| {"id":"FreyPackage","label":"FreyPackage","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"A Frey package \\((a,b,c,p)\\) is three nonzero pairwise-coprime integers \\(a\\), \\(b\\), \\(c\\), with \\(a\\equiv 3\\pmod4\\) and \\(b\\equiv 0\\pmod2\\), and a prime \\(p\\geq 5\\), such that \\(a^p+b^p=c^p\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage","latex_link":"ch_reductions.html#FreyPackage","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage","lean_decl_name":"FreyPackage","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"./FLT/Basic/FreyPackage.lean","lean4_code":"structure FreyPackage where\n /-- The integer `a` in the Frey package. -/\n a : ℤ\n /-- The integer `b` in the Frey package. -/\n b : ℤ\n /-- The integer `c` in the Frey package. -/\n c : ℤ\n ha0 : a ≠ 0\n hb0 : b ≠ 0\n hc0 : c ≠ 0\n /-- The prime number `p` in the Frey package. -/\n p : ℕ\n pp : Nat.Prime p\n hp5 : 5 ≤ p\n hFLT : a ^ p + b ^ p = c ^ p\n hgcdab : gcd a b = 1 -- same as saying a,b,c pairwise coprime\n ha4 : (a : ZMod 4) = 3\n hb2 : (b : ZMod 2) = 0\n\nnamespace FreyPackage\n\n","line":80,"end_line":100,"proof_status":"formalized"} |
| {"id":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","label":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage"],"nlp_text":"If Fermat’s Last Theorem is false for \\(p \\ge 5\\) and prime, then there exists a Frey package.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.of_not_FermatLastTheorem_p_ge_5","latex_link":"ch_reductions.html#FreyPackage.of_not_FermatLastTheorem_p_ge_5","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.of_not_FermatLastTheorem_p_ge_5","lean_decl_name":"FreyPackage.of_not_FermatLastTheorem_p_ge_5","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/FreyPackage.lean","lean4_code":"/-- Given a counterexample a^p+b^p=c^p to Fermat's Last Theorem with p>=5,\nthere exists a Frey package. -/\nlemma of_not_FermatLastTheorem_p_ge_5 {a b c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0)\n {p : ℕ} (pp : p.Prime) (hp5 : 5 ≤ p) (H : a ^ p + b ^ p = c ^ p) : Nonempty FreyPackage := by\n have p_odd := pp.odd_of_ne_two (by omega)\n -- First, show that we can make a,b coprime by dividing through by gcd a b\n have ⟨a, b, c, a0, b0, c0, ab, H⟩ :\n ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧ a^p + b^p = c^p := by\n obtain ⟨d, a', b', d0, cop, a_eq, b_eq⟩ :=\n Int.exists_gcd_one' (Int.gcd_pos_of_ne_zero_left b ha)\n simp only [a_eq, mul_pow, b_eq] at H\n rw [← add_mul, mul_comm] at H\n obtain ⟨c', rfl⟩ := (Int.pow_dvd_pow_iff pp.ne_zero).1 ⟨_, H.symm⟩\n rw [mul_pow] at H\n have a0' := left_ne_zero_of_mul (a_eq ▸ ha)\n have b0' := left_ne_zero_of_mul (b_eq ▸ hb)\n have c0' := right_ne_zero_of_mul hc\n exact ⟨a', b', c', a0', b0', c0', cop, mul_left_cancel₀ (pow_ne_zero _ (mod_cast d0.ne')) H⟩\n -- Then show that WLOG we can take b to be even,\n -- because at least one of a,b,c is even and we can permute if needed\n have ⟨a, b, c, a0, b0, c0, ab, eb, H⟩ :\n ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧ Even b ∧ a^p + b^p = c^p := by\n if eb : Even b then\n exact ⟨a, b, c, a0, b0, c0, ab, eb, H⟩\n else if ea : Even a then\n exact ⟨b, a, c, b0, a0, c0, Int.gcd_comm a b ▸ ab, ea, by rwa [add_comm]⟩\n else\n refine ⟨a, -c, -b, a0, neg_ne_zero.2 c0, neg_ne_zero.2 b0, ?_, even_neg.2 ?_, ?_⟩\n · refine Int.gcd_neg.trans (.trans (.symm ?_) ab)\n exact Nat.cast_inj.1 (gcdab_eq_gcdac pp.pos H)\n · refine ((Int.even_pow (n := p)).1 (H.symm ▸ Int.even_add.2 (iff_of_false ?_ ?_))).1\n · exact fun h => ea (Int.even_pow.1 h).1\n · exact fun h => eb (Int.even_pow.1 h).1\n · simp [p_odd.neg_pow, ← H]\n -- We can ensure additionally that a ≡ 3 [ZMOD 4] by negating everything if necessary\n have ⟨a, b, c, ha0, hb0, hc0, ab, ha3, eb, hFLT⟩ :\n ∃ (a b c : ℤ), a ≠ 0 ∧ b ≠ 0 ∧ c ≠ 0 ∧ Int.gcd a b = 1 ∧\n a ≡ 3 [ZMOD 4] ∧ Even b ∧ a^p + b^p = c^p := by\n -- Since b is even, a cannot also be even\n have a_odd' : ∀ {i}, a ≡ i [ZMOD 4] → ¬2 ∣ i := fun ai ei => by\n have ea := (dvd_sub_right ei).1 (.trans (by decide) (Int.modEq_iff_dvd.1 ai))\n simpa (config := {decide := true}) [gcd, ab] using dvd_gcd ea (even_iff_two_dvd.1 eb)\n mod_cases a_mod : a % 4\n · cases a_odd' a_mod (by decide)\n · exact ⟨-a, -b, -c, neg_ne_zero.2 a0, neg_ne_zero.2 b0, neg_ne_zero.2 c0,\n by rwa [Int.neg_gcd, Int.gcd_neg], a_mod.neg, eb.neg,\n by simp [p_odd.neg_pow, ← H, add_comm]⟩\n · cases a_odd' a_mod (by decide)\n · exact ⟨a, b, c, a0, b0, c0, ab, a_mod, eb, H⟩\n -- Build the Frey package from the assumptions\n exact ⟨{\n a, b, c, ha0, hb0, hc0, p, pp, hp5, hFLT\n hgcdab := by simp [gcd, ab]\n ha4 := (ZMod.intCast_eq_intCast_iff ..).2 ha3\n hb2 := (ZMod.intCast_zmod_eq_zero_iff_dvd ..).2 (even_iff_two_dvd.1 eb)\n }⟩\n","line":154,"end_line":209,"proof_status":"formalized"} |
| {"id":"FreyCurve","label":"FreyCurve","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage"],"nlp_text":"Given a Frey package \\((a,b,c,p)\\), the corresponding Frey curve (considered by Frey and, before him, Hellegouarch) is the elliptic curve over \\(\\mathbb {Q}\\) defined by the equation \\(Y^2=X(X-a^p)(X+b^p).\\)","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.freyCurve","latex_link":"ch_reductions.html#FreyCurve","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.freyCurve","lean_decl_name":"FreyPackage.freyCurve","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/FreyPackage.lean","lean4_code":"/-- The elliptic curve over `ℚ` associated to a Frey package. The conditions imposed\nupon a Frey package guarantee that the running hypotheses in\nSection 4.1 of [Serre] all hold. We put the curve into the form where the\nequation is semistable at 2, rather than the usual `Y^2=X(X-a^p)(X+b^p)` form.\nThe change of variables is `X=4x` and `Y=8y+4x`, and then divide through by 64. -/\ndef freyCurve (P : FreyPackage) : WeierstrassCurve ℚ where\n a₁ := 1\n -- a₂ is an integer because of the congruences assumed e.g. P.ha4\n a₂ := (P.b ^ P.p - 1 - P.a ^ P.p) / 4\n a₃ := 0\n a₄ := -(P.a ^ P.p) * (P.b ^ P.p) / 16 -- this is also an integer\n a₆ := 0\n","line":232,"end_line":243,"proof_status":"formalized"} |
| {"id":"FLT","label":"FLT","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyPackage.of_not_FermatLastTheorem_p_ge_5","FreyPackage.false","FermatLastTheorem.of_p_ge_5"],"nlp_text":"Fermat’s Last Theorem is true. In other words, there are no positive integers \\(a,b,c\\) and natural \\(n\\geq 3\\) such that \\(a^n+b^n=c^n\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Taylor_Wiles","latex_link":"ch_reductions.html#FLT","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Taylor_Wiles","lean_decl_name":"Wiles_Taylor_Wiles","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"/-- Fermat's Last Theorem is true -/\ntheorem Wiles_Taylor_Wiles : FermatLastTheorem := by\n -- Suppose Fermat's Last Theorem is false\n by_contra h\n -- then there exists a Frey package\n obtain ⟨P : FreyPackage⟩ := FreyPackage.of_not_FermatLastTheorem h\n -- but there is no Frey package\n exact P.false\n","line":68,"end_line":75,"proof_status":"formalized"} |
| {"id":"Wiles_Frey","label":"Wiles_Frey","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyCurve"],"nlp_text":"If \\(\\rho \\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\rho \\) is reducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","latex_link":"ch_reductions.html#Wiles_Frey","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Wiles_Frey","lean_decl_name":"Wiles_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Wiles_Frey (P : FreyPackage) :\n haveI : Fact P.p.Prime := ⟨P.pp⟩\n ¬ GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n FreyCurve.torsion_not_isIrreducible P\n","line":47,"end_line":50,"proof_status":"statement_formalized"} |
| {"id":"Mazur_Frey","label":"Mazur_Frey","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FreyCurve"],"nlp_text":"If \\(\\rho \\) is the mod \\(p\\) Galois representation associated to a Frey package \\((a,b,c,p)\\) then \\(\\rho \\) is irreducible.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Mazur_Frey","latex_link":"ch_reductions.html#Mazur_Frey","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/Mazur_Frey","lean_decl_name":"Mazur_Frey","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"FLT_ch2","source_path":"FLT/Basic/Reductions.lean","lean4_code":"theorem Mazur_Frey (P : FreyPackage) :\n haveI : Fact P.p.Prime := ⟨P.pp⟩\n GaloisRep.IsIrreducible (P.freyCurve.galoisRep P.p P.hppos) :=\n sorry\n","line":35,"end_line":38,"proof_status":"statement_formalized"} |
| {"id":"FreyPackage.false","label":"FreyPackage.false","background_color":"#9CEC8B","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["Wiles_Frey","Mazur_Frey"],"nlp_text":"There is no Frey package.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.false","latex_link":"ch_reductions.html#FreyPackage.false","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FreyPackage.false","lean_decl_name":"FreyPackage.false","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/Tactic/ITauto.lean","lean4_code":"@[match_pattern] def IProp.not (a : IProp) : IProp := a.imp .false\n\n/-- Constructor for `xor p q`. -/\n","line":115,"end_line":117,"proof_status":"formalized"} |
| {"id":"FermatLastTheorem.of_odd_primes","label":"FermatLastTheorem.of_odd_primes","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"If there is a counterexample to Fermat’s Last Theorem, then there is a counterexample \\(a^p+b^p=c^p\\) with \\(p\\) an odd prime.","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_odd_primes","latex_link":"ch_reductions.html#FermatLastTheorem.of_odd_primes","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_odd_primes","lean_decl_name":"FermatLastTheorem.of_odd_primes","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/leanprover-community/mathlib4/98379c78c5dd2e3614cd24ab7ad22f5713e0eb63/Mathlib/NumberTheory/FLT/Four.lean","lean4_code":"/--\nTo prove Fermat's Last Theorem, it suffices to prove it for odd prime exponents.\n-/\ntheorem FermatLastTheorem.of_odd_primes\n (hprimes : ∀ p : ℕ, Nat.Prime p → Odd p → FermatLastTheoremFor p) : FermatLastTheorem := by\n intro n h\n obtain hdvd | ⟨p, hpprime, hdvd, hpodd⟩ := Nat.four_dvd_or_exists_odd_prime_and_dvd_of_two_lt h\n <;> apply FermatLastTheoremWith.mono hdvd\n · exact fermatLastTheoremFour\n · exact hprimes p hpprime hpodd","line":278,"end_line":287,"proof_status":"formalized"} |
| {"id":"FermatLastTheorem.of_p_ge_5","label":"FermatLastTheorem.of_p_ge_5","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":["FermatLastTheorem.of_odd_primes","fermatLastTheoremThree"],"nlp_text":"If there is a counterexample to Fermat’s Last Theorem, then there is a counterexample \\(a^p+b^p=c^p\\) with \\(p\\) prime and \\(p\\geq 5\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_p_ge_5","latex_link":"ch_reductions.html#FermatLastTheorem.of_p_ge_5","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/FermatLastTheorem.of_p_ge_5","lean_decl_name":"FermatLastTheorem.of_p_ge_5","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/ImperialCollegeLondon/FLT/712e8a20d2647895feed1cbf5518dce4056f58cb/FLT/Basic/FreyPackage.lean","lean4_code":"/-- If Fermat's Last Theorem is true for primes `p ≥ 5`, then FLT is true. -/\nlemma FermatLastTheorem.of_p_ge_5 (H : ∀ p ≥ 5, p.Prime → FermatLastTheoremFor p) :\n FermatLastTheorem := by\n apply FermatLastTheorem.of_odd_primes -- this is Fermat's proof for n=4, plus reduction to\n -- the case n prime.\n intro p pp p_odd\n if hp5 : 5 ≤ p then\n exact H _ hp5 pp\n else\n have hp2 := pp.two_le\n interval_cases p\n · contradiction\n · exact fermatLastTheoremThree -- this is Euler's proof for n=3\n · contradiction","line":58,"end_line":71,"proof_status":"formalized"} |
| {"id":"fermatLastTheoremThree","label":"fermatLastTheoremThree","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://imperialcollegelondon.github.io/FLT/blueprint/dep_graph_chapter_2.html","dependencies":[],"nlp_text":"There are no solutions in positive integers to \\(a^3+b^3=c^3\\).","doc_url":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/fermatLastTheoremThree","latex_link":"ch_reductions.html#fermatLastTheoremThree","lean_link":"https://ImperialCollegeLondon.github.io/FLT/docs/find/#doc/fermatLastTheoremThree","lean_decl_name":"fermatLastTheoremThree","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/FLT/Three.lean","lean4_code":"theorem fermatLastTheoremThree : FermatLastTheoremFor 3 := by\n classical\n let K := CyclotomicField 3 ℚ\n let hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K\n have : NumberField K := IsCyclotomicExtension.numberField {3} ℚ _\n apply FermatLastTheoremForThree_of_FermatLastTheoremThreeGen hζ\n intro a b c u hc ha hb hcdvd coprime H\n let S' : FermatLastTheoremForThreeGen.Solution' hζ :=\n { a := a\n b := b\n c := c\n u := u\n ha := ha\n hb := hb\n hc := hc\n coprime := coprime\n hcdvd := hcdvd\n H := H }\n obtain ⟨S, -⟩ := FermatLastTheoremForThreeGen.exists_Solution_of_Solution' S'\n obtain ⟨Smin, hSmin⟩ := S.exists_minimal\n obtain ⟨Sfin, hSfin⟩ := Smin.exists_Solution_multiplicity_lt\n linarith [hSmin Sfin]\n","line":743,"end_line":764,"proof_status":"formalized"} |
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