## Code Snippet: Mathematical Theorems
### Overview
The image displays a code snippet, likely from a formal verification system or a programming language designed for mathematical proofs. It presents two theorems, `mathd_algebra_148` and `amc12a_2016_p3`, along with their definitions and proofs.
### Components/Axes
The code snippet is structured as follows:
* **Theorem Declarations:** Each theorem is declared with the keyword `theorem` followed by its name.
* **Variable Declarations:** Variables and their types are declared within parentheses, e.g., `(c : R)` indicates that `c` is a real number.
* **Function Definitions:** Functions are defined with their domain and codomain, e.g., `(f : R → R)` indicates that `f` is a function from real numbers to real numbers.
* **Hypotheses:** Hypotheses are introduced with labels like `h0` and `h1`, followed by a colon and the logical statement they represent.
* **Goal:** The statement to be proven is presented before the `:= by` keyword.
* **Proof:** The proof is initiated by `:= by` and followed by proof tactics like `linarith`, `norm_num`, `field_simp`, and `norm_cast`.
### Detailed Analysis or ### Content Details
**Theorem: `mathd_algebra_148`**
* **Variables:**
* `c`: A real number (R).
* `f`: A function from real numbers to real numbers (R → R).
* **Hypotheses:**
* `h0`: For all x, `f x = c * x^3 - 9 * x + 3`.
* `h1`: `f 2 = 9`.
* **Goal:** `c = 3`.
* **Proof:** The proof uses the `linarith` tactic with hypotheses `h0` and `2`.
**Theorem: `amc12a_2016_p3`**
* **Function:** `f`: A function from R x R → R
* **Hypotheses:**
* `h0`: For all x, for all y, if `y ≠ 0`, then `f x y = x - y * Int.floor (x / y)`.
* **Goal:** `f (3 / 8) (-(2 / 5)) = -(1 / 40)`.
* **Proof:** The proof uses the `norm_num`, `field_simp`, and `norm_cast` tactics with hypothesis `h0`.
### Key Observations
* The code uses a formal language to define and prove mathematical theorems.
* The theorems involve real numbers, functions, and logical quantifiers.
* The proofs rely on automated tactics to simplify expressions and derive the desired conclusions.
### Interpretation
The code snippet demonstrates the use of formal methods in mathematics. It shows how theorems can be precisely defined and rigorously proven using a combination of logical statements and automated proof tactics. The `mathd_algebra_148` theorem establishes a specific value for the coefficient `c` in a cubic function, given a function value at `x = 2`. The `amc12a_2016_p3` theorem evaluates a function `f` defined using the floor function. The use of tactics like `linarith`, `norm_num`, `field_simp`, and `norm_cast` suggests that the system is capable of performing algebraic manipulations and simplifying expressions automatically. This approach is valuable for ensuring the correctness of mathematical results and for automating the process of mathematical discovery.
