+++ /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/groups/".
-
-include "algebra/monoids.ma".
-include "nat/le_arith.ma".
-include "datatypes/bool.ma".
-include "nat/compare.ma".
-
-record PreGroup : Type ≝
- { premonoid:> PreMonoid;
- opp: 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)
- }.
-
-record Group : Type ≝
- { pregroup:> PreGroup;
- group_properties:> isGroup pregroup
- }.
-
-(*notation < "G"
-for @{ 'monoid $G }.
-
-interpretation "Monoid coercion" 'monoid G =
- (cic:/matita/algebra/groups/monoid.con G).*)
-
-notation < "G"
-for @{ 'type_of_group $G }.
-
-interpretation "Type_of_group coercion" 'type_of_group G =
- (cic:/matita/algebra/groups/Type_of_Group.con G).
-
-notation < "G"
-for @{ 'magma_of_group $G }.
-
-interpretation "magma_of_group coercion" 'magma_of_group G =
- (cic:/matita/algebra/groups/Magma_of_Group.con G).
-
-notation "hvbox(x \sup (-1))" with precedence 89
-for @{ 'gopp $x }.
-
-interpretation "Group inverse" 'gopp x =
- (cic:/matita/algebra/groups/opp.con _ x).
-
-definition left_cancellable ≝
- λT:Type. λop: T -> T -> T.
- ∀x. injective ? ? (op x).
-
-definition right_cancellable ≝
- λT:Type. λop: T -> T -> T.
- ∀x. injective ? ? (λz.op z x).
-
-theorem eq_op_x_y_op_x_z_to_eq:
- ∀G:Group. left_cancellable G (op G).
-intros;
-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))));
-apply eq_f;
-assumption.
-qed.
-
-
-theorem eq_op_x_y_op_z_y_to_eq:
- ∀G:Group. right_cancellable G (op G).
-intros;
-unfold right_cancellable;
-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 > 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.
-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: \forall n,m:nat. \forall P:bool \to Prop.
-(n < m \to (P true)) \to (n ≮ m \to (P false)) \to
-P (ltb n m).
-intros.
-cut
-(match (ltb n m) with
-[ true \Rightarrow n < m
-| false \Rightarrow n ≮ m] \to (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 \to 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 \to ∃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;
- |
- |
- ]
- ]
- | apply (ltn_to_ltO ? ? H4)
- ]
- | apply le_S;
- assumption
- ]
- | apply Hcut
- | apply Hcut1
- | apply le_S_S_to_le;
- rewrite < S_pred;
- exact H3
- ]
- (* TODO: caso complicato, ma simile al terzo *)
- | 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 foo:
- ∀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)
- ]
- ]
-| apply pigeonhole
-].
projects / helm.git / blobdiff