Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Rectangle disjoint singleton

rectangle_disjoint_singleton

analytic number theory prime numbers asymptotics rectangle

Plain-language statement

Let $R$ be the axis-parallel rectangle with opposite corners $z,w\in\mathbb C$ . If a point $p$ lies strictly to the left, right, below, or above both corners, then $p\notin R$ ; equivalently, the rectangle and the singleton $\{p\}$ are disjoint.

Exact Lean statement

lemma rectangle_disjoint_singleton {z w p : ℂ}
    (h : (p.re < z.re ∧ p.re < w.re) ∨ (p.im < z.im ∧ p.im < w.im) ∨
      (z.re < p.re ∧ w.re < p.re) ∨ (z.im < p.im ∧ w.im < p.im)) :
    Disjoint (Rectangle z w) {p}

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma rectangle_disjoint_singleton {z w p : ℂ}    (h : (p.re < z.re ∧ p.re < w.re) ∨ (p.im < z.im ∧ p.im < w.im) ∨      (z.re < p.re ∧ w.re < p.re) ∨ (z.im < p.im ∧ w.im < p.im)) :    Disjoint (Rectangle z w) {p} := by  refine disjoint_singleton_right.mpr (not_and_or.mpr ?_)  obtain h | h | h | h := h  · exact Or.inl (notMem_uIcc_of_lt h.1 h.2)  · exact Or.inr (notMem_uIcc_of_lt h.1 h.2)  · exact Or.inl (notMem_uIcc_of_gt h.1 h.2)  · exact Or.inr (notMem_uIcc_of_gt h.1 h.2)

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/Rectangle.lean:144-153

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