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theories/exp.v
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| have -> : 1 + x = limn (series (f x)). | ||
| by apply/esym/lim_near_cst => //; near=> n; apply: F; near: n. | ||
| apply: ler_lim; first by apply: is_cvg_near_cst; near=> n; apply: F; near: n. | ||
| pose f n (x : R) i := ((i == 0%nat)%:R + x ^+ n / n`!%:R *+ (i == n)). |
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| pose f n (x : R) i := ((i == 0%nat)%:R + x ^+ n / n`!%:R *+ (i == n)). | |
| pose f n (x : R) i := ((i == 0%N)%:R + x ^+ n / n`!%:R *+ (i == n)). |
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theories/exp.v
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| by apply/esym/lim_near_cst => //; near=> n; apply: F; near: n. | ||
| apply: ler_lim; first by apply: is_cvg_near_cst; near=> n; apply: F; near: n. | ||
| pose f n (x : R) i := ((i == 0%nat)%:R + x ^+ n / n`!%:R *+ (i == n)). | ||
| have F m : (n.+1 < m)%nat -> |
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| have F m : (n.+1 < m)%nat -> | |
| have F m : (n.+1 < m)%N -> |
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theories/exp.v
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| Local Lemma expR_ge1Dx_subproof x : 0 <= x -> 1 + x <= expR x. | ||
| Lemma expR_ge1Dxn x n : 0 <= x -> | ||
| 1 + x ^+ n.+1 / n.+1`!%:R <= expR x. |
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can be on the previous line without exceeding 80 cols?
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Looks good. Any awaiting application? |
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I used this lemma to prove a property of the gamma function, which I'll include in an upcoming PR: Lemma Gamma_add1 a : 1 <= a -> (Gamma (a + 1) = a%:E * Gamma a)%E. |
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I did not at first sight judge whether this rather special form of statement could be useful, but with the application given, I approve this PR. (However I cannot find the approval button now..) |
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* expR_ge1Dxn
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Motivation for this change
This lemma is a property of expR, which is generalized from a local lemma
0 <= x -> 1 + x <= expR x.Checklist
CHANGELOG_UNRELEASED.mdReference: How to document
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