Datasets:

phanerozoic
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Lean4-FLT

statement stringlengths 2 2.94k	proof stringlengths 0 6.15k	type stringclasses 9 values	symbolic_name stringlengths 4 131	library stringclasses 71 values	filename stringclasses 155 values	imports listlengths 0 12	deps listlengths 0 11	docstring stringlengths 0 1.4k	source_url stringclasses 1 value	commit stringclasses 1 value
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

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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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