DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Binary residue carrier and operations

Definition

Set

F_{2} : = Z / 2 Z = {[0]_{2}, [1]_{2}},

with addition and multiplication induced by the quotient operations

[r]_{2} + [s]_{2} = [r + s]_{2}, [r]_{2} [s]_{2} = [r s]_{2}

from Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$ . This item fixes the two residue classes and their operations. It does not assert the field axioms; those must be established separately before an argument uses field properties.

Facts & Assumptions

Given: The quotient-set and arithmetic definitions for the modulus $2$ .

[F1]

The congruence class $[a]_{n}$ and the quotient set $Z / n$ : two residue classes modulo $2$ are equal exactly when their representatives are congruent modulo $2$ .

[F2]

Addition and multiplication on $Z / n$ by $[a]_{n} + [b]_{n} = [a + b]_{n}$ and $[a]_{n} [b]_{n} = [a b]_{n}$ : addition and multiplication are the representative-independent operations $[r]_{2} + [s]_{2} = [r + s]_{2}$ and $[r]_{2} [s]_{2} = [r s]_{2}$ .

[F3]

Congruence modulo an integer: $a \equiv b (m o d n)$ when $n ∣ (a - b)$ , including the moduli $0$ and $1$ : $r \equiv s (m o d 2)$ means $2$ divides $r - s$ .

Verification

1.1givenF1F3inductionalgebra

Every integer $r$ has the form $r = 2 q + ε$ for some $q \in Z$ and $ε \in {0, 1}$ . For $r \geq 0$ , induction starts with $0 = 2 \cdot 0 + 0$ ; from $r = 2 q$ obtain $r + 1 = 2 q + 1$ , and from $r = 2 q + 1$ obtain $r + 1 = 2 (q + 1) + 0$ . For $r < 0$ , apply the nonnegative case to $- r = 2 q + ε$ : if $ε = 0$ , then $r = 2 (- q)$ , and if $ε = 1$ , then $r = 2 (- q - 1) + 1$ . By [F3], $r \equiv ε (m o d 2)$ , so [F1] gives $[r]_{2} = [ε]_{2}$ . Hence every quotient class is one of $[0]_{2}, [1]_{2}$ .

2.1givenF1F3step 1.1algebra

The classes $[0]_{2}$ and $[1]_{2}$ are distinct: equality would give $0 \equiv 1 (m o d 2)$ by [F1], hence $2 ∣ - 1$ by [F3], which is impossible because no integer $q$ satisfies $- 1 = 2 q$ . Together with step 1.1, this proves that the quotient has exactly the two listed classes.

3.1givenF1F2F3step 1.1step 2.1algebra∎

Applying [F2] to representatives $0$ and $1$ gives the addition table $[0] + [0] = [0]$ , $[0] + [1] = [1]$ , $[1] + [0] = [1]$ , $[1] + [1] = [0]$ , and the multiplication table $[0] [0] = [0]$ , $[0] [1] = [1] [0] = [0]$ , $[1] [1] = [1]$ . Since steps 1.1 and 2.1 show that these are all quotient classes, the tables are exhaustive; both operations take values in this carrier and are exactly the quotient operations.

Depends on

Used by

Gaussian elimination over F2 for Simon recovery Algorithm
Binary dot product as parity Definition
Simon hidden-XOR-period promise Definition
Simon independent-sample rank process Definition
The coordinate vector space over F2 Definition
The Simon annihilator is an (n−1)-dimensional binary subspace Lemma
F2 row reduction returns the unique Simon mask Theorem
Simon rank growth probability Theorem
The quotient Z/2 is the binary field Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ronald de Wolf, Quantum Computing: Lecture Notes (arXiv:1907.09415v6) (standard reference, not scraped)