pattern_to_nat_bound
The theorem establishes that any d-bit pattern, when summed as powers of two for its set bits, yields an integer strictly less than 2^d. Discrete mathematicians and computer scientists formalizing Gray code conversions would cite this bound to guarantee pattern values remain in the representable range. The proof is a one-line wrapper that applies the GrayCodeFacts hypothesis envelope.
claimFor every natural number $d$ and every bit pattern $p$ of length $d$, the integer value obtained by summing $2^k$ over positions $k$ where the $k$-th bit of $p$ is set satisfies $p < 2^d$.
background
The module Patterns.GrayCodeAxioms declares classical Gray code properties as axioms pending full bitwise formalization. A Pattern d denotes a bit vector of length d, i.e., a function from Fin d to Bool whose support determines which powers of two are included in the sum. The local theoretical setting stresses that all such properties possess multiple published proofs in the discrete mathematics literature, with references to Savage (1997) and Knuth (2011).
proof idea
The proof is a one-line wrapper that applies GrayCodeFacts.pattern_to_nat_bound. This directly invokes the hypothesis envelope class declared earlier in the same module, which collects the classical Gray code axioms.
why it matters in Recognition Science
The result supplies the numerical bound required by the GrayCodeFacts class, which is the immediate downstream consumer. It completes the elementary combinatorics step inside the Patterns module and supports axiomatic treatment of bit patterns that appear in Recognition Science encodings. The declaration aligns with the framework's reliance on discrete combinatorial objects that interface with the phi-ladder and forcing chain without introducing new hypotheses.
scope and limits
- Does not prove the single-bit difference property between consecutive Gray codes.
- Does not supply bitwise inversion or gray-to-natural conversion functions.
- Does not extend the bound to infinite bit strings or non-binary bases.
- Does not derive any connection to physical constants such as phi or the fine-structure interval.
Lean usage
example (d : ℕ) (p : Pattern d) : (∑ k : Fin d, if p k then 2^(k.val) else 0) < 2^d := pattern_to_nat_bound d pformal statement (Lean)
127theorem pattern_to_nat_bound :
128 ∀ (d : ℕ) (p : Pattern d),
129 (∑ k : Fin d, if p k then 2^(k.val) else 0) < 2^d :=
proof body
Term-mode proof.
130 GrayCodeFacts.pattern_to_nat_bound
131
132/-- **Classical Result**: Consecutive Gray codes differ in one bit.
133
134For any n < 2^d - 1, gray(n) and gray(n+1) differ in exactly one bit position.
135
136**Proof**:
137- gray(n) XOR gray(n+1) = [n XOR (n>>1)] XOR [(n+1) XOR ((n+1)>>1)]
138- This simplifies to a single power of 2 (bit at position of least significant 0 in n)
139
140**References**:
141- Savage (1997), Theorem 2.1
142- Knuth (2011), Theorem 7.2.1.1.A
143
144**Status**: Defining property of Gray codes
145-/