48 lines
1.4 KiB
Text
48 lines
1.4 KiB
Text
import algebra.group
|
|
open algebra
|
|
|
|
variables {A : Type}
|
|
variables [s : group A]
|
|
include s
|
|
namespace foo
|
|
set_option blast.strategy "ematch"
|
|
attribute inv_inv mul.left_inv mul.assoc one_mul mul_one [forward]
|
|
|
|
theorem mul.right_inv (a : A) : a * a⁻¹ = 1 :=
|
|
calc
|
|
a * a⁻¹ = (a⁻¹)⁻¹ * a⁻¹ : by blast
|
|
... = 1 : by blast
|
|
|
|
theorem mul.right_inv₂ (a : A) : a * a⁻¹ = 1 :=
|
|
by blast
|
|
|
|
theorem mul_inv_cancel_left (a b : A) : a * (a⁻¹ * b) = b :=
|
|
calc
|
|
a * (a⁻¹ * b) = a * a⁻¹ * b : by blast
|
|
... = 1 * b : by blast
|
|
... = b : by blast
|
|
|
|
theorem mul_inv_cancel_left₂ (a b : A) : a * (a⁻¹ * b) = b :=
|
|
by blast
|
|
|
|
theorem mul_inv (a b : A) : (a * b)⁻¹ = b⁻¹ * a⁻¹ :=
|
|
inv_eq_of_mul_eq_one
|
|
(calc
|
|
a * b * (b⁻¹ * a⁻¹) = a * (b * (b⁻¹ * a⁻¹)) : by blast
|
|
... = 1 : by blast)
|
|
|
|
theorem eq_of_mul_inv_eq_one {a b : A} (H : a * b⁻¹ = 1) : a = b :=
|
|
calc
|
|
a = a * b⁻¹ * b : by blast
|
|
... = 1 * b : by blast
|
|
... = b : by blast
|
|
|
|
|
|
-- This is another theorem that can be easily proved using superposition,
|
|
-- but cannot to be proved using E-matching.
|
|
-- To prove it using E-matching, we must provide the following auxiliary step using calc.
|
|
theorem eq_of_mul_inv_eq_one₂ {a b : A} (H : a * b⁻¹ = 1) : a = b :=
|
|
calc
|
|
a = a * b⁻¹ * b : by blast
|
|
... = b : by blast
|
|
end foo
|