Coq：不充分的对正错误

我是Coq新手，假设H3的理由不足。我试着重写了好几次，但错误依然存在。有人能解释一下原因吗？谢谢

Section GroupTheory.
Variable G: Set.
Variable operation: G -> G -> G.
Variable e : G.
Variable inv : G -> G.
Infix "*" := operation.

Hypothesis associativity : forall x y z : G, (x * y) * z = x * (y * z).
Hypothesis identity : forall x : G, exists e : G, (x * e = x) /\ (e * x = x).
Hypothesis inverse : forall x : G, (x * inv x = e) /\ (inv x * x = e).

Theorem latin_square_property : 
  forall a b : G, exists x : G, a * x = b.
proof.
  let a : G, b : G.
  take (inv a * b).
  have H1:(a * (inv a * b) = (a * inv a) * b) by associativity.
  have H2:(a * inv a = e) by inverse.
  have H3:(e * b = b) by identity.
  have (a * (inv a * b) = (a * inv a) * b) by H1.
                       ~= (e * b) by H2.
                       ~= (b) by H3.
hence thesis.
end proof.
Qed.
End GroupTheory.

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原因是您的

身份

公理独立于本节中定义的单位

e

，因为您已将

e

与

身份

公理定义中的存在量词绑定

我们可以修改

identity

，去掉定义中存在的

e

：

Hypothesis identity : forall x : G, (x * e = x) /\ (e * x = x).

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然后你就可以完成你的证明了。

就是这样。谢谢