[Merged by Bors] - feat(analysis/special_functions/complex/arg): review, golf, lemmas #10365
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Looks like a lemma you used does not exist? My review was too hasty. |
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Co-authored-by: Rémy Degenne <remydegenne@gmail.com>
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@RemyDegenne It exists in |
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…10365) * add `|z| * exp (arg z * I) = z`; * reorder theorems to help golfing; * prove formulas for `arg z` in terms of `arccos (re z / abs z)` in cases `0 < im z` and `im z < 0`; * use them to golf continuity of `arg`.
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…10365) * add `|z| * exp (arg z * I) = z`; * reorder theorems to help golfing; * prove formulas for `arg z` in terms of `arccos (re z / abs z)` in cases `0 < im z` and `im z < 0`; * use them to golf continuity of `arg`.
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…10365) * add `|z| * exp (arg z * I) = z`; * reorder theorems to help golfing; * prove formulas for `arg z` in terms of `arccos (re z / abs z)` in cases `0 < im z` and `im z < 0`; * use them to golf continuity of `arg`.
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…10365) * add `|z| * exp (arg z * I) = z`; * reorder theorems to help golfing; * prove formulas for `arg z` in terms of `arccos (re z / abs z)` in cases `0 < im z` and `im z < 0`; * use them to golf continuity of `arg`.
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Add lemmas about the value of `arg (-x)`: one each for positive and negative sign of `x.im`, two `iff` lemmas saying exactly when it's equal to `arg x - π` or `arg x + π`, and a simpler lemma giving the value of `(arg (-x) : real.angle)` for any nonzero `x`. These replace the previous lemmas `arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg` and `arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg`, which are strictly less general (they have a hypothesis `x.re < 0` that's not needed unless the imaginary part is 0). Those two lemmas are unused in current mathlib (they were used in the proof of `cos_arg` before the golfing in #10365) and it seems reasonable to me to replace them with the more general lemmas.
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Add lemmas about the value of `arg (-x)`: one each for positive and negative sign of `x.im`, two `iff` lemmas saying exactly when it's equal to `arg x - π` or `arg x + π`, and a simpler lemma giving the value of `(arg (-x) : real.angle)` for any nonzero `x`. These replace the previous lemmas `arg_eq_arg_neg_add_pi_of_im_nonneg_of_re_neg` and `arg_eq_arg_neg_sub_pi_of_im_neg_of_re_neg`, which are strictly less general (they have a hypothesis `x.re < 0` that's not needed unless the imaginary part is 0). Those two lemmas are unused in current mathlib (they were used in the proof of `cos_arg` before the golfing in #10365) and it seems reasonable to me to replace them with the more general lemmas.
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|z| * exp (arg z * I) = z;arg zin terms ofarccos (re z / abs z)in cases0 < im zandim z < 0;arg.