}.
record isGroup (G:PreGroup) : Prop ≝
- { is_monoid: isMonoid G;
+ { is_monoid:> isMonoid 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)
}.
unfold left_cancellable;
unfold injective;
intros (x y z);
-rewrite < (e_is_left_unit ? (is_monoid ? G));
-rewrite < (e_is_left_unit ? (is_monoid ? G) z);
+rewrite < (e_is_left_unit ? G);
+rewrite < (e_is_left_unit ? 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)));
+rewrite > (associative ? (is_semi_group ? ( G)));
+rewrite > (associative ? (is_semi_group ? ( G)));
apply eq_f;
assumption.
qed.
unfold injective;
simplify;fold simplify (op G);
intros (x y z);
-rewrite < (e_is_right_unit ? (is_monoid ? G));
-rewrite < (e_is_right_unit ? (is_monoid ? G) z);
+rewrite < (e_is_right_unit ? ( G));
+rewrite < (e_is_right_unit ? ( 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 < (associative ? (is_semi_group ? ( G)));
+rewrite < (associative ? (is_semi_group ? ( G)));
rewrite > H;
reflexivity.
qed.
∀G:Group. ∀x,y,z:G. x·y=z → x = z·y \sup -1.
intros;
apply (eq_op_x_y_op_z_y_to_eq ? y);
-rewrite > (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite > (associative ? G);
rewrite > (inv_is_left_inverse ? G);
-rewrite > (e_is_right_unit ? (is_monoid ? G));
+rewrite > (e_is_right_unit ? G);
assumption.
qed.
∀G:Group. ∀x,y,z:G. x·y=z → y = x \sup -1·z.
intros;
apply (eq_op_x_y_op_x_z_to_eq ? x);
-rewrite < (associative ? (is_semi_group ? (is_monoid ? G)));
+rewrite < (associative ? G);
rewrite > (inv_is_right_inverse ? G);
rewrite > (e_is_left_unit ? (is_monoid ? G));
assumption.
-qed.
\ No newline at end of file
+qed.
+
+theorem eq_inv_op_x_y_op_inv_y_inv_x:
+ ∀G:Group. ∀x,y:G. (x·y) \sup -1 = y \sup -1 · x \sup -1.
+intros;
+apply (eq_op_x_y_op_z_y_to_eq ? (x·y));
+rewrite > (inv_is_left_inverse ? G);
+rewrite < (associative ? G);
+rewrite > (associative ? G (y \sup -1));
+rewrite > (inv_is_left_inverse ? G);
+rewrite > (e_is_right_unit ? G);
+rewrite > (inv_is_left_inverse ? G);
+reflexivity.
+qed.
+
+(* Morphisms *)
+
+record morphism (G,G':Group) : Type ≝
+ { image: G → G';
+ f_morph: ∀x,y:G.image(x·y) = image x · image y
+ }.
+
+notation "hvbox(f˜ x)" with precedence 79
+for @{ 'morimage $f $x }.
+
+interpretation "Morphism image" 'morimage f x =
+ (cic:/matita/algebra/groups/image.con _ _ f x).
+
+theorem morphism_to_eq_f_1_1:
+ ∀G,G'.∀f:morphism G G'.f˜1 = 1.
+intros;
+apply (eq_op_x_y_op_z_y_to_eq G' (f˜1));
+rewrite > (e_is_left_unit ? G' ?);
+rewrite < (f_morph ? ? f);
+rewrite > (e_is_left_unit ? G);
+reflexivity.
+qed.
+
+theorem eq_image_inv_inv_image:
+ ∀G,G'.∀f:morphism G G'.
+ ∀x.f˜(x \sup -1) = (f˜x) \sup -1.
+intros;
+apply (eq_op_x_y_op_z_y_to_eq G' (f˜x));
+rewrite > (inv_is_left_inverse ? G');
+rewrite < (f_morph ? ? f);
+rewrite > (inv_is_left_inverse ? G);
+apply (morphism_to_eq_f_1_1 ? ? f).
+qed.
+
+record monomorphism (G,G':Group) : Type ≝
+ { morphism:> morphism G G';
+ injective: injective ? ? (image ? ? morphism)
+ }.
+
+(* Subgroups *)
+
+record subgroup (G:Group) : Type ≝
+ { group: Group;
+ embed:> monomorphism group G
+ }.
+
+notation "hvbox(x \sub H)" with precedence 79
+for @{ 'subgroupimage $H $x }.
+
+interpretation "Subgroup image" 'subgroupimage H x =
+ (cic:/matita/algebra/groups/image.con _ _
+ (cic:/matita/algebra/groups/morphism_of_subgroup.con _ H) x).
+
+definition member_of_subgroup ≝
+ λG.λH:subgroup G.λx:G.∃y.x=y \sub H.
+
+notation "hvbox(x break ∈ H)" with precedence 79
+for @{ 'member_of $x $H }.
+
+interpretation "Member of subgroup" 'member_of x H =
+ (cic:/matita/algebra/groups/member_of_subgroup.con _ H x).
+
+(* Left cosets *)
+
+record left_coset (G:Group) : Type ≝
+ { element: G;
+ subgrp: subgroup G
+ }.
+
+(* Here I would prefer 'magma_op, but this breaks something in the next definition *)
+interpretation "Left_coset" 'times x C =
+ (cic:/matita/algebra/groups/left_coset.ind#xpointer(1/1/1) _ x C).
+
+definition member_of_left_coset ≝
+ λG:Group.λC:left_coset G.λx:G.
+ ∃y.x=(element ? C)·y \sub (subgrp ? C).
+
+interpretation "Member of left_coset" 'member_of x C =
+ (cic:/matita/algebra/groups/member_of_left_coset.con _ C x).
+
+definition left_coset_eq ≝
+ λG.λC,C':left_coset G.
+ ∀x.((element ? C)·x \sub (subgrp ? C)) ∈ C'.
+
+interpretation "Left cosets equality" 'eq C C' =
+ (cic:/matita/algebra/groups/left_coset_eq.con _ C C').
+
+definition left_coset_disjoint ≝
+ λG.λC,C':left_coset G.
+ ∀x.¬(((element ? C)·x \sub (subgrp ? C)) ∈ C').
+
+notation "hvbox(a break ∥ b)"
+ non associative with precedence 45
+for @{ 'disjoint $a $b }.
+
+interpretation "Left cosets disjoint" 'disjoint C C' =
+ (cic:/matita/algebra/groups/left_coset_disjoint.con _ C C').
+
+(* The following should be a one-shot alias! *)
+alias symbol "member_of" (instance 0) = "Member of subgroup".
+theorem member_of_subgroup_op_inv_x_y_to_left_coset_eq:
+ ∀G.∀x,y.∀H:subgroup G. (x \sup -1 ·y) ∈ H → x*H = y*H.
+intros;
+unfold left_coset_eq;
+simplify in ⊢ (? → ? ? ? (? ? % ?));
+simplify in ⊢ (? → ? ? ? (? ? ? (? ? ? (? ? %) ?)));
+simplify in ⊢ (? % → ?);
+intros;
+unfold member_of_left_coset;
+simplify in ⊢ (? ? (λy:?.? ? ? (? ? ? (? ? ? (? ? %) ?))));
+simplify in ⊢ (? ? (λy:? %.?));
+simplify in ⊢ (? ? (λy:?.? ? ? (? ? % ?)));
+unfold member_of_subgroup in H1;
+elim H1;
+clear H1;
+exists;
+[ apply (a\sup-1 · x1)
+| rewrite > (f_morph ? ? (morphism ? ? H));
+ rewrite > (eq_image_inv_inv_image ? ?
+ rewrite < H2;
+ rewrite > (eq_inv_op_x_y_op_inv_y_inv_x ? ? ? ? H2);
+].
+qed.
+
+(*theorem foo:
+ \forall G:Group. \forall x1,x2:G. \forall H:subgroup G.
+ x1*x2^-1 \nin H \to x1*H does_not_overlap x2*H
+
+theorem foo:
+ \forall x:G. \forall H:subgroup G. x \in x*H
+
+definition disjoinct
+ (T: Type) (n:nat) (S: \forall x:nat. x < n -> {S:Type * (S -> T)})
+:=
+ \forall i,j:nat. i < n \to j < n \to ...
+
+
+check
+ (λG.λH,H':left_coset G.λx:Type_of_Group (group ? (subgrp ? H)). (embed ? (subgrp ? H) x)).
+
+definition left_coset_eq ≝
+ λG.λH,H':left_coset G.
+ ∀x:group ? (subgrp ? H).
+ ex (group ? (subgroup ? H')) (λy.
+ (element ? H)·(embed ? (subgrp ? H) x) =
+ (element ? H')·(embed ? (subgrp ? H') y)).
+
+(*record left_coset (G:Group) : Type ≝
+ { subgroup: Group;
+ subgroup_is_subgroup: subgroup ≤ G;
+ element: G
+ }.
+
+definition left_coset_eq ≝
+ λG.λH,H':left_coset G.
+ ∀x:subgroup ? H.
+ ex (subgroup ? H') (λy.
+ (element ? H)·(embed ? ? (subgroup_is_subgroup ? H) ˜ x) =
+ (element ? H')·(embed ? ? (subgroup_is_subgroup ? H') ˜ y)).
+*)
+*)
projects / helm.git / blobdiff