
- unfold lt in H6;
- apply le_S_S_to_le;
- assumption
- ]
- ]
- | unfold f';
- simplify;
- intro;
- apply (ltb_elim (f (S n1)) (f x1));
- simplify;
- intros;
- [ generalize in match (H2 x1);
- intro;
- change in match n1 with (pred (S n1));
- apply le_to_le_pred;
- apply H6;
- apply le_S;
- assumption
- | generalize in match (H2 (S n1) (le_n ?));
- intro;
- generalize in match (not_lt_to_le ? ? H4);
- intro;
- generalize in match (transitive_le ? ? ? H7 H6);
- intro;
- cut (f x1 ≠ f (S n1));
- [ generalize in match (not_eq_to_le_to_lt ? ? Hcut1 H7);
- intro;
- unfold lt in H9;
- generalize in match (transitive_le ? ? ? H9 H6);
- intro;
- apply le_S_S_to_le;
- assumption
- | unfold Not;
- intro;
- generalize in match (H1 ? ? ? ? H9);
- [ intro;
- rewrite > H10 in H5;
- apply (not_le_Sn_n ? H5)
- | apply le_S;
- assumption
- | apply le_n
- ]
- ]
- ]
- ]
- | intros 4;
- unfold f';
- simplify;
- apply (ltb_elim (f (S n1)) (f x1));
- simplify;
- apply (ltb_elim (f (S n1)) (f y));
- simplify;
- intros;
- [ cut (f x1 = f y);
- [ apply (H1 ? ? ? ? Hcut);
- apply le_S;
- assumption
- | apply eq_pred_to_eq;
- [ apply (ltn_to_ltO ? ? H7)
- | apply (ltn_to_ltO ? ? H6)
- | assumption
- ]
- ]
- | (* pred (f x1) = f y absurd since y ≠ S n1 and thus f y ≠ f (S n1)
- so that f y < f (S n1) < f x1; hence pred (f x1) = f y is absurd *)
- cut (y < S n1);
- [ generalize in match (lt_to_not_eq ? ? Hcut);
- intro;
- cut (f y ≠ f (S n1));
- [ cut (f y < f (S n1));
- [ rewrite < H8 in Hcut2;
- unfold lt in Hcut2;
- unfold lt in H7;
- generalize in match (le_S_S ? ? Hcut2);
- intro;
- generalize in match (transitive_le ? ? ? H10 H7);
- intros;
- rewrite < (S_pred (f x1)) in H11;
- [ elim (not_le_Sn_n ? H11)
- | fold simplify ((f (S n1)) < (f x1)) in H7;
- apply (ltn_to_ltO ? ? H7)
- ]
- | apply not_eq_to_le_to_lt;
- [ assumption
- | apply not_lt_to_le;
- assumption
- ]
- ]
- | unfold Not;
- intro;
- apply H9;
- apply (H1 ? ? ? ? H10);
- [ apply lt_to_le;
- assumption
- | constructor 1
- ]
- ]
- | unfold lt;
- apply le_S_S;
- assumption
- ]
- | (* f x1 = pred (f y) absurd since it implies S (f x1) = f y and
- f x1 ≤ f (S n1) < f y = S (f x1) so that f x1 = f (S n1); by
- injectivity x1 = S n1 that is absurd since x1 ≤ n1 *)
- generalize in match (eq_f ? ? S ? ? H8);
- intro;
- rewrite < S_pred in H9;
- [ rewrite < H9 in H6;
- generalize in match (not_lt_to_le ? ? H7);
- intro;
- unfold lt in H6;
- generalize in match (le_S_S ? ? H10);
- intro;
- generalize in match (antisym_le ? ? H11 H6);
- intro;
- generalize in match (inj_S ? ? H12);
- intro;
- generalize in match (H1 ? ? ? ? H13);
- [ intro;
- rewrite > H14 in H4;
- elim (not_le_Sn_n ? H4)
- | apply le_S;
- assumption
- | apply le_n
- ]
- | apply (ltn_to_ltO ? ? H6)
- ]
- | apply (H1 ? ? ? ? H8);
- apply le_S;
- assumption
- ]
- ]
-].
-qed.
-
-theorem finite_enumerable_SemiGroup_to_left_cancellable_to_right_cancellable_to_isMonoid:
- ∀G:finite_enumerable_SemiGroup.
- left_cancellable ? (op G) →
- right_cancellable ? (op G) →
- ∃e:G. isMonoid (mk_PreMonoid G e).
+theorem eq_opxy_z_to_eq_y_opinvxz:
+ ∀G:Group. ∀x,y,z:G. x·y=z → y = x \sup -1·z.
intros;
-letin f ≝ (λn.ι(G \sub O · G \sub n));
-cut (∀n.n ≤ order ? (is_finite_enumerable G) → ∃m.f m = n);
-[ letin EX ≝ (Hcut O ?);
- [ apply le_O_n
- | clearbody EX;
- clear Hcut;
- unfold f in EX;
- elim EX;
- clear EX;
- letin HH ≝ (eq_f ? ? (repr ? (is_finite_enumerable G)) ? ? H2);
- clearbody HH;
- rewrite > (repr_index_of ? (is_finite_enumerable G)) in HH;
- apply (ex_intro ? ? (G \sub a));
- letin GOGO ≝ (refl_eq ? (repr ? (is_finite_enumerable G) O));
- clearbody GOGO;
- rewrite < HH in GOGO;
- rewrite < HH in GOGO:(? ? % ?);
- rewrite > (associative ? G) in GOGO;
- letin GaGa ≝ (H ? ? ? GOGO);
- clearbody GaGa;
- clear GOGO;
- constructor 1;
- [ simplify;
- apply (semigroup_properties G)
- | unfold is_left_unit; intro;
- letin GaxGax ≝ (refl_eq ? (G \sub a ·x));
- clearbody GaxGax;
- rewrite < GaGa in GaxGax:(? ? % ?);
- rewrite > (associative ? (semigroup_properties G)) in GaxGax;
- apply (H ? ? ? GaxGax)
- | unfold is_right_unit; intro;
- letin GaxGax ≝ (refl_eq ? (x·G \sub a));
- clearbody GaxGax;
- rewrite < GaGa in GaxGax:(? ? % ?);
- rewrite < (associative ? (semigroup_properties G)) in GaxGax;
- apply (H1 ? ? ? GaxGax)
- ]
- ]
-| intros;
- elim (pigeonhole (order ? G) f ? ? ? H2);
- [ apply (ex_intro ? ? a);
- elim H3;
- assumption
- | intros;
- change in H5 with (ι(G \sub O · G \sub x) = ι(G \sub O · G \sub y));
- cut (G \sub (ι(G \sub O · G \sub x)) = G \sub (ι(G \sub O · G \sub y)));
- [ rewrite > (repr_index_of ? ? (G \sub O · G \sub x)) in Hcut;
- rewrite > (repr_index_of ? ? (G \sub O · G \sub y)) in Hcut;
- generalize in match (H ? ? ? Hcut);
- intro;
- generalize in match (eq_f ? ? (index_of ? G) ? ? H6);
- intro;
- rewrite > index_of_repr in H7;
- rewrite > index_of_repr in H7;
- assumption
- | apply eq_f;
- assumption
- ]
- | intros;
- apply index_of_sur
- ]
-].
+apply (eq_op_x_y_op_x_z_to_eq ? x);
+rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (inv_is_right_inverse ? G);
+rewrite > (e_is_left_unit ? (is_monoid ? G));
+assumption.
+qed.
\ No newline at end of file
projects / helm.git / commitdiff