The Abstraction and Reasoning Corpus: D. Deferred Proofs

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11 Mar 2024

This paper is available on arxiv under CC 4.0 license.

Authors:

(1) Mattia Atzeni, EPFL, Switzerland and [email protected];

(2) Mrinmaya Sachan, ETH Zurich, Switzerland;

(3) Andreas Loukas, Prescient Design, Switzerland.

D. Deferred Proofs

We prove both Theorem 3.1 and 3.2 by induction on the dimensionality of the hypercubic lattice m.

D.1. Base Case for Theorems 1 and 2

D.2. Inductive Step for Theorems 1 and 2

D.3. Proof of Corollary 1

The proof of Corollary 1 follows immediately from Theorem 3.2 and from the property of the Fourier transform according to which multiplying in the Fourier domain implements a convolution in the original domain.