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Eckmann—Hilton in Lean
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import tactic.interactive | |
universe u | |
namespace eckmann_hilton | |
variables (X : Type u) | |
local notation a `<`m`>` b := @has_mul.mul X m a b | |
class is_unital [m : has_mul X] [e : has_one X] : Prop := | |
(one_mul : ∀ x : X, (e.one <m> x) = x) | |
(mul_one : ∀ x : X, (x <m> e.one) = x) | |
attribute [simp] is_unital.one_mul | |
attribute [simp] is_unital.mul_one | |
variables {X} {m₁ : has_mul X} {e₁ : has_one X} {m₂ : has_mul X} {e₂ : has_one X} | |
@[extensionality] lemma mul.ext : m₁.mul = m₂.mul → m₁ = m₂ := | |
begin tactic.unfreeze_local_instances, cases m₁, cases m₂, intro h, cases h, refl end | |
variables (h₁ : @is_unital X m₁ e₁) (h₂ : @is_unital X m₂ e₂) | |
variables (distrib : ∀ a b c d, ((a <m₂> b) <m₁> (c <m₂> d)) = ((a <m₁> c) <m₂> (b <m₁> d))) | |
include h₁ h₂ distrib | |
lemma one : (e₁.one = e₂.one) := | |
by convert distrib e₂.one e₁.one e₁.one e₂.one; simp | |
set_option pp.all true | |
lemma mul : m₁ = m₂ := | |
begin | |
ext a b, | |
convert distrib a e₁.one e₁.one b; | |
{ simp <|> { rw [one h₁ h₂ distrib]; simp } <|> { rw ←(one h₁ h₂ distrib); simp } } | |
end | |
lemma mul_comm : is_commutative _ m₂.mul := | |
⟨λ a b, by convert distrib e₂.one a b e₂.one; try { rw mul h₁ h₂ distrib }; simp⟩ | |
lemma mul_assoc : is_associative _ m₂.mul := | |
⟨λ a b c, by convert distrib a b e₂.one c; try { rw mul h₁ h₂ distrib }; simp⟩ | |
instance : comm_monoid X := | |
{ mul_comm := (mul_comm h₁ h₂ distrib).comm, | |
mul_assoc := (mul_assoc h₁ h₂ distrib).assoc, | |
..e₂, | |
..m₂, | |
..h₂ } | |
end eckmann_hilton |
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