\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.