Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
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values | symbolic_name stringlengths 4 131 | library stringclasses 71
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PNat.pow_add_pow_ne_pow
(x y z : ℕ+)
(n : ℕ) (hn : n > 2) :
x^n + y^n ≠ z^n | PNat.pow_add_pow_ne_pow_of_FermatLastTheorem Wiles_Taylor_Wiles x y z n hn
/--
info: 'PNat.pow_add_pow_ne_pow' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]
-/
#guard_msgs in
#print axioms PNat.pow_add_pow_ne_pow | theorem | PNat.pow_add_pow_ne_pow | Root | FermatsLastTheorem.lean | [
"FLT"
] | [
"PNat.pow_add_pow_ne_pow_of_FermatLastTheorem",
"Wiles_Taylor_Wiles"
] | Fermat's Last Theorem for positive naturals. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
main (args : List String) : IO UInt32 | Informal.PreviewManifest.manualMainWithSharedPreviewManifest
(%doc FLTBlueprint.Blueprint)
args
(extensionImpls := by exact extension_impls%) | def | main | blueprint-verso | blueprint-verso/FLTBlueprintMain.lean | [
"VersoManual",
"VersoBlueprint.PreviewManifest",
"FLTBlueprint.Blueprint"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
knownin1980s {P : Prop} : P | axiom | knownin1980s | Assumptions | FLT/Assumptions/KnownIn1980s.lean | [] | [] | `knownin1980s` is a term which provides a proof of an
arbitrary proposition. In this current phase of the FLT project,
`knownin1980s` will be allowed as a proof of any theorem
which would have been easy for an expert to deduce from
the pre-1990 literature. This stretches from standard easy
statements about things like ... | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
"knownin1980s" : tactic => `(tactic| exact knownin1980s) | macro | knownin1980s | Assumptions | FLT/Assumptions/KnownIn1980s.lean | [] | [] | `knownin1980s` is a term which provides a proof of an
arbitrary proposition. In this current phase of the FLT project,
`knownin1980s` will be allowed as a proof of any theorem
which would have been easy for an expert to deduce from
the pre-1990 literature. This stretches from standard easy
statements about things like ... | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Mazur_statement (E : WeierstrassCurve ℚ) [E.IsElliptic] :
(AddCommGroup.torsion (E⁄ℚ).Point : Set (E⁄ℚ).Point).ncard ≤ 16 | axiom | Mazur_statement | Assumptions | FLT/Assumptions/Mazur.lean | [] | [] | Mazur's bound for the size of the torsion subgroup of an elliptic curve
over the rationals . | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Odlyzko_statement (K : Type*) [Field K] [NumberField K] [IsTotallyComplex K]
(hdim : finrank ℚ K ≥ 18) : |(discr K : ℝ)| ≥ 8.25 ^ finrank ℚ K | axiom | Odlyzko_statement | Assumptions | FLT/Assumptions/Odlyzko.lean | [] | [] | An "Odlyzko bound" for the root discriminant of a totally complex number field
of degree 18 and above. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Dfx | (D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ | abbrev | TotallyDefiniteQuaternionAlgebra.Dfx | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | `Dfx` is an abbreviation for $(D\otimes_F\mathbb{A}_F^\infty)^\times.$ | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
incl₁ : Dˣ →* Dfx F D | Units.map (Algebra.TensorProduct.includeLeftRingHom.toMonoidHom) | abbrev | TotallyDefiniteQuaternionAlgebra.incl₁ | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | incl₁ is an abbreviation for the inclusion
$D^\times\to(D\otimes_F\mathbb{A}_F^\infty)^\times.$ Remark: I wrote the `incl₁`
docstring in LaTeX and the `incl₂` one in unicode. Which is better? | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
incl₂ : (FiniteAdeleRing (𝓞 F) F)ˣ →* Dfx F D | Units.map (algebraMap _ _).toMonoidHom | abbrev | TotallyDefiniteQuaternionAlgebra.incl₂ | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | `incl₂` is he inclusion `𝔸_F^∞ˣ → (D ⊗ 𝔸_F^∞ˣ)`. Remark: I wrote the `incl₁`
docstring in LaTeX and the `incl₂` one in unicode. Which is better? | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
range_incl₂_le_center : MonoidHom.range (incl₂ F D) ≤ Subgroup.center (Dfx F D) | by
rintro x ⟨y, rfl⟩
refine Subgroup.mem_center_iff.mpr fun g ↦ Units.ext ?_
exact (Algebra.commutes _ _).symm | lemma | TotallyDefiniteQuaternionAlgebra.range_incl₂_le_center | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
WeightTwoAutomorphicForm
-- defined over R
(R : Type*) [AddCommMonoid R] where
/-- The function underlying an automorphic form. -/
toFun : Dfx F D → R
left_invt : ∀ (δ : Dˣ) (g : Dfx F D),
toFun (incl₁ F D δ * g) = (toFun g)
right_invt : ∃ (U : Subgroup (Dfx F D)),
IsOpen (U : Set (Dfx F D)) ∧
∀... | structure | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | This definition is made in mathlib-generality but is *not* the definition of a
weight 2 automorphic form unless `Dˣ` is compact mod centre at infinity.
This hypothesis will be true if `D` is a totally definite quaternion algebra
over a totally real field. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
ext (φ ψ : WeightTwoAutomorphicForm F D R) (h : ∀ x, φ x = ψ x) : φ = ψ | by
cases φ; cases ψ; simp only [mk.injEq]; ext; apply h | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.ext | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
zero : (WeightTwoAutomorphicForm F D R) | where
toFun := 0
left_invt δ _ := by simp
-- this used to be `by simp` but now it times out doing some crazy typeclass search for
-- `DiscreteTopology (D ⊗[F] FiniteAdeleRing (𝓞 F) F)ˣ`
right_invt := ⟨⊤, by simp only [Subgroup.coe_top, isOpen_univ, Subgroup.mem_top,
Pi.zero_apply, imp_self, implies_true,... | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.zero | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | The zero automorphic form for a totally definite quaterion algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
zero_apply (x : Dfx F D) :
(0 : WeightTwoAutomorphicForm F D R) x = 0 | rfl | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.zero_apply | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
neg (φ : WeightTwoAutomorphicForm F D R) : WeightTwoAutomorphicForm F D R | where
toFun x := - φ x
left_invt δ g := by simp [left_invt]
right_invt := by
obtain ⟨U, hU⟩ := φ.right_invt
simp_all only [neg_inj, right_invt]
trivial_central_char g z := by simp [trivial_central_char] | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.neg | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | Negation on the space of automorphic forms over a totally definite quaternion algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
neg_apply (φ : WeightTwoAutomorphicForm F D R) (x : Dfx F D) :
(-φ : WeightTwoAutomorphicForm F D R) x = -(φ x) | rfl | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.neg_apply | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
add (φ ψ : WeightTwoAutomorphicForm F D R) : WeightTwoAutomorphicForm F D R | where
toFun x := φ x + ψ x
left_invt := by simp [left_invt]
right_invt := by
obtain ⟨U, hU⟩ := φ.right_invt
obtain ⟨V, hV⟩ := ψ.right_invt
use U ⊓ V
simp_all only [Subgroup.coe_inf, IsOpen.inter, Subgroup.mem_inf, implies_true, and_self]
trivial_central_char := by simp [trivial_central_char] | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.add | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | Addition on the space of automorphic forms over a totally definite quaternion algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
add_apply (φ ψ : WeightTwoAutomorphicForm F D R) (x : Dfx F D) :
(φ + ψ) x = (φ x) + (ψ x) | rfl | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.add_apply | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
addCommGroup : AddCommGroup (WeightTwoAutomorphicForm F D R) | where
add := (· + ·)
add_assoc := by intros; ext; simp [add_assoc];
zero := 0
zero_add := by intros; ext; simp
add_zero := by intros; ext; simp
nsmul := nsmulRec
neg := (-·)
zsmul := zsmulRec
neg_add_cancel := by intros; ext; simp
add_comm := by intros; ext; simp [add_comm] | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.addCommGroup | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
_root_.ConjAct.isOpen_smul {G : Type*} [Group G] [TopologicalSpace G]
[IsTopologicalGroup G] {U : Subgroup G} (hU : IsOpen (U : Set G)) (g : ConjAct G) :
IsOpen ((g • U : Subgroup G) : Set G) | (Homeomorph.smul g).isOpen_image.mpr hU | lemma | ConjAct.isOpen_smul | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
group_smul (g : Dfx F D) (φ : WeightTwoAutomorphicForm F D R) :
WeightTwoAutomorphicForm F D R | where
toFun x := φ (x * g)
left_invt δ x := by simp [left_invt, mul_assoc]
right_invt := by
obtain ⟨U, hU⟩ := φ.right_invt
refine ⟨(toConjAct g) • U, ?_, ?_⟩
· replace hU := hU.1
exact isOpen_smul hU (toConjAct g)
· rintro k x ⟨u, hu, rfl⟩
simp only [MulDistribMulAction.toMonoidEnd_app... | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.group_smul | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | The adelic group action on the space of automorphic forms over a totally definite
quaternion algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
group_smul_apply (g : Dfx F D)
(φ : WeightTwoAutomorphicForm F D R) (x : Dfx F D) :
(g • φ) x = φ (x * g) | rfl | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.group_smul_apply | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
mulAction :
MulAction (Dfx F D) (WeightTwoAutomorphicForm F D R) | where
smul := group_smul
one_smul φ := by ext; simp only [group_smul_apply, mul_one]
mul_smul g h φ := by ext; simp only [group_smul_apply, mul_assoc] | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.mulAction | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
distribMulAction : DistribMulAction (Dfx F D)
(WeightTwoAutomorphicForm F D R) | where
__ := mulAction
smul_zero g := by ext; simp only [group_smul_apply, zero_apply]
smul_add g φ ψ := by ext; simp only [group_smul_apply, add_apply] | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.distribMulAction | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
ring_smul (r : R) (φ : WeightTwoAutomorphicForm F D R) :
WeightTwoAutomorphicForm F D R | where
toFun g := r • φ g
left_invt := by simp [left_invt]
right_invt := by
obtain ⟨U, hU⟩ := φ.right_invt
use U
simp_all only [smul_eq_mul, implies_true, and_self]
trivial_central_char g z := by simp only [trivial_central_char] | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.ring_smul | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | The scalar action on the space of weight 2 automorphic forms on a totally definite
quaternion algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
smul_apply (r : R) (φ : WeightTwoAutomorphicForm F D R)
(g : Dfx F D) :
(r • φ) g = r • (φ g) | rfl | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.smul_apply | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
module : Module R (WeightTwoAutomorphicForm F D R) | where
one_smul g := by ext; simp [smul_apply]
mul_smul r s g := by ext; simp [smul_apply, mul_assoc]
smul_zero r := by ext; simp [smul_apply]
smul_add r f g := by ext; simp [smul_apply, mul_add]
add_smul r s g := by ext; simp [smul_apply, add_mul]
zero_smul g := by ext; simp [smul_apply] | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.module | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
WeightTwoAutomorphicFormOfLevel (U : Subgroup (Dfx F D))
(R : Type*) [CommRing R] : Type _ | MulAction.FixedPoints U (WeightTwoAutomorphicForm F D R) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [
"MulAction.FixedPoints"
] | `WeightTwoAutomorphicFormOfLevel U R` is the `R`-valued weight 2 automorphic forms of a fixed
level `U` for a totally definite quaternion algebra over a totally real field. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
toFun (f : WeightTwoAutomorphicFormOfLevel U R)
(x : Dfx F D) : R | f.1.toFun x | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel.toFun | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | Enables coercion of automorphic forms to functions. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
ext ⦃f g : WeightTwoAutomorphicFormOfLevel U R⦄ (h : ∀ x, f x = g x) : f = g | Subtype.ext <| WeightTwoAutomorphicForm.ext _ _ h | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel.ext | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
left_invt (f : WeightTwoAutomorphicFormOfLevel U R) (δ : Dˣ) (g : Dfx F D) :
f ((incl₁ F D) δ * g) = f g | f.1.left_invt δ g | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel.left_invt | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
right_invt (f : WeightTwoAutomorphicFormOfLevel U R) (g : Dfx F D) (u : U) :
f (g * u) = f g | congr($(f.2 u) g) | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicFormOfLevel.right_invt | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/Defs.lean | [
"Mathlib.MeasureTheory.Integral.Bochner.Basic"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
WeightTwoAutomorphicForm.finiteDimensional
(hU : IsOpen (U : Set (Dfx F D))) :
FiniteDimensional K (WeightTwoAutomorphicFormOfLevel U K) | by
let H' : Subgroup (Dfx F D) := (incl₁ F D).range
-- We will define a free K-module with a basis indexed by
-- the elements of a double coset space which (in the totally
-- definite case) is finite)
let X := DoubleCoset.Quotient (Set.range (incl₁ F D)) U
borelize (D ⊗[F] AdeleRing (𝓞 F) F)
-- (the fini... | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.finiteDimensional | AutomorphicForm.QuaternionAlgebra | FLT/AutomorphicForm/QuaternionAlgebra/FiniteDimensional.lean | [
"FLT.DivisionAlgebra.Finiteness",
"Mathlib.RingTheory.PicardGroup"
] | [
"NumberField.FiniteAdeleRing.DivisionAlgebra.finiteDoubleCoset"
] | Let `D/F` be a totally definite quaterion algebra over a totally real
field. Then the space of `K`-valued weight 2 level `U` quaternionic automorphic forms
for `Dˣ` is finite-dimensional over `K`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
coe_zero : ((0 : fixedPoints G A) : A) = 0 | rfl | lemma | FixedPoints.coe_zero | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
coe_add (a b : fixedPoints G A) :
((a + b : fixedPoints G A) : A) = a + b | rfl | lemma | FixedPoints.coe_add | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
coe_smul [SMul R A] [SMulCommClass G R A] (r : R) (a : fixedPoints G A) :
((r • a : fixedPoints G A) : A) = r • a | rfl | lemma | FixedPoints.coe_smul | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
module [Ring R] [Module R A] [SMulCommClass G R A] : Module R (fixedPoints G A) | where
one_smul a := one_smul _ _
mul_smul r s a := mul_smul _ _ _
smul_zero a := by
ext
exact smul_zero _
smul_add r s a := by
ext
exact smul_add _ _ _
add_smul r s a := by
ext
exact add_smul _ _ _
zero_smul a := by
ext
exact zero_smul _ _ | instance | FixedPoints.module | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
Set.bijOn_smul (hu : u ∈ U) : Set.BijOn (fun x ↦ u • x) ((U : Set G) * X) (U * X) | by
refine ⟨?_, Set.injOn_of_injective (MulAction.injective u), ?_⟩
· rintro x ⟨u', hu', x, hx, rfl⟩
exact ⟨u * u', mul_mem hu hu', x, hx, by simp [mul_assoc]⟩
· rintro x ⟨u', hu', x, hx, rfl⟩
exact ⟨(u⁻¹ * u') * x, ⟨u⁻¹ * u', mul_mem (inv_mem hu) hu', x, hx, rfl⟩, by simp [mul_assoc]⟩ | lemma | Set.bijOn_smul | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
eq_finsum_quotient_out_of_bijOn' (a : fixedPoints V A)
{X : Set (G ⧸ V)}
{s : Set G} (hs : s.BijOn (QuotientGroup.mk : G → G ⧸ V) X) :
∑ᶠ g ∈ s, g • (a : A) = ∑ᶠ g ∈ Quotient.out '' X, g • (a : A) | by
let e (g : G) : G := Quotient.out (QuotientGroup.mk g : G ⧸ V)
have he₀ : Set.BijOn e s (Quotient.out '' X) := by
refine Set.BijOn.comp ?_ hs
exact Set.InjOn.bijOn_image <| Set.injOn_of_injective Quotient.out_injective
have he₁ : ∀ g ∈ s, g • (a : A) = (Quotient.out (QuotientGroup.mk g : G ⧸ V)) • a :=... | lemma | AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn' | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | If `a` is fixed by `V` then `∑ᶠ g ∈ s, g • a` is independent of the choice `s` of
coset representatives in `G` for a subset of `G ⧸ V` | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
HeckeOperator_toFun (a : fixedPoints V A) : fixedPoints U A | ⟨∑ᶠ gᵢ ∈ Quotient.out '' (QuotientGroup.mk '' (U * {g}) : Set (G ⧸ V)), gᵢ • a.1, by
rintro ⟨u, huU⟩
rw [smul_finsum_mem (h.image Quotient.out), ← eq_finsum_quotient_out_of_bijOn' a]
· rw [finsum_mem_eq_of_bijOn (fun g ↦ u • g)]
· exact Set.InjOn.bijOn_image <| Set.injOn_of_injective (MulAction.injective u)
... | def | AbstractHeckeOperator.HeckeOperator_toFun | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [
"Set.bijOn_smul"
] | The Hecke operator T_g = [UgV] : A^V → A^U associated to the double coset UgV. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
HeckeOperator_addMonoidHom : fixedPoints V A →+ fixedPoints U A | where
toFun := HeckeOperator_toFun h
map_zero' := by
ext
simp [HeckeOperator_toFun]
map_add' a b := by
ext
simp only [HeckeOperator_toFun, FixedPoints.coe_add, smul_add,
finsum_mem_add_distrib (h.image Quotient.out)] | def | AbstractHeckeOperator.HeckeOperator_addMonoidHom | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [
"FixedPoints.coe_add"
] | The Hecke operator `T_g = [UgV] : A^V → A^U` packaged as an additive monoid homomorphism. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
HeckeOperator : fixedPoints V A →ₗ[R] fixedPoints U A | where
toFun := HeckeOperator_toFun h
map_add' a b := by
ext
simp only [HeckeOperator_toFun, FixedPoints.coe_add, smul_add,
finsum_mem_add_distrib (h.image Quotient.out)]
map_smul' r a := by
ext
simp only [HeckeOperator_toFun, FixedPoints.coe_smul, smul_comm,
smul_finsum_mem (h.image Qu... | def | AbstractHeckeOperator.HeckeOperator | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [
"FixedPoints.coe_add",
"FixedPoints.coe_smul"
] | The Hecke operator `T_g = [UgV] : A^V → A^U` as an `R`-linear map, where `R` is any ring
acting on `A` and commuting with the `G`-action. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
HeckeOperator_apply (a : fixedPoints V A) :
(HeckeOperator (R := R) g U V h a : A) =
∑ᶠ (gᵢ ∈ Quotient.out '' (QuotientGroup.mk '' (U * {g}) : Set (G ⧸ V))), gᵢ • (a : A) | rfl | lemma | AbstractHeckeOperator.HeckeOperator_apply | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
comm {g₁ g₂ : G} (h₁ : (QuotientGroup.mk '' (U * {g₁}) : Set (G ⧸ U)).Finite)
(h₂ : (QuotientGroup.mk '' (U * {g₂}) : Set (G ⧸ U)).Finite)
(hcomm : ∃ s₁ s₂ : Set G,
Set.BijOn QuotientGroup.mk s₁ (QuotientGroup.mk '' (U * {g₁}) : Set (G ⧸ U)) ∧
Set.BijOn QuotientGroup.mk s₂ (QuotientGroup.mk '' (U * ... | by
ext a
rcases hcomm with ⟨s₁, s₂, hs₁, hs₂, hcomm⟩
simp only [LinearMap.coe_comp, Function.comp_apply]
nth_rw 1 [HeckeOperator_apply]
rw [← eq_finsum_quotient_out_of_bijOn' _ hs₁]
nth_rw 1 [HeckeOperator_apply]
rw [← eq_finsum_quotient_out_of_bijOn' _ hs₂]
nth_rw 1 [HeckeOperator_apply]
rw [← eq_fin... | theorem | AbstractHeckeOperator.comm | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Abstract.lean | [
"Mathlib.Algebra.BigOperators.GroupWithZero.Action",
"Mathlib.GroupTheory.GroupAction.Quotient"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
QuotientGroup.mk_image_finite_of_compact_of_open
(hU : IsCompact (U : Set G)) (hVopen : IsOpen (V : Set G)) :
(QuotientGroup.mk '' (U * {g}) : Set (G ⧸ V)).Finite | by
have : DiscreteTopology (G ⧸ V) := by
rw [discreteTopology_iff_forall_isOpen]
intro s
rw [← (isQuotientMap_mk V).isOpen_preimage, ← (QuotientGroup.mk_surjective).image_preimage s,
preimage_image_mk_eq_iUnion_image, iUnion_subtype]
conv in ⋃ x ∈ _, _ => change ⋃ x ∈ (V : Set G), _
rw [iUni... | lemma | QuotientGroup.mk_image_finite_of_compact_of_open | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U1 : Subgroup (D ⊗[F] (IsDedekindDomain.FiniteAdeleRing (𝓞 F) F))ˣ | Subgroup.map (Units.map r.symm.toMonoidHom) (GL2.TameLevel S) | abbrev | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.U1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | U1(S) | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
U1_compact : IsCompact (U1 r S : Set (D ⊗[F] (IsDedekindDomain.FiniteAdeleRing (𝓞 F) F))ˣ) | sorry | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.U1_compact | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U1_open : IsOpen (U1 r S : Set (D ⊗[F] (IsDedekindDomain.FiniteAdeleRing (𝓞 F) F))ˣ) | sorry | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.U1_open | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
T (v : HeightOneSpectrum (𝓞 F)) :
WeightTwoAutomorphicFormOfLevel (U1 r S) R →ₗ[R]
WeightTwoAutomorphicFormOfLevel (U1 r S) R | letI : DecidableEq (HeightOneSpectrum (𝓞 F)) := Classical.typeDecidableEq _
let g : (D ⊗[F] (IsDedekindDomain.FiniteAdeleRing (𝓞 F) F))ˣ :=
Units.map r.symm.toMonoidHom (Matrix.GeneralLinearGroup.diagonal
![FiniteAdeleRing.localUniformiserUnit F v, 1])
AbstractHeckeOperator.HeckeOperator (R := R) g (U1 r ... | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.T | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [
"AbstractHeckeOperator.HeckeOperator",
"Matrix.GeneralLinearGroup.diagonal",
"QuotientGroup.mk_image_finite_of_compact_of_open"
] | The Hecke operator T_v as an R-linear map from R-valued quaternionic weight 2
automorphic forms of level U_1(S). | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
diag :
(D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ | Units.mapEquiv r.symm.toMulEquiv
(FiniteAdeleRing.GL2.restrictedProduct.symm
(RestrictedProduct.mulSingle _ _ (Local.GL2.diag α hα))) | abbrev | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | The (global) matrix element `diag[α, 1]`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
unipotent_mul_diag (t : ↑(adicCompletionIntegers F v) ⧸ (Ideal.span {α})) :
(D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ | Units.mapEquiv r.symm.toMulEquiv
(FiniteAdeleRing.GL2.restrictedProduct.symm
(RestrictedProduct.mulSingle _ _
(Local.GL2.unipotent_mul_diag α hα (Quotient.out t : adicCompletionIntegers F v)))) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.unipotent_mul_diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | The (global) matrix element `(unipotent t) * (diag α hα) = !![α, t; 0, 1]`.
Here `t ∈ 𝒪ᵥ / α` and we lift it arbitrarily to `𝒪ᵥ`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
unipotent_mul_diag_image :
Set (D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ | (unipotent_mul_diag r α hα) '' ⊤ | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.unipotent_mul_diag_image | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [
"IsQuaternionAlgebra"
] | The set of elements `unipotent_mul_diag`, that is, the elements of `(D ⊗ 𝔸_F^∞)ˣ`
which are `(α t;0 1)` at `v` and the identity elsewhere, as `t` runs through a set
of coset reps of `𝓞ᵥ / α`. These will form a set of coset representatives for `U1 diag U1`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
unipotent_mul_diag_inj :
Set.InjOn (unipotent_mul_diag r α hα) ⊤ | by
intro t₁ h₁ t₂ h₂ h
simp only [unipotent_mul_diag, EmbeddingLike.apply_eq_iff_eq, RestrictedProduct.ext_iff] at h
let h' := h v; simp only [RestrictedProduct.mulSingle_eq_same, Units.ext_iff] at h'
rw [← Matrix.ext_iff] at h'
let h'' := h' 0 1
simpa [Local.GL2.unipotent_mul_diag, Matrix.GeneralLinearGrou... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.unipotent_mul_diag_inj | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [
"Matrix.GeneralLinearGroup.GL2.unipotent",
"Matrix.GeneralLinearGroup.diagonal"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U1diagU1 :
Set ((D ⊗[F] (FiniteAdeleRing (𝓞 F) F))ˣ ⧸ (U1 r S)) | QuotientGroup.mk '' ((U1 r S) * {diag r α hα}) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | The double coset space `U₁(S) diag(αᵥ,1) U₁(S)` as a set of left cosets. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
bijOn_unipotent_mul_diagU1_U1diagU1 :
(unipotent_mul_diag_image r α hα).BijOn QuotientGroup.mk (U1diagU1 r S α hα) | sorry | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.bijOn_unipotent_mul_diagU1_U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
unipotent_mul_diag_image_finite :
(unipotent_mul_diag_image r α hα).Finite | by
apply (Set.BijOn.finite_iff_finite (bijOn_unipotent_mul_diagU1_U1diagU1 r {v} α hα)).mpr
unfold U1diagU1
exact (QuotientGroup.mk_image_finite_of_compact_of_open (U1_compact r {v}) (U1_open r {v})) | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.unipotent_mul_diag_image_finite | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [
"QuotientGroup.mk_image_finite_of_compact_of_open"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
quot_top_finite (r : Rigidification F D) (α : v.adicCompletionIntegers F) (hα : α ≠ 0) :
(⊤ : Set ((adicCompletionIntegers F v) ⧸ (Ideal.span {α}))).Finite | by
apply Set.Finite.of_finite_image _ (unipotent_mul_diag_inj r α hα)
apply unipotent_mul_diag_image_finite | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.quot_top_finite | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U :
WeightTwoAutomorphicFormOfLevel (U1 r S) R →ₗ[R]
WeightTwoAutomorphicFormOfLevel (U1 r S) R | AbstractHeckeOperator.HeckeOperator (R := R) (diag r α hα) (U1 r S) (U1 r S)
(QuotientGroup.mk_image_finite_of_compact_of_open (U1_compact r S) (U1_open r S)) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [
"AbstractHeckeOperator.HeckeOperator",
"QuotientGroup.mk_image_finite_of_compact_of_open"
] | The Hecke operator U_{v,α} associated to the matrix (α 0;0 1) at v,
considered as an R-linear map from R-valued quaternionic weight 2
automorphic forms of level U_1(S). Here α is a nonzero element of 𝓞ᵥ.
We do not demand the condition v ∈ S, the bad primes, but this operator
should only be used in this setting. See al... | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
_root_.Ne.mul {M₀ : Type*} [Mul M₀] [Zero M₀] [NoZeroDivisors M₀] {a b : M₀}
(ha : a ≠ 0) (hb : b ≠ 0) : a * b ≠ 0 | mul_ne_zero ha hb | lemma | Ne.mul | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U_apply (a : WeightTwoAutomorphicFormOfLevel (U1 r S) R) :
((U r S R α hα) a).1 =
∑ᶠ (gᵢ : (D ⊗[F] FiniteAdeleRing (𝓞 F) F)ˣ) (_ : gᵢ ∈ Quotient.out '' (U1diagU1 r S α hα)),
gᵢ • a.1 | rfl | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_apply | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U_apply_eq_finsum_unipotent_mul_diag_image (a : WeightTwoAutomorphicFormOfLevel (U1 r S) R) :
((U r S R α hα) a).1 =
∑ᶠ (g : (D ⊗[F] FiniteAdeleRing (𝓞 F) F)ˣ) (_ : g ∈ unipotent_mul_diag_image r α hα),
g • a.1 | (eq_finsum_quotient_out_of_bijOn' a (bijOn_unipotent_mul_diagU1_U1diagU1 r S α hα)) ▸
U_apply r S R α hα a | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_apply_eq_finsum_unipotent_mul_diag_image | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U_mul_aux {v : HeightOneSpectrum (𝓞 F)}
{α β : v.adicCompletionIntegers F} (hα : α ≠ 0) (hβ : β ≠ 0)
(a : WeightTwoAutomorphicFormOfLevel (U1 r S) R) :
∑ᶠ (i : (adicCompletionIntegers F v) ⧸ Ideal.span {α})
(j : (adicCompletionIntegers F v) ⧸ Ideal.span {β}),
unipotent_mul_diag r α hα i • unipo... | sorry | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_mul_aux | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U_mul {v : HeightOneSpectrum (𝓞 F)}
{α β : v.adicCompletionIntegers F} (hα : α ≠ 0) (hβ : β ≠ 0) :
(U r S R α hα ∘ₗ U r S R β hβ) =
U r S R (α * β) (hα.mul hβ) | by
ext1 a
apply (Subtype.coe_inj).mp
simp only [U_apply_eq_finsum_unipotent_mul_diag_image,
LinearMap.coe_comp, Function.comp_apply,
smul_finsum_mem (unipotent_mul_diag_image_finite r β hβ)]
unfold unipotent_mul_diag_image
simp only [finsum_mem_image (unipotent_mul_diag_inj _ _ _)]
simpa using U_mul... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_mul | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U_comm {v : HeightOneSpectrum (𝓞 F)}
{α β : v.adicCompletionIntegers F} (hα : α ≠ 0) (hβ : β ≠ 0) :
U r S R α hα ∘ₗ U r S R β hβ =
U r S R β hβ ∘ₗ U r S R α hα | by
rw [U_mul, U_mul]
congr 1
rw [mul_comm] | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.U_comm | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
HeckeAlgebra : Type _ | (Algebra.adjoin R ({T r R v | v ∉ S} ∪
{φ | ∃ (v : HeightOneSpectrum (𝓞 F)) (_hv : v ∈ S)
(α : v.adicCompletionIntegers F) (hα : α ≠ 0), φ = U r S R α hα}) :
Subalgebra R (WeightTwoAutomorphicFormOfLevel (U1 r S) R →ₗ[R]
WeightTwoAutomorphicFormOfLevel (U1 r S) R)) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | `HeckeAlgebra F D r S R` is the Hecke algebra associated to the weight 2
`R`-valued automorphic forms associated to the discriminant 1 totally definite
quaternion algebra `D` over the totally real field `F`, of level `U₁(S)` where `S` is
a finite set of nonzero primes `v` of `𝓞 F`. To make sense of this definition we ... | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
instRing :
Ring (HeckeAlgebra F D r S R) | inferInstanceAs <|
Ring (Algebra.adjoin R _ : Subalgebra R (WeightTwoAutomorphicFormOfLevel (U1 r S) R →ₗ[R]
WeightTwoAutomorphicFormOfLevel (U1 r S) R)) | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra.instRing | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
instAlgebra :
Algebra R (HeckeAlgebra F D r S R) | inferInstanceAs <|
Algebra R (Algebra.adjoin R _ : Subalgebra R (WeightTwoAutomorphicFormOfLevel (U1 r S) R →ₗ[R]
WeightTwoAutomorphicFormOfLevel (U1 r S) R)) | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra.instAlgebra | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
instCommRing :
CommRing (HeckeAlgebra F D r S R) | where
__ := instRing F D r S R
mul_comm := sorry | instance | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra.instCommRing | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
T (v : HeightOneSpectrum (𝓞 F)) (hv : v ∉ S) : HeckeAlgebra F D r S R | ⟨HeckeOperator.T r R v, by
apply Algebra.subset_adjoin
left
use v⟩ | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra.T | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | The Hecke operator Tᵥ as an element of the Hecke algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
U (v : HeightOneSpectrum (𝓞 F)) (hv : v ∈ S) (α : v.adicCompletionIntegers F)
(hα : α ≠ 0) : HeckeAlgebra F D r S R | ⟨HeckeOperator.U r S R α hα, by
apply Algebra.subset_adjoin
right
use v, hv, α, hα⟩ | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeAlgebra.U | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Concrete.lean | [] | [] | The Hecke operator Uᵥ,ₐ as an element of the Hecke algebra. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
U1 : Subgroup (GL (Fin 2) (adicCompletion F v)) | GL2.localTameLevel v | abbrev | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | The subgroup `U1 v = GL2.localTameLevel v`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
diag : (GL (Fin 2) (adicCompletion F v)) | Matrix.GeneralLinearGroup.diagonal (![⟨(α : v.adicCompletion F),
(α : v.adicCompletion F)⁻¹, by
rw [mul_inv_cancel₀]
exact_mod_cast hα, by
rw [inv_mul_cancel₀]
exact_mod_cast hα⟩, 1]) | abbrev | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [
"Matrix.GeneralLinearGroup.diagonal"
] | The matrix element `diag[α, 1]`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
diag_def :
(diag α hα : Matrix (Fin 2) (Fin 2) (adicCompletion F v))
= !![↑α, 0; 0, 1] | by
rw[diag, Matrix.GeneralLinearGroup.diagonal]
ext i j; fin_cases i; all_goals fin_cases j
all_goals simp | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.diag_def | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [
"Matrix.GeneralLinearGroup.diagonal"
] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
conjBy_diag {a b c d : adicCompletion F v} :
(diag α hα)⁻¹ * !![a, b; c, d] * diag α hα
= !![a, (α : v.adicCompletion F)⁻¹ * b; c * α, d] | by
simp only [Matrix.coe_units_inv, diag_def, Matrix.inv_def, Matrix.det_fin_two_of, mul_one,
mul_zero, sub_zero, Ring.inverse_eq_inv', Matrix.adjugate_fin_two_of, neg_zero, Matrix.smul_of,
Matrix.smul_cons, smul_eq_mul, Matrix.smul_empty, Matrix.cons_mul, Nat.succ_eq_add_one,
Nat.reduceAdd, Matrix.vecMul... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.conjBy_diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
unipotent_mem_U1 (t : v.adicCompletionIntegers F) :
unipotent ↑t ∈ (U1 v) | by
unfold unipotent
constructor
· apply GL2.mem_localFullLevel_iff_v_le_one_and_v_det_eq_one.mpr
constructor
· intro i j
fin_cases i; all_goals fin_cases j
all_goals simp only [Matrix.unitOfDetInvertible, Fin.mk_one, Fin.isValue, Fin.zero_eta,
val_unitOfInvertible, Matrix.of_apply, Mat... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.unipotent_mem_U1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
unipotent_mul_diag (t : v.adicCompletionIntegers F) :
(GL (Fin 2) (adicCompletion F v)) | (unipotent (t : adicCompletion F v)) * (diag α hα) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.unipotent_mul_diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | The matrix element `(unipotent t) * (diag α hα) = !![α, t; 0, 1]`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
unipotent_mul_diag_inv_mul_unipotent_mul_diag (t₁ t₂ : v.adicCompletionIntegers F) :
(unipotent_mul_diag α hα t₁)⁻¹ * unipotent_mul_diag α hα t₂
= unipotent ((α : v.adicCompletion F)⁻¹ * ((t₂ + -t₁) : adicCompletion F v )) | by
ext i j
push_cast [unipotent_mul_diag, mul_inv_rev, unipotent_inv]
rw [← mul_assoc]; nth_rw 2 [mul_assoc]
rw_mod_cast [unipotent_mul]; push_cast [unipotent_def]
rw_mod_cast [conjBy_diag]
simp | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.GL2.unipotent_mul_diag_inv_mul_unipotent_mul_diag | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | `!![α t₁; 0 1]⁻¹ * [α t₂; 0 1] = [1 (t₂ - t₁) / α; 0 1]`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
apply_mem_integer (i j : Fin 2) :
(x i j) ∈ (adicCompletionIntegers F v) | GL2.v_le_one_of_mem_localFullLevel _ hx.left _ _ | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.apply_mem_integer | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
apply_zero_zero_sub_apply_one_one_mem_maximalIdeal :
(⟨(x 0 0), apply_mem_integer hx _ _⟩ - ⟨(x 1 1), apply_mem_integer hx _ _⟩)
∈ IsLocalRing.maximalIdeal (adicCompletionIntegers F v) | (mem_completionIdeal_iff _ v _).mpr hx.right.left | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.apply_zero_zero_sub_apply_one_one_mem_maximalIdeal | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
apply_one_zero_mem_maximalIdeal :
⟨(x 1 0), apply_mem_integer hx _ _⟩
∈ IsLocalRing.maximalIdeal (adicCompletionIntegers F v) | (mem_completionIdeal_iff _ v _).mpr hx.right.right | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.apply_one_zero_mem_maximalIdeal | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
apply_one_one_notMem_maximalIdeal :
⟨(x 1 1), apply_mem_integer hx _ _⟩
∉ IsLocalRing.maximalIdeal (adicCompletionIntegers F v) | by
by_contra mem_maximalIdeal
have det_mem_maximalIdeal :
⟨(x 0 0), apply_mem_integer hx _ _⟩ * ⟨(x 1 1), apply_mem_integer hx _ _⟩
- ⟨(x 0 1), apply_mem_integer hx _ _⟩ * ⟨(x 1 0), apply_mem_integer hx _ _⟩
∈ IsLocalRing.maximalIdeal (adicCompletionIntegers F v) :=
Ideal.sub_mem _
(Idea... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.apply_one_one_notMem_maximalIdeal | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
isUnit_apply_one_one :
IsUnit (⟨(x 1 1), apply_mem_integer hx _ _⟩ : adicCompletionIntegers F v) | (IsLocalRing.notMem_maximalIdeal.mp (apply_one_one_notMem_maximalIdeal hx)) | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.isUnit_apply_one_one | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
conjBy_diag_mem_U1_iff_apply_zero_one_mem_ideal :
(diag α hα)⁻¹ * x * diag α hα ∈ U1 v
↔ ⟨(x 0 1), apply_mem_integer hx _ _⟩ ∈ (Ideal.span {α}) | by
let a : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 0 0⟩
let b : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 0 1⟩
let c : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 1 0 ⟩
let d : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 1 1⟩
have hx₁ : x = !![(a : adi... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1.conjBy_diag_mem_U1_iff_apply_zero_one_mem_ideal | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
U1diagU1 :
Set ((GL (Fin 2) (adicCompletion F v)) ⧸ (U1 v)) | (QuotientGroup.mk '' ((U1 v) * {diag α hα})) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | The double coset space `U1 diag U1` as a set of left cosets. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
unipotent_mul_diagU1
(t : ↑(adicCompletionIntegers F v) ⧸ (Ideal.span {α})) :
((GL (Fin 2) (adicCompletion F v)) ⧸ ↑(U1 v)) | QuotientGroup.mk (unipotent_mul_diag α hα (Quotient.out t : adicCompletionIntegers F v)) | def | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.unipotent_mul_diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | For each `t ∈ O_v / αO_v`, the left coset `unipotent_mul_diag U1`
for a lift of t to `O_v`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
mapsTo_unipotent_mul_diagU1_U1diagU1 :
Set.MapsTo (unipotent_mul_diagU1 v α hα) ⊤ (U1diagU1 v α hα) | (fun t _ => Set.mem_image_of_mem QuotientGroup.mk
(Set.mul_mem_mul (unipotent_mem_U1 (Quotient.out t)) rfl)) | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.mapsTo_unipotent_mul_diagU1_U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | `unipotent_mul_diagU1` is contained in `U1diagU1` for all t. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
injOn_unipotent_mul_diagU1 :
Set.InjOn (unipotent_mul_diagU1 v α hα) ⊤ | by
intro t₁ h₁ t₂ h₂ h
/- If `unipotent_mul_diagU1 t₁ = unipotent_mul_diagU1 t₂`,
then `(unipotent_mul_diag t₁)⁻¹ * (unipotent_mul_diag t₂)` is in `U1 v`.
Note `unipotent_mul_diag_inv_mul_unipotent_mul_diag` tells us that
`(unipotent_mul_diag t₁)⁻¹ * (unipotent_mul_diag t₂)` is `unipotent`. -/
have unipoten... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.injOn_unipotent_mul_diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | Distinct t give distinct `unipotent_mul_diagU1`, i.e. we have a disjoint union. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
surjOn_unipotent_mul_diagU1_U1diagU1 :
Set.SurjOn (unipotent_mul_diagU1 v α hα) ⊤ (U1diagU1 v α hα) | by
rintro _ ⟨_, ⟨x, hx, _, rfl, rfl⟩, rfl⟩
/- Each element of `U1diagU1` can be written as `x * diag`,
where `x = !![a,b;c,d]` is viewed as a matrix over `O_v`. -/
let a : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 0 0⟩
let b : (adicCompletionIntegers F v) := ⟨_, apply_mem_integer hx 0 1⟩
let ... | lemma | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.surjOn_unipotent_mul_diagU1_U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | Each coset in `U1diagU1` is of the form `unipotent_mul_diagU1` for some `t ∈ O_v`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
bijOn_unipotent_mul_diagU1_U1diagU1 :
Set.BijOn (unipotent_mul_diagU1 v α hα) ⊤ (U1diagU1 v α hα) | ⟨mapsTo_unipotent_mul_diagU1_U1diagU1 α hα,
injOn_unipotent_mul_diagU1 α hα,
surjOn_unipotent_mul_diagU1_U1diagU1 α hα⟩ | theorem | TotallyDefiniteQuaternionAlgebra.WeightTwoAutomorphicForm.HeckeOperator.Local.bijOn_unipotent_mul_diagU1_U1diagU1 | AutomorphicForm.QuaternionAlgebra.HeckeOperators | FLT/AutomorphicForm/QuaternionAlgebra/HeckeOperators/Local.lean | [
"FLT.DedekindDomain.AdicValuation"
] | [] | The double coset space `U1diagU1` is the disjoint union of
`unipotent_mul_diagU1` as t ranges over `O_v / αO_v`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
Hurwitz : Type where
/-- The coefficient of `1` in the basis `1, ω, i, ωi`. -/
re : ℤ -- 1
/-- The coefficient of `ω = (-1+i+j+k)/2` in the basis `1, ω, i, ωi`. -/
im_o : ℤ -- ω
/-- The coefficient of `i` in the basis `1, ω, i, ωi`. -/
im_i : ℤ -- i
/-- The coefficient of `ωi = (-1-i+j-k)/2` in the basis ... | structure | Hurwitz | Data | FLT/Data/Hurwitz.lean | [] | [] | Hurwitz integers in the quaternions. ℤ-Basis 1, ω=(-1+i+j+k)/2, i and
ωi=(-1-i+j-k)/2. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
toQuaternion (z : 𝓞) : ℍ | where
re := z.re - 2⁻¹ * z.im_o - 2⁻¹ * z.im_oi
imI := z.im_i + 2⁻¹ * z.im_o - 2⁻¹ * z.im_oi
imJ := 2⁻¹ * z.im_o + 2⁻¹ * z.im_oi
imK := 2⁻¹ * z.im_o - 2⁻¹ * z.im_oi | def | Hurwitz.toQuaternion | Data | FLT/Data/Hurwitz.lean | [] | [] | The embedding of the Hurwitz integers into the rational quaternions. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
fromQuaternion (z : ℍ) : 𝓞 | where
re := Int.floor <| z.re + z.imJ
im_o := Int.floor <| z.imJ + z.imK
im_i := Int.floor <| z.imI - z.imK
im_oi := Int.floor <| z.imJ - z.imK | def | Hurwitz.fromQuaternion | Data | FLT/Data/Hurwitz.lean | [] | [] | Rounds a quaternion to a nearby Hurwitz integer; serves as a left inverse to `toQuaternion`. | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
leftInverse_fromQuaternion_toQuaternion :
Function.LeftInverse fromQuaternion toQuaternion | by
intro z
simp only [fromQuaternion, toQuaternion, sub_add_add_cancel, sub_add_cancel, Int.floor_intCast,
add_add_sub_cancel, ← two_mul, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
mul_inv_cancel_left₀, sub_sub_sub_cancel_right, add_sub_cancel_right, add_sub_sub_cancel] | lemma | Hurwitz.leftInverse_fromQuaternion_toQuaternion | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
toQuaternion_injective : Function.Injective toQuaternion | leftInverse_fromQuaternion_toQuaternion.injective | lemma | Hurwitz.toQuaternion_injective | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
leftInvOn_toQuaternion_fromQuaternion :
Set.LeftInvOn toQuaternion fromQuaternion
{ q : ℍ | ∃ a b c d : ℤ, q = ⟨a, b, c, d⟩ ∨ q = ⟨a + 2⁻¹, b + 2⁻¹, c + 2⁻¹, d + 2⁻¹⟩ } | by
have h₀ (x y : ℤ) : (x + 2 ⁻¹ : ℝ) + (y + 2⁻¹) = ↑(x + y + 1) := by
field_simp; norm_cast; ring
intro q hq
simp only [Set.mem_setOf] at hq
simp only [toQuaternion, fromQuaternion]
obtain ⟨a, b, c, d, rfl|rfl⟩ := hq <;>
ext <;>
simp only [h₀, add_sub_add_right_eq_sub, Int.floor_sub_intCast, Int.floo... | lemma | Hurwitz.leftInvOn_toQuaternion_fromQuaternion | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
fromQuaternion_injOn :
Set.InjOn fromQuaternion
{ q : ℍ | ∃ a b c d : ℤ, q = ⟨a, b, c, d⟩ ∨ q = ⟨a + 2⁻¹, b + 2⁻¹, c + 2⁻¹, d + 2⁻¹⟩ } | leftInvOn_toQuaternion_fromQuaternion.injOn | lemma | Hurwitz.fromQuaternion_injOn | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
zero : 𝓞 | ⟨0, 0, 0, 0⟩ | def | Hurwitz.zero | Data | FLT/Data/Hurwitz.lean | [] | [] | The Hurwitz number 0 | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
zero_re : re (0 : 𝓞) = 0 | rfl | lemma | Hurwitz.zero_re | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
zero_im_o : im_o (0 : 𝓞) = 0 | rfl | lemma | Hurwitz.zero_im_o | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 | |
zero_im_i : im_i (0 : 𝓞) = 0 | rfl | lemma | Hurwitz.zero_im_i | Data | FLT/Data/Hurwitz.lean | [] | [] | https://github.com/ImperialCollegeLondon/FLT | 6cffefeb368ca4cfabc907f86f96783a49ae4033 |
End of preview. Expand in Data Studio
Lean4-FLT
Structured dataset from FLT — Formalization of Fermat's Last Theorem.
Source
- Repository: https://github.com/ImperialCollegeLondon/FLT
- Commit:
6cffefeb368ca4cfabc907f86f96783a49ae4033 - Files: 179
- License: apache-2.0
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 1,754
- With proof: 1,712 (97.6%)
- With docstring: 700 (39.9%)
- Libraries: 71
By type
| Type | Count |
|---|---|
| lemma | 784 |
| def | 467 |
| theorem | 297 |
| instance | 88 |
| abbrev | 75 |
| class | 25 |
| structure | 14 |
| axiom | 3 |
| macro | 1 |
Example
PNat.pow_add_pow_ne_pow
(x y z : ℕ+)
(n : ℕ) (hn : n > 2) :
x^n + y^n ≠ z^n
PNat.pow_add_pow_ne_pow_of_FermatLastTheorem Wiles_Taylor_Wiles x y z n hn
/--
info: 'PNat.pow_add_pow_ne_pow' depends on axioms: [propext, sorryAx, Classical.choice, Quot.sound]
-/
#guard_msgs in
#print axioms PNat.pow_add_pow_ne_pow
- type: theorem | symbolic_name:
PNat.pow_add_pow_ne_pow| FermatsLastTheorem.lean
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{lean4_flt_dataset,
title = {Lean4-FLT},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/ImperialCollegeLondon/FLT, commit 6cffefeb368c},
url = {https://huggingface.co/datasets/phanerozoic/Lean4-FLT}
}
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