Skip to content

feat: prove Real.sqrt_le_self.mp #21

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
YaelDillies merged 1 commit into from
Jul 1, 2026
Merged

feat: prove Real.sqrt_le_self.mp #21

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
12 changes: 5 additions & 7 deletions APAP/Mathlib/Analysis/Real/Sqrt.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -7,16 +7,14 @@ public section
namespace Real
variable {x : ℝ}

@[simp] lemma sqrt_le_self : √x ≤ x ↔ x = 0 ∨ 1 ≤ x where
mp := sorry
mpr := by
rintro (rfl | hx)
· simp
· exact sqrt_le_iff.2 ⟨zero_le_one.trans hx, le_self_pow₀ hx two_ne_zero⟩
@[simp] lemma sqrt_le_self : √x ≤ x ↔ x = 0 ∨ 1 ≤ x := by
obtain hx | hx := le_or_gt x 0
· simp [sqrt_eq_zero_of_nonpos hx, le_antisymm_iff, hx, not_le.2 (hx.trans_lt zero_lt_one)]
· simp [sqrt_le_iff, sq, le_mul_iff_one_le_right hx, hx.le, hx.ne']

@[simp] lemma le_sqrt_self : x ≤ √x ↔ x ≤ 1 := by
obtain hx | hx := le_or_gt x 0
· exact iff_of_true (hx.trans x.sqrt_nonneg) (hx.trans zero_le_one)
· simp [hx.trans]
· rw [le_sqrt' hx, sq, mul_le_iff_le_one_left hx]

end Real
Loading