## 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.

\n ## Screenshot: Mathematical Theorem Proofs ### Overview The image is a screenshot of a terminal window displaying two mathematical theorem proofs, likely generated by an automated theorem prover. The proofs are presented in a formal, symbolic notation. The background is dark, and the text is light-colored for readability. ### Components/Axes There are no axes or traditional chart components. The content is purely textual, representing mathematical statements and proof steps. The screen displays two theorems: `mathd_algebra_148` and `amc12a_2016_p3`. Each theorem is followed by its assumptions (labeled `h0` and `h1`) and the resulting conclusion. The proof steps are indicated by `:= by` followed by the tactic used (e.g., `linarith`, `norm_num`, `field_simp`, `norm_cast`). ### Detailed Analysis or Content Details **Theorem 1: `mathd_algebra_148`** * **Variables:** * `c : ℝ` (c is a real number) * `f : ℝ → ℝ` (f is a function from real numbers to real numbers) * **Assumptions:** * `h₀ : ∀ x, f x = c * x^3 - 9 * x + 3` (For all x, f(x) equals c times x cubed minus 9 times x plus 3) * `h₁ : f 2 = 9` (f of 2 equals 9) * **Conclusion:** * `c = 3 := by linarith [h₀ 2]` (c equals 3, proven by linear arithmetic using assumption h0 with x=2) **Theorem 2: `amc12a_2016_p3`** * **Variables:** * `f : ℝ → ℝ` (f is a function from real numbers to real numbers) * **Assumptions:** * `h₀ : ∀ x, ∀ (y : y ≠ 0), f x y = x - y * Int.floor(x / y)` (For all x and y (where y is not equal to 0), f(x, y) equals x minus y times the integer floor of x divided by y) * **Conclusion:** * `f (3 / 8) / (-2 / 5) = -(1 / 40) := by norm_num [h₀]` (f(3/8, -2/5) divided by (-2/5) equals -1/40, proven by numerical normalization using assumption h0) * **Tactics used:** * `norm_num` * `field_simp` * `norm_cast` ### Key Observations The screenshot presents formal mathematical proofs. The theorems involve functions and real numbers. The proofs are concise and rely on automated tactics like `linarith`, `norm_num`, `field_simp`, and `norm_cast` to derive the conclusions from the assumptions. The use of `Int.floor` in the second theorem suggests the involvement of integer division or floor functions. ### Interpretation The image demonstrates the use of an automated theorem prover to verify mathematical statements. The prover takes assumptions as input and applies a series of logical steps (tactics) to arrive at a conclusion. The output shows the theorem statement, the assumptions, and the proof steps. The theorems themselves appear to be algebraic in nature, involving polynomial functions and functional equations. The use of tactics like `linarith` and `norm_num` indicates that the prover is capable of performing linear arithmetic and numerical simplification. The screenshot provides a glimpse into the process of formal mathematical reasoning and the role of automation in verifying mathematical proofs.

\n ## Screenshot: Formal Mathematical Proof Code ### Overview The image displays a dark-themed code editor window (resembling a macOS interface with red, yellow, and green window control dots in the top-left corner) containing two formal mathematical theorem statements and their proofs. The code is written in a formal proof language, likely Lean, with syntax highlighting. The content consists of two distinct theorems: one algebraic and one from a mathematics competition. ### Components/Axes * **UI Elements**: A dark gray editor window with a subtle drop shadow, set against a light gray-blue background. The window has three colored circles (red, yellow, green) in its top-left corner. * **Code Structure**: The code is presented in a monospaced font with syntax highlighting. Key colors include: * **Blue/Cyan**: Theorem names, type declarations (`ℝ`, `→`). * **White/Light Gray**: General code text, parentheses, colons. * **Red**: Numerical constants (e.g., `3`, `9`, `2`, `5`, `40`). * **Green**: Proof tactics (`by`, `linarith`, `norm_num`, `field_simp`, `norm_cast`). * **Content Layout**: The two theorem blocks are separated by a blank line. Each follows a similar structure: a `theorem` declaration, a list of parameters and hypotheses in parentheses, a conclusion, and a proof starting with `:= by`. ### Detailed Analysis The image contains two complete, self-contained formal proofs. **Theorem 1: `mathd_algebra_148`** * **Declaration**: `theorem mathd_algebra_148` * **Parameters & Hypotheses**: * `(c : ℝ)` - `c` is a real number. * `(f : ℝ → ℝ)` - `f` is a function from real numbers to real numbers. * `(h₀ : ∀ x, f x = c * x^3 - 9 * x + 3)` - Hypothesis `h₀`: For all `x`, `f(x) = c*x³ - 9*x + 3`. * `(h₁ : f 2 = 9)` - Hypothesis `h₁`: `f(2) = 9`. * **Conclusion**: `c = 3` * **Proof**: `:= by linarith [h₀ 2]` * The proof uses the `linarith` (linear arithmetic) tactic, supplied with the instantiation of hypothesis `h₀` at `x=2` (i.e., `f(2) = c*2³ - 9*2 + 3`). Combined with `h₁` (`f(2)=9`), this allows the tactic to solve for `c`. **Theorem 2: `amc12a_2016_p3`** * **Declaration**: `theorem amc12a_2016_p3` * **Parameters & Hypotheses**: * `(f : ℝ → ℝ → ℝ)` - `f` is a function taking two real numbers and returning a real number. * `(h₀ : ∀ x, ∀ (y) (_ : y ≠ 0), f x y = x - y * Int.floor (x / y))` - Hypothesis `h₀`: For all `x` and all `y` where `y ≠ 0`, `f(x, y) = x - y * floor(x / y)`. This defines `f` as the remainder operation (modulo). * **Conclusion**: `f (3 / 8) (-(2 / 5)) = -(1 / 40)` * **Proof**: `:= by norm_num [h₀] field_simp norm_cast` * The proof uses a sequence of tactics: `norm_num` (normalizes numerical expressions using hypothesis `h₀`), `field_simp` (simplifies expressions in a field), and `norm_cast` (manages type casts between numeric types). ### Key Observations 1. **Formal Verification**: The code represents machine-checkable proofs of mathematical statements, not just informal descriptions. 2. **Proof Strategy Contrast**: The first proof is very concise, relying on a single powerful tactic (`linarith`) applied to a specific instance of a hypothesis. The second proof uses a sequence of more specialized simplification and normalization tactics. 3. **Mathematical Content**: The first theorem is a straightforward algebraic solve for a coefficient. The second theorem formalizes a property of the modulo operation (specifically, the remainder when dividing `3/8` by `-2/5`) and is sourced from the 2016 AMC 12A competition (Problem 3). 4. **Syntax Highlighting**: The color scheme aids readability by distinguishing between theorem names, types, numerical values, and proof commands. ### Interpretation This image is a snapshot of formal mathematics in practice. It demonstrates how abstract mathematical problems—solving for a constant in a cubic function and evaluating a specific modular arithmetic expression—can be translated into a precise, logical language for computer verification. The first theorem (`mathd_algebra_148`) shows the power of automated reasoning (`linarith`) for solving linear constraints derived from algebraic definitions. The second theorem (`amc12a_2016_p3`) is more intricate, requiring the formal definition of the floor function (`Int.floor`) and the manipulation of rational numbers. The proof tactics suggest the challenge lies in correctly simplifying the expression `3/8 - (-2/5) * floor((3/8)/(-2/5))` to `-1/40`. The presence of a competition problem (`amc12a_2016_p3`) indicates this code might be part of a repository for formalizing olympiad or contest problems, serving as both a verification tool and an educational resource. The clean, highlighted presentation suggests it is meant for human reading as well as machine execution, bridging the gap between human mathematical intuition and formal logical rigor.

## Screenshot: Functional Programming Theorem Proofs ### Overview The image shows a code editor interface displaying two formal theorem proofs written in a functional programming language (likely Coq or Lean). The text uses syntax highlighting with multiple colors, and the editor has standard macOS window controls (red, yellow, green circles) in the top-left corner. ### Components/Axes - **Window Controls**: Red (close), yellow (minimize), green (maximize) circles in the top-left corner. - **Text Content**: - **Theorem 1**: `theorem mathd_algebra_148` - Hypotheses: - `h0 : ∀ x, f x = c * x^3 - 9 * x + 3` - `h1 : f 2 = 9` - Definitions: - `c = 3 := by` - `linarith [h0 2]` - **Theorem 2**: `theorem amc12a_2016_p3` - Hypotheses: - `h0 : ∀ x, ∀ y, (y ≠ 0 → f x y = x - y * Int.floor(x / y))` - Calculations: - `f (3 / 8) (-(2 / 5)) = -(1 / 40) := by` - Tactics: - `norm_num [h0]` - `field_simp` - `norm_cast` ### Detailed Analysis 1. **Theorem 1 (`mathd_algebra_148`)**: - Defines a cubic polynomial `f x = c * x^3 - 9 * x + 3` with `c = 3`. - Proves `f 2 = 9` using linear arithmetic (`linarith`) with hypothesis `h0`. 2. **Theorem 2 (`amc12a_2016_p3`)**: - Defines a function `f` with a conditional: `f x y = x - y * Int.floor(x / y)` when `y ≠ 0`. - Computes `f (3/8) (-2/5)` and proves the result equals `-1/40` using: - Normalization (`norm_num`) - Field simplification (`field_simp`) - Type casting (`norm_cast`) ### Key Observations - **Color Coding**: - Blue: Keywords (`theorem`, `by`, `linarith`, `field_simp`) - Red: Hypotheses (`h0`, `h1`) and function arguments (`3/8`, `-2/5`) - Green: Operators (`*`, `-`, `/`, `=`) and constants (`3`, `9`) - Yellow: Function definitions (`f x y = ...`) - **Syntax Highlighting**: Distinguishes between types, variables, operators, and tactics. - **Tactics**: Explicit proof strategies (`linarith`, `norm_num`, `field_simp`) indicate automated theorem proving. ### Interpretation This code demonstrates formal verification of mathematical properties using a proof assistant. The theorems: 1. Prove a cubic polynomial evaluates to a specific value at `x = 2`. 2. Verify a piecewise function's behavior under division, leveraging integer flooring and field properties. The use of `linarith` suggests the first theorem relies on linear arithmetic reasoning, while the second theorem combines normalization, field axioms, and type casting to handle rational numbers and integer operations. The structured proof approach (`hypotheses → definitions → tactics`) reflects formal methods for ensuring mathematical correctness in computational systems.