## Diagram: Formal Theorem Blocks ### Overview The image presents a diagram consisting of four distinct rectangular blocks, each representing a formal theorem or lemma from different software projects: PFR, SciLean, Coxeter, and MiniF2F. Each block contains a formal statement of a theorem or lemma, along with associated code or commands related to its proof or definition. The blocks are arranged in a roughly square configuration. ### Components/Axes The diagram does not have traditional axes or a legend. It consists of four labeled blocks, each containing text. The blocks are visually separated by green borders. The labels (PFR, SciLean, Coxeter, MiniF2F) are positioned above each block. ### Detailed Analysis or Content Details **1. PFR Block (Top-Left)** * **Label:** PFR * **Theorem/Lemma:** `lemma condRho_of_translate` * **Formal Statement:** `condRho (fun w -> X ω + s) Y A = condRho X Y A := by` `simp only [condRho, rho_of_translate]` * **Types/Parameters:** * `Ω : Type*` * `[MeasureSpace Ω]` * `(X : Ω -> G) (Y : Ω -> S) (A : Finset G) (s:G)` **2. SciLean Block (Bottom-Left)** * **Label:** SciLean * **Theorem/Lemma:** `theorem re_float (a : Float)` * **Formal Statement:** `RCLike.re a = a := by` `exact RCLike.re_eq_self_of_le_le_rfl` **3. Coxeter Block (Bottom-Center)** * **Label:** Coxeter * **Theorem/Lemma:** `lemma invmap_of_eq` * **Formal Statement:** `invmap S s = s := by` `simp [CoxeterSystem.Presentation.invmap]` `unfold CoxeterSystem.toMatrix` `apply CoxeterSystem.monoidLift.mapLift.of` * **Types/Parameters:** * `(S:Set G) [CoxeterSystem G S] {s :S}` **4. MiniF2F Block (Top-Right)** * **Label:** MiniF2F * **Theorem/Lemma 1:** `theorem induction_12dvd4expnp1p20` * **Formal Statement:** ` (n : N) :` `12 * 4^ (n+1) + 20 := by` `omega` `norm_num` `induction 'n with n hn` `simp` * **Theorem/Lemma 2:** `theorem amc12a_2002_p6` * **Formal Statement:** `(n : N)` `(h₀ : 0 < n) :` `∃ m, (m > n ∧ ∃ p, m * p ≤ n + p)` := by `lift n to N+ using h₀` `cases 'n with n` `exact (_, lt_add_of_pos_right _ zero_lt_one, 1, by simp)` **5. Formal Book Block (Bottom-Right)** * **Label:** Formal Book * **Theorem/Lemma:** `theorem wedderburn (h: Fintype R): IsField R` * **Formal Statement:** `:= by` `apply Field.toIsField` ### Key Observations The diagram presents snippets of formal mathematical proofs or definitions. Each block represents a self-contained theorem or lemma. The content within each block is highly technical and specific to the respective software project. There is no apparent relationship between the theorems/lemmas presented in different blocks. ### Interpretation The diagram serves as a visual catalog of formal theorems or lemmas from different formalization projects. It highlights the use of formal methods in mathematics and computer science. The diversity of the projects (PFR, SciLean, Coxeter, MiniF2F, Formal Book) suggests a broad application of formal verification and proof techniques. The content is not intended for general consumption but rather for researchers and developers working in the field of formal methods. The diagram doesn't demonstrate a specific relationship between the theorems, but rather showcases the independent work being done in different areas of formalization. The use of specific commands like `simp`, `unfold`, `apply`, `induction`, and `omega` indicates the use of interactive theorem provers or proof assistants.