include "algebra/monoids.ma".
include "nat/le_arith.ma".
+include "datatypes/bool.ma".
+include "nat/compare.ma".
-record isGroup (M:Monoid) (opp: M -> M) : Prop ≝
- { opp_is_left_inverse: is_left_inverse M opp;
- opp_is_right_inverse: is_right_inverse M opp
+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 ≝
- { monoid: Monoid;
- opp: monoid -> monoid;
- group_properties: isGroup ? opp
+ { pregroup:> PreGroup;
+ group_properties:> isGroup pregroup
}.
-coercion cic:/matita/algebra/groups/monoid.con.
-
-notation < "G"
+(*notation < "G"
for @{ 'monoid $G }.
interpretation "Monoid coercion" 'monoid G =
- (cic:/matita/algebra/groups/monoid.con G).
+ (cic:/matita/algebra/groups/monoid.con G).*)
notation < "G"
for @{ 'type_of_group $G }.
(cic:/matita/algebra/groups/Type_of_Group.con G).
notation < "G"
-for @{ 'semigroup_of_group $G }.
+for @{ 'magma_of_group $G }.
-interpretation "Semigroup_of_group coercion" 'semigroup_of_group G =
- (cic:/matita/algebra/groups/SemiGroup_of_Group.con 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 }.
unfold left_cancellable;
unfold injective;
intros (x y z);
-rewrite < (e_is_left_unit ? ? (monoid_properties G));
-rewrite < (e_is_left_unit ? ? (monoid_properties G) z);
-rewrite < (opp_is_left_inverse ? ? (group_properties G) x);
-rewrite > (semigroup_properties G);
-rewrite > (semigroup_properties G);
+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.
unfold injective;
simplify;fold simplify (op G);
intros (x y z);
-rewrite < (e_is_right_unit ? ? (monoid_properties G));
-rewrite < (e_is_right_unit ? ? (monoid_properties G) z);
-rewrite < (opp_is_right_inverse ? ? (group_properties G) x);
-rewrite < (semigroup_properties G);
-rewrite < (semigroup_properties G);
+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.
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
+
+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).
+ (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 _).
-theorem repr_inj:
- ∀T:Type. ∀H:finite_enumerable T.
- ∀n,n'. n ≤ order ? H → n' ≤ order ? H →
- repr ? H n = repr ? H n' → n=n'.
+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;
-rewrite < (index_of_repr ? ? ? H1);
-rewrite > H3;
-apply index_of_repr;
+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:SemiGroup.
- finite_enumerable (carrier G) →
- left_cancellable (carrier G) (op G) →
- right_cancellable (carrier G) (op G) →
- ∃e:G. isMonoid ? e.
-intros (G H);
-letin f ≝ (λn.index_of ? H ((repr ? H O)·(repr ? H n)));
-cut (∀n.n ≤ order ? H → ∃m.f m = n);
+ ∀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;
- letin HH ≝ (eq_f ? ? (repr ? H) ? ? H3);
+ clear EX;
+ letin HH ≝ (eq_f ? ? (repr ? (is_finite_enumerable G)) ? ? H2);
clearbody HH;
- rewrite > (repr_index_of ? H) in HH;
- apply (ex_intro ? ? (repr ? H a));
- letin GOGO ≝ (refl_eq ? (repr ? H O));
+ 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 > (semigroup_properties G) in GOGO;
- letin GaGa ≝ (H1 ? ? ? GOGO);
+ rewrite > (associative ? G) in GOGO;
+ letin GaGa ≝ (H ? ? ? GOGO);
clearbody GaGa;
clear GOGO;
constructor 1;
- [ unfold is_left_unit; intro;
- letin GaxGax ≝ (refl_eq ? ((repr ? H a)·x));
+ [ 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 > (semigroup_properties G) in GaxGax;
- apply (H1 ? ? ? GaxGax)
+ rewrite > (associative ? (semigroup_properties G)) in GaxGax;
+ apply (H ? ? ? GaxGax)
| unfold is_right_unit; intro;
- letin GaxGax ≝ (refl_eq ? (x·(repr ? H a)));
+ letin GaxGax ≝ (refl_eq ? (x·G \sub a));
clearbody GaxGax;
rewrite < GaGa in GaxGax:(? ? % ?);
- rewrite < (semigroup_properties G) in GaxGax;
- apply (H2 ? ? ? GaxGax)
+ rewrite < (associative ? (semigroup_properties G)) in GaxGax;
+ apply (H1 ? ? ? GaxGax)
+ ]
]
-|
-].
\ No newline at end of file
+| apply pigeonhole
+].
projects / helm.git / blobdiff