[Merged by Bors] - feat(data/rat/basic): Add nat num and denom inv lemmas #8581
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| lemma inv_coe_nat_num {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.num = 1 := | ||
| begin | ||
| rw [rat.inv_def', rat.coe_nat_num, rat.coe_nat_denom], | ||
| suffices : (((1 : ℤ) : ℚ) / (a : ℤ)).num = 1, | ||
| exact_mod_cast this, | ||
| apply num_div_eq_of_coprime, | ||
| { assumption_mod_cast }, | ||
| { simp only [nat.coprime_one_left_iff, int.nat_abs_one] } | ||
| end |
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I think this proof is simpler if you prove it for int first:
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| lemma inv_coe_nat_num {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.num = 1 := | |
| begin | |
| rw [rat.inv_def', rat.coe_nat_num, rat.coe_nat_denom], | |
| suffices : (((1 : ℤ) : ℚ) / (a : ℤ)).num = 1, | |
| exact_mod_cast this, | |
| apply num_div_eq_of_coprime, | |
| { assumption_mod_cast }, | |
| { simp only [nat.coprime_one_left_iff, int.nat_abs_one] } | |
| end | |
| lemma inv_coe_int_num {a : ℤ} (ha0 : 0 < a) : (a : ℚ)⁻¹.num = 1 := | |
| begin | |
| rw [rat.inv_def', rat.coe_int_num, rat.coe_int_denom, nat.cast_one, ←int.cast_one], | |
| apply num_div_eq_of_coprime ha0, | |
| rw int.nat_abs_one, | |
| exact nat.coprime_one_left _, | |
| end | |
| lemma inv_coe_nat_num {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.num = 1 := | |
| inv_coe_int_num (by exact_mod_cast ha0 : 0 < (a : ℤ)) |
Can you do the same for the lemma below?
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Yeah, thanks for the feedback! I updated your golf and added inv_coe_int_denom. The nat version of the proof uses an additional exact_mod_cast, because of casting the denom, which is a nat.
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Add `inv_coe_nat_num` and `inv_coe_nat_denom` lemmas. These lemmas show that the denominator and numerator of `1/ n` given `0 < n`, are equal to `n` and `1` respectively. Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Add
inv_coe_nat_numandinv_coe_nat_denomlemmas.These lemmas show that the denominator and numerator of
1/ ngiven0 < n, are equal tonand1respectively.Hello All👋, this is my first PR. I am working a project where I formalized the results for thomae's function. This lemma was necessary and seemed fundamental enough to rationals, to make a PR.
The lemmas are the following, 0 < n, 1/n .denom = n and 1/n.num = 1
Was done with collaboration with Xena and in particular Kevin buzzard.