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fixes #1733 (ln_eq0) #1744

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Merged
merged 2 commits into from
Nov 6, 2025
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fixes #1733 (ln_eq0) #1744

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2aa252e
fixes #1733
affeldt-aist Oct 30, 2025
595ff9b
address comment
affeldt-aist Nov 6, 2025
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2 changes: 2 additions & 0 deletions CHANGELOG_UNRELEASED.md
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Original file line number Diff line number Diff line change
Expand Up @@ -81,6 +81,8 @@
+ lemmas `open_disjoint_itv_open`, `open_disjoint_itv_is_interval`,
`open_disjoint_itv_trivIset`, `open_disjoint_itv_bigcup`

- in `exp.v`:
+ lemma `ln_eq0`

### Changed

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23 changes: 13 additions & 10 deletions theories/exp.v
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@@ -1,10 +1,9 @@
(* mathcomp analysis (c) 2025 Inria and AIST. License: CeCILL-C. *)
From mathcomp Require Import all_ssreflect ssralg ssrint ssrnum matrix.
From mathcomp Require Import interval rat.
From mathcomp Require Import interval rat interval_inference.
From mathcomp Require Import boolp classical_sets functions mathcomp_extra.
From mathcomp Require Import unstable reals ereal interval_inference.
From mathcomp Require Import topology tvs normedtype landau sequences derive.
From mathcomp Require Import realfun interval_inference convex.
From mathcomp Require Import unstable reals topology ereal tvs normedtype.
From mathcomp Require Import landau sequences derive realfun convex.

(**md**************************************************************************)
(* # Theory of exponential/logarithm functions *)
Expand Down Expand Up @@ -785,6 +784,13 @@ have [x0|x0 x1] := leP x 0; first by rewrite ln0.
by rewrite -ler_expR expR0 lnK.
Qed.

Lemma ln_eq0 x : 0 < x -> (ln x == 0) = (x == 1).
Proof.
move=> x0; apply/idP/idP => [/eqP lnx0|/eqP ->]; last by rewrite ln1.
have [| |//] := ltgtP x 1; last by move/ln_gt0; rewrite lnx0 ltxx.
by move/(conj x0)/andP/ln_lt0; rewrite lnx0 ltxx.
Qed.

Lemma continuous_ln x : 0 < x -> {for x, continuous ln}.
Proof.
move=> x_gt0; rewrite -[x]lnK//.
Expand Down Expand Up @@ -1246,12 +1252,9 @@ Proof. by rewrite 2!ltNge lne_ge0. Qed.

Lemma lne_eq0 x : (lne x == 0) = (x == 1).
Proof.
apply/idP/idP => /eqP; last by move=> ->; rewrite lne1.
have [|x0] := leP x 0.
by case: x => [r| |]//; rewrite lee_fin => /= ->.
rewrite -lne1 => /lne_inj.
rewrite ?in_itv/= ?leey ?andbT// => /(_ _ _)/eqP.
by apply => //; exact: ltW.
rewrite /lne; move: x => [r| |] //; case: ifPn => r0.
by apply/esym; rewrite lt_eqF// lte_fin (le_lt_trans r0).
by rewrite !eqe ln_eq0// ltNge.
Qed.

Lemma lne_gt0 x : (0 < lne x) = (1 < x).
Expand Down