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Grid indexing constructs the scalar whole-cell certificate

For all $\htmlData{mx=1}{x, \mathit{anchor}, \mathit{width} \in \htmlData{mx=0}{\mathbb{R}}}$ , if $0 \mathrel{\htmlData{mx=1}{<}} \htmlData{mx=0}{\mathit{width}}$ , then

\left(\htmlData{mx=16}{\left\lvert \htmlData{mx=15}{\htmlData{mx=0}{x} - \left(\htmlData{mx=14}{\htmlData{mx=1}{\mathit{anchor}} + \htmlData{mx=13}{\htmlData{mx=2}{\mathit{width}} \cdot \left(\htmlData{mx=12}{\htmlData{mx=10}{\htmlData{mx=9}{\htmlData{mx=8}{\left\lfloor \htmlData{mx=7}{\frac{\htmlData{mx=5}{\htmlData{mx=3}{x} - \htmlData{mx=4}{\mathit{anchor}}}}{\htmlData{mx=6}{\mathit{width}}}} \right\rfloor}}} + \htmlData{mx=11}{\frac{1}{2}}}\right)}}\right)} \right\rvert} \mathrel{\htmlData{mx=19}{\leq}} \htmlData{mx=18}{\frac{\htmlData{mx=17}{\mathit{width}}}{2}}\right)

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: x, anchor, width range over real numbers: the statement must hold for every choice. ((x anchor width : ℝ))
<Less than: 0 is strictly smaller than width. (0 < width)
|x|Absolute value: The size of x − (anchor + width · (⌊(x − anchor)/width⌋ + 1/2)) without its sign. (|x-(anchor+width*(((⌊(x-anchor)/width⌋ …)
≤At most: |x − (anchor + width · (⌊(x − anchor)/width⌋ + 1/2))| is less than or equal to width/2. (|x-(anchor+width*(((⌊(x-anchor)/width⌋ …)
·Multiplication: width times ⌊(x − anchor)/width⌋ + 1/2. (width*(((⌊(x-anchor)/width⌋ : ℤ) : ℝ)+1…)
(x : T)As real numbers: ⌊(x − anchor)/width⌋ read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. (((⌊(x-anchor)/width⌋ : ℤ) : ℝ))
+Addition: ⌊(x − anchor)/width⌋ plus 1/2. (((⌊(x-anchor)/width⌋ : ℤ) : ℝ)+1/2)
(x : T)As integers: ⌊(x − anchor)/width⌋ read as an element of integers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((⌊(x-anchor)/width⌋ : ℤ))
⌊x⌋Floor: The largest integer not above (x − anchor)/width. (⌊(x-anchor)/width⌋)
a/bDivision: x − anchor divided by width. In Lean, dividing by 0 gives 0. ((x-anchor)/width)
-Subtraction: x minus anchor. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (x-anchor)

theorem grid-floor-center-certificate (x anchor width : ℝ) (hw : 0 < width) :
    (|x-(anchor+width*(((⌊(x-anchor)/width⌋ : ℤ) : ℝ)+1/2))| ≤ width/2)

For the actual floor-based cell center and positive width, derives the half-width coordinate bound without assuming that certificate as a binder. Vector dot contraction and the lifecycle data structure remain paper steps.

How it was proved

Proved: 6 October 2026
By: Ajax
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 16 lines of Lean

Grid indexing constructs the scalar whole-cell certificate

How it was proved

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