## Screenshot: Code Snippet from Number Theory Theorems
### Overview
The image displays a code snippet with two mathematical theorems (`mathd_numbertheory_293` and `mathd_numbertheory_233`) written in a programming language with syntax highlighting. The code defines variables, hypotheses, and conclusions using formal logic and modular arithmetic.
### Components/Axes
- **Theorems**:
- `theorem mathd_numbertheory_293`
- `theorem mathd_numbertheory_233`
- **Variables**:
- `n : ℕ` (natural number)
- `h₀ : n ≤ 9` (hypothesis 0)
- `h₁ : 11/20 * 100 + 10 * n + 7` (hypothesis 1)
- `b : ZMod(11^2)` (modular integer)
- `h₀ : b = 24⁻¹` (hypothesis 0 for theorem 233)
- **Operators/Syntax**:
- `:=` (assignment)
- `*` (multiplication)
- `^` (exponentiation)
- `ZMod` (modular arithmetic)
- `exact` (proof keyword)
### Detailed Analysis
#### Theorem `mathd_numbertheory_293`
- **Variables**:
- `n` is constrained to natural numbers (`ℕ`).
- `h₀` restricts `n` to values ≤ 9.
- `h₁` computes `11/20 * 100 + 10 * n + 7`.
- **Conclusion**:
- `n = 5` is derived via `omega` (likely a proof tactic for termination).
#### Theorem `mathd_numbertheory_233`
- **Variables**:
- `b` is defined as `ZMod(11^2)`, equivalent to integers modulo 121.
- `h₀` asserts `b = 24⁻¹` (modular inverse of 24 modulo 121).
- **Conclusion**:
- `b = 116` is proven using `exact h₀`, confirming the modular inverse relationship.
### Key Observations
1. **Modular Arithmetic**:
- `ZMod(11^2)` implies operations are performed modulo 121.
- `24⁻¹ ≡ 116 (mod 121)` because `24 * 116 = 2784 ≡ 1 (mod 121)`.
2. **Hypothesis Constraints**:
- `h₀` in theorem 293 limits `n` to a small range (0–9), simplifying the search for solutions.
3. **Syntax Highlighting**:
- Keywords (`theorem`, `h₀`, `exact`) are in red.
- Variables (`n`, `b`) and values (`5`, `116`) are in blue/orange.
### Interpretation
This code demonstrates formal verification of number theory results using a proof assistant (e.g., Lean or Coq). The theorems:
1. **Theorem 293**: Solves for `n` under constraints, showing `n = 5` satisfies the equation `11/20 * 100 + 10n + 7` within the hypothesis bounds.
2. **Theorem 233**: Computes the modular inverse of 24 modulo 121, yielding `116`, verified via `exact h₀`.
The use of `ZMod` and modular inverses highlights applications in cryptography or cyclic group theory. The structured hypotheses and conclusions suggest a focus on automated theorem proving, where constraints (`h₀`, `h₁`) guide the proof search.
