full_collection/FormalBook.jsonl

{"id":"ch19proposition5","label":"ch19proposition5","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let the infinite set \\(M\\) have cardinality \\(\\mathfrak {m}\\), and let \\(M\\) be well ordered according to the initial ordinal number \\(\\omega _{\\mathfrak {m}}\\). Then \\(M\\) has no last element.","doc_url":null,"latex_link":"sect0019.html#ch19proposition5","lean_link":null,"lean_decl_name":"ch19proposition5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch19proposition6","label":"ch19proposition6","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose \\(\\{ A_\\alpha \\} \\) is a family of size \\(\\mathfrak {m}\\) of countable sets \\(A_\\alpha \\), where \\(\\mathfrak {m}\\) is an infinite cardinal. Then the union \\(\\bigcup _\\alpha A_\\alpha \\) has size at most \\(\\mathfrak {m}\\).","doc_url":null,"latex_link":"sect0019.html#ch19proposition6","lean_link":null,"lean_decl_name":"ch19proposition6","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch11_theorem1","label":"ch11_theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In any configuration of \\(n\\) points in the plane, not all on a line, there is a line which contains exactly two of the points.","doc_url":null,"latex_link":"sect0011.html#ch11_theorem1","lean_link":null,"lean_decl_name":"ch11_theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch11_theorem2","label":"ch11_theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(P\\) be a set of \\(n\\ge 3\\) points in the plane, not all on a line. Then the set \\(\\mathcal{L}\\) of lines passing through at least two points contains at least \\(n\\) lines.","doc_url":null,"latex_link":"sect0011.html#ch11_theorem2","lean_link":null,"lean_decl_name":"ch11_theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch19theorem3","label":"ch19theorem3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The set \\(\\mathbb {R}^2\\) of all ordered pairs of real numbers (that is, the real plane) has the same size as \\(\\mathbb {R}\\).","doc_url":null,"latex_link":"sect0019.html#ch19theorem3","lean_link":null,"lean_decl_name":"ch19theorem3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lyusternik_shnirelman","label":"lyusternik_shnirelman","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["borsuk_ulam"],"nlp_text":"If the \\(d\\)-sphere \\(S^d\\) is covered by \\(d + 1\\) sets, \\[ S^d = U_1 \\cup \\dots \\cup U_d \\cup U_{d+1}, \\]","doc_url":null,"latex_link":"sect0040.html#lyusternik_shnirelman","lean_link":null,"lean_decl_name":"lyusternik_shnirelman","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch37theorem1","label":"ch37theorem1","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch37fact_a","ch37fact_b","ch37fact_c"],"nlp_text":"Let \\(M = (m_{ij})\\) be an \\(n \\times n\\) matrix with entries in \\(\\{ 0, 1\\} \\), and let \\(d_1, \\dots , d_n\\) be the row sums of \\(M\\), that is, \\(d_i = \\sum _{j=1}^{n} m_{ij}\\). Then \\[ \\operatorname {per} M \\leq \\prod _{i=1}^{n} (d_i!)^{1/d_i}. \\]","doc_url":null,"latex_link":"sect0034.html#ch37theorem1","lean_link":null,"lean_decl_name":"ch37theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26lemma_c","label":"ch26lemma_c","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Both \\(f\\) and \\(g\\) are odd functions, that is we have \\(f(-x) = -f(x)\\) and \\(g(-x) = -g(x)\\) for all \\(x\\in \\mathbb {R}\\setminus \\mathbb {Z}\\).","doc_url":null,"latex_link":"sect0025.html#ch26lemma_c","lean_link":null,"lean_decl_name":"ch26lemma_c","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26","label":"ch26","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch26lemma_c","ch26lemma_b","ch26lemma_a","ch26lemma_e","ch26lemma_d"],"nlp_text":"\\[ \\pi \\cot {\\pi x} = \\frac{1}{x} + \\sum _{n = 1}^\\infty \\left(\\frac{1}{x + n} + \\frac{1}{x - n}\\right) \\]","doc_url":null,"latex_link":"sect0025.html#ch26","lean_link":null,"lean_decl_name":"ch26","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:fourth_proof","label":"fourth_proof","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The prime counting function is unbounded","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₄","latex_link":"sect0001.html#thm:fourth_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₄","lean_decl_name":"infinity_of_primes₄","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₄ : Tendsto π atTop atTop := by\n  -- two parts:\n  -- (1) log x ≤ π x + 1\n  -- (2) This implies that it is not bounded\n  have H_log_le_primeCountingReal_add_one (n : ℕ) (x : ℝ) (hxge : x ≥ n) (hxlt : x < n + 1) :\n      Real.log x ≤ primeCountingReal x + 1 :=\n    calc\n      Real.log x ≤ ∑ k ∈ Icc 1 n, (k : ℝ)⁻¹ := by sorry\n      _ ≤ (∑' m : (S₁ x), (m : ℝ)⁻¹) := by sorry\n      _ ≤ (∏ p ∈ primesBelow ⌊x⌋.natAbs, (∑' k : ℕ, (p ^ k : ℝ)⁻¹)) := by sorry\n      _ ≤ (∏ k ∈ Icc 1 (primeCountingReal x), (nth Nat.Prime k) / ((nth Nat.Prime k) - 1)) := by sorry\n      _ ≤ (∏ k ∈ Icc 1 (primeCountingReal x), (k + 1) / k) := by sorry\n      _ ≤ primeCountingReal x + 1 := by sorry\n  sorry","line":180,"end_line":193,"proof_status":"statement_formalized"}
{"id":"thm:infinity_of_primes","label":"infinity_of_primes","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["thm:fourth_proof","thm:fifth_proof","thm:euclids_proof","thm:second_proof","thm:infty_proof","thm:third_proof","thm:sixth_proof"],"nlp_text":"There are infinitely many primes. (Six + infinitely many proofs)","doc_url":null,"latex_link":"sect0001.html#thm:infinity_of_primes","lean_link":null,"lean_decl_name":"infinity_of_primes","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:estimate_factorials","label":"estimate_factorials","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(n \\in \\mathbb {N}\\) \\[ e\\left(\\frac{n}{e}\\right)^n {\\lt} n! {\\lt} en\\left(\\frac{n}{e}\\right)^n. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.bound_factorial","latex_link":"sect0002.html#thm:estimate_factorials","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.bound_factorial","lean_decl_name":"chapter2.bound_factorial","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_02.lean","lean4_code":"theorem bound_factorial {n : ℕ} (hn : 1 < n):\n  n.factorial > e * (n / e) ^ n := by sorry","line":220,"end_line":221,"proof_status":"statement_formalized"}
{"id":"hilberts_third","label":"hilberts_third","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["pearl_lemma","bricard_condition","cone_lemma","example2","example3","example1"],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0010.html#hilberts_third","lean_link":null,"lean_decl_name":"hilberts_third","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"pearl_lemma","label":"pearl_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(P\\) and \\(Q\\) are equidecomposable, then one can place a positive number of pearls (that is, assign positive integers) to all the segments of the decompositions \\(P = P_1 \\cup \\cdots \\cup P_n\\) and \\(Q = Q_1 \\cup \\cdots \\cup Q_n\\) in such a way that each edge of a piece \\(P_k\\) receives the same number of pearls as the corresponding edge of \\(Q_k\\).","doc_url":null,"latex_link":"sect0010.html#pearl_lemma","lean_link":null,"lean_decl_name":"pearl_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"argand_inequality","label":"argand_inequality","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p(z) = \\sum _{k=0}^n c_k z^k\\) be a complex polynomial of degree \\(n\\ge 1\\). If \\(p(a)\\ne 0\\), then every disk \\(D\\) around \\(a\\) contains an interior point \\(b\\) with \\(|p(b)| {\\lt} |p(a)|\\)","doc_url":null,"latex_link":"sect0020.html#argand_inequality","lean_link":null,"lean_decl_name":"argand_inequality","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"fundamental_theorem_of_algbra","label":"fundamental_theorem_of_algbra","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["argand_inequality"],"nlp_text":"Every nonconstant polynomial with complex coefficients has at least one root in the field of complex numbers.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/fundamental_theorem_of_algebra","latex_link":"sect0020.html#fundamental_theorem_of_algbra","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/fundamental_theorem_of_algebra","lean_decl_name":"fundamental_theorem_of_algebra","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_21.lean","lean4_code":"theorem fundamental_theorem_of_algebra (p : Polynomial ℂ) (hp : p.natDegree > 0) :\n    ∃ z : ℂ, p.eval z = 0 := by\n  obtain ⟨z0, hfz0_le⟩ := p.exists_forall_norm_le\n  use z0\n  let R1 := ‖z0‖ + 1\n  have R1_pos : 0 < R1 := by linarith only [norm_nonneg z0]\n  by_contra! hz0_ne0\n  obtain ⟨b, hb1, hb2⟩ := dalembert_lemma hp hz0_ne0 R1_pos\n  linarith only [hb2, hfz0_le b]","line":206,"end_line":214,"proof_status":"statement_formalized"}
{"id":"ch22lemma1","label":"ch22lemma1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any blue point \\(p_0 = (x_b, y_b)\\), green point \\((x_g, y_g)\\), and red point \\((x_r, y_r)\\), the \\(v\\)-value of the determinant \\[ \\det \\begin{bmatrix} x_b & y_b & 1 \\\\ x_g & y_g & 1 \\\\ x_r & y_r & 1 \\end{bmatrix} \\]","doc_url":null,"latex_link":"sect0021.html#ch22lemma1","lean_link":null,"lean_decl_name":"ch22lemma1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch22corollary","label":"ch22corollary","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch22lemma1","rainbow_triangle"],"nlp_text":"Any line of the plane receives at most two different colors. The area of a rainbow triangle cannot be \\(0\\), and it cannot be \\(\\frac{1}{n}\\) for odd \\(n\\).","doc_url":null,"latex_link":"sect0021.html#ch22corollary","lean_link":null,"lean_decl_name":"ch22corollary","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"monsky_theorem","label":"monsky_theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch22corollary","ch22lemma2"],"nlp_text":"It is not possible to dissect a square into an odd number of triangles of equal algebra area.","doc_url":null,"latex_link":"sect0021.html#monsky_theorem","lean_link":null,"lean_decl_name":"monsky_theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26lemma_b","label":"ch26lemma_b","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Both \\(f\\) and \\(g\\) are periodic of period \\(1\\), that is \\(f(x + 1) = f(x)\\) and \\(g(x + 1) = g(x)\\) hold for all \\(x\\in \\mathbb {R}\\setminus \\mathbb {Z}\\).","doc_url":null,"latex_link":"sect0025.html#ch26lemma_b","lean_link":null,"lean_decl_name":"ch26lemma_b","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch35lemma1","label":"ch35lemma1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every nonzero polynomial \\(p(x) \\in F[x_1, \\dots , x_n]\\) of degree \\(d\\) has at most \\(dq^{n-1}\\) roots in \\(F^n\\).","doc_url":null,"latex_link":"sect0032.html#ch35lemma1","lean_link":null,"lean_decl_name":"ch35lemma1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"kakeya","label":"kakeya","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch35lemma1","ch35lemma2"],"nlp_text":"Let \\(K \\subseteq F^n\\) be a Kakeya set. Then \\[ |K| \\geq \\binom {|F| + n - 1}{n} \\geq \\frac{|F|^n}{n!}. \\]","doc_url":null,"latex_link":"sect0032.html#kakeya","lean_link":null,"lean_decl_name":"kakeya","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"little_lemma","label":"little_lemma","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any \\(n \\ge 1\\) the integer \\(n!\\) contains the prime factor \\(2\\) at most \\(n-1\\) times — with equality if (and only if) \\(n\\) is a power of two, \\(n = 2^m\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.little_lemma","latex_link":"sect0008.html#little_lemma","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.little_lemma","lean_decl_name":"book.irrational.little_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"/--\n\"For any `n ≥ 1` the integer `n!` contains the prime factor `2` at most `n − 1` times —\nwith equality if (and only if) `n` is a power of two, `n = 2 ^ m`.\"\n-/\nlemma little_lemma (n : ℕ) (h_n : n ≠ 0) :\n  ¬ (2 ^ n ∣ n.factorial) ∧ (2 ^ (n - 1) ∣ n.factorial ↔ ∃ m : ℕ, n = 2 ^ m) := by\n  sorry","line":70,"end_line":76,"proof_status":"statement_formalized"}
{"id":"e_pow_4_irrational","label":"e_pow_4_irrational","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["little_lemma","e_pow_2_irrational"],"nlp_text":"\\(e^4\\) is irrational.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_pow_4_irrational","latex_link":"sect0008.html#e_pow_4_irrational","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_pow_4_irrational","lean_decl_name":"book.irrational.e_pow_4_irrational","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"theorem e_pow_4_irrational : Irrational (e ^ 4) := by\n  sorry","line":78,"end_line":79,"proof_status":"statement_formalized"}
{"id":"ch14theorem1","label":"ch14theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a link consists of disjoint perfect circles that are pairwise not linked, then the link is trivial","doc_url":null,"latex_link":"sect0015.html#ch14theorem1","lean_link":null,"lean_decl_name":"ch14theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"borromean","label":"borromean","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch14theorem1","borromean_nontrivial"],"nlp_text":"The Borromean rings cannot be build from three perfect circles","doc_url":null,"latex_link":"sect0015.html#borromean","lean_link":null,"lean_decl_name":"borromean","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch42theorem","label":"ch42theorem","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Whenever \\(T = \\{ v^{(1)}, \\dots , v^{(m)}\\} \\) is an orthonormal representation of \\(G\\) with constant \\(\\sigma _T\\), then \\[ \\Theta (G) \\leq \\frac{1}{\\sigma _T}. \\]","doc_url":null,"latex_link":"sect0039.html#ch42theorem","lean_link":null,"lean_decl_name":"ch42theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26lemma_a","label":"ch26lemma_a","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The functions \\(f\\) and \\(g\\) are defined for all non-integral values and are continuous there.","doc_url":null,"latex_link":"sect0025.html#ch26lemma_a","lean_link":null,"lean_decl_name":"ch26lemma_a","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch35lemma2","label":"ch35lemma2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every set \\(E \\subseteq F^n\\) of size \\(|E| {\\lt} \\binom {n+d}{d}\\) there is a nonzero polynomial \\(p(x) \\in F[x_1, \\dots , x_n]\\) of degree at most \\(d\\) that vanishes on \\(E\\).","doc_url":null,"latex_link":"sect0032.html#ch35lemma2","lean_link":null,"lean_decl_name":"ch35lemma2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch19theorem1","label":"ch19theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The set of \\(\\mathbb {Q}\\) of rational numbers is countable.","doc_url":null,"latex_link":"sect0019.html#ch19theorem1","lean_link":null,"lean_decl_name":"ch19theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch28theorem","label":"ch28theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["double_counting"],"nlp_text":"If the graph \\(G\\) on \\(n\\) vertices contains no \\(4\\)-cycles, then \\[ |E| \\le \\lfloor \\frac{n}{4}(1 + \\sqrt{4n - 3})\\rfloor \\]","doc_url":null,"latex_link":"chapter28.html#ch28theorem","lean_link":null,"lean_decl_name":"ch28theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch41proof2","label":"ch41proof2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a graph \\(G = (V, E)\\) on \\(n\\) vertices has no \\(p\\)-clique, \\(p \\geq 2\\), then \\[ |E| \\leq \\left(1 - \\frac{1}{p - 1}\\right) \\frac{n^2}{2}. \\tag {1} \\]","doc_url":null,"latex_link":"sect0038.html#ch41proof2","lean_link":null,"lean_decl_name":"ch41proof2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"sperner","label":"sperner","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose that some “big” triangle with vertices \\(V_1, V_2, V_3\\) is triangulated (that is, decomposed into a finite number of “small” triangles that fit together edge-by-edge). Assume that the vertices in the triangulation get “colors” from the set \\(\\{ 1, 2, 3\\} \\) such that \\(V_i\\) receives the color \\(i\\) (for each \\(i\\)), and only the colors \\(i\\) and \\(j\\) are used for vertices along the edge from \\(V_i\\) to \\(V_j\\) (for \\(i \\neq j\\)), while the interior vertices are colored arbitrarily with 1, 2, or 3. Then in the triangulation there must be a small “tricolored” triangle, which has all three different vertex colors.","doc_url":null,"latex_link":"chapter28.html#sperner","lean_link":null,"lean_decl_name":"sperner","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"brower","label":"brower","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["sperner"],"nlp_text":"Every continuous function \\(f : B^2 \\longrightarrow B^2\\) of an \\(2\\)-dimensional ball to itself has a fixed point (a point \\(x \\in B^2\\) with \\(f(x) = x\\)).","doc_url":null,"latex_link":"chapter28.html#brower","lean_link":null,"lean_decl_name":"brower","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"e_irrational","label":"e_irrational","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(e\\) is irrational.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_irrational","latex_link":"sect0008.html#e_irrational","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_irrational","lean_decl_name":"book.irrational.e_irrational","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"theorem e_irrational : Irrational e := by\n  sorry","line":64,"end_line":65,"proof_status":"statement_formalized"}
{"id":"e_pow_2_irrational","label":"e_pow_2_irrational","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["e_irrational"],"nlp_text":"\\(e^2\\) is irrational.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_pow_2_irrational","latex_link":"sect0008.html#e_pow_2_irrational","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.e_pow_2_irrational","lean_decl_name":"book.irrational.e_pow_2_irrational","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"theorem e_pow_2_irrational : Irrational (e ^ 2) := by\n  sorry","line":67,"end_line":68,"proof_status":"statement_formalized"}
{"id":"bricard_condition","label":"bricard_condition","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0010.html#bricard_condition","lean_link":null,"lean_decl_name":"bricard_condition","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"borsuk_conjecture","label":"borsuk_conjecture","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(q = p^m\\) be a prime power, \\(n := 4q - 2\\), and \\(d := \\binom {n}{2} = (2q - 1)(4q - 3)\\). Then there is a set \\(S \\subseteq \\{ +1, -1\\} ^d\\) of \\(2^{n-2}\\) points in \\(\\mathbb {R}^d\\) such that every partition of \\(S\\), whose parts have smaller diameter than \\(S\\), has at least \\[ \\frac{2^{n-2}}{\\sum _{i=0}^{q-2} \\binom {n-1}{i}} \\]","doc_url":null,"latex_link":"sect0018.html#borsuk_conjecture","lean_link":null,"lean_decl_name":"borsuk_conjecture","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch28claim3","label":"ch28claim3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"In any sequence \\(a_1, a_2, \\ldots , a_{mn+1}\\) of \\(mn+1\\) distinct real numbers, there exists an increasing subsequence \\[ a_{i_1} {\\lt} a_{i_2} {\\lt} \\cdots {\\lt} a_{i_{m+1}} \\quad (i_1 {\\lt} i_2 {\\lt} \\cdots {\\lt} i_{m+1}) \\]","doc_url":null,"latex_link":"chapter28.html#ch28claim3","lean_link":null,"lean_decl_name":"ch28claim3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"gale_theorem","label":"gale_theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is an arrangement of \\(2k + d\\) points on \\(S^d\\) such that every open hemisphere contains at least \\(k\\) of these points.","doc_url":null,"latex_link":"sect0040.html#gale_theorem","lean_link":null,"lean_decl_name":"gale_theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"kneser_conjecture","label":"kneser_conjecture","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["gale_theorem"],"nlp_text":"We have \\[ \\chi (K(2k + d, k)) = d + 2. \\]","doc_url":null,"latex_link":"sect0040.html#kneser_conjecture","lean_link":null,"lean_decl_name":"kneser_conjecture","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch23corollary","label":"ch23corollary","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p(x)\\) be a real polynomial of degree \\(n \\geq 1\\) with leading coefficient 1, and suppose that \\(|p(x)| \\leq 2\\) for all \\(x\\) in the interval \\([a, b]\\). Then \\(b - a \\leq 4\\).","doc_url":null,"latex_link":"sect0022.html#ch23corollary","lean_link":null,"lean_decl_name":"ch23corollary","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch23theorem2","label":"ch23theorem2","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch23corollary","chebyshev"],"nlp_text":"Let \\(p(x)\\) be a real polynomial of degree \\(n \\geq 1\\) with leading coefficient 1, and all roots real. Then the set \\(\\mathcal{P} = \\{ x \\in \\mathbb {R} : |p(x)| \\leq 2\\} \\) can be covered by intervals of total length at most 4.","doc_url":null,"latex_link":"sect0022.html#ch23theorem2","lean_link":null,"lean_decl_name":"ch23theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch23theorem1","label":"ch23theorem1","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch23theorem2"],"nlp_text":"Let \\(f(z)\\) be a complex polynomial of degree at least 1 and leading coefficient 1. Set \\(C = \\{ z \\in \\mathbb {C} : |f(z)| \\leq 2 \\} \\) and let \\(\\mathcal{R}\\) be the orthogonal projection of \\(C\\) onto the real axis. Then there are intervals \\(I_1, \\dots , I_t\\) on the real line which together cover \\(\\mathcal{R}\\) and satisfy \\[ \\ell (I_1) + \\cdots + \\ell (I_t) \\leq 4. \\]","doc_url":null,"latex_link":"sect0022.html#ch23theorem1","lean_link":null,"lean_decl_name":"ch23theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch32lemma","label":"ch32lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G = (V, E)\\) be a finite weighted acyclic directed graph, \\(A = \\{ A_1, \\dots , A_n\\} \\) and \\(\\mathcal{B} = \\{ B_1, \\dots , B_n\\} \\) two \\(n\\)-sets of vertices, and \\(M\\) the path matrix from \\(A\\) to \\(\\mathcal{B}\\). Then \\[ \\det M = \\sum _{\\mathcal{P} \\text{ vertex-disjoint path system}} \\text{sign}(\\mathcal{P}) \\, w(\\mathcal{P}). \\tag {3} \\]","doc_url":null,"latex_link":"sect0029.html#ch32lemma","lean_link":null,"lean_decl_name":"ch32lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch32theorem","label":"ch32theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch32lemma"],"nlp_text":"Let \\(G = (V, E)\\) be a finite weighted acyclic directed graph, \\(A = \\{ A_1, \\dots , A_n\\} \\) and \\(\\mathcal{B} = \\{ B_1, \\dots , B_n\\} \\) two \\(n\\)-sets of vertices, and \\(M\\) the path matrix from \\(A\\) to \\(\\mathcal{B}\\). Then \\[ \\det M = \\sum _{\\mathcal{P} \\text{ vertex-disjoint path system}} \\text{sign}(\\mathcal{P}) \\, w(\\mathcal{P}). \\tag {3} \\]","doc_url":null,"latex_link":"sect0029.html#ch32theorem","lean_link":null,"lean_decl_name":"ch32theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch36lemma2","label":"ch36lemma2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(P\\) be a partial Latin square of order \\(n\\) with at most \\(n - 1\\) cells filled and at most \\(\\frac{n}{2}\\) distinct elements, then \\(P\\) can be completed to a Latin square of order \\(n\\).","doc_url":null,"latex_link":"sect0033.html#ch36lemma2","lean_link":null,"lean_decl_name":"ch36lemma2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"smetaniuk_theorem","label":"smetaniuk_theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch36lemma2","ch36lemma1"],"nlp_text":"Any partial Latin square of order \\(n\\) with at most \\(n - 1\\) filled cells can be completed to a Latin square of the same order.","doc_url":null,"latex_link":"sect0033.html#smetaniuk_theorem","lean_link":null,"lean_decl_name":"smetaniuk_theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch04.proposition1","label":"ch04.proposition1","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch04.lemma₁"],"nlp_text":"Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can be written as \\(p = x^2 + y^2\\) for some natural numbers \\(x,y \\in \\mathbb {N}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.theorem₁","latex_link":"sect0004.html#ch04.proposition1","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.theorem₁","lean_decl_name":"ch04.theorem₁","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_04.lean","lean4_code":"theorem theorem₁  {p : ℕ} [h : Fact p.Prime] (hp : p % 4 = 1) : ∃ a b : ℕ, a ^ 2 + b ^ 2 = p := by\n  sorry","line":84,"end_line":85,"proof_status":"statement_formalized"}
{"id":"sum_of_two_squares","label":"sum_of_two_squares","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch04.proposition1","ch04.proposition3","ch04.proposition2","ch04.lemma2"],"nlp_text":"A natural number \\(n\\) can be represented as a sum of two squares if and only if every prime factor of the form \\(p = 4m + 3\\) appears with an even exponent in the prime decomposition of \\(n\\).","doc_url":null,"latex_link":"sect0004.html#sum_of_two_squares","lean_link":null,"lean_decl_name":"sum_of_two_squares","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"tiling_rectangles2","label":"tiling_rectangles2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Whenever a rectangle is tiled by rectangles all of which have at least one side of integer length, then the tiled rectangle has at least one side of integer length.","doc_url":null,"latex_link":"sect0027.html#tiling_rectangles2","lean_link":null,"lean_decl_name":"tiling_rectangles2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch38definition1","label":"ch38definition1","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\vec{G} = (V, E)\\) be a directed graph. A kernel \\(K \\subseteq V\\) is a subset of the vertices such that \\(K\\) is independent in \\(G\\), and for every \\(u \\notin K\\) there exists a vertex \\(v \\in K\\) with an edge \\(u \\rightarrow v\\).","doc_url":null,"latex_link":"sect0035.html#ch38definition1","lean_link":null,"lean_decl_name":"ch38definition1","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch38lemma1","label":"ch38lemma1","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch38definition1"],"nlp_text":"Let \\(\\vec{G} = (V, E)\\) be a directed graph, and suppose that for each vertex \\(v \\in V\\) we have a color set \\(C(v)\\) that is larger than the outdegree, \\(|C(v)| \\geq d^+(v) + 1\\). If every induced subgraph of \\(\\vec{G}\\) possesses a kernel, then there exists a list coloring of \\(G\\) with a color from \\(C(v)\\) for each \\(v\\).","doc_url":null,"latex_link":"sect0035.html#ch38lemma1","lean_link":null,"lean_decl_name":"ch38lemma1","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch38theorem","label":"ch38theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch38lemma1","ch38lemma2"],"nlp_text":"We have \\(\\chi _\\ell (S_n) = n\\) for all \\(n\\).","doc_url":null,"latex_link":"sect0035.html#ch38theorem","lean_link":null,"lean_decl_name":"ch38theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"mult_cyclic","label":"mult_cyclic","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The multiplicative group of a finite field is cyclic.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.mult_cyclic","latex_link":"sect0005.html#mult_cyclic","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.mult_cyclic","lean_decl_name":"book.quadratic_reciprocity.mult_cyclic","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"lemma mult_cyclic (K : Type _) [Field K] [Fintype K] : ∃ ζ : Kˣ, ∀ α : Kˣ, ∃ k : ℤ, α = ζ ^ k := by\n  sorry","line":97,"end_line":98,"proof_status":"statement_formalized"}
{"id":"fact_B","label":"fact_B","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["mult_cyclic","fermats_little"],"nlp_text":"For the field \\(F\\) defined in (A), there exists an element \\(\\zeta \\in F\\) of multiplicative order \\(p\\), that is, \\(\\zeta ^p = 1\\). Moreover, we have a polynomial decomposition \\[ x^p - 1 = (x - \\zeta ) (x - \\zeta ^2) \\cdots (x - \\zeta ^p). \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fact_B","latex_link":"sect0005.html#fact_B","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fact_B","lean_decl_name":"book.quadratic_reciprocity.fact_B","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"lemma fact_B (p : ℕ) [Fact (Prime p)] (K : Type _) [Field K] (ζ : Kˣ) (h_1 : ζ ^ p = 1)\n  (h_2 : ζ ≠ 1) :\n  X  ^ (p - 1) - 1  = ∏ i ∈ Icc 1 p, (X - (Polynomial.C (ζ : K)) ^ i) := by\n  sorry","line":110,"end_line":113,"proof_status":"statement_formalized"}
{"id":"quadratic_reciprocity2","label":"quadratic_reciprocity2","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["fact_B","fact_A"],"nlp_text":"Let \\(p\\) and \\(q\\) be different odd primes. Then \\[ (\\frac{q}{p})(\\frac{p}{q}) = (-1)^{\\frac{p-1}{2} \\frac{q-1}{2}}. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.quadratic_reciprocity_2","latex_link":"sect0005.html#quadratic_reciprocity2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.quadratic_reciprocity_2","lean_decl_name":"book.quadratic_reciprocity.quadratic_reciprocity_2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"theorem quadratic_reciprocity_2 (p q : ℕ) (hp : p ≠ 2) (hq : q ≠ 2)\n  [Fact (Nat.Prime p)] [Fact (Nat.Prime q)] :\n  (legendre_sym p q) * (legendre_sym q p) = -1 ^ ((p-1) / 2 * (q - 1) / 2 ) := by\n  sorry","line":115,"end_line":118,"proof_status":"statement_formalized"}
{"id":"ch16theorem2","label":"ch16theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(d \\ge 1\\), we have \\(f(d){\\lt} 2^{d+1}\\).","doc_url":null,"latex_link":"sect0016.html#ch16theorem2","lean_link":null,"lean_decl_name":"ch16theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch25theorem","label":"ch25theorem","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(a_1, \\dots , a_n\\) be vectors in \\(\\mathbb {R}^d\\), each of length at least 1, and let \\(R_1, \\dots , R_k\\) be \\(k\\) open regions of \\(\\mathbb {R}^d\\), where \\(|x - y| {\\lt} 2\\) for any \\(x, y\\) that lie in the same region \\(R_i\\). Then the number of linear combinations \\(\\sum _{i=1}^{n} \\epsilon _i a_i\\), \\(\\epsilon _i \\in \\{ 1, -1\\} \\), that can lie in the union \\(\\bigcup _i R_i\\) of the regions is at most the sum of the \\(k\\) largest binomial coefficients \\(\\binom {n}{j}\\). In particular, we get the bound \\(\\binom {\\lfloor n/2 \\rfloor }{n}\\) for \\(k = 1\\).","doc_url":null,"latex_link":"sect0024.html#ch25theorem","lean_link":null,"lean_decl_name":"ch25theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"pick_theorem","label":"pick_theorem","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_formula","pick_lemma"],"nlp_text":"The area of any (not necessarily convex) polygon \\(Q\\subset \\mathbb {R}^2\\) with integral vertices is given by \\[ A(Q) = n_{int} + \\frac{1}{2}n_{bd} - 1 \\]","doc_url":null,"latex_link":"sect0013.html#pick_theorem","lean_link":null,"lean_decl_name":"pick_theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch16theorem1","label":"ch16theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every \\(d\\ge 2\\), there is a family of \\(2^d\\) pairwise touching \\(d\\)-simplices in \\(\\mathbb {R}^d\\) together with a transversal line that hits the interior of every single on of them.","doc_url":null,"latex_link":"sect0016.html#ch16theorem1","lean_link":null,"lean_decl_name":"ch16theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"tiling_rectangles1","label":"tiling_rectangles1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Whenever a rectangle is tiled by rectangles all of which have at least one side of integer length, then the tiled rectangle has at least one side of integer length.","doc_url":null,"latex_link":"sect0027.html#tiling_rectangles1","lean_link":null,"lean_decl_name":"tiling_rectangles1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:fifth_proof","label":"fifth_proof","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The set of primes \\(\\mathbb {P}\\) is infinite.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₅","latex_link":"sect0001.html#thm:fifth_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₅","lean_decl_name":"infinity_of_primes₅","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₅ : { p : ℕ | p.Prime }.Infinite := by\n  let TopoSpace : TopologicalSpace ℤ := {\n    IsOpen := isOpen\n    isOpen_univ := Or.inr fun a _ ↦ ⟨1, Int.zero_lt_one, Set.subset_univ _⟩\n    isOpen_sUnion := by\n      refine fun S hS ↦ Or.inr fun z hz ↦ ?_\n      obtain ⟨t, tS, zt⟩ := hz\n      rcases (hS t tS) with empty | ha\n      · aesop\n      obtain ⟨b, hb⟩ := ha z zt\n      refine ⟨b, hb.1, Set.subset_sUnion_of_subset S t hb.2 tS⟩\n    isOpen_inter := by\n      intro O₁ O₂ hO₁ hO₂\n      rcases hO₁ with rfl | hO₁\n      · unfold isOpen; aesop\n      rcases hO₂ with rfl | hO₂\n      · unfold isOpen; aesop\n      refine Or.inr fun a ⟨haO₁, haO₂⟩ ↦ ?_\n      obtain ⟨b₁, hb₁, hNab₁⟩ := hO₁ a haO₁\n      obtain ⟨b₂, hb₂, hNab₂⟩ := hO₂ a haO₂\n      refine ⟨b₁*b₂, mul_pos hb₁ hb₂,\n        Set.subset_inter (subset_trans ?_ hNab₁) (subset_trans ?_ hNab₂)⟩\n      <;> simp only [N, Set.setOf_subset_setOf, forall_exists_index, forall_apply_eq_imp_iff,\n        add_right_inj]\n      · refine fun k ↦ ⟨b₂*k, by ring⟩\n      · refine fun k ↦ ⟨b₁*k, by ring⟩\n  }\n  have Infinite_of_NonemptyOpen {O : Set ℤ} (hnO : Set.Nonempty O)\n      (hO : TopoSpace.IsOpen O): Set.Infinite O := by\n    have Infinite_N {a b : ℤ} (ha : 0 < b ) : Set.Infinite (N a b) := by\n      have : Function.Injective (fun k ↦ a + b*k) := by\n        apply Function.Injective.comp (add_right_injective a)\n        refine fun _ _ ↦ mul_left_cancel₀ (Int.ne_of_lt ha).symm\n      apply Set.infinite_of_injective_forall_mem this\n      unfold N; refine fun x ↦ ⟨x, by ring⟩\n    rcases hO with _ | hO\n    · aesop\n    · obtain ⟨a, ha⟩ := hnO\n      obtain ⟨b, hb, hOb⟩ := hO a ha\n      apply Set.Infinite.mono hOb (Infinite_N hb)\n\n  have IsClosed_N (a b : ℤ) (hb : 0 < b) : IsClosed (N a b):= by\n    refine isOpen_compl_iff.1 (Or.inr fun n hn ↦ ⟨b, hb, fun k hk ↦ ?_⟩)\n    simp only [N, Set.mem_compl_iff, Set.mem_setOf_eq, not_exists] at *\n    intro b₁ hb₁\n    obtain ⟨m, hm⟩ := hk\n    apply hn (b₁ - m)\n    rw [sub_mul, add_sub, hb₁, ← hm]\n    ring\n\n  have : ⋃ p ∈ { p : ℕ | Nat.Prime p }, N 0 p = {-1, 1}ᶜ := by\n    have (n : ℤ) (n_ne_one : n ≠ 1) (n_ne_negone : n ≠ -1):\n        ∃ p, Nat.Prime p ∧ ∃m, m * (p : ℤ) = n:= by\n      use n.natAbs.minFac\n      constructor\n      · refine Nat.minFac_prime ?_\n        have := @Int.natAbs_eq_iff_sq_eq n 1\n        aesop\n      use n / n.natAbs.minFac\n      rw [Int.ediv_mul_cancel]\n      rw [Int.ofNat_dvd_left]\n      exact (Nat.minFac_dvd (Int.natAbs n))\n    ext n\n    simp only [Set.mem_setOf_eq, N, zero_add, Set.mem_iUnion, exists_prop, Int.reduceNeg,\n      Set.mem_compl_iff, Set.mem_insert_iff, Set.mem_singleton_iff, not_or]\n    constructor\n    · intro ⟨p, hp, ⟨k, hk⟩⟩\n      have hp := Prime.not_dvd_one (Nat.prime_iff_prime_int.1 hp)\n      constructor <;>  (intro h; rw [h] at hk; apply hp)\n      · use -k\n        nlinarith\n      · use k\n        nlinarith\n    · refine fun hn ↦ this n hn.2 hn.1\n\n  intro primes_finite\n  have H : IsClosed (⋃ p ∈ { p : ℕ | Nat.Prime p }, N 0 p) := by\n    refine Set.Finite.isClosed_biUnion primes_finite (fun p prime_p ↦ ?_)\n    exact IsClosed_N 0 p (by exact_mod_cast Nat.Prime.pos prime_p)\n  rw [this] at H\n  rw [isClosed_compl_iff] at H\n  have contradiction : Set.Infinite {-1, 1} :=\n    Infinite_of_NonemptyOpen (show Set.Nonempty {-1, 1} by aesop) H\n  exact contradiction (show Set.Finite {-1, 1} by aesop)","line":251,"end_line":334,"proof_status":"statement_formalized"}
{"id":"chebyshev","label":"chebyshev","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch23fact1","ch23fact2"],"nlp_text":"Let \\(p(x)\\) be a real polynomial of degree \\(n \\geq 1\\) with leading coefficient 1. Then \\[ \\max _{-1 \\leq x \\leq 1} |p(x)| \\geq \\frac{1}{2^{n-1}}. \\]","doc_url":null,"latex_link":"sect0022.html#chebyshev","lean_link":null,"lean_decl_name":"chebyshev","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch38lemma2","label":"ch38lemma2","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch38definition2"],"nlp_text":"A stable matching always exists.","doc_url":null,"latex_link":"sect0035.html#ch38lemma2","lean_link":null,"lean_decl_name":"ch38lemma2","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"euler_consequence_c","label":"euler_consequence_c","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_formula"],"nlp_text":"Let \\(G\\) be any simple plane graph with \\(n{\\gt}2\\) vertices. If the edges of \\(G\\) are two-colored, then there is a vertex of \\(G\\) with at most two color-changes in the cyclic order of the edges around the vertex.","doc_url":null,"latex_link":"sect0013.html#euler_consequence_c","lean_link":null,"lean_decl_name":"euler_consequence_c","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"monochromatic_lines","label":"monochromatic_lines","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_consequence_c"],"nlp_text":"Given any finite configuration of “black” and “white” points in the plane, not all on one line, there is always a “monochromatic” line: a line that contains at least two points of one color and none of the other.","doc_url":null,"latex_link":"sect0013.html#monochromatic_lines","lean_link":null,"lean_decl_name":"monochromatic_lines","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:euclids_proof","label":"euclids_proof","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A finite set \\(\\{ p_1, \\dots , p_r\\} \\) cannot be the collection of all prime numbers.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₁","latex_link":"sect0001.html#thm:euclids_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₁","lean_decl_name":"infinity_of_primes₁","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₁ (S : Finset ℕ) (h : ∀ q ∈ S, Nat.Prime q):\n  ∃ (p : ℕ), Nat.Prime p ∧ p ∉ S := by\n  let n := 1 + ∏ q ∈ S, q\n  /- \"This `n` has a prime divisor\":\n  we pick the minimal one, the argument works with any prime divisor -/\n  let p := n.minFac\n  use p\n  have hp : Nat.Prime p := Nat.minFac_prime <| Nat.ne_of_gt <| lt_add_of_pos_right 1\n    (Finset.prod_pos fun q hq ↦ Prime.pos <| h q hq)\n  refine ⟨hp, ?_⟩\n  by_contra a\n  have h_p_div_prod : p ∣ ∏ q ∈  S, q := dvd_prod_of_mem (fun (i : ℕ) ↦ i) a\n  have h_p_div_diff : p ∣ n - ∏ q ∈ S, q := dvd_sub (minFac_dvd n) h_p_div_prod\n  have h_p_div_one : p ∣ 1 := by aesop\n  exact Nat.Prime.not_dvd_one hp h_p_div_one","line":29,"end_line":43,"proof_status":"formalized"}
{"id":"thm:second_proof","label":"second_proof","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Any two Fermat numbers \\(F_n := 2^{2^n} + 1\\) are relatively prime.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₂","latex_link":"sect0001.html#thm:second_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₂","lean_decl_name":"infinity_of_primes₂","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₂  (k n : ℕ) (h : k < n) : Coprime (F n) (F k) := by\n  let m := (F n).gcd (F k)\n  have h_n : m ∣ F n := (F n).gcd_dvd_left (F k)\n  have h_k : m ∣ F k := (F n).gcd_dvd_right (F k)\n  have h_m : m ∣ 2 :=  by\n    have h_m_prod : m ∣ (∏ k ∈ range n, F k) :=\n      dvd_trans h_k (dvd_prod_of_mem F (mem_range.mpr h))\n    have h_prod : (∏ k ∈ range n, F k) + 2 = F n := by\n      rw [fermatProduct, Nat.sub_add_cancel]\n      refine' le_of_lt _\n      simp [two_lt_fermatNumber]\n    exact (Nat.dvd_add_right h_m_prod).mp (h_prod ▸ h_n)\n  rcases (dvd_prime prime_two).mp h_m with h_one | h_two\n  · exact h_one\n  · by_contra\n    rw [h_two] at h_n\n    exact (not_even_iff_odd.mpr <| odd_fermatNumber n) (even_iff_two_dvd.mpr h_n)","line":67,"end_line":83,"proof_status":"formalized"}
{"id":"book.irrational.Theorem_2","label":"book.irrational.Theorem_2","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["lem_aux_ii","lem_aux_iii"],"nlp_text":"\\(\\pi ^2\\) is irrational.","doc_url":null,"latex_link":"sect0008.html#book.irrational.Theorem_2","lean_link":null,"lean_decl_name":"book.irrational.Theorem_2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"book.irrational.Theorem_3","label":"book.irrational.Theorem_3","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["book.irrational.Theorem_2"],"nlp_text":"For every odd integer \\(n \\ge 3\\), the number \\[ A(n) := \\frac{1}{\\pi } \\arccos \\left(\\frac{1}{\\sqrt{n}}\\right) \\]","doc_url":null,"latex_link":"sect0008.html#book.irrational.Theorem_3","lean_link":null,"lean_decl_name":"book.irrational.Theorem_3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"euler_series_2","label":"euler_series_2","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\sum _{k \\ge 0}\\frac{1}{(2k+1)^2} = \\frac{\\pi ^2}{8} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series'","latex_link":"sect0009.html#euler_series_2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series'","lean_decl_name":"euler_series'","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_09.lean","lean4_code":"theorem euler_series' :\n   ∑' (k : ℕ), (1 : ℝ) / (2 * k + 1) ^ 2  = π ^ 2  / 8 := by\n  sorry","line":613,"end_line":615,"proof_status":"statement_formalized"}
{"id":"four_proofs_euler_series","label":"four_proofs_euler_series","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_series_2","euler_series_4","euler_series","euler_series_3"],"nlp_text":"Collecting the proofs from the chapter.","doc_url":null,"latex_link":"sect0009.html#four_proofs_euler_series","lean_link":null,"lean_decl_name":"four_proofs_euler_series","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"sylvester_gallai2","label":"sylvester_gallai2","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_consequence_b"],"nlp_text":"Given any set of \\(n \\ge 3\\) points in the plane, not all on one line, there is always a line that contains exactly two of the points.","doc_url":null,"latex_link":"sect0013.html#sylvester_gallai2","lean_link":null,"lean_decl_name":"sylvester_gallai2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch28claim1","label":"ch28claim1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Consider the numbers \\(1, 2, 3, \\dots 2n\\), and take away \\(n + 1\\) of them. Then there are two among these \\(n + 1\\) numbers which are relatively prime.","doc_url":null,"latex_link":"chapter28.html#ch28claim1","lean_link":null,"lean_decl_name":"ch28claim1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"valuation","label":"valuation","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["valuation_on_reals"],"nlp_text":"","doc_url":null,"latex_link":"sect0021.html#valuation","lean_link":null,"lean_decl_name":"valuation","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Topology/Algebra/Valued/NormedValued.lean","lean4_code":"def valuation : Valuation K ℝ≥0 where\n  toFun           := nnnorm\n  map_zero'       := nnnorm_zero\n  map_one'        := nnnorm_one\n  map_mul'        := nnnorm_mul\n  map_add_le_max' := IsUltrametricDist.norm_add_le_max\n\n","line":40,"end_line":46,"proof_status":"statement_formalized"}
{"id":"three_coloring","label":"three_coloring","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["valuation"],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0021.html#three_coloring","lean_link":null,"lean_decl_name":"three_coloring","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"rainbow_triangle","label":"rainbow_triangle","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["three_coloring"],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0021.html#rainbow_triangle","lean_link":null,"lean_decl_name":"rainbow_triangle","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch23fact1","label":"ch23fact1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(b\\) is a multiple root of \\(p'(x)\\), then \\(b\\) is also a root of \\(p(x)\\).","doc_url":null,"latex_link":"sect0022.html#ch23fact1","lean_link":null,"lean_decl_name":"ch23fact1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch23fact2","label":"ch23fact2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We have \\(p'(x)^2 \\geq p(x)p''(x)\\) for all \\(x \\in \\mathbb {R}\\).","doc_url":null,"latex_link":"sect0022.html#ch23fact2","lean_link":null,"lean_decl_name":"ch23fact2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch37fact_a","label":"ch37fact_a","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ H(X)\\le \\log _2(|\\operatorname{supp}{X}). \\]","doc_url":null,"latex_link":"sect0034.html#ch37fact_a","lean_link":null,"lean_decl_name":"ch37fact_a","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem_aux_ii","label":"lem_aux_ii","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(0 {\\lt} x {\\lt} 1\\) we have \\(0 {\\lt} f(x) {\\lt} \\frac{1}{n!}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_ii","latex_link":"sect0008.html#lem_aux_ii","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_ii","lean_decl_name":"book.irrational.lem_aux_ii","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"lemma lem_aux_ii (n : ℕ) (x : ℝ) (h_1 : 0 < x) (h_2 : x < 0) :\n  (0 < f_aux n x) ∧ (f_aux n x < (1 : ℝ) / n.factorial) := by\n  constructor <;> linarith","line":100,"end_line":102,"proof_status":"statement_formalized"}
{"id":"book.irrational.Theorem_1","label":"book.irrational.Theorem_1","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["lem_aux_ii","lem_aux_i","lem_aux_iii"],"nlp_text":"\\(e^r\\) is irrational for every \\(r \\in \\mathbb {Q} \\setminus \\{ 0\\} \\).","doc_url":null,"latex_link":"sect0008.html#book.irrational.Theorem_1","lean_link":null,"lean_decl_name":"book.irrational.Theorem_1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:infty_proof","label":"infty_proof","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If the sequence \\(S = (s_1, s_2, s_3, \\dots )\\) is almost injective and of subexponential growth, then the set \\(\\mathbb {P}_S\\) of primes that divide some member of \\(S\\) is infinite.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/Asymptotics.infinitely_many_more_proofs","latex_link":"sect0001.html#thm:infty_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/Asymptotics.infinitely_many_more_proofs","lean_decl_name":"Asymptotics.infinitely_many_more_proofs","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinitely_many_more_proofs (S : ℕ → ℤ)\n  (h₁ : AlmostInjective S) (h₂ : ofSubexponentialGrowth S) :\n  {p : Nat.Primes | ∃ n : ℕ, (p : ℤ) ∣ S n}.Finite := by\n  sorry","line":365,"end_line":368,"proof_status":"statement_formalized"}
{"id":"euler_formula","label":"euler_formula","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(G\\) is a connected plane graph with \\(n\\) vertices, \\(e\\) edges and \\(f\\) faces, then \\[ n - e + f = 2. \\]","doc_url":null,"latex_link":"sect0013.html#euler_formula","lean_link":null,"lean_decl_name":"euler_formula","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"euler_consequence_a","label":"euler_consequence_a","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_formula"],"nlp_text":"Let \\(G\\) be any simple plane graph with \\(n{\\gt}2\\) vertices. Then \\(G\\) has at most \\(3*n - 6\\) edges.","doc_url":null,"latex_link":"sect0013.html#euler_consequence_a","lean_link":null,"lean_decl_name":"euler_consequence_a","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"euler_consequence_b","label":"euler_consequence_b","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_consequence_a"],"nlp_text":"Let \\(G\\) be any simple plane graph with \\(n{\\gt}2\\) vertices. Then \\(G\\) has a vertex of degree at most \\(5\\).","doc_url":null,"latex_link":"sect0013.html#euler_consequence_b","lean_link":null,"lean_decl_name":"euler_consequence_b","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch17theorem2","label":"ch17theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every \\(d\\ge 2\\), there is a set \\(S\\subset \\{ 0, 1\\} ^d\\) of \\(2\\lfloor \\frac{sqrt{6}}{9}(\\frac{2}{\\sqrt(3)})^d\\rfloor \\) points in \\(\\mathbb {R}^n\\) (vertices of the unit \\(d\\)-cube) that determine only acute angels. In particular, in dimension \\(d = 34\\) there is a set of \\(72 {\\gt} 2*34 - 1\\) points with only acute angels.","doc_url":null,"latex_link":"sect0017.html#ch17theorem2","lean_link":null,"lean_decl_name":"ch17theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch30theorem1","label":"ch30theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The size of a largest antichain of an \\(n\\)-set is \\(\\binom {n}{\\lfloor n/2\\rfloor }\\).","doc_url":null,"latex_link":"chapter30.html#ch30theorem1","lean_link":null,"lean_decl_name":"ch30theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:third_proof","label":"third_proof","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There is no largest prime.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₃","latex_link":"sect0001.html#thm:third_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₃","lean_decl_name":"infinity_of_primes₃","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₃:\n  ¬ (∃ (p : ℕ), Nat.Prime p ∧ (∀ (q : ℕ), (Nat.Prime q) → q ≤ p)) := by\n  simp only [not_exists, not_and, not_forall, not_le, exists_prop]\n  intros p hp\n  have : Fact (Nat.Prime p) := by exact { out := hp }\n  let m := mersenne p\n  -- This m has a prime factor;\n  -- we pick the minimal one, the argument works with any prime factor\n  let q := m.minFac\n  have hq : q.Prime := minFac_prime <| Nat.ne_of_gt <| one_lt_mersenne.mpr <| Prime.one_lt hp\n  have : Fact (Nat.Prime q) := by exact { out := hq }\n  have h_mod_q : 2 ^ p  ≡ 1 [MOD q] := by\n    have : (2^p - 1) % q = 0 :=  mod_eq_zero_of_dvd (minFac_dvd m)\n    change (2^p - 1) ≡ 0 [MOD q] at this\n    rw [modEq_iff_dvd, dvd_iff_exists_eq_mul_left] at *\n    obtain ⟨c, hc⟩ := this\n    use c\n    simp only [CharP.cast_eq_zero, zero_sub] at hc\n    simp [cast_one, cast_pow, cast_ofNat, hc.symm]\n  have h_mod_q' : (2 : (ZMod q)) ^ p = 1 := by\n    have := (ZMod.natCast_eq_natCast_iff _ _ _).mpr h_mod_q\n    norm_cast at this\n    rw [← this, cast_pow, cast_ofNat]\n  have : (2 : (ZMod q)) * (2 ^ (p - 1)) = 1 := by\n    convert h_mod_q'\n    nth_rw 1 [← pow_one 2]\n    rw [← pow_add 2 1 (p - 1)]\n    congr\n    exact add_sub_of_le <| Prime.pos hp\n  let two := Units.mkOfMulEqOne (2 : (ZMod q)) (2 ^ (p - 1)) this\n  have two_desc : ↑two = (2 : (ZMod q)) := by\n    convert Units.val_mkOfMulEqOne this\n  have h_two : two ^ p = 1 := by\n    ext\n    push_cast\n    rw [two_desc]\n    exact h_mod_q'\n  have two_ne_one : two ≠ 1 := by\n    by_contra h\n    rw [Units.ext_iff, two_desc] at h\n    exact (ZMod.two_ne_one q) h\n  have h_piv_div_q_sub_one : p ∣ q - 1 := by\n    -- The following shorter proof would work, but we want to use Lagrange's theorem\n    -- convert ZMod.orderOf_units_dvd_card_sub_one two\n    -- exact (orderOf_eq_prime h_two two_ne_one).symm\n\n    -- Using Lagrange's theorem here!\n    convert Subgroup.card_subgroup_dvd_card (Subgroup.zpowers (two))\n    · rw [← orderOf_eq_prime h_two two_ne_one, card_eq_fintype_card]\n      exact Fintype.card_zpowers.symm\n    · rw [card_eq_fintype_card, ZMod.card_units_eq_totient]\n      exact (totient_prime hq).symm\n  refine ⟨q, minFac_prime <| Nat.ne_of_gt ?_, ?_⟩\n  · calc 1 < 2^2 - 1 := one_lt_succ_succ 1\n        _  ≤ 2^p - 1 := sub_one_le_sub_one <| Nat.pow_le_pow_right (succ_pos 1) (Prime.two_le hp)\n  · have h2q : 2 ≤ q := Prime.two_le <| minFac_prime <| Nat.ne_of_gt <| lt_of_succ_lt <|\n      Nat.sub_le_sub_right ((Nat.pow_le_pow_right (succ_pos 1) (Prime.two_le hp))) 1\n    exact lt_of_le_of_lt (Nat.le_of_dvd  (Nat.sub_pos_of_lt <| h2q) h_piv_div_q_sub_one)\n      <| pred_lt <| Nat.ne_of_gt <| Nat.le_of_lt h2q","line":104,"end_line":162,"proof_status":"formalized"}
{"id":"thm:estimate_integral","label":"estimate_integral","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(n \\in \\mathbb {N}\\) \\[ \\log n + \\frac1n {\\lt} H_n {\\lt} \\log n + 1. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.harmonic_number_bounds","latex_link":"sect0002.html#thm:estimate_integral","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.harmonic_number_bounds","lean_decl_name":"chapter2.harmonic_number_bounds","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_02.lean","lean4_code":"theorem harmonic_number_bounds {n : ℕ} (hn : 1 < n):\n    Real.log n + 1 / n < harmonic n ∧\n    harmonic n <  Real.log n + 1 := by sorry","line":213,"end_line":215,"proof_status":"statement_formalized"}
{"id":"thm:hadamard_order","label":"hadamard_order","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a Hadamard matrix of size \\(n \\times n\\) exists for \\(n {\\gt} 2\\), then \\(n\\) must be a multiple of \\(4\\).","doc_url":null,"latex_link":"sect0007.html#thm:hadamard_order","lean_link":null,"lean_decl_name":"hadamard_order","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cauchy_rigidity","label":"cauchy_rigidity","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["arm_lemma"],"nlp_text":"If two \\(3\\)-dimensional convex polyhedra \\(P\\) and \\(P'\\) are combinatorially equivalent with corresponding pairs of adjacent congruent, then also the angels between corresponding pairs of adjacent facets are equal (and thus \\(P\\) is congruent to \\(P'\\)).","doc_url":null,"latex_link":"sect0014.html#cauchy_rigidity","lean_link":null,"lean_decl_name":"cauchy_rigidity","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"borromean_nontrivial","label":"borromean_nontrivial","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The Borromean rings are nontrivial, and they are also not equivalent to Tait’s link No.&#160;18","doc_url":null,"latex_link":"sect0015.html#borromean_nontrivial","lean_link":null,"lean_decl_name":"borromean_nontrivial","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"buffon_needle","label":"buffon_needle","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a short needle, of length \\(\\ell \\), is dropped on paper that is ruled with equally spaced lines of distance \\(d \\geq \\ell \\), then the probability that the needle comes to lie in a position where it crosses one of the lines is exactly \\[ p = \\frac{2 \\ell }{\\pi d}. \\]","doc_url":null,"latex_link":"sect0026.html#buffon_needle","lean_link":null,"lean_decl_name":"buffon_needle","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch41proof1","label":"ch41proof1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a graph \\(G = (V, E)\\) on \\(n\\) vertices has no \\(p\\)-clique, \\(p \\geq 2\\), then \\[ |E| \\leq \\left(1 - \\frac{1}{p - 1}\\right) \\frac{n^2}{2}. \\tag {1} \\]","doc_url":null,"latex_link":"sect0038.html#ch41proof1","lean_link":null,"lean_decl_name":"ch41proof1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"turan_graph","label":"turan_graph","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch41proof1","ch41proof4","ch41proof5"],"nlp_text":"Collecting the proofs from the chapter...","doc_url":null,"latex_link":"sect0038.html#turan_graph","lean_link":null,"lean_decl_name":"turan_graph","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch41proof4","label":"ch41proof4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a graph \\(G = (V, E)\\) on \\(n\\) vertices has no \\(p\\)-clique, \\(p \\geq 2\\), then \\[ |E| \\leq \\left(1 - \\frac{1}{p - 1}\\right) \\frac{n^2}{2}. \\tag {1} \\]","doc_url":null,"latex_link":"sect0038.html#ch41proof4","lean_link":null,"lean_decl_name":"ch41proof4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26lemma_e","label":"ch26lemma_e","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"By setting \\(h(x) := 0\\) for \\(x \\in \\mathbb {Z}\\), \\(h\\) becomes a continuous function on all of \\(\\mathbb {R}\\) that shares the properties given in 26.2, 26.3, 26.4.","doc_url":null,"latex_link":"sect0025.html#ch26lemma_e","lean_link":null,"lean_decl_name":"ch26lemma_e","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch17theorem1","label":"ch17theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every \\(d\\), one has the following chain of inequalities: \\begin{align} 2^d & \\leq _{\\text{(1)}} \\max \\left\\{ \\# S \\, |\\, S \\subseteq \\mathbb {R}^d, \\angle (s_i, s_j, s_k) \\leq \\frac{\\pi }{2} \\text{ for every } \\{ s_i, s_j, s_k\\} \\subseteq S \\right\\} \\\\ & \\leq _{\\text{(2)}} \\max \\left\\{ \\# S \\, |\\, S \\subseteq \\mathbb {R}^d \\text{ such that for any two points } \\{ s_i, s_j\\} \\subseteq S, \\right. \\nonumber \\\\ & \\qquad \\qquad \\left. \\text{there is a strip } S(i, j) \\text{ that contains } S, \\text{ with } s_i \\text{ and } s_j \\text{ lying in the parallel boundary hyperplanes of } S(i, j) \\right\\} \\\\ & =_{\\text{(3)}} \\max \\left\\{ \\# S \\, |\\, S \\subseteq \\mathbb {R}^d \\text{ such that the translates } P - s_i, s_i \\in S, \\text{ of the convex hull } P := \\text{conv}(S) \\right. \\nonumber \\\\ & \\qquad \\qquad \\left. \\text{intersect in a common point, but they only touch} \\right\\} \\\\ & \\leq _{\\text{(4)}} \\max \\left\\{ \\# S \\, |\\, S \\subseteq \\mathbb {R}^d \\text{ such that the translates } Q + s_i \\text{ of some d-dimensional convex polytope } Q \\subseteq \\mathbb {R}^d \\text{ touch pairwise} \\right\\} \\\\ & =_{\\text{(5)}} \\max \\left\\{ \\# S \\, |\\, S \\subseteq \\mathbb {R}^d \\text{ such that the translates } Q^* + s_i \\text{ of some d-dimensional centrally symmetric convex polytope } Q^* \\subseteq \\mathbb {R}^d \\text{ touch pairwise} \\right\\} \\\\ & \\leq _{\\text{(6)}} 2^d. \\end{align}","doc_url":null,"latex_link":"sect0017.html#ch17theorem1","lean_link":null,"lean_decl_name":"ch17theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"gauss_lemma","label":"gauss_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_criterion"],"nlp_text":"Suppose \\(a \\not\\equiv 0 \\pmod{p}\\). Take the numbers \\(1a, 2a, \\dots , \\frac{p-1}{2}a\\) and reduce them modulo \\(p\\) to the residue system smallest in absolute value, \\(ia \\equiv r_i \\pmod{p}\\) with \\(-\\frac{p-1}{2} \\le r_i \\le \\frac{p-1}{2}\\) for all \\(i\\). Then \\[ (\\frac{a}{p}) = (-1)^s, \\quad \\text{where } s = \\# \\{ i : r_i {\\lt} 0\\} . \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.lemma_of_Gauss","latex_link":"sect0005.html#gauss_lemma","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.lemma_of_Gauss","lean_decl_name":"book.quadratic_reciprocity.lemma_of_Gauss","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"lemma lemma_of_Gauss (p : ℕ) [Fact (Nat.Prime p)] (a : ℤ) (h_a : (a : ZMod p) ≠ 0)\n  ( r : ℤ → ℤ ) (h_r : (∀ i, (- (p: ℤ) - 1)/2 ≤ r i ∧ r i ≤ ((p : ℤ) - 1)/2))\n  ( H : ∀ i, (r i : ℤ) = (a * i : ZMod p) ) :\n   -- TODO: check why this is needed after porting to lean4\n   have : LocallyFiniteOrder ℤ := by sorry\n   legendre_sym p a =\n   Finset.card ((Icc (1 : ℤ) (((p : ℤ)-1)/2)).image r ∩ (Icc (-((p: ℤ) - 1)/2) (-1))) := by\n  sorry","line":72,"end_line":79,"proof_status":"statement_formalized"}
{"id":"quadratic_reciprocity1","label":"quadratic_reciprocity1","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["gauss_lemma"],"nlp_text":"Let \\(p\\) and \\(q\\) be different odd primes. Then \\[ (\\frac{q}{p})(\\frac{p}{q}) = (-1)^{\\frac{p-1}{2} \\frac{q-1}{2}}. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.quadratic_reciprocity_1","latex_link":"sect0005.html#quadratic_reciprocity1","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.quadratic_reciprocity_1","lean_decl_name":"book.quadratic_reciprocity.quadratic_reciprocity_1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"theorem quadratic_reciprocity_1 (p q : ℕ) (hp : p ≠ 2) (hq : q ≠ 2)\n  [Fact (Nat.Prime p)] [Fact (Nat.Prime q)] :\n  (legendre_sym p q) * (legendre_sym q p) = -1 ^ ((p-1) / 2 * (q - 1) / 2 ) :=\n  sorry","line":81,"end_line":84,"proof_status":"formalized"}
{"id":"lem_aux_i","label":"lem_aux_i","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For some fixed \\(n \\ge 1\\), let \\[ f(x) = \\frac{x^n (1-x)^n}{n!}. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_i","latex_link":"sect0008.html#lem_aux_i","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_i","lean_decl_name":"book.irrational.lem_aux_i","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"lemma lem_aux_i (n : ℕ) (x : ℝ) : ∃ c : ℕ → ℤ, f_aux n x = ∑ i ∈ Icc n (2 * n), (c i) * x ^ i := by\n  sorry","line":97,"end_line":98,"proof_status":"statement_formalized"}
{"id":"cone_lemma","label":"cone_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a system of homogeneous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution.","doc_url":null,"latex_link":"sect0010.html#cone_lemma","lean_link":null,"lean_decl_name":"cone_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch04.proposition3","label":"ch04.proposition3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can be written as \\(p = x^2 + y^2\\) for some natural numbers \\(x,y \\in \\mathbb {N}\\).","doc_url":null,"latex_link":"sect0004.html#ch04.proposition3","lean_link":null,"lean_decl_name":"ch04.proposition3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"example2","label":"example2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0010.html#example2","lean_link":null,"lean_decl_name":"example2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"museum_guards","label":"museum_guards","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any museum with \\(n\\) walls, \\(\\lfloor \\frac{n}{3} \\rfloor \\) guards suffice.","doc_url":null,"latex_link":"sect0037.html#museum_guards","lean_link":null,"lean_decl_name":"museum_guards","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch41proof5","label":"ch41proof5","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a graph \\(G = (V, E)\\) on \\(n\\) vertices has no \\(p\\)-clique, \\(p \\geq 2\\), then \\[ |E| \\leq \\left(1 - \\frac{1}{p - 1}\\right) \\frac{n^2}{2}. \\tag {1} \\]","doc_url":null,"latex_link":"sect0038.html#ch41proof5","lean_link":null,"lean_decl_name":"ch41proof5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"friendship","label":"friendship","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose that \\(G\\) is a finite graph in which any two vertices have precisely one common neighbor. Then there is a vertex which is adjacent to all other vertices.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter44.friendship_theorem","latex_link":"sect0041.html#friendship","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter44.friendship_theorem","lean_decl_name":"chapter44.friendship_theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Archive/Wiedijk100Theorems/FriendshipGraphs.lean","lean4_code":"theorem friendship_theorem [Nonempty V] : ExistsPolitician G := by\n  by_contra npG\n  rcases hG.isRegularOf_not_existsPolitician npG with ⟨d, dreg⟩\n  rcases lt_or_ge d 3 with dle2 | dge3\n  · exact npG (hG.existsPolitician_of_degree_le_two dreg (Nat.lt_succ_iff.mp dle2))\n  · exact hG.false_of_three_le_degree dreg dge3\n\nend\n\n","line":335,"end_line":343,"proof_status":"statement_formalized"}
{"id":"handshaking","label":"handshaking","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a finite simple graph with vertex set \\(V\\) and edge set \\(E\\) and let \\(d(v)\\) denote the degree of a vertex \\(v\\). Then \\[ \\sum _{v\\in V}d(v) = 2|E|. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter28.handshaking","latex_link":"chapter28.html#handshaking","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter28.handshaking","lean_decl_name":"chapter28.handshaking","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_28.lean","lean4_code":"lemma handshaking : ∑ v, d(v) = 2 * #E := by\n  calc  ∑ v, d(v)\n    _ = ∑ v, #I(v)             := by simp [G.card_incidenceFinset_eq_degree]\n    _ = ∑ v, #{e ∈ E | v ∈ e}  := by simp [G.incidenceFinset_eq_filter]\n    _ = ∑ e ∈ E, #{v | v ∈ e}  := Finset.sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow _\n    -- FIXME: was (G.card_filter_mem_of_mem_edgeFinset e he)) but is commented out currently in Mathlib.EdgeFinset\n    _ = ∑ e ∈ E, 2             := Finset.sum_congr rfl (λ e he ↦ sorry)\n    _ = 2 * ∑ e ∈ E, 1         := (Finset.mul_sum E (λ _ ↦ 1) 2).symm\n    _ = 2 * #E                 := by rw [Finset.card_eq_sum_ones E]","line":58,"end_line":66,"proof_status":"formalized"}
{"id":"ch19proposition2","label":"ch19proposition2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Any two ordinal numbers \\(\\mu \\) and \\(\\nu \\) satisfy precisely one of the relations \\(\\mu {\\lt} \\nu \\), \\(\\mu = \\nu \\), or \\(\\mu {\\gt} \\nu \\).","doc_url":null,"latex_link":"sect0019.html#ch19proposition2","lean_link":null,"lean_decl_name":"ch19proposition2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"fermats_little","label":"fermats_little","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(a \\not\\equiv 0 \\pmod{p}\\), \\[ a^{p - 1} \\equiv 1 \\pmod{p} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fermat_little","latex_link":"sect0005.html#fermats_little","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fermat_little","lean_decl_name":"book.quadratic_reciprocity.fermat_little","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"/--\nFermat's little theorem: If `a` is nonzero modulo the odd prime `p`, then `a ^ (p - 1) = -1`\nmodulo `p`.\n-/\nlemma fermat_little (a : ℤ): (a : ZMod p) ≠ 0 → a ^ (p - 1) = (-1 : ZMod p) := by\n  let units_finset := (Finset.univ : Finset (ZMod p)).erase 0\n  let image_finset := (units_finset).image (fun x : ZMod p => (a : ZMod p) * x)\n  have : units_finset = image_finset := by sorry\n  sorry","line":46,"end_line":54,"proof_status":"statement_formalized"}
{"id":"euler_criterion","label":"euler_criterion","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["fermats_little"],"nlp_text":"For \\(a \\not\\equiv 0 \\pmod{p}\\), \\[ (\\frac{a}{p}) \\equiv a ^{\\frac{p-1}{2}} \\pmod{p} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.euler_criterion","latex_link":"sect0005.html#euler_criterion","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.euler_criterion","lean_decl_name":"book.quadratic_reciprocity.euler_criterion","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/LegendreSymbol/Basic.lean","lean4_code":"theorem euler_criterion {a : ZMod p} (ha : a ≠ 0) : IsSquare (a : ZMod p) ↔ a ^ (p / 2) = 1 := by\n  apply (iff_congr _ (by simp [Units.ext_iff])).mp (euler_criterion_units p (Units.mk0 a ha))\n  simp only [Units.ext_iff, sq, Units.val_mk0, Units.val_mul]\n  constructor\n  · rintro ⟨y, hy⟩; exact ⟨y, hy.symm⟩\n  · rintro ⟨y, rfl⟩\n    have hy : y ≠ 0 := by\n      rintro rfl\n      simp [mul_zero, ne_eq] at ha\n    refine ⟨Units.mk0 y hy, ?_⟩; simp\n\n/-- If `a : ZMod p` is nonzero, then `a^(p/2)` is either `1` or `-1`. -/\n","line":58,"end_line":69,"proof_status":"statement_formalized"}
{"id":"product_rule","label":"product_rule","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["euler_criterion"],"nlp_text":"\"> \\begin{equation} \\label{eq:product_rule} (\\frac{ab}{p}) = (\\frac{a}{p}) \\cdot (\\frac{b}{p}) \\end{equation}","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.product_rule","latex_link":"sect0005.html#product_rule","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.product_rule","lean_decl_name":"book.quadratic_reciprocity.product_rule","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"lemma product_rule (a b : ℤ) :\n  legendre_sym p (a * b) = (legendre_sym p a) * (legendre_sym p b) := by\n  sorry","line":61,"end_line":63,"proof_status":"statement_formalized"}
{"id":"euler_series_4","label":"euler_series_4","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\sum _{n\\ge 1}\\frac{1}{n^2} = \\frac{\\pi ^2}{6} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series_4","latex_link":"sect0009.html#euler_series_4","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series_4","lean_decl_name":"euler_series_4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_09.lean","lean4_code":"theorem euler_series_4 :\n  ∑' (n : ℕ+), (1 : ℝ) / n = π ^ 2  / 6 := by\n  sorry","line":621,"end_line":623,"proof_status":"statement_formalized"}
{"id":"ch19theorem4","label":"ch19theorem4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If each of two sets \\(M\\) and \\(N\\) can be mapped injectively into the other, then there is a bijection from \\(M\\) to \\(N\\), that is \\(|M| = |N|\\).","doc_url":null,"latex_link":"sect0019.html#ch19theorem4","lean_link":null,"lean_decl_name":"ch19theorem4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch30theorem2","label":"ch30theorem2","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch30lemma"],"nlp_text":"The largest size of an intersection \\(k\\)-family in an \\(n\\)-set is \\(\\binom {n - 1}{k - 1}\\).","doc_url":null,"latex_link":"chapter30.html#ch30theorem2","lean_link":null,"lean_decl_name":"ch30theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch45theorem1","label":"ch45theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every family of at most \\(2^{d-1}\\) \\(d\\)-sets is \\(2\\)-colorable, that is, \\(m(d) {\\gt} 2^{d-1}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter45.theorem_1","latex_link":"sect0042.html#ch45theorem1","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter45.theorem_1","lean_decl_name":"chapter45.theorem_1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./MathlibTest/Explode.lean","lean4_code":"theorem theorem_1 : ∀ (p : Prop), p → p :=\n  fun (p : Prop) ↦ (fun hP : p ↦ hP)\n/--\ninfo: theorem_1 : ∀ (p : Prop), p → p\n\n0│     │ p  ├ Prop\n1│     │ hP ├ p\n2│0,1,1│ ∀I │ ∀ (p : Prop), p → p\n-/\n#guard_msgs in #explode theorem_1\n\n","line":55,"end_line":65,"proof_status":"statement_formalized"}
{"id":"ch30lemma","label":"ch30lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(n \\ge 2k\\), and suppose we are given \\(t\\) distinct arcs \\(A_1, \\dots A_t\\) of length \\(k\\), such that any two arcs have an edge in common. Then \\(t \\le k\\).","doc_url":null,"latex_link":"chapter30.html#ch30lemma","lean_link":null,"lean_decl_name":"ch30lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch04.proposition2","label":"ch04.proposition2","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every prime of the form \\(p = 4m + 1\\) is a sum of two squares, that is, it can be written as \\(p = x^2 + y^2\\) for some natural numbers \\(x,y \\in \\mathbb {N}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.theorem₂","latex_link":"sect0004.html#ch04.proposition2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.theorem₂","lean_decl_name":"ch04.theorem₂","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_04.lean","lean4_code":"theorem theorem₂ {p : ℕ} [h : Fact p.Prime] (hp : p % 4 = 1) :\n    ∃ a b : ℕ, a ^ 2 + b ^ 2 = p := by\n  rw [← div_add_mod p 4, hp] at h ⊢\n  let k := p / 4\n  have ⟨a, b, h⟩ := sq_add_sq_of_nonempty_fixedPoints k <| trivialInvo_fixedPoints k\n  refine ⟨a.natAbs, b.natAbs, ?_⟩\n  zify\n  simpa only [sq_abs] using h","line":291,"end_line":298,"proof_status":"statement_formalized"}
{"id":"ch19proposition3","label":"ch19proposition3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every set of ordinal numbers (ordered according to magnitude) is well-ordered.","doc_url":null,"latex_link":"sect0019.html#ch19proposition3","lean_link":null,"lean_decl_name":"ch19proposition3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"root_of_unity","label":"root_of_unity","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The \\(n\\)-th roots of unity are \\[ \\lambda _k = e^{\\frac{2k\\pi i}{n}} = \\cos (2k\\pi /n) + i \\sin (2k\\pi /n), \\quad 0 \\le k \\le n - 1. \\]","doc_url":null,"latex_link":"sect0006.html#root_of_unity","lean_link":null,"lean_decl_name":"root_of_unity","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"fact_A","label":"fact_A","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) and \\(q\\) be distinct odd primes, and consider the finite field \\(F\\) with \\(q^{p-1}\\) elements. Then for any \\(a, b \\in F\\), \\((a + b)^q = a^q + b^q\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fact_A","latex_link":"sect0005.html#fact_A","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.quadratic_reciprocity.fact_A","lean_decl_name":"book.quadratic_reciprocity.fact_A","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_05.lean","lean4_code":"lemma fact_A (p q : ℕ) (hp : p ≠ 2) (hq : q ≠ 2) [Fact (Nat.Prime p)] [Fact (Nat.Prime q)]\n  (h_pq : p ≠ q) (K : Type _) [Field K] [Fintype K] (H : Fintype.card K = q ^ (p - 1)) :\n  ∀ a b : K, (a + b) ^ q = a ^ q + b ^ q :=\n  sorry","line":101,"end_line":104,"proof_status":"statement_formalized"}
{"id":"ch11_theorem3","label":"ch11_theorem3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(X\\) be a set of \\(n\\ge 3\\) elements, and let \\(A_1, \\dots , A_m\\) be proper subsets of \\(X\\), such that every pair of elements of \\(X\\) is contained in precisely one set \\(A_i\\). Then \\(m\\ge n\\) holds.","doc_url":null,"latex_link":"sect0011.html#ch11_theorem3","lean_link":null,"lean_decl_name":"ch11_theorem3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch30theorem3","label":"ch30theorem3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(A_1, \\dots A_n\\) be a collection of subset of a finite set \\(X\\). Then there exists a system of distinct representatives if and only if the union of any \\(m\\) sets \\(A_i\\) contains at least \\(m\\) elements, for \\(1 \\le m \\le n\\).","doc_url":null,"latex_link":"chapter30.html#ch30theorem3","lean_link":null,"lean_decl_name":"ch30theorem3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch45theorem2","label":"ch45theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every family of at most \\(2^{d-1}\\) \\(d\\)-sets is \\(2\\)-colorable, that is, \\(m(d) {\\gt} 2^{d-1}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter45.ramsey_exists","latex_link":"sect0042.html#ch45theorem2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter45.ramsey_exists","lean_decl_name":"chapter45.ramsey_exists","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_45.lean","lean4_code":"lemma ramsey_exists (m n : ℕ) (h_m : m ≥ 2) (h_n : n ≥ 2) : ∃ N, ramsey_property m n N := by\n  sorry","line":265,"end_line":266,"proof_status":"statement_formalized"}
{"id":"gurvit","label":"gurvit","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(p(x)\\in \\mathbb {R}_+[x_1, \\dots , x_n]\\) is a \\(H\\)-stable and homogeneous of degree \\(n\\), then either \\(p'\\cong 0\\), or \\(p'\\) is \\(H\\)-stable and homogeneous of degree \\(n - 1\\). In either case \\[ \\operatorname{cap}(p') \\ge \\operatorname{cap}\\cdot g(\\deg _np). \\]","doc_url":null,"latex_link":"sect0023.html#gurvit","lean_link":null,"lean_decl_name":"gurvit","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"vanderwaerden","label":"vanderwaerden","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["gurvit"],"nlp_text":"Let \\(M = (m_{ij})\\) be a doubly stochastic \\(n \\times n\\) matrix. Then \\[ \\operatorname{per}M \\ge \\frac{n!}{n^n} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/Matrix.permanent_conjecture","latex_link":"sect0023.html#vanderwaerden","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/Matrix.permanent_conjecture","lean_decl_name":"Matrix.permanent_conjecture","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_24.lean","lean4_code":"theorem permanent_conjecture (M : Matrix (Fin n) (Fin n) ℝ) :\n    M ∈ doublyStochastic ℝ (Fin n) → permanent M ≥ (n.factorial)/(n ^ n) := sorry","line":37,"end_line":38,"proof_status":"statement_formalized"}
{"id":"ch20theorem2","label":"ch20theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f(x)\\) be a real polynomial of degree \\(n \\ge 2\\) with only real roots, such that \\(f(x){\\gt} 0\\) for \\( -1 {\\lt} x {\\lt} 1\\) amd \\(f(-1) = f(1) = 0\\). Then \\[ \\frac{2}{3}T \\le A \\le \\frac{2}{3}R, \\]","doc_url":null,"latex_link":"chapter20.html#ch20theorem2","lean_link":null,"lean_decl_name":"ch20theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch45theorem4","label":"ch45theorem4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(G\\) be a simple graph with \\(n\\) vertices and \\(m\\) edges, where \\(m \\ge 4n\\). Then \\[ \\operatorname{cr}(G) \\ge \\frac{1}{64}\\frac{m^3}{n^2}. \\]","doc_url":null,"latex_link":"sect0042.html#ch45theorem4","lean_link":null,"lean_decl_name":"ch45theorem4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch38definition2","label":"ch38definition2","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A matching \\(M\\) of \\(G = (X \\cup Y, E)\\) is called stable if the following condition holds: Whenever \\(uv \\in E \\setminus M\\), \\(u \\in X\\), \\(v \\in Y\\), then either \\(uy \\in M\\) with \\(y {\\gt} v\\) in \\(N(u)\\) or \\(xv \\in M\\) with \\(x {\\gt} u\\) in \\(N(v)\\), or both.","doc_url":null,"latex_link":"sect0035.html#ch38definition2","lean_link":null,"lean_decl_name":"ch38definition2","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch19proposition1","label":"ch19proposition1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mu \\) be an ordinal number and denote by \\(W_\\mu \\) the set of ordinal numbers smaller than \\(\\mu \\). Then the following holds: The elements of \\(W_\\mu \\) are pairwise comparable. If we order \\(W_\\mu \\) according to their magnitude, then \\(W_\\mu \\) is well-ordered and has ordinal number \\(\\mu \\).","doc_url":null,"latex_link":"sect0019.html#ch19proposition1","lean_link":null,"lean_decl_name":"ch19proposition1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"sylvester","label":"sylvester","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all positive natural \\(n, k\\) such that \\(n \\ge 2k\\), at least one of the numbers \\(n, n - 1, \\dots , n - k + 1\\) has a prime divisor \\(p\\) greater than \\(k\\), or, equivalently the binomial coefficient \\(\\binom {n}{k}\\) always has a prime factor \\(p {\\gt} k\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter3.sylvester","latex_link":"sect0003.html#sylvester","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter3.sylvester","lean_decl_name":"chapter3.sylvester","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/RingTheory/Polynomial/Resultant/Basic.lean","lean4_code":"def sylvester (f g : R[X]) (m n : ℕ) : Matrix (Fin (n + m)) (Fin (n + m)) R :=\n  .of fun i j ↦ j.addCases\n    (fun j₁ ↦ if (i : ℕ) ∈ Set.Icc (j₁ : ℕ) (j₁ + m) then f.coeff (i - j₁) else 0)\n    (fun j₁ ↦ if (i : ℕ) ∈ Set.Icc (j₁ : ℕ) (j₁ + n) then g.coeff (i - j₁) else 0)\n\nvariable (f g : R[X]) (m n : ℕ)\n\n","line":48,"end_line":54,"proof_status":"statement_formalized"}
{"id":"binomial_never_powers","label":"binomial_never_powers","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["sylvester"],"nlp_text":"The equation \\(\\binom n k = m ^l\\) has no integer solutions with \\(l \\ge 2\\) and \\(4 \\le k \\le n - 4\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter3.binomials_coefficients_never_powers","latex_link":"sect0003.html#binomial_never_powers","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter3.binomials_coefficients_never_powers","lean_decl_name":"chapter3.binomials_coefficients_never_powers","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_03.lean","lean4_code":"theorem binomials_coefficients_never_powers (k l m n : ℕ) (h_2lel : 2 ≤ l) (h_4lek : 4 ≤ k)\n(h_klen4 : k ≤ n - 4) : choose n k ≠ m^l := by\n  -- Assumption that n ≥ 2k\n  have h_wlog :\n      ∀ (k' : ℕ) (h_4lek' : 4 ≤ k') (h_klen4' : k' ≤ n - 4), 2*k' ≤ n → choose n k' ≠ m^l := by\n    clear h_4lek h_klen4 k\n    intros k h_4lek h_klen4 h\n    -- inequalities needed\n    have h_klen : k ≤ n := le_trans h_klen4 (Nat.sub_le n 4)\n    have h_1lel : 1 ≤ l := le_of_succ_le (h_2lel)\n    by_contra H\n    -- main proof here proceeding in four steps\n    -- Step (1)\n    have h₁: ∃ p, p.Prime ∧ p^l ≤ n ∧ k^l < p^l ∧ k^2 ≤ k^l := by\n      obtain ⟨p, ⟨h_klp, ⟨h_p, h_p_div_binom⟩⟩⟩ := sylvester k n h (lt_of_lt_of_le (Nat.zero_lt_succ 3) h_4lek)\n      use p\n      refine' ⟨h_p,  ⟨_, _⟩⟩\n      -- prove p^l ≤ n\n      · have h_pl_div_desc: p^l ∣ n.descFactorial k :=\n            prime_div_descFactorial n k m l p  h_klen h_p h_p_div_binom H\n        have h_klp_pow_dvd := factor_in_descFactorial n k p l h_klen (gt_iff_lt.mp h_klp) (h_p)\n            h_pl_div_desc h_1lel\n        -- working with them\n        rcases h_klp_pow_dvd with ⟨i, hi⟩\n        rcases hi with ⟨hi_left, hi_right⟩\n        have : p^l ≤ n - i := by\n          refine' Nat.le_of_dvd _ hi_right\n          simp only [tsub_pos_iff_lt]\n          have h_ilk : i < k := Iff.mpr (lt_iff_le_pred (lt_of_lt_of_le four_pos h_4lek)) hi_left\n          exact lt_of_lt_of_le h_ilk h_klen\n        have h_klen4i : n - i ≤ n := Nat.sub_le n i\n        exact le_trans this h_klen4i\n      · exact ⟨\n            -- prove k^l < p^l\n            Nat.pow_lt_pow_left (h_klp) (Nat.ne_of_gt <| lt_of_le_of_lt' h_2lel two_pos),\n            Nat.pow_le_pow_of_le (by omega) h_2lel -- prove k² ≤ k^l\n            ⟩\n\n    -- Step (2) : aⱼ only have prime divisors ≤ k ; aᵢ ≠ aⱼ\n    --have h₂ : ∀ j, (j ≤ k - 1) ∧ (∀ (q : ℕ), q ∣ (aFct l n j) ∧ prime q → q ≤ k) ∧\n    --    (∀ i ≤ k - 1, i ≠ j → (aFct l n i) ≠ (aFct l n j)) := by\n    -- sorry\n    -- Step (3) : a_i are integers 1..k\n    --have h₃ : a_values l n k = s_1tok k := by\n    -- divide in two cases\n    cases em (l = 2)\n    -- Special Case l = 2 by Contradiction\n    ·  sorry\n    -- STEP (4) : l ≥ 3 by Contradiction\n    -- case l ≥ 3\n    · have h_3lel : 3 ≤ l := by\n        sorry\n      -- main work : n < k³\n      have h₄ : n < k^3 := by\n        sorry\n\n      sorry\n\n  rcases em (n ≥ 2*k) with h_2k | h\n  · exact h_wlog k h_4lek h_klen4 h_2k\n  · -- transform ¬(n ≥ 2 * k) into (n < 2 * k)\n    simp only [not_le] at h\n    -- transform (n.choose k) into (n.choose (n - k))\n    have h_klen : k ≤ n := le_trans h_klen4 (Nat.sub_le n 4)\n    rw [← choose_symm h_klen]\n    -- define k' as n - k, such that k' can be used for h_wlog as k'\n    -- satisfies all required features\n    let k' := n - k\n    have h_k'_def : k' = n - k := by rfl\n    -- third requirement: 2 * k' ≤ n\n    have h_2k'len : 2 * k' ≤ n := by\n      sorry\n    -- second requirement: k ≤ n - 4\n    have h_k'len4 : k' ≤ n - 4 := by\n      simp only [h_k'_def, tsub_le_iff_right]\n      have help : k + k ≤ n - 4 + k := add_le_add_left h_klen4 k\n      rw [← (two_mul k)] at help\n      exact le_trans (le_of_lt h) help\n    -- first requirement: 4 ≤ k\n    have h_4lek' : 4 ≤ k' := Iff.mp (le_tsub_iff_le_tsub h_klen (le_trans (le_trans h_4lek h_klen4)\n      (Nat.sub_le n 4))) h_klen4\n    -- now we can use h_wlog\n    exact h_wlog k' h_4lek' h_k'len4 h_2k'len","line":102,"end_line":184,"proof_status":"statement_formalized"}
{"id":"euler_series","label":"euler_series","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\sum _{n \\ge 1} \\frac{1}{n^2} = \\frac{\\pi ^2}{6} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series","latex_link":"sect0009.html#euler_series","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series","lean_decl_name":"euler_series","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_09.lean","lean4_code":"theorem euler_series : ∑' n : ℕ, ((n : ℝ) ^ 2)⁻¹ = π ^ 2 / 6 := by\n  convert_to ∑' n : ℕ, (n : ℝ)⁻¹ ^ 2 = _ using 3 with n\n  · simp\n  -- Change the index from n to n + 1 to avoid division by zero\n  convert_to ∑' n : ℕ, ((n : ℝ) + 1)⁻¹ ^ 2 = _ using 1\n  . apply tsum_eq_tsum_of_ne_zero_bij fun n => n.1 + 1\n    · simp [Function.Injective]\n    · simp; norm_cast; simp; omega\n    · simp\n  convert_to ∑' n : ℕ, (∫ x : ℝ in 0..1, x ^ n) ^ 2 = _ using 1\n  · simp\n  -- As every term is nonnegative, we will use ℝ≥0∞ Lebesgue integrals to\n  -- facilitate the exchange of sum and integral later with Tonelli's theorem.\n  convert_to (∑' n : ℕ, ofReal (∫ x : ℝ in 0..1, x ^ n) ^ 2).toReal = _ using 1\n  . rw [tsum_toReal_eq]\n    · congr 1 with n\n      rw [← ofReal_pow, toReal_ofReal]\n      · positivity\n      · exact intervalIntegral.integral_nonneg (by norm_num) fun x hx => pow_nonneg hx.1 n\n    · finiteness\n  convert_to _ = (ofReal (π ^ 2 / 6)).toReal using 1\n  · rw [toReal_ofReal]\n    positivity\n  congr 1\n  convert_to ∑' n : ℕ, (∫⁻ x : ℝ in Ioo 0 1, ofReal (x ^ n)) ^ 2 = _ using 4 with n\n  . rw [intervalIntegral_eq_integral_uIoc]\n    norm_num\n    rw [← integral_Icc_eq_integral_Ioc, integral_Icc_eq_integral_Ioo]\n    rw [ofReal_integral_eq_lintegral_ofReal]\n    · apply IntegrableOn.integrable\n      rw [←intervalIntegrable_iff_integrableOn_Ioo_of_le]\n      · exact intervalIntegrable_pow n\n      · norm_num\n    · filter_upwards [ae_restrict_mem measurableSet_Ioo] with x hx\n      exact pow_nonneg hx.1.le n\n  convert_to ∑' n : ℕ, (∫⁻ x : ℝ in Ioo 0 1, ofReal x ^ n) ^ 2 = _ using 4 with n\n  . refine setLIntegral_congr_fun measurableSet_Ioo fun x hx => ?_\n    rw [ofReal_pow hx.1.le]\n  -- Introduce indicator functions to extend the integral to all ℝ.\n  -- This will facilitate later uses of Tonelli's theorem.\n  convert_to ∑' n : ℕ, (∫⁻ x : ℝ, (Ioo 0 1).indicator 1 x * ofReal x ^ n) ^ 2 = _\n    using 4 with n\n  · simp [←lintegral_indicator, indicator]\n  convert_to ∑' n : ℕ, (∫⁻ x : ℝ, (Ioo 0 1).indicator 1 x * ofReal x ^ n) *\n    (∫⁻ y : ℝ, (Ioo 0 1).indicator 1 y * ofReal y ^ n) = _ using 3 with n\n  · ring_nf\n  convert_to ∑' n : ℕ, ∫⁻ x : ℝ, (Ioo 0 1).indicator 1 x * ofReal x ^ n *\n    ∫⁻ y : ℝ, (Ioo 0 1).indicator 1 y * ofReal y ^ n = _ using 3 with n\n  . rw [lintegral_mul_const _ (by clear * -; measurability)]\n  convert_to ∑' n : ℕ, ∫⁻ x : ℝ, ∫⁻ y : ℝ,\n    (Ioo 0 1).indicator 1 x * (Ioo 0 1).indicator 1 y * (ofReal x * ofReal y) ^ n = _\n      using 5 with n x\n  . rw [←lintegral_const_mul _ (by clear * -; measurability)]\n    ring_nf\n    -- Now we exchange the sum and the integrals using Tonelli's theorem twice.\n    -- Using ℝ≥0∞ integrals saves us from checking integrability conditions.\n  convert_to ∫⁻ x : ℝ, ∑' n : ℕ, ∫⁻ y : ℝ,\n    (Ioo 0 1).indicator 1 x * (Ioo 0 1).indicator 1 y * (ofReal x * ofReal y) ^ n = _\n      using 1\n  . rw [lintegral_tsum (by clear * -; measurability)]\n  convert_to ∫⁻ x : ℝ, ∫⁻ y : ℝ, ∑' n : ℕ,\n    (Ioo 0 1).indicator 1 x * (Ioo 0 1).indicator 1 y * (ofReal x * ofReal y) ^ n = _\n      using 3 with x\n  . rw [lintegral_tsum (by clear * -; measurability)]\n  convert_to ∫⁻ x : ℝ, ∫⁻ y : ℝ, (Ioo 0 1).indicator 1 x * (Ioo 0 1).indicator 1 y *\n    ((1 - ofReal x * ofReal y)⁻¹) = _ using 5 with x y\n  · rw [ENNReal.tsum_mul_left, tsum_geometric]\n  -- The integration set in xy-coordinates\n  let T := {xy : ℝ × ℝ | xy.1 ∈ Ioo 0 1 ∧ xy.2 ∈ Ioo 0 1}\n  -- the 2D change of variables function x = u - v, y = u + v\n  let f (uv : ℝ × ℝ) : ℝ × ℝ := (uv.1 - uv.2, uv.1 + uv.2)\n  -- the inverse of the 2D change of variables function u = (x + y) / 2, v = (y - x) / 2\n  let finv (xy : ℝ × ℝ) : ℝ × ℝ := ((xy.1 + xy.2) / 2, (xy.2 - xy.1) / 2)\n  -- The jacobian matrix of the change of variables\n  let J : Matrix (Fin 2) (Fin 2) ℝ := ![![1, -1], ![1, 1]]\n  -- The linear map associated to the jacobian matrix\n  let f' (_ : ℝ × ℝ) : ℝ × ℝ →L[ℝ] ℝ × ℝ :=\n    let i := LinearEquiv.finTwoArrow ℝ ℝ\n    (i ∘ₗ J.toLin' ∘ₗ i.symm).toContinuousLinearMap\n  -- First region of integration in uv-coordinates\n  let S1 := {uv : ℝ × ℝ | uv.1 ∈ Ioo 0 2⁻¹ ∧ uv.2 ∈ Ioo (-uv.1) (uv.1)}\n  -- Second region of integration in uv-coordinates\n  let S2 := {uv : ℝ × ℝ | uv.1 ∈ Ioo 2⁻¹ 1 ∧ uv.2 ∈ Ioo (-(1 - uv.1)) (1 - uv.1)}\n  -- Separating line of measure zero in uv-coordinates, to ensure symmetry\n  let S3 : Set (ℝ × ℝ) := {2⁻¹} ×ˢ Ioo (-2⁻¹) 2⁻¹\n  -- The full integration set in uv-coordinates\n  let S := S1 ∪ S2 ∪ S3\n  have bijective_f : f.Bijective := by\n    refine Function.bijective_iff_has_inverse.mpr ?_\n    use finv\n    unfold Function.RightInverse Function.LeftInverse f finv\n    split_ands\n    · simp\n    · intro a; ext <;> linarith only\n  have fS_eq_T : f '' S = T := by\n    ext xy\n    simp only [mem_setOf_eq, mem_image, T, f, S]\n    constructor\n    · intro ⟨uv, h1, h2⟩\n      set u := uv.1\n      set v := uv.2\n      have hx : xy.1 = u - v := by simp [←h2]\n      have hy : xy.2 = u + v := by simp [←h2]\n      simp [S1, S2, S3] at h1 ⊢\n      grind only\n    · intro h1\n      let u := (xy.1 + xy.2) / 2\n      let v := (xy.2 - xy.1) / 2\n      use (u, v)\n      split_ands\n      · simp [S1, S2, S3] at h1 ⊢\n        grind only\n      · ext <;> simp [u, v] <;> linarith only\n  have isOpen_T : IsOpen T := by\n    refine IsOpen.and (IsOpen.and ?_ ?_) (IsOpen.and ?_ ?_) <;> apply isOpen_lt <;> fun_prop\n  have isOpen_S : IsOpen S := by\n    convert_to IsOpen (f ⁻¹' T) using 1\n    · have : f ⁻¹' T = S := ((eq_preimage_iff_image_eq bijective_f).mpr fS_eq_T).symm\n      rw [this]\n    refine IsOpen.preimage ?_ ?_\n    · unfold f; fun_prop\n    · exact isOpen_T\n  have measurableSet_S : MeasurableSet S := by\n    unfold S S1 S2 S3\n    clear * -; measurability\n  convert_to ∫⁻ x : ℝ, ∫⁻ y : ℝ, T.indicator 1 (x, y) * ((1 - ofReal x * ofReal y)⁻¹) = _\n    using 6 with x y\n  . simp [T, indicator]\n    grind only\n  -- Using Tonelli's theorem, we couple integrals over ℝ × ℝ to prepare for a change of variables.\n  convert_to ∫⁻ xy : ℝ × ℝ, T.indicator 1 xy * ((1 - ofReal xy.1 * ofReal xy.2)⁻¹) = _\n    using 1\n  . rw [←lintegral_prod fun xy => T.indicator 1 xy * ((1 - ofReal xy.1 * ofReal xy.2)⁻¹)]\n    · rw [←Measure.volume_eq_prod]\n    · unfold T\n      clear * -; measurability\n  convert_to ∫⁻ xy : ℝ × ℝ in T, (1 - ofReal xy.1 * ofReal xy.2)⁻¹ = _ using 1\n  . rw [←lintegral_indicator]\n    · simp [indicator]\n    · unfold T\n      clear * -; measurability\n  -- Now we perform the change of variables x = u - v, y = u + v.\n  rw [← fS_eq_T]\n  convert_to ∫⁻ uv : ℝ × ℝ in S, 2 * ((1 - ofReal (uv.1 - uv.2) * ofReal (uv.1 + uv.2))⁻¹) = _\n    using 1\n  . rw [lintegral_image_eq_lintegral_abs_det_fderiv_mul (f' := f')]\n    · apply setLIntegral_congr_fun\n      · exact measurableSet_S\n      · intro uv huv\n        beta_reduce\n        congr 1\n        norm_num [f', Matrix.det_fin_two, J]\n    · exact measurableSet_S\n    · intro uv huv\n      refine (hasFDerivWithinAt_of_mem_nhds ?_).mpr ?_\n      · rw [mem_nhds_iff]\n        use S, by simp, isOpen_S, huv\n      · rw [hasFDerivAt_iff_isLittleO_nhds_zero]\n        suffices ∀ h, f (uv + h) - f uv - (f' uv) h = 0 by simp [this]\n        intro h\n        simp [f, f', J, Matrix.mulVec]\n        split_ands <;> ring\n    · exact bijective_f.injective.injOn\n  clear! f finv J f'\n  -- Pull the ofReal upwards to prepare going back to regular ℝ valued interval based integrals.\n  convert_to ∫⁻ uv : ℝ × ℝ in S, 2 * ((ofReal 1 - ofReal ((uv.1 - uv.2) * (uv.1 + uv.2)))⁻¹) = _\n    using 1\n  . refine setLIntegral_congr_fun measurableSet_S fun (u, v) huv => ?_\n    congr 3\n    · norm_num\n    rw [ofReal_mul]\n    simp [S, S1, S2, S3] at huv\n    grind only\n  convert_to ∫⁻ uv : ℝ × ℝ in S, 2 * (ofReal (1 - (uv.1 - uv.2) * (uv.1 + uv.2)))⁻¹ = _ using 1\n  . refine setLIntegral_congr_fun measurableSet_S fun (u, v) huv => ?_\n    congr 2\n    rw [ofReal_sub]\n    suffices 0 < u - v ∧ 0 < u + v by nlinarith only [this]\n    simp [S, S1, S2, S3] at huv\n    grind only\n  convert_to 2 * ∫⁻ uv : ℝ × ℝ in S, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹) = _ using 1\n  . rw [←lintegral_const_mul]\n    · refine setLIntegral_congr_fun measurableSet_S fun (u, v) huv => ?_\n      congr 1\n      rw [ofReal_inv_of_pos]\n      suffices 0 < u - v ∧ u - v < 1 ∧ 0 < u + v ∧ u + v < 1 by nlinarith only [this]\n      simp [S, S1, S2, S3] at huv\n      grind only\n    · clear * -; measurability\n    -- Split the integral over two simpler regions that will allow us to separate the integrals.\n  convert_to 2 * ((∫⁻ uv : ℝ × ℝ in S1, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)) +\n            (∫⁻ uv : ℝ × ℝ in S2, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)) +\n            (∫⁻ uv : ℝ × ℝ in S3, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹))) = _ using 2\n  . unfold S\n    rw [lintegral_union, lintegral_union]\n    · unfold S2\n      clear * -; measurability\n    · grind\n    · unfold S3\n      clear * -; measurability\n    · grind\n  convert_to (2 * ∫⁻ uv : ℝ × ℝ in S1, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)) +\n        2 * ∫⁻ uv : ℝ × ℝ in S2, ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹) = _ using 1\n  . rw [setLIntegral_measure_zero S3]\n    · ring\n    · simp [S3, volume]\n  convert_to\n    (2 * ∫⁻ uv : ℝ × ℝ, S1.indicator 1 uv * ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)) +\n     2 * ∫⁻ uv : ℝ × ℝ, S2.indicator 1 uv * ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹) = _\n    using 3\n  · rw [←lintegral_indicator]\n    · simp [indicator]\n    · unfold S1\n      clear * -; measurability\n  · rw [←lintegral_indicator]\n    · simp [indicator]\n    · unfold S2\n      clear * -; measurability\n  -- Separate the integrals over u and v using Tonelli's theorem again.\n  convert_to\n    (2 * ∫⁻ u : ℝ, ∫⁻ v : ℝ, S1.indicator 1 (u, v) * ofReal ((1 - (u - v) * (u + v))⁻¹)) +\n     2 * ∫⁻ u : ℝ, ∫⁻ v : ℝ, S2.indicator 1 (u, v) * ofReal ((1 - (u - v) * (u + v))⁻¹) = _\n    using 3\n  · rw [←lintegral_prod fun uv => S1.indicator 1 uv * ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)]\n    · rw [←Measure.volume_eq_prod]\n    · unfold S1\n      clear * -; measurability\n  · rw [←lintegral_prod fun uv => S2.indicator 1 uv * ofReal ((1 - (uv.1 - uv.2) * (uv.1 + uv.2))⁻¹)]\n    · rw [←Measure.volume_eq_prod]\n    · unfold S2\n      clear * -; measurability\n  -- Get rid of the indicator functions\n  convert_to\n    (2 * ∫⁻ u : ℝ, ∫⁻ v : ℝ, (Ioo 0 2⁻¹).indicator 1 u * (Ioo (-u) u).indicator 1 v *\n      ofReal ((1 - (u - v) * (u + v))⁻¹)) +\n    2 * ∫⁻ u : ℝ, ∫⁻ v : ℝ, (Ioo 2⁻¹ 1).indicator 1 u * (Ioo (-(1 - u)) (1 - u)).indicator 1 v *\n      ofReal ((1 - (u - v) * (u + v))⁻¹) = _ using 8 with u v\n  · simp [S1, indicator]\n    grind only\n  · simp [S2, indicator]\n    grind only\n  clear! S1 S2 S3 S T\n  convert_to\n    (2 * ∫⁻ u : ℝ, (Ioo 0 2⁻¹).indicator 1 u * ∫⁻ v : ℝ, (Ioo (-u) u).indicator 1 v *\n      ofReal ((1 - (u - v) * (u + v))⁻¹)) +\n    2 * ∫⁻ u : ℝ, (Ioo 2⁻¹ 1).indicator 1 u * ∫⁻ v : ℝ, (Ioo (-(1 - u)) (1 - u)).indicator 1 v *\n      ofReal ((1 - (u - v) * (u + v))⁻¹) = _ using 1\n  . congr 2\n    all_goals\n    · congr! 2 with u\n      rw [←lintegral_const_mul]\n      · ring_nf\n      · clear *-; measurability\n  convert_to\n    (2 * ∫⁻ u : ℝ in Ioo 0 2⁻¹, ∫⁻ v : ℝ in Ioo (-u) u,\n      ofReal ((1 - (u - v) * (u + v))⁻¹)) +\n    2 * ∫⁻ u : ℝ in Ioo 2⁻¹ 1, ∫⁻ v : ℝ in Ioo (-(1 - u)) (1 - u),\n      ofReal ((1 - (u - v) * (u + v))⁻¹) = _ using 1\n  . simp [←lintegral_indicator, indicator]\n  -- Pull the ofReal upwards again to convert the inner integrals to ℝ.\n  convert_to\n    (2 * ∫⁻ u : ℝ in Ioo 0 2⁻¹,\n      ofReal (∫ v : ℝ in Ioo (-u) u, (1 - (u - v) * (u + v))⁻¹)) +\n    2 * ∫⁻ u : ℝ in Ioo 2⁻¹ 1,\n      ofReal (∫ v : ℝ in Ioo (-(1 - u)) (1 - u), (1 - (u - v) * (u + v))⁻¹) = _ using 1\n  . congr 2\n    all_goals\n    · refine setLIntegral_congr_fun measurableSet_Ioo fun u hu => ?_\n      simp at hu\n      rw [ofReal_integral_eq_lintegral_ofReal]\n      · apply IntegrableOn.integrable\n        rw [←integrableOn_Icc_iff_integrableOn_Ioo]\n        apply ContinuousOn.integrableOn_Icc\n        apply ContinuousOn.inv₀\n        · fun_prop\n        · intro v hv\n          simp at hv\n          nlinarith only [hu, hv]\n      · filter_upwards [ae_restrict_mem measurableSet_Ioo] with v hv\n        simp at hv ⊢\n        nlinarith only [hu, hv]\n  -- Now compute the inner integrals explicitly.\n  -- This will facilitate checking integrability conditions necessary to convert\n  -- the outer integrals back to ℝ.\n  convert_to\n    (2 * ∫⁻ u : ℝ in Ioo 0 2⁻¹, ofReal (∫ v : ℝ in (-u)..u, (1 - (u - v) * (u + v))⁻¹)) +\n    2 * ∫⁻ u : ℝ in Ioo 2⁻¹ 1, ofReal (∫ v : ℝ in -(1 - u)..(1 - u), (1 - (u - v) * (u + v))⁻¹) = _\n      using 3\n  · refine setLIntegral_congr_fun measurableSet_Ioo fun u hu => ?_\n    have hu2 : -u ≤ u := by linarith only [hu.1, hu.2]\n    simp only [intervalIntegral_eq_integral_uIoc, hu2, uIoc_of_le hu2, ←integral_Ioc_eq_integral_Ioo]\n    simp\n  · refine setLIntegral_congr_fun measurableSet_Ioo fun u hu => ?_\n    have hu2 : -(1 - u) ≤ 1 - u := by linarith only [hu.1, hu.2]\n    simp only [intervalIntegral_eq_integral_uIoc, hu2, uIoc_of_le hu2, ←integral_Ioc_eq_integral_Ioo]\n    simp\n  convert_to\n    (2 * ∫⁻ u : ℝ in Ioo 0 2⁻¹, ofReal (∫ v : ℝ in (-u)..u, (√(1 - u ^ 2) ^ 2 + v ^ 2)⁻¹)) +\n    2 * ∫⁻ u : ℝ in Ioo 2⁻¹ 1, ofReal (∫ v : ℝ in -(1 - u)..(1 - u), (√(1 - u ^ 2) ^ 2 + v ^ 2)⁻¹)\n      = _ using 1\n  . congr 2\n    all_goals\n    · refine setLIntegral_congr_fun measurableSet_Ioo fun u hu => ?_\n      congr 1\n      refine intervalIntegral.integral_congr fun v hv => ?_\n      congr 1\n      calc\n        1 - (u - v) * (u + v) = 1 - u ^ 2 + v ^ 2 := by ring_nf\n        _ = _ := by rw [sq_sqrt (by nlinarith only [hu.1, hu.2])]\n  convert_to\n    (2 * ∫⁻ u : ℝ in Ioo 0 2⁻¹, ofReal (2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2)))) +\n    2 * ∫⁻ u : ℝ in Ioo 2⁻¹ 1, ofReal (2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2)))\n      = _ using 1\n  . congr 2\n    all_goals\n    · refine setLIntegral_congr_fun measurableSet_Ioo fun u hu => ?_\n      congr 1\n      rw [integral_inv_sq_add_sq]\n      swap\n      · rw [sqrt_ne_zero] <;> nlinarith only [hu.1, hu.2]\n      simp_rw [sub_eq_add_neg, ←arctan_neg]\n      ring_nf\n  have integrable_deriv_g_explicit : IntervalIntegrable\n      (fun u ↦ 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2))) volume 0 2⁻¹ := by\n    rw [intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num)]\n    apply ContinuousOn.integrableOn_Icc\n    apply ContinuousOn.mul\n    apply ContinuousOn.mul\n    · exact continuousOn_const\n    · apply ContinuousOn.inv₀ (by fun_prop)\n      intro u hu; simp at hu; rw [sqrt_ne_zero] <;> nlinarith only [hu]\n    · apply continuous_arctan.comp_continuousOn'\n      apply ContinuousOn.div (by fun_prop) (by fun_prop)\n      intro u hu; simp at hu; rw [sqrt_ne_zero] <;> nlinarith only [hu]\n  have integrable_deriv_h_explicit : IntervalIntegrable\n      (fun u ↦ 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2))) volume 2⁻¹ 1 := by\n    have h1 : IntervalIntegrable (fun u ↦ π * (√(1 - u ^ 2))⁻¹) volume 2⁻¹ 1 :=\n      intervalIntegrable_inv_sqrt_one_sub_sq.const_mul π\n    apply IntervalIntegrable.mono_fun h1 (by clear * -; measurability)\n    filter_upwards [ae_restrict_mem measurableSet_uIoc] with u hu\n    replace hu : 2⁻¹ < u ∧ u ≤ 1 := by simp [uIoc] at hu; grind only\n    have h2 : 0 ≤ (√(1 - u ^ 2))⁻¹ := by simp\n    rw [norm_of_nonneg, norm_of_nonneg]\n    · calc\n        _ = 2 * arctan ((1 - u) / √(1 - u ^ 2)) * (√(1 - u ^ 2))⁻¹ := by ring\n        _ ≤ 2 * (π / 2) * (√(1 - u ^ 2))⁻¹ := by gcongr 2; apply le_of_lt (arctan_lt_pi_div_two _)\n        _ = _ := by field_simp\n    · nlinarith only [h2, pi_nonneg]\n    · have h3 : 0 ≤ arctan ((1 - u) / √(1 - u ^ 2)) := by\n        rw [arctan_nonneg]\n        apply div_nonneg (by linarith only [hu]) (by simp)\n      nlinarith only [h2, h3]\n  -- Convert the outer integrals back to ℝ.\n  convert_to\n    (2 * ofReal (∫ u : ℝ in Ioo 0 2⁻¹, 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2)))) +\n    2 * ofReal (∫ u : ℝ in Ioo 2⁻¹ 1, 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2))) = _\n      using 3\n  · rw [ofReal_integral_eq_lintegral_ofReal]\n    · apply IntegrableOn.integrable\n      rw [←intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n      exact integrable_deriv_g_explicit\n    · filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu\n      simp only [Pi.zero_apply]\n      have h1 : (√(1 - u ^ 2))⁻¹ ≥ 0 := inv_nonneg_of_nonneg (by simp)\n      have h2 : 0 ≤ arctan (u / √(1 - u ^ 2)) := by\n        simpa [arctan_nonneg] using div_nonneg (by linarith only [hu.1]) (by simp)\n      nlinarith only [h1, h2]\n  · rw [ofReal_integral_eq_lintegral_ofReal]\n    · apply IntegrableOn.integrable\n      rw [←intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n      exact integrable_deriv_h_explicit\n    · filter_upwards [ae_restrict_mem measurableSet_Ioo] with u hu\n      simp only [Pi.zero_apply]\n      have h1 : (√(1 - u ^ 2))⁻¹ ≥ 0 := inv_nonneg_of_nonneg (by simp)\n      have h3 : 0 ≤ arctan ((1 - u) / √(1 - u ^ 2)) := by\n        simpa [arctan_nonneg] using div_nonneg (by nlinarith only [hu.1, hu.2]) (by simp)\n      nlinarith only [h1, h3]\n  convert_to\n    ofReal (2 * (∫ u : ℝ in 0..2⁻¹, 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2)))) +\n    ofReal (2 * (∫ u : ℝ in 2⁻¹..1, 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2)))) = _\n      using 1\n  . congr 1\n    all_goals\n    · rw [←integral_Ioc_eq_integral_Ioo, intervalIntegral_eq_integral_uIoc]\n      norm_num\n  convert_to\n    ofReal (2 * (∫ u : ℝ in 0..2⁻¹, 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2))) +\n    2 * (∫ u : ℝ in 2⁻¹..1, 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2)))) = _ using 1\n  . rw [ofReal_add]\n    all_goals\n    · simp only [Nat.ofNat_pos, mul_nonneg_iff_of_pos_left]\n      apply intervalIntegral.integral_nonneg\n      · norm_num\n      · intro u hu\n        have h1 : (√(1 - u ^ 2))⁻¹ ≥ 0 := inv_nonneg_of_nonneg (by simp)\n        have h2 : 0 ≤ arctan (u / √(1 - u ^ 2)) := by\n          simpa [arctan_nonneg] using div_nonneg (by linarith only [hu.1]) (by simp)\n        have h3 : 0 ≤ arctan ((1 - u) / √(1 - u ^ 2)) := by\n          simpa [arctan_nonneg] using div_nonneg (by linarith only [hu.2]) (by simp)\n        nlinarith only [h1, h2, h3]\n  -- Now let's go back to ℝ, and compute the remaining integrals.\n  congr 1\n  -- Primitive appearing in the first integral after substitution\n  let g := fun u : ℝ => arctan (u / √(1 - u ^ 2)) ^ 2\n  have deriv_g_eq (u : ℝ) (hu : u ∈ Ioo 0 1) :\n      deriv g u = 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2)) := by\n    simp at hu\n    have h1 : √(1 - u ^ 2) ≠ 0 := by rw [sqrt_ne_zero] <;> nlinarith only [hu]\n    have h2 : 1 - u ^ 2 ≠ 0 := by nlinarith only [hu]\n    have h3 : DifferentiableAt ℝ (fun u ↦ √(1 - u ^ 2)) u := by\n      apply DifferentiableAt.comp\n      · exact (hasDerivAt_sqrt h2).differentiableAt\n      · simp\n    have h4 : DifferentiableAt ℝ (fun u ↦ u / √(1 - u ^ 2)) u := by\n      apply DifferentiableAt.div\n      · simp\n      · exact h3\n      · exact h1\n    show deriv (fun u => ((arctan ∘ fun u => _ / ((sqrt ∘ fun u => _) u)) ^ 2) u) u = _\n    erw [deriv_fun_pow, deriv_comp, deriv_fun_div, deriv_comp, deriv_fun_sub]\n    erw [deriv_sqrt, deriv_fun_pow]\n    erw [Real.deriv_arctan, deriv_id'', deriv_const]\n    · simp\n      field_simp\n    · simp\n    · simp\n    · exact h2\n    · simp\n    · simp\n    · exact (hasDerivAt_sqrt h2).differentiableAt\n    · simp\n    · simp\n    · exact h3\n    · exact h1\n    · apply differentiableAt_arctan\n    · exact h4\n    · apply (differentiableAt_arctan _).comp _ h4\n  have integrable_deriv_g : IntervalIntegrable (deriv g) volume 0 2⁻¹ := by\n    rw [intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n    convert_to IntegrableOn (fun u => 2 * (√(1 - u ^ 2))⁻¹ * arctan (u / √(1 - u ^ 2))) (Ioo 0 2⁻¹)\n      using 0\n    · refine integrableOn_congr_fun (fun u hu => ?_) measurableSet_Ioo\n      simp at hu\n      rw [deriv_g_eq u (by split_ands <;> linarith only [hu])]\n    rw [←intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n    exact integrable_deriv_g_explicit\n  have differentiableAt_uIoo_g u (hu : u ∈ uIoo 0 2⁻¹) : DifferentiableAt ℝ g u := by\n    simp at hu\n    refine differentiableAt_of_deriv_ne_zero ?_\n    rw [deriv_g_eq u (by split_ands <;> linarith only [hu])]\n    have h1 : √(1 - u ^ 2) ≠ 0 := by rw [sqrt_ne_zero] <;> nlinarith only [hu]\n    have h2 : arctan (u / √(1 - u ^ 2)) ≠ 0 := by simp [h1]; linarith\n    grind only\n  have continuousOn_uIcc_g : ContinuousOn g (uIcc 0 2⁻¹) := by\n    convert_to ContinuousOn g (Icc 0 2⁻¹) using 1\n    · grind [uIcc]\n    apply ContinuousOn.pow\n    apply continuous_arctan.comp_continuousOn'\n    apply ContinuousOn.div (by fun_prop) (by fun_prop)\n    intro u hu\n    simp at hu\n    rw [sqrt_ne_zero] <;> nlinarith only [hu]\n  -- Primitive appearing in the second integral after substitution\n  let h := fun u : ℝ => -2 * arctan ((1 - u) / √(1 - u ^ 2)) ^ 2\n  have deriv_h_eq (u : ℝ) (hu : u ∈ Ioo 0 1) :\n      deriv h u = 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2)) := by\n    simp at hu\n    have h1 : √(1 - u ^ 2) ≠ 0 := by rw [sqrt_ne_zero] <;> nlinarith only [hu]\n    have h2 : 1 - u ^ 2 ≠ 0 := by nlinarith only [hu]\n    have h3 : DifferentiableAt ℝ (fun u ↦ √(1 - u ^ 2)) u := by\n      apply DifferentiableAt.comp\n      · exact (hasDerivAt_sqrt h2).differentiableAt\n      · simp\n    have h5 : DifferentiableAt ℝ (HSub.hSub 1) u := DifferentiableAt.sub (by simp) (by simp)\n    have h4 : DifferentiableAt ℝ (fun u ↦ (1 - u) / √(1 - u ^ 2)) u := by\n      apply DifferentiableAt.div\n      · exact h5\n      · exact h3\n      · exact h1\n    show deriv (fun u => -2 * ((arctan ∘ fun u => _ / ((sqrt ∘ fun u => _) u)) ^ 2) u) u = _\n    erw [deriv_const_mul]\n    erw [deriv_fun_pow, deriv_comp, deriv_fun_div, deriv_comp, deriv_fun_sub]\n    erw [deriv_sqrt, deriv_sub, deriv_fun_pow]\n    erw [Real.deriv_arctan, deriv_id'', deriv_const]\n    · simp\n      field_simp\n      rw [sq_sqrt (by nlinarith only [hu])]\n      ring\n    · simp\n    · simp\n    · simp\n    · simp\n    · exact h2\n    · simp\n    · simp\n    · exact h5\n    · simp\n    · exact h5\n    · exact h3\n    · exact h1\n    · apply differentiableAt_arctan\n    · exact h4\n    · apply (differentiableAt_arctan _).comp _ h4\n    · apply (differentiableAt_pow _).comp _ ((differentiableAt_arctan _).comp _ h4)\n  have integrable_deriv_h : IntervalIntegrable (deriv h) volume 2⁻¹ 1 := by\n    rw [intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n    convert_to IntegrableOn (fun u => 2 * (√(1 - u ^ 2))⁻¹ * arctan ((1 - u) / √(1 - u ^ 2)))\n      (Ioo 2⁻¹ 1) using 0\n    · refine integrableOn_congr_fun (fun u ⟨hu1, hu2⟩ => ?_) measurableSet_Ioo\n      exact deriv_h_eq u (by split_ands <;> linarith only [hu1, hu2])\n    rw [←intervalIntegrable_iff_integrableOn_Ioo_of_le (by norm_num)]\n    exact integrable_deriv_h_explicit\n  have continuousOn_uIcc_h : ContinuousOn h (uIcc 2⁻¹ 1) := by\n    convert_to ContinuousOn h (Icc 2⁻¹ 1) using 1\n    · grind [uIcc]\n    apply ContinuousOn.mul\n    · exact continuousOn_const\n    apply ContinuousOn.pow\n    apply continuous_arctan.comp_continuousOn'\n    set f1 := fun u => (1 - u) / √(1 - u ^ 2)\n    set f2 := fun u => √((1 - u) / (1 + u))\n    convert_to ContinuousOn f2 (Icc 2⁻¹ 1) using 0\n    · refine continuousOn_congr fun u hu => ?_\n      unfold f1 f2\n      simp at hu\n      calc\n        _ = √((1 - u) ^ 2) / √(1 - u ^ 2) := by rw [sqrt_sq]; linarith only [hu]\n        _ = √(((1 - u) * (1 - u)) / ((1 - u) * (1 + u))) := by\n          rw [←sqrt_div]\n          · ring_nf\n          positivity\n        _ = _ := by field_simp\n    apply ContinuousOn.sqrt\n    apply ContinuousOn.div (by fun_prop) (by fun_prop)\n    intro u hu\n    simp at hu\n    linarith only [hu]\n  have differentiableAt_uIoo_h u (hu : u ∈ uIoo 2⁻¹ 1) : DifferentiableAt ℝ h u := by\n    replace hu : (2⁻¹ : ℝ) < u ∧ u < 1 := by grind [uIoo]\n    refine differentiableAt_of_deriv_ne_zero ?_\n    rw [deriv_h_eq u (by split_ands <;> linarith only [hu])]\n    have h1 : √(1 - u ^ 2) ≠ 0 := by rw [sqrt_ne_zero] <;> nlinarith only [hu]\n    have h2 : arctan ((1 - u) / √(1 - u ^ 2)) ≠ 0 := by simp [h1]; linarith\n    grind only\n  convert_to (2 * ∫ u : ℝ in 0..2⁻¹, deriv g u) + 2 * (∫ u : ℝ in 2⁻¹..1, deriv h u) = _\n    using 3\n  · iterate 2 rw [intervalIntegral_eq_integral_uIoc]\n    norm_num\n    iterate 2 rw [integral_Ioc_eq_integral_Ioo]\n    refine setIntegral_congr_fun measurableSet_Ioo fun u ⟨hu1, hu2⟩ => ?_\n    rw [deriv_g_eq u]\n    split_ands <;> nlinarith only [hu1, hu2]\n  · iterate 2 rw [intervalIntegral_eq_integral_uIoc]\n    norm_num\n    iterate 2 rw [integral_Ioc_eq_integral_Ioo]\n    refine setIntegral_congr_fun measurableSet_Ioo fun u ⟨hu1, hu2⟩ => ?_\n    rw [deriv_h_eq u]\n    split_ands <;> nlinarith only [hu1, hu2]\n  convert_to (2 * (g 2⁻¹ - g 0)) + (2 * (h 1 - h 2⁻¹)) = _ using 3\n  · rw [integral_deriv_eq_sub_uIoo]\n    · exact continuousOn_uIcc_g\n    · exact differentiableAt_uIoo_g\n    · exact integrable_deriv_g\n  · rw [integral_deriv_eq_sub_uIoo]\n    · exact continuousOn_uIcc_h\n    · exact differentiableAt_uIoo_h\n    · exact integrable_deriv_h\n  unfold g h\n  norm_num\n  cancel_denoms\n  rw [arctan_inv_of_pos (by norm_num), arctan_sqrt_three]\n  field_simp\n  norm_num","line":38,"end_line":610,"proof_status":"statement_formalized"}
{"id":"double_counting","label":"double_counting","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose that we are given two finite sets \\(R\\) and \\(C\\) and a subset \\(S \\subseteq R \\times C\\). Whenever \\((p, q) \\in S\\), then we say \\(p\\) and \\(q\\) are incident. If \\(r_p\\) denotes the number of elements that are incident to \\(p \\in R\\), and \\(c_q\\) denotes the number of elements that are incident to \\(q \\in C\\), then \\[ \\sum _{p \\in R} r_p = |S| = \\sum _{q \\in C} c_q. \\tag {3} \\]","doc_url":null,"latex_link":"chapter28.html#double_counting","lean_link":null,"lean_decl_name":"double_counting","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cayley_formala_proof4","label":"cayley_formala_proof4","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["double_counting"],"nlp_text":"There are \\(n^{n - 2}\\) different labeled trees on \\(n\\) nodes.","doc_url":null,"latex_link":"sect0030.html#cayley_formala_proof4","lean_link":null,"lean_decl_name":"cayley_formala_proof4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch26lemma_d","label":"ch26lemma_d","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The two functions \\(f\\) and \\(g\\) satisfy the same functional equation: \\(f(\\frac{x}{2}) + f(\\frac{x + 1}{2}) = 2f(x)\\) and \\(g(\\frac{x}{2}) + g(\\frac{x + 1}{2}) = gf(x)\\).","doc_url":null,"latex_link":"sect0025.html#ch26lemma_d","lean_link":null,"lean_decl_name":"ch26lemma_d","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"pigeon_hole_principle","label":"pigeon_hole_principle","background_color":"#1CAC78","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(n\\) objects are placed in \\(r\\) boxes, where \\(r {\\lt} n\\), then at least one of the boxes contains more than one object.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter28.pigeon_hole_principle","latex_link":"chapter28.html#pigeon_hole_principle","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter28.pigeon_hole_principle","lean_decl_name":"chapter28.pigeon_hole_principle","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_28.lean","lean4_code":"theorem pigeon_hole_principle (n r : ℕ) (h : r < n) (object_to_boxes : Fin n → Fin r) :\n  ∃ box : Fin r, ∃ object₁ object₂ : Fin n,\n  object₁ ≠ object₂ ∧\n  object_to_boxes object₁ = box ∧\n  object_to_boxes object₂ = box := by\n  have ⟨object₁, object₂, h_object⟩ :=\n      Fintype.exists_ne_map_eq_of_card_lt object_to_boxes (by convert h <;> simp)\n  use object_to_boxes object₁\n  use object₁\n  use object₂\n  tauto","line":36,"end_line":46,"proof_status":"formalized"}
{"id":"cayley_formala_proof3","label":"cayley_formala_proof3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There are \\(n^{n - 2}\\) different labeled trees on \\(n\\) nodes.","doc_url":null,"latex_link":"sect0030.html#cayley_formala_proof3","lean_link":null,"lean_decl_name":"cayley_formala_proof3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch37fact_b","label":"ch37fact_b","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ H(X, Y) = H(X) + H(Y|X). \\]","doc_url":null,"latex_link":"sect0034.html#ch37fact_b","lean_link":null,"lean_decl_name":"ch37fact_b","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch31lemma","label":"ch31lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\mathbb {Q} : \\mathfrak {S}_n \\longrightarrow \\mathbb {R}\\) be any probability distribution that defines a shuffling process \\(\\mathbb {Q}^*k\\) with a strong uniform stopping rule whose stopping time is \\(T\\). Then for all \\(k \\geq 0\\), \\[ ||\\mathbb {Q}^*k - \\mathbb {U}|| \\leq \\text{Prob}[T {\\gt} k]. \\]","doc_url":null,"latex_link":"sect0028.html#ch31lemma","lean_link":null,"lean_decl_name":"ch31lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch31theorem1","label":"ch31theorem1","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch31lemma"],"nlp_text":"Let \\(c \\geq 0\\) and \\(k := \\lceil n \\log n + cn \\rceil \\). Then after performing \\(k\\) top-in-at-random shuffles on a deck of \\(n\\) cards, the variation distance from the uniform distribution satisfies \\[ d(k) := ||\\text{Top}^*k - \\mathbb {U}|| \\leq e^{-c}. \\]","doc_url":null,"latex_link":"sect0028.html#ch31theorem1","lean_link":null,"lean_decl_name":"ch31theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch20theorem1","label":"ch20theorem1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose all roots fo the polynomial \\(x^n + a_{n - 1}x^{n - 1} + \\dots +a_0\\) are real. Then the roots of are contained in the interval with the endpoints \\[ -\\frac{n_{n-1}}{n}\\pm \\frac{n - 1}{n}\\sqrt{a^n_{n - 1} - \\frac{2n}{n - 1}a_{n - 2}}. \\]","doc_url":null,"latex_link":"chapter20.html#ch20theorem1","lean_link":null,"lean_decl_name":"ch20theorem1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch28claim2","label":"ch28claim2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose again \\(A\\subset \\{ 1, 2, \\dots , 2n\\} \\) with \\(|A| = n + 1\\). Then there are always two numbers in \\(A\\) such that one divides the other.","doc_url":null,"latex_link":"chapter28.html#ch28claim2","lean_link":null,"lean_decl_name":"ch28claim2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch45theorem3","label":"ch45theorem3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every \\(k \\geq 2\\), there exists a graph \\(G\\) with chromatic number \\(\\chi (G) {\\gt} k\\) and girth \\(\\gamma (G) {\\gt} k\\).","doc_url":null,"latex_link":"sect0042.html#ch45theorem3","lean_link":null,"lean_decl_name":"ch45theorem3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch04.lemma2","label":"ch04.lemma2","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"No number \\(n = 4m + 3\\) is a sum of two squares.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.lemma₂","latex_link":"sect0004.html#ch04.lemma2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.lemma₂","lean_decl_name":"ch04.lemma₂","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_04.lean","lean4_code":"lemma lemma₂ (n m : ℕ) (hn : n = 4 * m + 3) :\n  ¬ ∃ a b, n = a ^2 + b ^2 := by\n  push_neg\n  intro a b\n  by_contra h\n  have : (n : ZMod 4) =  a ^ 2 + b ^ 2 := by\n    rw [h]\n    simp only [Nat.cast_add, Nat.cast_pow]\n  rw [hn] at this\n  simp only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] at this\n  rw [mul_eq_zero_of_left (by rfl) (m : ZMod 4), zero_add] at this\n  have hx : ∀ (x : ZMod 4),  x ^2 ∈ ({0, 1} : Finset (ZMod 4)) := by\n    intro x\n    fin_cases x <;> simp\n    · exact Or.inl rfl\n    · exact Or.inr rfl\n  have ha := hx <| a\n  have hb := hx  <| b\n  generalize hA: (a : ZMod 4) ^ 2 = A\n  generalize hB: (b : ZMod 4) ^ 2 = B\n  rw [hA, hB] at this\n  rw [hA] at ha\n  rw [hB] at hb\n  fin_cases A <;> fin_cases B <;> norm_num at this <;> tauto","line":56,"end_line":79,"proof_status":"formalized"}
{"id":"ch19theorem2","label":"ch19theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The set \\(\\mathbb {R}\\) of real numbers is not countable","doc_url":null,"latex_link":"sect0019.html#ch19theorem2","lean_link":null,"lean_decl_name":"ch19theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch20theorem3proof1","label":"ch20theorem3proof1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose \\(G\\) is a graph on \\(n\\) vertices without triangles. Then \\(G\\) has at most \\(\\frac{n^2}{4}\\) edges, and equality holds only when \\(n\\) is even and \\(G\\) is the complete bipartite graph \\(K_{n/2, n/2}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/mantel","latex_link":"chapter20.html#ch20theorem3proof1","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/mantel","lean_decl_name":"mantel","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_20.lean","lean4_code":"theorem mantel (h: G.CliqueFree 3) : #E ≤ (n^2 / 4) := by\n\n  -- The degrees of two adjacent vertices cannot sum to more than n\n  have adj_degree_bnd (i j : α) (hij: G.Adj i j) : d(i) + d(j) ≤ n := by\n    -- Assume the contrary ...\n    by_contra hc; simp at hc\n\n    -- ... then by pigeonhole there would exist a vertex k adjacent to both i and j ...\n    obtain ⟨k, h⟩ := Finset.inter_nonempty_of_card_lt_card_add_card (by simp) (by simp) hc\n    simp at h\n    obtain ⟨hik, hjk⟩ := h\n\n    -- ... but then i, j, k would form a triangle, contradicting that G is triangle-free\n    exact h {k, j, i} ⟨by aesop (add safe G.adj_symm), by simp [hij.ne', hik.ne', hjk.ne']⟩\n\n  -- We need to define the sum of the degrees of the vertices of an edge ...\n  let sum_deg (e : Sym2 α) : ℕ := Sym2.lift ⟨λ x y ↦ d(x) + d(y), by simp [Nat.add_comm]⟩ e\n\n  -- ... and establish a variant of adj_degree_bnd ...\n  have adj_degree_bnd' (e : Sym2 α) (he: e ∈ E) : sum_deg e ≤ n := by\n    induction e with | _ v w => simp at he; exact adj_degree_bnd v w (by simp [he])\n\n  -- ... and the identity for the sum of the squares of the degrees ...\n  have sum_sum_deg_eq_sum_deg_sq : ∑ e ∈ E, sum_deg e = ∑ v ∈ V, d(v)^2 := by\n    calc  ∑ e ∈ E, sum_deg e\n      _ = ∑ e ∈ E, ∑ v ∈ e, d(v)                  := Finset.sum_congr rfl (λ e he ↦ by induction e with | _ v w => simp at he; simp [sum_deg, he.ne])\n      _ = ∑ e ∈ E, ∑ v ∈ {v' ∈ V | v' ∈ e}, d(v)  := Finset.sum_congr rfl (by intro e _; exact congrFun (congrArg Finset.sum (by ext; simp)) _)\n      _ = ∑ v ∈ V, ∑ _ ∈ {e ∈ E | v ∈ e}, d(v)    := Finset.sum_sum_bipartiteAbove_eq_sum_sum_bipartiteBelow _ _\n      _ = ∑ v ∈ V, ∑ _ ∈ I(v), d(v)               := Finset.sum_congr rfl (λ v ↦ by simp [G.incidenceFinset_eq_filter v])\n      _ = ∑ v ∈ V, d(v)^2                         := by simp [Nat.pow_two]\n\n  -- We now slightly modify the main argument to avoid division by a potentially zero n ...\n  have := calc #E * n^2\n    _ = (n * (∑ e ∈ E, 1)) * n               := by simp [Nat.pow_two, Nat.mul_assoc, Nat.mul_comm]\n    _ = (∑ _ ∈ E, n) * n                     := by rw [Finset.mul_sum]; simp\n    _ ≥ (∑ e ∈ E, sum_deg e) * n             := Nat.mul_le_mul_right n (Finset.sum_le_sum adj_degree_bnd')\n    _ = (∑ v ∈ V, d(v)^2) * (∑ v ∈ V, 1^2)   := by simp [sum_sum_deg_eq_sum_deg_sq]\n    _ ≥ (∑ v ∈ V, d(v) * 1)^2                := (Finset.sum_mul_sq_le_sq_mul_sq V (λ v ↦ d(v)) 1)\n    _ = (2 * #E)^2                           := by simp [G.sum_degrees_eq_twice_card_edges]\n    _ = 4 * #E^2                             := by ring\n\n  -- .. and clean up the inequality.\n  rw [Nat.pow_two (#E)] at this\n  rw [(Nat.mul_assoc 4 (#E) (#E)).symm] at this\n  rw [Nat.mul_comm (4 * #E) (#E)] at this\n\n  -- Now we can show #E ≤ n^2 / 4 by \"simply\" dividing by 4 * #E\n  by_cases hE : #E = 0\n  · simp [hE]\n  · apply Nat.zero_lt_of_ne_zero at hE\n    apply Nat.le_of_mul_le_mul_left this at hE\n    rw [Nat.mul_comm] at hE\n    exact (Nat.le_div_iff_mul_le (Nat.zero_lt_succ 3)).mpr hE","line":137,"end_line":189,"proof_status":"statement_formalized"}
{"id":"ch20theorem3proof2","label":"ch20theorem3proof2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose \\(G\\) is a graph on \\(n\\) vertices without triangles. Then \\(G\\) has at most \\(\\frac{n^2}{4}\\) edges, and equality holds only when \\(n\\) is even and \\(G\\) is the complete bipartite graph \\(K_{n/2, n/2}\\).","doc_url":null,"latex_link":"chapter20.html#ch20theorem3proof2","lean_link":null,"lean_decl_name":"ch20theorem3proof2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch41proof3","label":"ch41proof3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If a graph \\(G = (V, E)\\) on \\(n\\) vertices has no \\(p\\)-clique, \\(p \\geq 2\\), then \\[ |E| \\leq \\left(1 - \\frac{1}{p - 1}\\right) \\frac{n^2}{2}. \\tag {1} \\]","doc_url":null,"latex_link":"sect0038.html#ch41proof3","lean_link":null,"lean_decl_name":"ch41proof3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch22lemma2","label":"ch22lemma2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every dissection of the unit square \\(S = [0, 1]^2\\) into finitely many triangles contains an odd number of rainbow triangles, and thus at least one.","doc_url":null,"latex_link":"sect0021.html#ch22lemma2","lean_link":null,"lean_decl_name":"ch22lemma2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"valuation_lemma","label":"valuation_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A proper subring \\(R\\subset K\\) is a valuation ring with respect to some valuation \\(v\\) into some ordered group \\(G\\) if and only if \\(K = R \\cup R^{-1}\\).","doc_url":null,"latex_link":"sect0021.html#valuation_lemma","lean_link":null,"lean_decl_name":"valuation_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"valuation_on_reals","label":"valuation_on_reals","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["valuation_lemma"],"nlp_text":"The field of real numbers \\(\\mathbb {R}\\) has a non-Archimedean valuation to an ordered abelian group \\[ v: \\mathbb {R} \\to \\{ 0\\} \\cup G \\]","doc_url":null,"latex_link":"sect0021.html#valuation_on_reals","lean_link":null,"lean_decl_name":"valuation_on_reals","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch19proposition4","label":"ch19proposition4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every cardinal number \\(\\mathfrak {m}\\), there is a definite next larger cardinal number.","doc_url":null,"latex_link":"sect0019.html#ch19proposition4","lean_link":null,"lean_decl_name":"ch19proposition4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:bertrands_postulate","label":"bertrands_postulate","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any positive natural number, there is a prime which is greater than it, but no more than twice as large.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.exists_prime_lt_and_le_two_mul","latex_link":"sect0002.html#thm:bertrands_postulate","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.exists_prime_lt_and_le_two_mul","lean_decl_name":"chapter2.exists_prime_lt_and_le_two_mul","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/Bertrand.lean","lean4_code":"theorem exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) :\n    ∃ p, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n := by\n  -- Split into cases whether `n` is large or small\n  rcases lt_or_ge 511 n with h | h\n  -- If `n` is large, apply the lemma derived from the inequalities on the central binomial\n  -- coefficient.\n  · exact exists_prime_lt_and_le_two_mul_eventually n h\n  replace h : n < 521 := h.trans_lt (by norm_num1)\n  revert h\n  -- For small `n`, supply a list of primes to cover the initial cases.\n  open Lean Elab Tactic in\n  run_tac do\n    for i in [317, 163, 83, 43, 23, 13, 7, 5, 3, 2] do\n      let i : Term := quote i\n      evalTactic <| ←\n        `(tactic| refine exists_prime_lt_and_le_two_mul_succ $i (by norm_num1) (by norm_num1) ?_)\n  exact fun h2 => ⟨2, prime_two, h2, Nat.mul_le_mul_left 2 (Nat.pos_of_ne_zero hn0)⟩\n\nalias bertrand := Nat.exists_prime_lt_and_le_two_mul\n\nend Nat\n\n","line":211,"end_line":232,"proof_status":"formalized"}
{"id":"ch20theoremIIproof1","label":"ch20theoremIIproof1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(a_1, \\dots a_n\\) be positive real numbers, then \\[ \\frac{n}{\\frac{1}{a_1}+\\dots +\\frac{1}{n_n}} \\le \\sqrt[n]{a_1a_2\\dots a_n} \\le \\frac{a_1\\dots a_n}{n} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₁","latex_link":"chapter20.html#ch20theoremIIproof1","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₁","lean_decl_name":"harmonic_geometric_arithmetic₁","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_20.lean","lean4_code":"theorem harmonic_geometric_arithmetic₁ (n : ℕ) (hn : 1 ≤ n)\n  (a : Finset.Icc 1 n → ℝ) (hpos : ∀ i, 0 < a i) :\n  let harmonic := n / (∑ i : Finset.Icc 1 n, 1 / (a i))\n  let geometric := (∏ i : Finset.Icc 1 n, a i) ^ ((1 : ℝ) / n)\n  let arithmetic := (∑ i : Finset.Icc 1 n, a i) / n\n  let all_equal := ∀ i : Finset.Icc 1 n, a i = a ⟨1, Finset.mem_Icc.mpr  ⟨NeZero.one_le, hn⟩⟩\n  harmonic ≤ geometric ∧ geometric ≤ arithmetic ∧\n  ((harmonic = geometric) ↔ all_equal) ∧\n  ((geometric = arithmetic) ↔ all_equal) := by sorry","line":87,"end_line":95,"proof_status":"statement_formalized"}
{"id":"ch11_theorem4","label":"ch11_theorem4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(K_n\\) is decomposed into complete bipartite subgraphs \\(H_1, \\dots , H_m\\), then \\(m \\ge n - 1\\).","doc_url":null,"latex_link":"sect0011.html#ch11_theorem4","lean_link":null,"lean_decl_name":"ch11_theorem4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cayley_formala_proof2","label":"cayley_formala_proof2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There are \\(n^{n - 2}\\) different labeled trees on \\(n\\) nodes.","doc_url":null,"latex_link":"sect0030.html#cayley_formala_proof2","lean_link":null,"lean_decl_name":"cayley_formala_proof2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch34theorem","label":"ch34theorem","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\prod _{k\\ge 1}(1 - x^k) = 1 + \\sum _{j\\ge 1}(-1)^j(x^{\\frac{3j^2 - j}{2}} + x^{3j^2 + j}{2}). \\]","doc_url":null,"latex_link":"sect0031.html#ch34theorem","lean_link":null,"lean_decl_name":"ch34theorem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem_aux_iii","label":"lem_aux_iii","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The derivatives \\(f^{(k)}(0)\\) and \\(f^{(k)}(1)\\) are integers for all \\(k \\ge 0\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_iii","latex_link":"sect0008.html#lem_aux_iii","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/book.irrational.lem_aux_iii","lean_decl_name":"book.irrational.lem_aux_iii","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_08.lean","lean4_code":"lemma lem_aux_iii (n : ℕ) (k : ℕ): iteratedDeriv k (f_aux n) 0 ∈  Set.range (fun (q : ℚ) ↦ (q : ℝ)) ∧\n  iteratedDeriv k (f_aux n) 1 ∈ Set.range (fun (q : ℚ) ↦ (q : ℝ))  := by\n  sorry","line":108,"end_line":110,"proof_status":"statement_formalized"}
{"id":"diagonalize_real_symmetric","label":"diagonalize_real_symmetric","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":["ch7.lemma"],"nlp_text":"For every real symmetric matrix \\(A\\) there is a real orthogonal matrix \\(Q\\) such that \\(Q^{T}AQ\\) is diagonal.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₁","latex_link":"sect0007.html#diagonalize_real_symmetric","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₁","lean_decl_name":"chapter7.Theorem₁","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_07.lean","lean4_code":"theorem Theorem₁ (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (h : IsHermitian A) :\n    ∃ Q : Matrix (Fin n) (Fin n) ℝ,\n    Q ∈ Matrix.orthogonalGroup (Fin n) ℝ ∧\n    ∃ (d : (Fin n) → ℝ), diagonal d = (Q.conjTranspose * A * Q) := by\n  sorry","line":28,"end_line":32,"proof_status":"formalized"}
{"id":"example3","label":"example3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0010.html#example3","lean_link":null,"lean_decl_name":"example3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:sixth_proof","label":"sixth_proof","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The series \\(\\sum _{p\\in \\mathbb {P}}\\frac1p\\) diverges.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₆","latex_link":"sect0001.html#thm:sixth_proof","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/infinity_of_primes₆","lean_decl_name":"infinity_of_primes₆","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_01.lean","lean4_code":"theorem infinity_of_primes₆ :\n  Tendsto (fun n ↦ ∑ p ∈ Finset.filter (fun p ↦ Nat.Prime p) (range n), 1 / (p : ℝ))\n      atTop atTop := by\n  sorry","line":342,"end_line":345,"proof_status":"statement_formalized"}
{"id":"thm:estimate_binomial_coefficient","label":"estimate_binomial_coefficient","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\binom {n}{k} \\le \\frac{n^k}{k!} \\le \\frac{n^k}{2^{k - 1}} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.bound_binomial_coeff","latex_link":"sect0002.html#thm:estimate_binomial_coefficient","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter2.bound_binomial_coeff","lean_decl_name":"chapter2.bound_binomial_coeff","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_02.lean","lean4_code":"theorem bound_binomial_coeff {k n : ℕ} :\n  Nat.choose n k ≤ n ^ k / (k.factorial : ℚ)∧\n  n ^ k / (k.factorial : ℚ) ≤ n ^ k / 2 ^ (k - 1) := by sorry","line":223,"end_line":225,"proof_status":"statement_formalized"}
{"id":"ch19theorem5","label":"ch19theorem5","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(c {\\gt} \\aleph _1\\), then every family \\(\\{ f_\\alpha \\} \\) satisfying \\((P_0)\\) is countable. If, on the other hand, \\(c = \\aleph _1\\), then there exists some family \\(\\{ f_\\alpha \\} \\) with property \\(P_0\\) which has size \\(c\\).","doc_url":null,"latex_link":"sect0019.html#ch19theorem5","lean_link":null,"lean_decl_name":"ch19theorem5","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch37theorem2","label":"ch37theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The number \\(L(n)\\) of Latin squares of order \\(n\\) is bounded by \\[ \\frac{n!^{2n}}{n^{n^2}} \\le L(n) \\le \\prod _{k=1}^{n}k!^{n/k} \\]","doc_url":null,"latex_link":"sect0034.html#ch37theorem2","lean_link":null,"lean_decl_name":"ch37theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"wedderburn","label":"wedderburn","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every finite division ring is commutative.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/wedderburn","latex_link":"sect0006.html#wedderburn","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/wedderburn","lean_decl_name":"wedderburn","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_06.lean","lean4_code":"theorem wedderburn (h: Fintype R): IsField R := by\n  -- Z is a finite field ...\n  let Z := center R\n\n  -- .. and we can view R as a vector space of dimension n over R\n  obtain ⟨n, h_card⟩ := VectorSpace.card_fintype Z R\n\n  have H : (∃ x y : R, x ≠ y) := by exact exists_pair_ne R\n\n  have h_n : n ≠ 0 := by\n    by_contra h\n    subst h\n    simp only [_root_.pow_zero] at h_card\n    rcases H with ⟨x, y, H⟩\n    absurd H\n    exact Fintype.card_le_one_iff.mp (Nat.le_of_eq h_card) x y\n\n  set q := Fintype.card Z\n\n  -- conjugacy classes with more than one element\n  -- indexed from 1 to t in the book, here we use \"S\".\n  have finclassa: ∀ (A : ConjClasses Rˣ), Fintype ↑(ConjClasses.carrier A) :=\n    fun _ ↦ ConjClasses.instFintypeElemCarrier\n\n  have : ∀ (A :  ConjClasses Rˣ), Fintype ↑(Set.centralizer {Quotient.out A}) :=\n    fun _ ↦ setFintype (Set.centralizer {Quotient.out _})\n\n  letI fintypea : ∀ (A :  ConjClasses Rˣ), Fintype ↑{A |\n      have := finclassa A; Fintype.card ↑(ConjClasses.carrier A) > 1} :=\n        fun A ↦\n          setFintype {A | let_fun this := finclassa A; Fintype.card ↑(ConjClasses.carrier A) > 1}\n\n  have : Fintype ↑{A |\n      have := finclassa A;  Fintype.card ↑(ConjClasses.carrier A) > 1} :=\n    setFintype {A |\n                  let_fun this := finclassa A;\n                  Fintype.card ↑(ConjClasses.carrier A) > 1}\n\n  let S' := ConjClasses.noncenter Rˣ\n  haveI : Fintype S' := Fintype.ofFinite ↑S'\n  let S := S'.toFinset\n  --This was wrong: n_k should be the dimension of the centralizer( in `R`), not the cardinality\n  let n_k : S' → ℕ := sorry -- fun A => Fintype.card\n    --(Set.centralizer ({(Quotient.out' (A : ConjClasses Rˣ))} : Set Rˣ))\n\n  have h_R: Fintype.card Rˣ = q ^ n - 1 := by\n    have : Fintype.card Rˣ + 1 = Fintype.card R := (Fintype.card_eq_card_units_add_one R).symm\n    rw [← h_card, ← this]\n    simp only [ge_iff_le, add_le_iff_nonpos_left, nonpos_iff_eq_zero, Fintype.card_ne_zero,\n    add_tsub_cancel_right]\n\n  have h_Z : Fintype.card Zˣ = q - 1 := by\n    have h : Fintype.card Zˣ + 1 = Fintype.card Z := (Fintype.card_eq_card_units_add_one _).symm\n    have : Fintype.card Z = q := rfl\n    rw [← this, ← h]\n    simp only [center_toSubsemiring, Subsemiring.center_toSubmonoid, ge_iff_le,\n      add_le_iff_nonpos_left, nonpos_iff_eq_zero, Fintype.card_ne_zero, add_tsub_cancel_right]\n\n  --class  formula (1)\n\n  have H1:= (Group.card_center_add_sum_card_noncenter_eq_card Rˣ).symm\n  let e := Subgroup.centerUnitsEquivUnitsCenter R\n  --Note: This is fishy, it seems like it's just tautological\n  let f : { x // x ∈ Submonoid.center R }ˣ ≃ { x // x ∈ Z }ˣ :=\n      Equiv.inv { x // x ∈ Submonoid.center R }ˣ\n  rw [h_R, Fintype.card_congr (e.toEquiv.trans f), h_Z] at H1\n\n  -- Orbit stabilizer formula for non-singleton conjugacy classes\n  have : ∀ A : S', (Fintype.card <| ConjClasses.carrier (A : ConjClasses Rˣ)) * (q ^ (n_k A) - 1)\n      = q ^ n - 1 := by\n    sorry\n\n  have h1 : (q ^ n - 1) = q - 1  + ∑ A : S', (q ^ n - 1) / (q ^ (n_k A) - 1) := by\n    convert H1\n    sorry\n  have hZ : Nonempty <| @Subtype R fun x => x ∈ Z := Zero.instNonempty\n  have hq_pow_pos : ∀ m,  1 ≤ q ^ m := fun m ↦ one_le_pow m q Fintype.card_pos\n\n  have h_n_k_A_dvd: ∀ A : S', (n_k A ∣ n) := by sorry\n  --rest of proof\n  have h_phi_dvd_q_sub_one : (phi n).eval (q : ℤ) ∣ (((q - (1 : ℕ)) : ℕ ) : ℤ) := by\n    have hq : q = (Fintype.card { x // x ∈ center R }) := by rfl\n    have h₁_dvd : (phi n).eval (q : ℤ) ∣ ((X : ℤ[X])  ^ n - 1).eval (q : ℤ)  := by\n      exact eval_dvd <| phi_dvd n\n    have h₂_dvd :\n        (phi n).eval (q : ℤ) ∣ ∑ A : S', (((q ^ n - 1) : ℕ):ℤ) / ((q ^ (n_k A) - 1) : ℕ):= by\n      refine Finset.dvd_sum fun A hs ↦ (Int.dvd_div_of_mul_dvd ?_)\n      have h_one_neq: 1 ≠ n_k A := by sorry\n      have h_k_n_lt_n: n_k A < n := by sorry\n      have h_noneval := phi_div_2 n (n_k A) (h_n_k_A_dvd A) h_k_n_lt_n\n      have := @eval_dvd ℤ _ _ _ q h_noneval\n      simp only [eval_mul, eval_sub, eval_pow, eval_X, eval_one, IsUnit.mul_iff] at this\n      rw [← hq] at *\n      convert this\n      · simp [hq_pow_pos <| n_k A]\n      · simp [hq_pow_pos n]\n    simp only [eval_sub, eval_pow, eval_X, eval_one] at h₁_dvd\n    have h1' :  (((q:ℤ) ^ n - (1 : ℕ)) : ℤ) =\n        ((q - (1 :ℕ) : ℕ):ℤ) + ∑ A : S', (q ^ n - 1) / (q ^ (n_k A) - 1) := by\n      have : ((q ^ n - 1 : ℕ) : ℤ ) = (q - 1 + ∑ A : S', (q ^ n - 1) / (q ^ n_k A - 1) : ℕ) :=\n        congrArg Nat.cast h1\n      rw [cast_add] at this\n      rw [← this]\n      simp [hq_pow_pos n]\n    --rw [hq] at h1'\n    norm_num at h1'\n    simp [h1'] at h₁_dvd\n    refine (Int.dvd_add_left h₂_dvd).mp ?_\n    convert h₁_dvd\n\n  by_contra\n\n  have g : Polynomial.map (Int.castRingHom ℂ) (phi n) =\n      ∏ lamb ∈ (primitiveRoots n ℂ), (X - C lamb) := by\n    dsimp only [phi]\n    simp only [map_cyclotomic]\n    have := isPrimitiveRoot_exp n h_n\n    rw [cyclotomic_eq_prod_X_sub_primitiveRoots this]\n\n  have : 2 ≤ q := Fintype.one_lt_card_iff.mpr (exists_pair_ne { x // x ∈ Z })\n  -- here the book uses h_lamb_gt_q_sub_one from above\n  have h_gt : ((cyclotomic n ℤ).eval ↑q).natAbs > q - 1 := by\n    have hn : 1 < n := by\n      sorry\n    have hq : q ≠ 1 := by exact Nat.ne_of_gt this\n    exact Polynomial.sub_one_lt_natAbs_cyclotomic_eval hn hq\n\n  have h_q_sub_one : 0 ≠ (q : ℤ) - 1 := by\n    have h1 : (q : ℤ) - 1 = (q - 1 : ℕ) := by\n      rw [Int.ofNat_sub $ le_of_lt this]\n      norm_num\n    rw [h1]\n    exact_mod_cast Nat.ne_of_lt <| Nat.sub_pos_of_lt this\n\n  have hq_eq : (q : ℤ) - 1 = (q - 1 : ℕ) := by\n    have : 1 ≤ q := Fintype.card_pos\n    simp [this]\n\n  rw [← hq_eq] at h_phi_dvd_q_sub_one\n\n  have : 1 ≤ Fintype.card { x // x ∈ center R } :=\n    Fintype.card_pos_iff.mpr (⟨1, Subring.one_mem (center R)⟩)\n  have h_q : |((q : ℤ) - 1)| = q - 1 := by\n    norm_num\n    exact this\n  have h_norm := le_abs_of_dvd h_q_sub_one h_phi_dvd_q_sub_one\n  rw [h_q] at h_norm\n\n  refine' not_le_of_gt h_gt _\n  have : (q - 1 : ℕ) = (q : ℤ) - 1 := Int.natCast_pred_of_pos this\n  rw [← this] at h_norm\n  have : Int.natAbs (eval (↑q) (cyclotomic n ℤ)) = |eval (↑q) (phi n)| := by\n    simp only [Int.natCast_natAbs]\n    rfl\n  rw [← this] at h_norm\n  exact_mod_cast h_norm","line":173,"end_line":328,"proof_status":"statement_formalized"}
{"id":"ch04.lemma₁","label":"ch04.lemma₁","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For primes \\(p = 4m + 1\\) the equation \\(s^2 \\equiv -1 (\\mod p)\\) has two solutions \\(s \\in \\{ 1, 2, \\dots , p - 1\\} \\), for \\(p = 2\\) there is one such solution, while for primes of the form \\(p = 4m + 3\\) there is no solution.","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.lemma₁","latex_link":"sect0004.html#ch04.lemma₁","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/ch04.lemma₁","lean_decl_name":"ch04.lemma₁","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_04.lean","lean4_code":"lemma lemma₁ {p : ℕ} [h : Fact p.Prime] :\n    let num_solutions := Finset.card { s : ZMod p | s ^ 2 = - 1 }\n    (∃ m, p = 4 * m + 1 → num_solutions = 2) ∧\n    (p = 2 → num_solutions = 1) ∧\n    (∃ m, p = 4 * m + 1 → num_solutions = 0) := by\n  constructor\n  · sorry\n  · constructor\n    · intro hp\n      -- TODO: figure out how to easily write `use 1` here to follow more closely the book\n      aesop\n    · sorry","line":42,"end_line":53,"proof_status":"statement_formalized"}
{"id":"thm:hadamard_existence","label":"hadamard_existence","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Hadamard matrices exist for all \\(n = 2^m\\).","doc_url":null,"latex_link":"sect0007.html#thm:hadamard_existence","lean_link":null,"lean_decl_name":"hadamard_existence","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"slope_problem","label":"slope_problem","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(n\\ge 3\\) points in the plane do not lie on one single line, then they determine at least \\(n - 1\\) different slopes, where equality is possible only if \\(n\\) is odd and \\(n \\ge 5\\).","doc_url":null,"latex_link":"sect0012.html#slope_problem","lean_link":null,"lean_decl_name":"slope_problem","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch20theoremI","label":"ch20theoremI","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\langle a, b \\rangle \\) be an inner product on a real vector space \\(V\\) (with the norm \\(|a|^2 := \\langle a, a \\rangle \\)). Then \\[ \\langle a, b \\rangle ^2 \\leq |a|^2 |b|^2 \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/cauchy_schwarz_inequality","latex_link":"chapter20.html#ch20theoremI","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/cauchy_schwarz_inequality","lean_decl_name":"cauchy_schwarz_inequality","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_20.lean","lean4_code":"theorem cauchy_schwarz_inequality (a b : V) : ⟪ a, b ⟫ ^ 2 ≤ ‖a‖ ^ 2 * ‖b‖ ^ 2 := by\n  have h: ∀ (x : ℝ), ‖x • a + b‖ ^ 2 = x ^ 2 * ‖a‖ ^ 2 + 2 * x * ⟪a, b⟫ + ‖b‖ ^ 2 := by\n    simp only [pow_two, ← (real_inner_self_eq_norm_mul_norm _)]\n    simp only [inner_add_add_self, inner_smul_right, inner_smul_left, conj_trivial,\n        add_left_inj, real_inner_comm]\n    intro x\n    ring_nf\n  by_cases ha : a = 0\n  · rw [ha]\n    simp\n  · by_cases hl : (∃ (l : ℝ),  b = l • a)\n    · obtain ⟨l, hb⟩ := hl\n      rw [hb]\n      simp only [pow_two, ← (real_inner_self_eq_norm_mul_norm _)]\n      simp only [inner_smul_right, inner_smul_left, conj_trivial]\n      ring_nf\n      rfl\n    · have : ∀ (x : ℝ), 0 < ‖x • a + b‖ := by\n        intro x\n        by_contra hx\n        simp only [norm_pos_iff, ne_eq, Decidable.not_not] at hx\n        absurd hl\n        use -x\n        rw [← add_zero (-x•a), ← hx]\n        simp only [neg_smul, neg_add_cancel_left]\n      have : ∀ (x : ℝ), 0 < ‖x • a + b‖ ^ 2 := by\n        exact fun x ↦ sq_pos_of_pos (this x)\n      have : ∀ (x : ℝ), 0 <  x ^ 2 * ‖a‖ ^ 2 + 2 * x * ⟪a, b⟫ + ‖b‖ ^ 2 := by\n        convert this\n        exact (h _).symm\n      have : ∀ (x : ℝ), 0 <  ‖a‖ ^ 2 * (x * x)  + 2 * ⟪a, b⟫ * x + ‖b‖ ^ 2 := by\n        intro x\n        calc\n          0 <  x ^ 2 * ‖a‖ ^ 2 + 2 * x * ⟪a, b⟫ + ‖b‖ ^ 2 := this x\n          _ = ‖a‖ ^ 2 * (x * x)  + 2 * ⟪a, b⟫ * x + ‖b‖ ^ 2  := by ring_nf\n      have ha_sq : ‖a‖ ^ 2 ≠ 0 := by aesop\n      have := discrim_lt_zero ha_sq this\n      unfold discrim at this\n      have  : (2 * inner _ a b) ^ 2 < 4 * ‖a‖ ^ 2 * ‖b‖ ^ 2 := by linarith\n      linarith","line":45,"end_line":84,"proof_status":"formalized"}
{"id":"tiling_rectangles3","label":"tiling_rectangles3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Whenever a rectangle is tiled by rectangles all of which have at least one side of integer length, then the tiled rectangle has at least one side of integer length.","doc_url":null,"latex_link":"sect0027.html#tiling_rectangles3","lean_link":null,"lean_decl_name":"tiling_rectangles3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch37fact_c","label":"ch37fact_c","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ H(Y|X)\\le \\sum _{j=1}^d \\operatorname{Prop}(X\\in E_j)\\log _2 j. \\]","doc_url":null,"latex_link":"sect0034.html#ch37fact_c","lean_link":null,"lean_decl_name":"ch37fact_c","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"borsuk_ulam","label":"borsuk_ulam","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every continuous map \\(f : S^d \\to \\mathbb {R}^d\\) from \\(d\\)-sphere to \\(d\\)-space, there are antipodal points \\(x^*\\), \\(-x^*\\) that are mapped to the same point \\(f(x^*) = f(-x^*)\\).","doc_url":null,"latex_link":"sect0040.html#borsuk_ulam","lean_link":null,"lean_decl_name":"borsuk_ulam","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"pick_lemma","label":"pick_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Every elementary triangle \\(\\Delta = \\operatorname{conv}\\{ p_0, p_1, p_2\\} \\subset \\mathbb {R}^2\\) has area \\(A(\\Delta )=12\\)","doc_url":null,"latex_link":"sect0013.html#pick_lemma","lean_link":null,"lean_decl_name":"pick_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch36lemma1","label":"ch36lemma1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Any \\((r \\times n)\\)-Latin rectangle, \\(r {\\lt} n\\), can be extended to an \\(((r+1) \\times n)\\)-Latin rectangle and hence can be completed to a Latin square.","doc_url":null,"latex_link":"sect0033.html#ch36lemma1","lean_link":null,"lean_decl_name":"ch36lemma1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"example1","label":"example1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0010.html#example1","lean_link":null,"lean_decl_name":"example1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"ch20theoremIIproof3","label":"ch20theoremIIproof3","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(a_1, \\dots a_n\\) be positive real numbers, then \\[ \\frac{n}{\\frac{1}{a_1}+\\dots +\\frac{1}{n_n}} \\le \\sqrt[n]{a_1a_2\\dots a_n} \\le \\frac{a_1\\dots a_n}{n} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₃","latex_link":"chapter20.html#ch20theoremIIproof3","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₃","lean_decl_name":"harmonic_geometric_arithmetic₃","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_20.lean","lean4_code":"theorem harmonic_geometric_arithmetic₃ (n : ℕ) (hn : 1 ≤ n)\n  (a : Finset.Icc 1 n → ℝ) (hpos : ∀ i, 0 < a i) :\n  let harmonic := n / (∑ i : Finset.Icc 1 n, 1 / (a i))\n  let geometric := (∏ i : Finset.Icc 1 n, a i) ^ ((1 : ℝ) / n)\n  let arithmetic := (∑ i : Finset.Icc 1 n, a i) / n\n  let all_equal := ∀ i : Finset.Icc 1 n, a i = a ⟨1, Finset.mem_Icc.mpr  ⟨NeZero.one_le, hn⟩⟩\n  harmonic ≤ geometric ∧ geometric ≤ arithmetic ∧\n  ((harmonic = geometric) ↔ all_equal) ∧\n  ((geometric = arithmetic) ↔ all_equal) := by sorry","line":110,"end_line":118,"proof_status":"statement_formalized"}
{"id":"ch31theorem2","label":"ch31theorem2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"After performing \\(k\\) riffle shuffles on a deck of \\(n\\) cards, the variation distance from a uniform distribution satisfies \\[ ||\\text{Rif}^*k - \\mathbb {U}|| \\leq 1 - \\prod _{i=1}^{n-1} \\left( 1 - \\frac{i}{2^k} \\right). \\]","doc_url":null,"latex_link":"sect0028.html#ch31theorem2","lean_link":null,"lean_decl_name":"ch31theorem2","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"cayley_formala_proof1","label":"cayley_formala_proof1","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There are \\(n^{n - 2}\\) different labeled trees on \\(n\\) nodes.","doc_url":null,"latex_link":"sect0030.html#cayley_formala_proof1","lean_link":null,"lean_decl_name":"cayley_formala_proof1","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"arm_lemma","label":"arm_lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"TODO","doc_url":null,"latex_link":"sect0014.html#arm_lemma","lean_link":null,"lean_decl_name":"arm_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:det_greater_n_fact","label":"det_greater_n_fact","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exists an \\(n \\times n\\) matrix with entries \\(\\pm 1\\) whose determinant is greater than \\(\\sqrt{n!}\\).","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₂","latex_link":"sect0007.html#thm:det_greater_n_fact","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₂","lean_decl_name":"chapter7.Theorem₂","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_07.lean","lean4_code":"theorem Theorem₂ (n : ℕ) (hn : 1 < n) : ∃ (M : Matrix (Fin n) (Fin n) ℤ),\n    (∀ i j, M i j = -1 ∨ M i j = 1) ∧\n    M.det > Real.sqrt n.factorial := by\n  sorry","line":35,"end_line":38,"proof_status":"statement_formalized"}
{"id":"ch7.lemma","label":"ch7.lemma","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(A\\) is a real symmetric \\(n \\times n\\) matrix that is not diagonal, that is, \\(\\operatorname{Od}(A) {\\gt} 0\\), then there exists \\(U \\in O(n)\\) such that \\(\\operatorname{Od}(U^T A U) {\\lt} \\operatorname{Od}(A)\\).","doc_url":null,"latex_link":"sect0007.html#ch7.lemma","lean_link":null,"lean_decl_name":"ch7.lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/NumberTheory/LSeries/SumCoeff.lean","lean4_code":"private theorem lemma₁ (hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l))\n    {s : ℝ} (hs : 1 < s) :\n    IntegrableOn (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * (t : ℂ) ^ (-(s : ℂ) - 1)) (Set.Ici 1) := by\n  have h₁ : LocallyIntegrableOn (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * (t : ℂ) ^ (-(s : ℂ) - 1))\n        (Set.Ici 1) := by\n    simp_rw [mul_comm]\n    refine locallyIntegrableOn_mul_sum_Icc f zero_le_one ?_\n    refine ContinuousOn.locallyIntegrableOn (fun t ht ↦ ?_) measurableSet_Ici\n    exact (continuousAt_ofReal_cpow_const _ _ <|\n      Or.inr (zero_lt_one.trans_le ht).ne').continuousWithinAt\n  have h₂ : (fun t : ℝ ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k) =O[atTop] fun t ↦ t ^ (1 : ℝ) := by\n    simp_rw [Real.rpow_one]\n    refine IsBigO.trans_isEquivalent ?_ isEquivalent_nat_floor\n    have : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ((n : ℝ) ^ (1 : ℝ) : ℝ)) atTop (𝓝 l) := by\n      simpa using hlim\n    simpa using (isBigO_atTop_natCast_rpow_of_tendsto_div_rpow this).comp_tendsto\n        tendsto_nat_floor_atTop\n  refine h₁.integrableOn_of_isBigO_atTop (g := fun t ↦ t ^ (-s)) ?_ ?_\n  · refine IsBigO.mul_atTop_rpow_of_isBigO_rpow 1 (-s - 1) _ h₂ ?_ (by linarith)\n    exact (norm_ofReal_cpow_eventually_eq_atTop _).isBigO.of_norm_left\n  · rwa [integrableAtFilter_rpow_atTop_iff, neg_lt_neg_iff]\n\nprivate theorem lemma₂ {s T ε : ℝ} {S : ℝ → ℂ} (hs : 1 < s)\n    (hS₁ : LocallyIntegrableOn (fun t ↦ S t) (Set.Ici 1)) (hS₂ : ∀ t ≥ T, ‖S t‖ ≤ ε * t) :\n    IntegrableOn (fun t : ℝ ↦ ‖S t‖ * (t ^ (-s - 1))) (Set.Ici 1) := by\n  have h : LocallyIntegrableOn (fun t : ℝ ↦ ‖S t‖ * (t ^ (-s - 1))) (Set.Ici 1) := by\n    refine hS₁.norm.mul_continuousOn ?_ isLocallyClosed_Ici\n    exact fun t ht ↦ (Real.continuousAt_rpow_const _ _\n      <| Or.inl (zero_lt_one.trans_le ht).ne').continuousWithinAt\n  refine h.integrableOn_of_isBigO_atTop (g := fun t ↦ t ^ (-s)) (isBigO_iff.mpr ⟨ε, ?_⟩) ?_\n  · filter_upwards [eventually_ge_atTop T, eventually_gt_atTop 0] with t ht ht'\n    simpa [abs_of_nonneg, Real.rpow_nonneg, ht'.le, Real.rpow_sub ht', mul_assoc, ht'.ne',\n      mul_div_cancel₀] using mul_le_mul_of_nonneg_right (hS₂ t ht) (norm_nonneg <| t ^ (-s - 1))\n  · exact integrableAtFilter_rpow_atTop_iff.mpr <| neg_lt_neg_iff.mpr hs\n\nend lemmas\n\nsection proof\n-- See `LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₃` for the strategy of proof\n\nprivate theorem LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₁\n    (hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l)) {ε : ℝ} (hε : 0 < ε) :\n    ∀ᶠ t : ℝ in atTop, ‖(∑ k ∈ Icc 1 ⌊t⌋₊, f k) - l * t‖ < ε * t := by\n  have h_lim' : Tendsto (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k : ℂ) / t) atTop (𝓝 l) := by\n    refine (mul_one l ▸ ofReal_one ▸ ((hlim.comp tendsto_nat_floor_atTop).mul <|\n      tendsto_ofReal_iff.mpr <| tendsto_nat_floor_div_atTop)).congr' ?_\n    filter_upwards [eventually_ge_atTop 1] with t ht\n    simp [div_mul_div_cancel₀ (show (⌊t⌋₊ : ℂ) ≠ 0 by simpa)]\n  filter_upwards [eventually_gt_atTop 0, Metric.tendsto_nhds.mp h_lim' ε hε] with t ht₁ ht₂\n  rwa [dist_eq_norm, div_sub' (ne_zero_of_re_pos ht₁), norm_div, norm_real,\n","line":168,"end_line":217,"proof_status":"statement_formalized"}
{"id":"euler_series_3","label":"euler_series_3","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\[ \\sum _{n\\ge 1}\\frac{1}{n^2} = \\frac{\\pi ^2}{6} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series_3","latex_link":"sect0009.html#euler_series_3","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/euler_series_3","lean_decl_name":"euler_series_3","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_09.lean","lean4_code":"theorem euler_series_3 :\n  ∑' (n : ℕ+), (1 : ℝ) / n = π ^ 2  / 6 := by\n  sorry","line":617,"end_line":619,"proof_status":"statement_formalized"}
{"id":"ch20theoremIIproof2","label":"ch20theoremIIproof2","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(a_1, \\dots a_n\\) be positive real numbers, then \\[ \\frac{n}{\\frac{1}{a_1}+\\dots +\\frac{1}{n_n}} \\le \\sqrt[n]{a_1a_2\\dots a_n} \\le \\frac{a_1\\dots a_n}{n} \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₂","latex_link":"chapter20.html#ch20theoremIIproof2","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/harmonic_geometric_arithmetic₂","lean_decl_name":"harmonic_geometric_arithmetic₂","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_20.lean","lean4_code":"theorem harmonic_geometric_arithmetic₂ (n : ℕ) (hn : 1 ≤ n)\n  (a : Finset.Icc 1 n → ℝ) (hpos : ∀ i, 0 < a i) :\n  let harmonic := n / (∑ i : Finset.Icc 1 n, 1 / (a i))\n  let geometric := (∏ i : Finset.Icc 1 n, a i) ^ ((1 : ℝ) / n)\n  let arithmetic := (∑ i : Finset.Icc 1 n, a i) / n\n  let all_equal := ∀ i : Finset.Icc 1 n, a i = a ⟨1, Finset.mem_Icc.mpr  ⟨NeZero.one_le, hn⟩⟩\n  harmonic ≤ geometric ∧ geometric ≤ arithmetic ∧\n  ((harmonic = geometric) ↔ all_equal) ∧\n  ((geometric = arithmetic) ↔ all_equal) := by sorry","line":99,"end_line":107,"proof_status":"statement_formalized"}
{"id":"thm:hadamard_inequality","label":"hadamard_inequality","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For any real \\(n \\times n\\) matrix \\(A = (a_{ij})\\) with \\(|a_{ij}| \\le 1\\), \\[ |\\det A| \\le n^{n/2}. \\]","doc_url":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₂","latex_link":"sect0007.html#thm:hadamard_inequality","lean_link":"https://mo271.github.io/FormalBook/docs/find/#doc/chapter7.Theorem₂","lean_decl_name":"chapter7.Theorem₂","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/mo271/FormalBook/842bcbbeb3b0930fb0a37f285ab03d22b0ae8104/FormalBook/Chapter_07.lean","lean4_code":"theorem Theorem₂ (n : ℕ) (hn : 1 < n) : ∃ (M : Matrix (Fin n) (Fin n) ℤ),\n    (∀ i j, M i j = -1 ∨ M i j = 1) ∧\n    M.det > Real.sqrt n.factorial := by\n  sorry","line":35,"end_line":38,"proof_status":"statement_formalized"}
{"id":"ch28claim4","label":"ch28claim4","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Suppose we are given \\(n\\) integers \\(a_1, \\dots , a_n\\), which need not be distinct. Then there is always a set of consecutive numbers \\(a_{k+1}, a_{k+2}, \\dots , a_{\\ell }\\) whose sum \\(\\sum _{i=k+1}^{\\ell } a_i\\) is a multiple of \\(n\\).","doc_url":null,"latex_link":"chapter28.html#ch28claim4","lean_link":null,"lean_decl_name":"ch28claim4","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"five_colorable","label":"five_colorable","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://firsching.ch/FormalBook/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"All planar graphs \\(G\\) can be \\(5\\)-colored: \\[ \\chi _{\\ell }(G)\\le 5. \\]","doc_url":null,"latex_link":"sect0036.html#five_colorable","lean_link":null,"lean_decl_name":"five_colorable","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}