40 lines
1.3 KiB
Text
40 lines
1.3 KiB
Text
import data.nat
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open nat eq.ops
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theorem lcm_dvd {m n k : nat} (H1 : (m | k)) (H2 : (n | k)) : (lcm m n | k) :=
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match eq_zero_or_pos k with
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| @or.inl _ _ kzero :=
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begin
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rewrite kzero,
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apply dvd_zero
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end
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| @or.inr _ _ kpos :=
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obtain (p : nat) (km : k = m * p), from exists_eq_mul_right_of_dvd H1,
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obtain (q : nat) (kn : k = n * q), from exists_eq_mul_right_of_dvd H2,
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begin
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have mpos : m > 0, from pos_of_dvd_of_pos H1 kpos,
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have npos : n > 0, from pos_of_dvd_of_pos H2 kpos,
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have gcd_pos : gcd m n > 0, from gcd_pos_of_pos_left n mpos,
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have ppos : p > 0,
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begin
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apply pos_of_mul_pos_left,
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apply (eq.rec_on km),
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exact kpos
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end,
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have qpos : q > 0, from pos_of_mul_pos_left (kn ▸ kpos),
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have H3 : p * q * (m * n * gcd p q) = p * q * (gcd m n * k),
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begin
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apply sorry
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end,
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have H4 : m * n * gcd p q = gcd m n * k, from
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!eq_of_mul_eq_mul_left (mul_pos ppos qpos) H3,
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have H5 : gcd m n * (lcm m n * gcd p q) = gcd m n * k,
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begin
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rewrite [-mul.assoc, gcd_mul_lcm],
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exact H4
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end,
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have H6 : lcm m n * gcd p q = k, from
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!eq_of_mul_eq_mul_left gcd_pos H5,
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exact (dvd.intro H6)
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end
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end
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