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September 5, 2024 06:40
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| example {x:Rat} : 3 * x / 0.2 = (2 + 1/2) / (3 + 1/3) -> x = 1/20 := by | |
| let is1 : (0.2:Rat) = 1/5 := by rfl | |
| rw [is1] | |
| let is2 : 3 * x / (1/5) = 3 * x * 5 := by | |
| simp_all only [one_div, div_inv_eq_mul] | |
| rw [is2] | |
| let is3 : 3 * x * 5 = 15 * x := by | |
| calc | |
| 3 * x * 5 = 3 * 5 * x := mul_right_comm 3 x 5 | |
| 3 * 5 * _ = 15 * x := by | |
| simp_all only [mul_eq_mul_right_iff] | |
| apply Or.inl | |
| rfl | |
| rw [is3] | |
| let is4 : (2:Rat) + 1/2 = 5/2 := by rfl | |
| rw [is4] | |
| let is5 : (3:Rat) + 1/3 = 10/3 := by rfl | |
| rw [is5] | |
| let is6 : (5:Rat)/2/(10/3) = 5/2 * (3/10) := by rfl | |
| rw [is6] | |
| let is7: ((5:Rat)/2) * (3/10) = 15/20 := by rfl | |
| rw [is7] | |
| let is8: 15*x=15/20 <-> 15*x*(1/15)=15/20*(1/15):= by | |
| simp_all only [one_div, mul_eq_mul_right_iff, inv_eq_zero,OfNat.ofNat_ne_zero, or_false] | |
| rw [is8] | |
| let is9: 15*x*(1/15) = x := by | |
| refine (div_eq_iff_mul_eq ?hb).mp ?_ | |
| refine one_div_ne_zero ?hb.h | |
| trivial | |
| let is1 : (x:Rat) / (1/15) = x * 15 := by aesop | |
| rw [is1] | |
| rw [Rat.mul_comm] | |
| rw [is9] | |
| let is10 : (15:Rat)/20 * (1/15) = 1/20 := by aesop | |
| rw [is10] | |
| apply id |
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