some algebra
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1 changed files with 62 additions and 44 deletions
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@ -2,69 +2,87 @@
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section Magma
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variable (G : ★);
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def binop := G → G → G;
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variable (G H : ★);
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def binop (A : ★) := A → A → A;
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def = := eq G;
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infixl 1 =;
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variable (* : binop);
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def ~ := eq H;
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infixl 1 ~;
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variable (* : binop G);
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infixl 20 *;
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variable (+ : binop H);
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infixl 20 +;
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variable (f : G → H);
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hypothesis (f_homo : forall (a b : G), f (a * b) ~ f a * f b);
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def *> (a b : G) := b * a;
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infixl 20 *>;
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section Semigroup
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def assoc := forall (a b c : G), a * (b * c) = a * b * c;
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hypothesis (Hassoc : assoc);
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hypothesis (assocG : forall (a b c : G), a * (b * c) = a * b * c);
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hypothesis (assocH : forall (a b c : H), a + (b + c) ~ a + b + c);
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def assoc_op (a b c : G) : a *> (b *> c) = a *> b *> c :=
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eq_sym G (a *> b *> c) (a *> (b *> c)) (Hassoc c b a);
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eq_sym G (a *> b *> c) (a *> (b *> c)) (assocG c b a);
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section Monoid
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variable (e : G);
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def id_l := forall (a : G), e * a = a;
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def id_r := forall (a : G), a * e = a;
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variable (eH : H);
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hypothesis (Hid_l : id_l);
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hypothesis (Hid_r : id_r);
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hypothesis (id_lG : forall (a : G), e * a = a);
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hypothesis (id_rG : forall (a : G), a * e = a);
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def id_l_op : forall (a : G), e *> a = a := Hid_r;
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def id_r_op : forall (a : G), a *> e = a := Hid_l;
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hypothesis (id_lH : forall (a : H), e * a = a);
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hypothesis (id_rH : forall (a : H), a * e = a);
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def left_id_unique (a : G) : #id_l G (*) a → a = e :=
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fun (H : #id_l G (*) a) =>
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eq_trans G a (a * e) e
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(eq_sym G (a * e) a (Hid_r a))
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(H e);
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def id_l_op : forall (a : G), e *> a = a := id_rG;
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def id_r_op : forall (a : G), a *> e = a := id_lG;
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def right_id_unique (a : G) : #id_r G (*) a → a = e := #left_id_unique G (*>) e Hid_l a;
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def left_id_unique (a : G) (H : forall (b : G), a * b = b) : a = e :=
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eq_trans G a (a * e) e (eq_sym G (a * e) a (id_rG a)) (H e);
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def right_id_unique (a : G) (H : forall (b : G), b * a = b) : a = e := #left_id_unique G (*>) e id_lG a H;
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section Group
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variable (i : G → G);
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hypothesis (Hinv_l : forall (a : G), i a * a = e);
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hypothesis (Hinv_r : forall (a : G), a * i a = e);
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hypothesis (inv_lG : forall (a : G), i a * a = e);
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hypothesis (inv_rG : forall (a : G), a * i a = e);
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variable (j : H → H);
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hypothesis (inv_lH : forall (a : H), j a * a ~ eH);
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hypothesis (inv_rH : forall (a : H), a * j a ~ eH);
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def left_inv_unique (a b : G) (h : b * a = e) : b = i a :=
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eq_trans G b (b * e) (i a)
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(eq_sym G (b * e) b (Hid_r b))
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(eq_sym G (b * e) b (id_rG b))
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(eq_trans G (b * e) (b * (a * i a)) (i a)
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(eq_cong G G e (a * i a) ((*) b)
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(eq_sym G (a * i a) e (Hinv_r a)))
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(eq_sym G (a * i a) e (inv_rG a)))
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(eq_trans G (b * (a * i a)) (b * a * i a) (i a)
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(Hassoc b a (i a))
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(assocG b a (i a))
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(eq_trans G (b * a * i a) (e * i a) (i a)
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(eq_cong G G (b * a) e ([z : G] z * i a) h)
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(Hid_l (i a)))));
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(id_lG (i a)))));
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def right_inv_unique (a b : G) (h : a * b = e) : b = i a :=
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#left_inv_unique G (*>) assoc_op e Hid_r Hid_l i Hinv_l a b h;
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#left_inv_unique G (*>) assoc_op e id_rG id_lG i inv_lG a b h;
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def homo_preserve_inv (a : G) : f (i a) ~ j (f a) :=
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#left_inv_unique H (+) assocH eH id_lH id_rH j inv_rH (f a) (f (i a))
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(eq_trans H (f a * f (i a)) (f (a * i a))
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)
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-- WTS: f a * f (i a) ~ eH
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def inverse_involutive (a : G) : i (i a) = a :=
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eq_sym G a (i (i a)) (right_inv_unique (i a) a (Hinv_l a));
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eq_sym G a (i (i a)) (right_inv_unique (i a) a (inv_lG a));
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def shoes_and_socks (a b : G) : i (a * b) = i b * i a :=
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eq_sym G (i b * i a) (i (a * b))
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@ -73,34 +91,34 @@ def shoes_and_socks (a b : G) : i (a * b) = i b * i a :=
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(under_ai (x y : G) (h : x = y) := eq_cong G G x y ([z : G] z * (i a)) h)
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in
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eq_trans G (a * b * (i b * i a)) (a * b * i b * i a) e
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(Hassoc (a * b) (i b) (i a))
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(assocG (a * b) (i b) (i a))
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(eq_trans G (a * b * i b * i a) (a * (b * i b) * i a) e
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(under_ai (a * b * i b) (a * (b * i b)) (eq_sym G (a * (b * i b)) (a * b * i b) (Hassoc a b (i b))))
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(under_ai (a * b * i b) (a * (b * i b)) (eq_sym G (a * (b * i b)) (a * b * i b) (assocG a b (i b))))
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(eq_trans G (a * (b * i b) * i a) (a * e * i a) e
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(eq_cong G G (b * i b) e (fun (x : G) => (a * x * i a)) (Hinv_r b))
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(eq_cong G G (b * i b) e (fun (x : G) => (a * x * i a)) (inv_rG b))
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(eq_trans G (a * e * i a) (a * i a) e
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(under_ai (a * e) a (Hid_r a))
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(Hinv_r a))))
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(under_ai (a * e) a (id_rG a))
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(inv_rG a))))
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end));
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def cancel_l (a b c : G) (h : a * b = a * c) : b = c :=
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eq_trans G b (e * b) c
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(eq_sym G (e * b) b (Hid_l b))
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(eq_sym G (e * b) b (id_lG b))
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(eq_trans G (e * b) (i a * a * b) c
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(eq_cong G G e (i a * a) ([x : G] x * b)
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(eq_sym G (i a * a) e (Hinv_l a)))
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(eq_sym G (i a * a) e (inv_lG a)))
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(eq_trans G (i a * a * b) (i a * (a * b)) c
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(eq_sym G (i a * (a * b)) (i a * a * b) (Hassoc (i a) a b))
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(eq_sym G (i a * (a * b)) (i a * a * b) (assocG (i a) a b))
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(eq_trans G (i a * (a * b)) (i a * (a * c)) c
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(eq_cong G G (a * b) (a * c) ((*) (i a)) h)
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(eq_trans G (i a * (a * c)) (i a * a * c) c
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(Hassoc (i a) a c)
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(assocG (i a) a c)
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(eq_trans G (i a * a * c) (e * c) c
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(eq_cong G G (i a * a) e ([x : G] x * c) (Hinv_l a))
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(Hid_l c))))));
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(eq_cong G G (i a * a) e ([x : G] x * c) (inv_lG a))
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(id_lG c))))));
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def cancel_r (a b c : G) (h : b * a = c * a) : b = c :=
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#cancel_l G (*>) assoc_op e Hid_r i Hinv_r a b c h;
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#cancel_l G (*>) assoc_op e id_rG i inv_rG a b c h;
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def abelian : ★ := forall (a b : G), a * b = b * a;
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@ -108,16 +126,16 @@ def left_right_cancel : (forall (x y z : G), x * y = z * x → y = z) → abelia
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fun (h : forall (x y z : G), x * y = z * x → y = z) (a b : G) =>
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h (i a) (a * b) (b * a)
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(eq_trans G (i a * (a * b)) (i a * a * b) (b * a * i a)
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(Hassoc (i a) a b)
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(assocG (i a) a b)
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(eq_trans G (i a * a * b) (e * b) (b * a * i a)
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(eq_cong G G (i a * a) e ([x : G] x * b) (Hinv_l a))
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(eq_cong G G (i a * a) e ([x : G] x * b) (inv_lG a))
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(eq_trans G (e * b) b (b * a * i a)
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(Hid_l b)
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(id_lG b)
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(eq_trans G b (b * e) (b * a * i a)
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(eq_sym G (b * e) b (Hid_r b))
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(eq_sym G (b * e) b (id_rG b))
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(eq_trans G (b * e) (b * (a * i a)) (b * a * i a)
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(eq_cong G G e (a * i a) ((*) b) (eq_sym G (a * i a) e (Hinv_r a)))
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(Hassoc b a (i a)))))));
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(eq_cong G G e (a * i a) ((*) b) (eq_sym G (a * i a) e (inv_rG a)))
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(assocG b a (i a)))))));
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def inv_distrib_abelian : (forall (a b : G), i (a * b) = i a * i b) → abelian :=
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fun (h : forall (a b : G), i (a * b) = i a * i b) (a b : G) =>
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