--- /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
+ }.
+
+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 *)
+
+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;
+ unfold f' in H5;
+ 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;
+ unfold f' in H5;
+ 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_to_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
+ | alias id "eq_pred_to_eq" = "cic:/matita/nat/relevant_equations/eq_pred_to_eq.con".
+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.
+(* demo *)
+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 > (op_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; (* demo *)
+ rewrite < GaGa in GaxGax:(? ? % ?);
+ rewrite > (op_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 < (op_associative ? (semigroup_properties G)) in GaxGax;
+ apply (H1 ? ? ? GaxGax)
+ ]
+ ]
+| intros;
+ elim (pigeonhole (order ? G) f ? ? ? H2);
+ [ apply (ex_intro ? ? a);
+ elim H3;
+ assumption
+ | intros;
+ simplify in H5;
+ 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;
+ unfold f;
+ apply index_of_sur
+ ]
+].
+qed.
projects / helm.git / blobdiff