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