Split (2)

train · 10.3k rows

Context stringlengths 57 6.04k	file_name stringlengths 21 79	start int64 14 1.49k	end int64 18 1.5k	theorem stringlengths 25 1.55k	proof stringlengths 5 7.36k	goals listlengths 1 224	goals_before listlengths 0 221	new_goals listlengths 0 92
import Mathlib.SetTheory.Cardinal.Finite #align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04" namespace Set variable {α β : Type*} {s t : Set α} noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s) @[simp] theorem encard_uni...	Mathlib/Data/Set/Card.lean	116	117	theorem encard_insert_of_not_mem {a : α} (has : a ∉ s) : (insert a s).encard = s.encard + 1 := by	rw [← union_singleton, encard_union_eq (by simpa), encard_singleton]	[ " univ.encard = s.encard", " univ.encard = PartENat.withTopEquiv (PartENat.card α)", " s.encard = ↑h.toFinset.card", " s.encard = ↑s.toFinset.card", " (↑s).encard = ↑s.card", " ↑⋯.toFinset.card = ↑s.card", " s.encard = ⊤", " s.encard = 0 ↔ s = ∅", " ∅.encard = 0", " s.Nonempty", " s.encard ≠ 0 ↔...	[ " univ.encard = s.encard", " univ.encard = PartENat.withTopEquiv (PartENat.card α)", " s.encard = ↑h.toFinset.card", " s.encard = ↑s.toFinset.card", " (↑s).encard = ↑s.card", " ↑⋯.toFinset.card = ↑s.card", " s.encard = ⊤", " s.encard = 0 ↔ s = ∅", " ∅.encard = 0", " s.Nonempty", " s.encard ≠ 0 ↔...	[ " Disjoint s {a}" ]
import Mathlib.Algebra.Order.Group.Abs import Mathlib.Algebra.Order.Group.Basic import Mathlib.Algebra.Order.Group.OrderIso import Mathlib.Algebra.Order.Ring.Defs import Mathlib.Data.Int.Cast.Lemmas import Mathlib.Order.Interval.Set.Basic import Mathlib.Logic.Pairwise #align_import data.set.intervals.group from "lean...	Mathlib/Algebra/Order/Interval/Set/Group.lean	219	221	theorem pairwise_disjoint_Ico_zpow : Pairwise (Disjoint on fun n : ℤ => Ico (b ^ n) (b ^ (n + 1))) := by	simpa only [one_mul] using pairwise_disjoint_Ico_mul_zpow 1 b	[ " Pairwise (Disjoint on fun n => Ioc (a * b ^ n) (a * b ^ (n + 1)))", " Pairwise fun x y => Disjoint (Ioc (a * b ^ x) (a * b ^ (x + 1))) (Ioc (a * b ^ y) (a * b ^ (y + 1)))", " Pairwise fun x y => Ioc (a * b ^ x) (a * b ^ (x + 1)) ∩ Ioc (a * b ^ y) (a * b ^ (y + 1)) ⊆ ∅", " x ∈ ∅", " m = n", " 1 < b", "...	[ " Pairwise (Disjoint on fun n => Ioc (a * b ^ n) (a * b ^ (n + 1)))", " Pairwise fun x y => Disjoint (Ioc (a * b ^ x) (a * b ^ (x + 1))) (Ioc (a * b ^ y) (a * b ^ (y + 1)))", " Pairwise fun x y => Ioc (a * b ^ x) (a * b ^ (x + 1)) ∩ Ioc (a * b ^ y) (a * b ^ (y + 1)) ⊆ ∅", " x ∈ ∅", " m = n", " 1 < b", "...	[]
import Mathlib.Algebra.Algebra.Subalgebra.Unitization import Mathlib.Analysis.RCLike.Basic import Mathlib.Topology.Algebra.StarSubalgebra import Mathlib.Topology.ContinuousFunction.ContinuousMapZero import Mathlib.Topology.ContinuousFunction.Weierstrass #align_import topology.continuous_function.stone_weierstrass fro...	Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean	137	143	theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ))) (f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A := by	convert inf_mem_subalgebra_closure A f g apply SetLike.ext' symm erw [closure_eq_iff_isClosed] exact h	[ " (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound = ↑((Polynomial.aeval f) g)", " ((g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound) a✝ = ↑((Polynomial.aeval f) g) a✝", " Polynomial.eval (↑((↑f).attachBound a✝)) g = Polynomial.eval (↑f a✝) g", " (g.toContinuousMapOn (Set.Icc (...	[ " (g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound = ↑((Polynomial.aeval f) g)", " ((g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound) a✝ = ↑((Polynomial.aeval f) g) a✝", " Polynomial.eval (↑((↑f).attachBound a✝)) g = Polynomial.eval (↑f a✝) g", " (g.toContinuousMapOn (Set.Icc (...	[ " A = A.topologicalClosure", " ↑A = ↑A.topologicalClosure", " ↑A.topologicalClosure = ↑A", " IsClosed ↑A" ]
import Mathlib.NumberTheory.Zsqrtd.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Data.Complex.Basic import Mathlib.Data.Real.Archimedean #align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9" open Zsqrtd Complex open sc...	Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean	81	81	theorem toComplex_def' (x y : ℤ) : ((⟨x, y⟩ : ℤ[i]) : ℂ) = x + y * I := by	simp [toComplex_def]	[ " I * I = ↑(-1)", " toComplex { re := x, im := y } = ↑x + ↑y * I" ]	[ " I * I = ↑(-1)", " toComplex { re := x, im := y } = ↑x + ↑y * I" ]	[]
import Mathlib.Analysis.Calculus.Deriv.Inv import Mathlib.Analysis.NormedSpace.BallAction import Mathlib.Analysis.SpecialFunctions.ExpDeriv import Mathlib.Analysis.InnerProductSpace.Calculus import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Geometry.Manifold.Algebra.LieGroup import Mathlib.Geometry.Manifol...	Mathlib/Geometry/Manifold/Instances/Sphere.lean	163	167	theorem hasFDerivAt_stereoInvFunAux_comp_coe (v : E) : HasFDerivAt (stereoInvFunAux v ∘ ((↑) : (ℝ ∙ v)ᗮ → E)) (ℝ ∙ v)ᗮ.subtypeL 0 := by	have : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((ℝ ∙ v)ᗮ.subtypeL 0) := hasFDerivAt_stereoInvFunAux v convert this.comp (0 : (ℝ ∙ v)ᗮ) (by apply ContinuousLinearMap.hasFDerivAt)	[ " ContDiffOn ℝ ⊤ (stereoToFun v) {x \| ((innerSL ℝ) v) x ≠ 1}", " ContDiffOn ℝ ⊤ (fun x => 2 / (1 - ((innerSL ℝ) v) x)) {x \| ((innerSL ℝ) v) x ≠ 1}", " ContDiffOn ℝ ⊤ (fun x => 1 - ((innerSL ℝ) v) x) {x \| ((innerSL ℝ) v) x ≠ 1}", " ∀ x ∈ {x \| ((innerSL ℝ) v) x ≠ 1}, 1 - ((innerSL ℝ) v) x ≠ 0", " False", " ...	[ " ContDiffOn ℝ ⊤ (stereoToFun v) {x \| ((innerSL ℝ) v) x ≠ 1}", " ContDiffOn ℝ ⊤ (fun x => 2 / (1 - ((innerSL ℝ) v) x)) {x \| ((innerSL ℝ) v) x ≠ 1}", " ContDiffOn ℝ ⊤ (fun x => 1 - ((innerSL ℝ) v) x) {x \| ((innerSL ℝ) v) x ≠ 1}", " ∀ x ∈ {x \| ((innerSL ℝ) v) x ≠ 1}, 1 - ((innerSL ℝ) v) x ≠ 0", " False", " ...	[ " HasFDerivAt ⇑(Submodule.span ℝ {v})ᗮ.subtypeL ?m.122574 0" ]
import Mathlib.SetTheory.Ordinal.Arithmetic import Mathlib.Tactic.TFAE import Mathlib.Topology.Order.Monotone #align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da" noncomputable section universe u v open Cardinal Order Topology namespace Ordina...	Mathlib/SetTheory/Ordinal/Topology.lean	64	65	theorem nhds_left_eq_nhds (a : Ordinal) : 𝓝[≤] a = 𝓝 a := by	rw [← nhds_left_sup_nhds_right', nhds_right', sup_bot_eq]	[ " IsOpen {a} ↔ ¬a.IsLimit", " False", " IsOpen {a}", " IsOpen {0}", " IsOpen (Set.Iio (succ ⊥))", " IsOpen {succ b}", " IsOpen (Set.Ioo b (succ (succ b)))", " 𝓝[<] a = 𝓝[≠] a", " 𝓝[≤] a = 𝓝 a" ]	[ " IsOpen {a} ↔ ¬a.IsLimit", " False", " IsOpen {a}", " IsOpen {0}", " IsOpen (Set.Iio (succ ⊥))", " IsOpen {succ b}", " IsOpen (Set.Ioo b (succ (succ b)))", " 𝓝[<] a = 𝓝[≠] a", " 𝓝[≤] a = 𝓝 a" ]	[]
import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar import Mathlib.MeasureTheory.Integral.Layercake #align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" noncomputable section op...	Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean	100	139	theorem finite_integral_one_add_norm {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) : (∫⁻ x : E, ENNReal.ofReal ((1 + ‖x‖) ^ (-r)) ∂μ) < ∞ := by	have hr : 0 < r := lt_of_le_of_lt (finrank ℝ E).cast_nonneg hnr -- We start by applying the layer cake formula have h_meas : Measurable fun ω : E => (1 + ‖ω‖) ^ (-r) := -- Porting note: was `by measurability` (measurable_norm.const_add _).pow_const _ have h_pos : ∀ x : E, 0 ≤ (1 + ‖x‖) ^ (-r) := fun x ...	[ " √(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖", " 0 ≤ 1 + ‖x‖", " 1 + ‖x‖ ^ 2 ≤ (1 + ‖x‖) ^ 2", " 1 + ‖x‖ ≤ √2 * √(1 + ‖x‖ ^ 2)", " 1 + ‖x‖ ≤ √(2 * (1 + ‖x‖ ^ 2))", " (1 + ‖x‖) ^ 2 ≤ 2 * (1 + ‖x‖ ^ 2)", " (1 + ‖x‖ ^ 2) ^ (-r / 2) = 2 ^ (r / 2) * ((√2 * √(1 + ‖x‖ ^ 2)) ^ r)⁻¹", " √2 ^ r ≠ 0", " 0 ≤ √(1 + ‖x‖ ^ 2)", "...	[ " √(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖", " 0 ≤ 1 + ‖x‖", " 1 + ‖x‖ ^ 2 ≤ (1 + ‖x‖) ^ 2", " 1 + ‖x‖ ≤ √2 * √(1 + ‖x‖ ^ 2)", " 1 + ‖x‖ ≤ √(2 * (1 + ‖x‖ ^ 2))", " (1 + ‖x‖) ^ 2 ≤ 2 * (1 + ‖x‖ ^ 2)", " (1 + ‖x‖ ^ 2) ^ (-r / 2) = 2 ^ (r / 2) * ((√2 * √(1 + ‖x‖ ^ 2)) ^ r)⁻¹", " √2 ^ r ≠ 0", " 0 ≤ √(1 + ‖x‖ ^ 2)", "...	[ " 0 ≤ (1 + ‖x‖) ^ (-r)", " ∫⁻ (t : ℝ) in Ioi 0, μ {a \| t ≤ (1 + ‖a‖) ^ (-r)} < ⊤", " μ {a \| t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))", " {a \| t ≤ (1 + ‖a‖) ^ (-r)} = Metric.closedBall 0 (t ^ (-r⁻¹) - 1)", " x ∈ {a \| t ≤ (1 + ‖a‖) ^ (-r)} ↔ x ∈ Metric.closedBall 0 (t ^ (-r⁻¹) - 1)", ...
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms import Mathlib.CategoryTheory.Limits.Shapes.Kernels import Mathlib.CategoryTheory.Abelian.Basic import Mathlib.CategoryTheory.Subobject.Lattice import Mathlib.Order.Atoms #align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011...	Mathlib/CategoryTheory/Simple.lean	103	107	theorem mono_to_simple_zero_of_not_iso {X Y : C} [Simple Y] {f : X ⟶ Y} [Mono f] (w : IsIso f → False) : f = 0 := by	classical by_contra h exact w (isIso_of_mono_of_nonzero h)	[ " IsIso f ↔ f ≠ 0", " IsIso f → f ≠ 0", " False", " IsIso (f ≫ i.hom)", " f ≠ 0 → IsIso f", " IsIso f", " f ≫ i.hom ≠ 0", " f = 0", " IsIso ((f ≫ i.hom) ≫ i.inv)", " kernel.ι f = 0", " Epi f", " Epi (factorThruImage f ≫ image.ι f)" ]	[ " IsIso f ↔ f ≠ 0", " IsIso f → f ≠ 0", " False", " IsIso (f ≫ i.hom)", " f ≠ 0 → IsIso f", " IsIso f", " f ≫ i.hom ≠ 0", " f = 0", " IsIso ((f ≫ i.hom) ≫ i.inv)", " kernel.ι f = 0", " Epi f", " Epi (factorThruImage f ≫ image.ι f)" ]	[]
import Mathlib.Analysis.SpecialFunctions.PolarCoord import Mathlib.Analysis.SpecialFunctions.Gamma.Basic open Real Set MeasureTheory MeasureTheory.Measure section real theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) : ∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q +...	Mathlib/MeasureTheory/Integral/Gamma.lean	59	63	theorem integral_exp_neg_rpow {p : ℝ} (hp : 0 < p) : ∫ x in Ioi (0:ℝ), exp (- x ^ p) = Gamma (1 / p + 1) := by	convert (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1 · simp_rw [rpow_zero, one_mul] · rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))]	[ " ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * ((q + 1) / p).Gamma", " ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =\n ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))", " ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-(x ^ (1 / p)) ^ p)) =\n ...	[ " ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * ((q + 1) / p).Gamma", " ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =\n ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))", " ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-(x ^ (1 / p)) ^ p)) =\n ...	[ " ∫ (x : ℝ) in Ioi 0, rexp (-x ^ p) = ∫ (x : ℝ) in Ioi 0, x ^ 0 * rexp (-x ^ p)", " (1 / p + 1).Gamma = 1 / p * ((0 + 1) / p).Gamma" ]
import Mathlib.MeasureTheory.Measure.Dirac set_option autoImplicit true open Set open scoped ENNReal Classical variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α} noncomputable section namespace MeasureTheory.Measure def count : Measure α := sum dirac #align measure_theory.measure.count MeasureTheo...	Mathlib/MeasureTheory/Measure/Count.lean	44	44	theorem count_empty : count (∅ : Set α) = 0 := by	rw [count_apply MeasurableSet.empty, tsum_empty]	[ " count s = ∑' (i : ↑s), 1", " count ∅ = 0" ]	[ " count s = ∑' (i : ↑s), 1", " count ∅ = 0" ]	[]
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor import Mathlib.CategoryTheory.Monoidal.Functor #align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055" noncomputable section open scoped Classical namespace CategoryTheory open Cat...	Mathlib/CategoryTheory/Monoidal/Preadditive.lean	57	58	theorem zero_tensor {W X Y Z : C} (f : Y ⟶ Z) : (0 : W ⟶ X) ⊗ f = 0 := by	simp [tensorHom_def]	[ " f ⊗ 0 = 0", " 0 ⊗ f = 0" ]	[ " f ⊗ 0 = 0", " 0 ⊗ f = 0" ]	[]
import Mathlib.Topology.Order.ProjIcc import Mathlib.Topology.CompactOpen import Mathlib.Topology.UnitInterval #align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section open scoped Classical open Topology Filter unitInterval Set Fun...	Mathlib/Topology/Connected/PathConnected.lean	194	200	theorem symm_range {a b : X} (γ : Path a b) : range γ.symm = range γ := by	ext x simp only [mem_range, Path.symm, DFunLike.coe, unitInterval.symm, SetCoe.exists, comp_apply, Subtype.coe_mk] constructor <;> rintro ⟨y, hy, hxy⟩ <;> refine ⟨1 - y, mem_iff_one_sub_mem.mp hy, ?_⟩ <;> convert hxy simp	[ " γ₁ = γ₂", " { toContinuousMap := toContinuousMap✝, source' := source'✝, target' := target'✝ } = γ₂", " { toContinuousMap := toContinuousMap✝¹, source' := source'✝¹, target' := target'✝¹ } =\n { toContinuousMap := toContinuousMap✝, source' := source'✝, target' := target'✝ }", " Continuous ⇑γ.toContinuousM...	[ " γ₁ = γ₂", " { toContinuousMap := toContinuousMap✝, source' := source'✝, target' := target'✝ } = γ₂", " { toContinuousMap := toContinuousMap✝¹, source' := source'✝¹, target' := target'✝¹ } =\n { toContinuousMap := toContinuousMap✝, source' := source'✝, target' := target'✝ }", " Continuous ⇑γ.toContinuousM...	[ " x ∈ range ⇑γ.symm ↔ x ∈ range ⇑γ", " (∃ x_1, ∃ (h : x_1 ∈ I), γ.toFun ⟨1 - x_1, ⋯⟩ = x) ↔ ∃ x_1, ∃ (h : x_1 ∈ I), γ.toFun ⟨x_1, h⟩ = x", " (∃ x_1, ∃ (h : x_1 ∈ I), γ.toFun ⟨1 - x_1, ⋯⟩ = x) → ∃ x_1, ∃ (h : x_1 ∈ I), γ.toFun ⟨x_1, h⟩ = x", " (∃ x_1, ∃ (h : x_1 ∈ I), γ.toFun ⟨x_1, h⟩ = x) → ∃ x_1, ∃ (h : x_1 ...
import Mathlib.Data.List.OfFn import Mathlib.Data.List.Range #align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" universe u namespace List variable {α : Type u} @[simp] theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ...	Mathlib/Data/List/FinRange.lean	44	47	theorem ofFn_eq_pmap {n} {f : Fin n → α} : ofFn f = pmap (fun i hi => f ⟨i, hi⟩) (range n) fun _ => mem_range.1 := by	rw [pmap_eq_map_attach] exact ext_get (by simp) fun i hi1 hi2 => by simp [get_ofFn f ⟨i, hi1⟩]	[ " map Fin.val (finRange n) = range n", " map (fun a => a) (range n) = range n", " finRange n.succ = 0 :: map Fin.succ (finRange n)", " map Fin.val (finRange n.succ) = map Fin.val (0 :: map Fin.succ (finRange n))", " 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)", ...	[ " map Fin.val (finRange n) = range n", " map (fun a => a) (range n) = range n", " finRange n.succ = 0 :: map Fin.succ (finRange n)", " map Fin.val (finRange n.succ) = map Fin.val (0 :: map Fin.succ (finRange n))", " 0 :: map (Nat.succ ∘ Fin.val) (finRange n) = 0 :: map (Fin.val ∘ Fin.succ) (finRange n)", ...	[ " ofFn f = map (fun x => f ⟨↑x, ⋯⟩) (range n).attach", " (ofFn f).length = (map (fun x => f ⟨↑x, ⋯⟩) (range n).attach).length", " (ofFn f).get ⟨i, hi1⟩ = (map (fun x => f ⟨↑x, ⋯⟩) (range n).attach).get ⟨i, hi2⟩" ]
import Mathlib.Algebra.Group.Fin import Mathlib.LinearAlgebra.Matrix.Symmetric #align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1" variable {α β m n R : Type*} namespace Matrix open Function open Matrix def circulant [Sub n] (v : n → α)...	Mathlib/LinearAlgebra/Matrix/Circulant.lean	142	151	theorem circulant_mul_comm [CommSemigroup α] [AddCommMonoid α] [Fintype n] [AddCommGroup n] (v w : n → α) : circulant v * circulant w = circulant w * circulant v := by	ext i j simp only [mul_apply, circulant_apply, mul_comm] refine Fintype.sum_equiv ((Equiv.subLeft i).trans (Equiv.addRight j)) _ _ ?_ intro x simp only [Equiv.trans_apply, Equiv.subLeft_apply, Equiv.coe_addRight, add_sub_cancel_right, mul_comm] congr 2 abel	[ " Injective circulant", " v = w", " v k = w k", " Injective fun v => circulant v", " (circulant v)ᵀ = circulant fun i => v (-i)", " (circulant v)ᵀ i✝ j✝ = circulant (fun i => v (-i)) i✝ j✝", " (circulant v)ᴴ = circulant (star fun i => v (-i))", " (circulant v)ᴴ i✝ j✝ = circulant (star fun i => v (-i))...	[ " Injective circulant", " v = w", " v k = w k", " Injective fun v => circulant v", " (circulant v)ᵀ = circulant fun i => v (-i)", " (circulant v)ᵀ i✝ j✝ = circulant (fun i => v (-i)) i✝ j✝", " (circulant v)ᴴ = circulant (star fun i => v (-i))", " (circulant v)ᴴ i✝ j✝ = circulant (star fun i => v (-i))...	[ " (circulant v * circulant w) i j = (circulant w * circulant v) i j", " ∑ x : n, v (i - x) * w (x - j) = ∑ x : n, w (i - x) * v (x - j)", " ∀ (x : n),\n v (i - x) * w (x - j) =\n w (i - ((Equiv.subLeft i).trans (Equiv.addRight j)) x) * v (((Equiv.subLeft i).trans (Equiv.addRight j)) x - j)", " v (i - ...
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology import Mathlib.Analysis.SpecialFunctions.Arsinh import Mathlib.Geometry.Euclidean.Inversion.Basic #align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c" noncomputable section ope...	Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean	45	47	theorem sinh_half_dist (z w : ℍ) : sinh (dist z w / 2) = dist (z : ℂ) w / (2 * √(z.im * w.im)) := by	rw [dist_eq, mul_div_cancel_left₀ (arsinh _) two_ne_zero, sinh_arsinh]	[ " (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))" ]	[ " (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))" ]	[]
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence import Mathlib.Algebra.ContinuedFractions.TerminatedStable import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.Ring #align_import algebra.continued_fractions.convergents_equiv from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40a...	Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean	114	117	theorem squashSeq_nth_of_not_terminated {gp_n gp_succ_n : Pair K} (s_nth_eq : s.get? n = some gp_n) (s_succ_nth_eq : s.get? (n + 1) = some gp_succ_n) : (squashSeq s n).get? n = some ⟨gp_n.a, gp_n.b + gp_succ_n.a / gp_succ_n.b⟩ := by	simp [*, squashSeq]	[ " squashSeq s n = s", " (squashSeq s n).get? n = some { a := gp_n.a, b := gp_n.b + gp_succ_n.a / gp_succ_n.b }" ]	[ " squashSeq s n = s", " (squashSeq s n).get? n = some { a := gp_n.a, b := gp_n.b + gp_succ_n.a / gp_succ_n.b }" ]	[]
import Mathlib.Data.Nat.Defs import Mathlib.Data.Option.Basic import Mathlib.Data.List.Defs import Mathlib.Init.Data.List.Basic import Mathlib.Init.Data.List.Instances import Mathlib.Init.Data.List.Lemmas import Mathlib.Logic.Unique import Mathlib.Order.Basic import Mathlib.Tactic.Common #align_import data.list.basic...	Mathlib/Data/List/Basic.lean	87	91	theorem _root_.Decidable.List.eq_or_ne_mem_of_mem [DecidableEq α] {a b : α} {l : List α} (h : a ∈ b :: l) : a = b ∨ a ≠ b ∧ a ∈ l := by	by_cases hab : a = b · exact Or.inl hab · exact ((List.mem_cons.1 h).elim Or.inl (fun h => Or.inr ⟨hab, h⟩))	[ " a = b ∨ a ≠ b ∧ a ∈ l" ]	[ " a = b ∨ a ≠ b ∧ a ∈ l" ]	[]
import Aesop import Mathlib.Algebra.Group.Defs import Mathlib.Data.Nat.Defs import Mathlib.Data.Int.Defs import Mathlib.Logic.Function.Basic import Mathlib.Tactic.Cases import Mathlib.Tactic.SimpRw import Mathlib.Tactic.SplitIfs #align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a...	Mathlib/Algebra/Group/Basic.lean	323	325	theorem mul_right_eq_self : a * b = a ↔ b = 1 := calc a * b = a ↔ a * b = a * 1 := by	rw [mul_one] _ ↔ b = 1 := mul_left_cancel_iff	[ " a * b = a ↔ a * b = a * 1" ]	[ " a * b = a ↔ a * b = a * 1" ]	[]
import Mathlib.LinearAlgebra.LinearPMap import Mathlib.Topology.Algebra.Module.Basic #align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" open Topology variable {R E F : Type*} variable [CommRing R] [AddCommGroup E] [AddCommGroup F] vari...	Mathlib/Topology/Algebra/Module/LinearPMap.lean	89	92	theorem IsClosable.existsUnique {f : E →ₗ.[R] F} (hf : f.IsClosable) : ∃! f' : E →ₗ.[R] F, f.graph.topologicalClosure = f'.graph := by	refine exists_unique_of_exists_of_unique hf fun _ _ hy₁ hy₂ => eq_of_eq_graph ?_ rw [← hy₁, ← hy₂]	[ " g.IsClosable", " g.graph.topologicalClosure ≤ f'.graph", " g.graph.topologicalClosure ≤ f.graph.topologicalClosure", " g.graph.topologicalClosure = g.graph.topologicalClosure.toLinearPMap.graph", " ∀ x ∈ g.graph.topologicalClosure, x.1 = 0 → x.2 = 0", " ∃! f', f.graph.topologicalClosure = f'.graph", "...	[ " g.IsClosable", " g.graph.topologicalClosure ≤ f'.graph", " g.graph.topologicalClosure ≤ f.graph.topologicalClosure", " g.graph.topologicalClosure = g.graph.topologicalClosure.toLinearPMap.graph", " ∀ x ∈ g.graph.topologicalClosure, x.1 = 0 → x.2 = 0", " ∃! f', f.graph.topologicalClosure = f'.graph" ]	[ " x✝¹.graph = x✝.graph" ]
import Mathlib.Algebra.Algebra.Defs import Mathlib.Algebra.Order.Group.Basic import Mathlib.Algebra.Order.Ring.Basic import Mathlib.RingTheory.Localization.Basic import Mathlib.SetTheory.Game.Birthday import Mathlib.SetTheory.Surreal.Basic #align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9...	Mathlib/SetTheory/Surreal/Dyadic.lean	85	86	theorem birthday_half : birthday (powHalf 1) = 2 := by	rw [birthday_def]; simp	[ " (powHalf n).LeftMoves = PUnit.{u_1 + 1}", " (powHalf 0).LeftMoves = PUnit.{u_1 + 1}", " (powHalf (n✝ + 1)).LeftMoves = PUnit.{u_1 + 1}", " (powHalf n).moveLeft i = 0", " (powHalf 0).moveLeft i = 0", " (powHalf (n✝ + 1)).moveLeft i = 0", " (powHalf 0).moveLeft PUnit.unit = 0", " (powHalf (n✝ + 1)).mo...	[ " (powHalf n).LeftMoves = PUnit.{u_1 + 1}", " (powHalf 0).LeftMoves = PUnit.{u_1 + 1}", " (powHalf (n✝ + 1)).LeftMoves = PUnit.{u_1 + 1}", " (powHalf n).moveLeft i = 0", " (powHalf 0).moveLeft i = 0", " (powHalf (n✝ + 1)).moveLeft i = 0", " (powHalf 0).moveLeft PUnit.unit = 0", " (powHalf (n✝ + 1)).mo...	[ " max (Ordinal.lsub fun i => ((powHalf 1).moveLeft i).birthday)\n (Ordinal.lsub fun i => ((powHalf 1).moveRight i).birthday) =\n 2" ]
import Mathlib.Topology.Sheaves.Sheaf import Mathlib.CategoryTheory.Sites.Limits import Mathlib.CategoryTheory.Limits.FunctorCategory #align_import topology.sheaves.limits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" noncomputable section universe v u open CategoryTheory open ...	Mathlib/Topology/Sheaves/Limits.lean	41	49	theorem isSheaf_of_isLimit [HasLimits C] {X : TopCat} (F : J ⥤ Presheaf.{v} C X) (H : ∀ j, (F.obj j).IsSheaf) {c : Cone F} (hc : IsLimit c) : c.pt.IsSheaf := by	let F' : J ⥤ Sheaf C X := { obj := fun j => ⟨F.obj j, H j⟩ map := fun f => ⟨F.map f⟩ } let e : F' ⋙ Sheaf.forget C X ≅ F := NatIso.ofComponents fun _ => Iso.refl _ exact Presheaf.isSheaf_of_iso ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e) (limit F').2	[ " c.pt.IsSheaf" ]	[ " c.pt.IsSheaf" ]	[]
import Mathlib.Data.ENNReal.Inv #align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open Set NNReal ENNReal namespace ENNReal section iInf variable {ι : Sort*} {f g : ι → ℝ≥0∞} variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0} theorem toNNReal_iInf (hf : ∀ i, f ...	Mathlib/Data/ENNReal/Real.lean	564	569	theorem toNNReal_sSup (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) : (sSup s).toNNReal = sSup (ENNReal.toNNReal '' s) := by	have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs -- Porting note: `← sSup_image'` had to be replaced by `← image_eq_range` as the lemmas are used -- in a different order. simpa only [← sSup_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iSup hf)	[ " (iInf f).toNNReal = ⨅ i, (f i).toNNReal", " (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal", " (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)", " (iSup f).toNNReal = ⨆ i, (f i).toNNReal", " (⨆ i, ↑(f i)).toNNReal = ⨆ i, ((fun i => ↑(f i)) i).toNNReal", " (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i...	[ " (iInf f).toNNReal = ⨅ i, (f i).toNNReal", " (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i => ↑(f i)) i).toNNReal", " (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)", " (iSup f).toNNReal = ⨆ i, (f i).toNNReal", " (⨆ i, ↑(f i)).toNNReal = ⨆ i, ((fun i => ↑(f i)) i).toNNReal", " (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i...	[]
import Mathlib.Algebra.GCDMonoid.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.Ideal.Basic import Mathlib.RingTheory.PrincipalIdealDomain #align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6" section open Euclidean...	Mathlib/RingTheory/EuclideanDomain.lean	50	55	theorem right_div_gcd_ne_zero {p q : R} (hq : q ≠ 0) : q / GCDMonoid.gcd p q ≠ 0 := by	obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq) nth_rw 1 [hr] rw [mul_comm, mul_div_cancel_right₀ _ pq0] exact r0	[ " p / GCDMonoid.gcd p q ≠ 0", " GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0", " r ≠ 0", " q / GCDMonoid.gcd p q ≠ 0" ]	[ " p / GCDMonoid.gcd p q ≠ 0", " GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0", " r ≠ 0", " q / GCDMonoid.gcd p q ≠ 0" ]	[]
import Mathlib.Algebra.Algebra.Defs import Mathlib.Algebra.CharP.ExpChar import Mathlib.FieldTheory.Separable #align_import field_theory.separable_degree from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15" noncomputable section namespace Polynomial open scoped Classical open Polynomial...	Mathlib/RingTheory/Polynomial/SeparableDegree.lean	78	82	theorem IsSeparableContraction.dvd_degree' {g} (hf : IsSeparableContraction q f g) : ∃ m : ℕ, g.natDegree * q ^ m = f.natDegree := by	obtain ⟨m, rfl⟩ := hf.2 use m rw [natDegree_expand]	[ " ∃ m, g.natDegree * q ^ m = f.natDegree", " ∃ m_1, g.natDegree * q ^ m_1 = ((expand F (q ^ m)) g).natDegree", " g.natDegree * q ^ m = ((expand F (q ^ m)) g).natDegree" ]	[ " ∃ m, g.natDegree * q ^ m = f.natDegree" ]	[ " ∃ m_1, g.natDegree * q ^ m_1 = ((expand F (q ^ m)) g).natDegree", " g.natDegree * q ^ m = ((expand F (q ^ m)) g).natDegree" ]
import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Lemmas import Mathlib.Algebra.Polynomial.Div #align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8" noncomputable ...	Mathlib/Algebra/Polynomial/RingDivision.lean	468	477	theorem rootMultiplicity_eq_natTrailingDegree' {p : R[X]} : p.rootMultiplicity 0 = p.natTrailingDegree := by	by_cases h : p = 0 · simp only [h, rootMultiplicity_zero, natTrailingDegree_zero] refine le_antisymm ?_ ?_ · rw [rootMultiplicity_le_iff h, map_zero, sub_zero, X_pow_dvd_iff, not_forall] exact ⟨p.natTrailingDegree, fun h' ↦ trailingCoeff_nonzero_iff_nonzero.2 h <\| h' <\| Nat.lt.base _⟩ · rw [le_root...	[ " n ≤ rootMultiplicity a p ↔ (X - C a) ^ n ∣ p", " (∀ m < n, ¬¬(X - C a) ^ (m + 1) ∣ p) ↔ (X - C a) ^ n ∣ p", " (∀ m < n, (X - C a) ^ (m + 1) ∣ p) ↔ (X - C a) ^ n ∣ p", " (X - C a) ^ n ∣ p", " (X - C a) ^ 0 ∣ p", " 1 ∣ p", " (X - C a) ^ (n + 1) ∣ p", " rootMultiplicity a p ≤ n ↔ ¬(X - C a) ^ (n + 1) ∣...	[ " n ≤ rootMultiplicity a p ↔ (X - C a) ^ n ∣ p", " (∀ m < n, ¬¬(X - C a) ^ (m + 1) ∣ p) ↔ (X - C a) ^ n ∣ p", " (∀ m < n, (X - C a) ^ (m + 1) ∣ p) ↔ (X - C a) ^ n ∣ p", " (X - C a) ^ n ∣ p", " (X - C a) ^ 0 ∣ p", " 1 ∣ p", " (X - C a) ^ (n + 1) ∣ p", " rootMultiplicity a p ≤ n ↔ ¬(X - C a) ^ (n + 1) ∣...	[ " rootMultiplicity 0 p ≤ p.natTrailingDegree", " ∃ x, ¬(x < p.natTrailingDegree + 1 → p.coeff x = 0)", " p.natTrailingDegree ≤ rootMultiplicity 0 p", " ∀ d < p.natTrailingDegree, p.coeff d = 0" ]
import Mathlib.Algebra.Order.Ring.Abs #align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e" open Function Nat namespace Int variable {a b : ℤ} {n : ℕ} theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by rw [← a...	Mathlib/Data/Int/Order/Lemmas.lean	35	37	theorem natAbs_lt_iff_mul_self_lt {a b : ℤ} : a.natAbs < b.natAbs ↔ a * a < b * b := by	rw [← abs_lt_iff_mul_self_lt, abs_eq_natAbs, abs_eq_natAbs] exact Int.ofNat_lt.symm	[ " a.natAbs = b.natAbs ↔ a * a = b * b", " a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs", " a.natAbs < b.natAbs ↔ a * a < b * b", " a.natAbs < b.natAbs ↔ ↑a.natAbs < ↑b.natAbs" ]	[ " a.natAbs = b.natAbs ↔ a * a = b * b", " a.natAbs = b.natAbs ↔ ↑a.natAbs = ↑b.natAbs", " a.natAbs < b.natAbs ↔ a * a < b * b" ]	[ " a.natAbs < b.natAbs ↔ ↑a.natAbs < ↑b.natAbs" ]
import Mathlib.Data.Finsupp.Lex import Mathlib.Data.Finsupp.Multiset import Mathlib.Order.GameAdd #align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded" namespace Relation open Multiset Prod variable {α : Type*} def CutExpand (r : α → α → Prop) (s' s : Multise...	Mathlib/Logic/Hydra.lean	109	121	theorem cutExpand_fibration (r : α → α → Prop) : Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 := by	rintro ⟨s₁, s₂⟩ s ⟨t, a, hr, he⟩; dsimp at he ⊢ classical obtain ⟨ha, rfl⟩ := add_singleton_eq_iff.1 he rw [add_assoc, mem_add] at ha obtain h \| h := ha · refine ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, ?_⟩, ?_⟩ · rw [add_comm, ← add_assoc, singleton_add, cons_erase h] · rw [add_assoc s₁, eras...	[ " CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp", " InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t", " (toFinsupp s) b = (toFinsupp t) b", " (fun {i} x x_1 => x < x_1) ((toFinsupp s) a) ((toFinsupp t) a)", " count b...	[ " CutExpand r ≤ InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) ⇑toFinsupp", " InvImage (Finsupp.Lex (rᶜ ⊓ fun x x_1 => x ≠ x_1) fun x x_1 => x < x_1) (⇑toFinsupp) s t", " (toFinsupp s) b = (toFinsupp t) b", " (fun {i} x x_1 => x < x_1) ((toFinsupp s) a) ((toFinsupp t) a)", " count b...	[ " ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ (fun s => s.1 + s.2) a' = s", " ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = s", " ∃ a', GameAdd (CutExpand r) (CutExpand r) a' (s₁, s₂) ∧ a'.1 + a'.2 = (s₁ + s₂ + t).erase a", " s₁.erase a + t + {a} = s₁ + t", " (s₁.erase a ...
import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.BigOperators.NatAntidiagonal import Mathlib.Algebra.CharZero.Lemmas import Mathlib.Data.Finset.NatAntidiagonal import Mathlib.Data.Nat.Choose.Central import Mathlib.Data.Tree.Basic import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.GCongr import Mathlib...	Mathlib/Combinatorics/Enumerative/Catalan.lean	79	79	theorem catalan_one : catalan 1 = 1 := by	simp [catalan_succ]	[ " catalan 0 = 1", " catalan (n + 1) = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)", " catalan (n + 1) = ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2", " catalan 1 = 1" ]	[ " catalan 0 = 1", " catalan (n + 1) = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)", " catalan (n + 1) = ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2", " catalan 1 = 1" ]	[]
import Mathlib.Data.Set.Image import Mathlib.Data.List.InsertNth import Mathlib.Init.Data.List.Lemmas #align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4" open List variable {α β γ : Type*} namespace List theorem injOn_insertNth_index_of_not_mem (l : List...	Mathlib/Data/List/Lemmas.lean	44	52	theorem foldr_range_subset_of_range_subset {f : β → α → α} {g : γ → α → α} (hfg : Set.range f ⊆ Set.range g) (a : α) : Set.range (foldr f a) ⊆ Set.range (foldr g a) := by	rintro _ ⟨l, rfl⟩ induction' l with b l H · exact ⟨[], rfl⟩ · cases' hfg (Set.mem_range_self b) with c hgf cases' H with m hgf' rw [foldr_cons, ← hgf, ← hgf'] exact ⟨c :: m, rfl⟩	[ " Set.InjOn (fun k => insertNth k x l) {n \| n ≤ l.length}", " Set.InjOn (fun k => insertNth k x []) {n \| n ≤ [].length}", " n = m", " Set.InjOn (fun k => insertNth k x (hd :: tl)) {n \| n ≤ (hd :: tl).length}", " 0 = m", " n✝ + 1 = m", " 0 = 0", " 0 = n✝ + 1", " n✝ + 1 = 0", " n✝¹ + 1 = n✝ + 1", ...	[ " Set.InjOn (fun k => insertNth k x l) {n \| n ≤ l.length}", " Set.InjOn (fun k => insertNth k x []) {n \| n ≤ [].length}", " n = m", " Set.InjOn (fun k => insertNth k x (hd :: tl)) {n \| n ≤ (hd :: tl).length}", " 0 = m", " n✝ + 1 = m", " 0 = 0", " 0 = n✝ + 1", " n✝ + 1 = 0", " n✝¹ + 1 = n✝ + 1", ...	[ " foldr f a l ∈ Set.range (foldr g a)", " foldr f a [] ∈ Set.range (foldr g a)", " foldr f a (b :: l) ∈ Set.range (foldr g a)", " g c (foldr g a m) ∈ Set.range (foldr g a)" ]
import Mathlib.CategoryTheory.Monoidal.Category import Mathlib.CategoryTheory.Adjunction.FullyFaithful import Mathlib.CategoryTheory.Products.Basic #align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042" open CategoryTheory universe v₁ v₂ v₃ u...	Mathlib/CategoryTheory/Monoidal/Functor.lean	164	167	theorem LaxMonoidalFunctor.left_unitality_inv (F : LaxMonoidalFunctor C D) (X : C) : (λ_ (F.obj X)).inv ≫ F.ε ▷ F.obj X ≫ F.μ (𝟙_ C) X = F.map (λ_ X).inv := by	rw [Iso.inv_comp_eq, F.left_unitality, Category.assoc, Category.assoc, ← F.toFunctor.map_comp, Iso.hom_inv_id, F.toFunctor.map_id, comp_id]	[ " (F.map f ⊗ F.map g) ≫ F.μ Y Y' = F.μ X X' ≫ F.map (f ⊗ g)", " { obj := F.obj, map := fun {X Y} => F.map, map_id := ⋯, map_comp := ⋯ }.map f ▷\n { obj := F.obj, map := fun {X Y} => F.map, map_id := ⋯, map_comp := ⋯ }.obj X' ≫\n μ Y✝ X' =\n μ X✝ X' ≫ { obj := F.obj, map := fun {X Y} => F.map, map_i...	[ " (F.map f ⊗ F.map g) ≫ F.μ Y Y' = F.μ X X' ≫ F.map (f ⊗ g)", " { obj := F.obj, map := fun {X Y} => F.map, map_id := ⋯, map_comp := ⋯ }.map f ▷\n { obj := F.obj, map := fun {X Y} => F.map, map_id := ⋯, map_comp := ⋯ }.obj X' ≫\n μ Y✝ X' =\n μ X✝ X' ≫ { obj := F.obj, map := fun {X Y} => F.map, map_i...	[]
import Mathlib.Data.Num.Lemmas import Mathlib.Data.Nat.Prime import Mathlib.Tactic.Ring #align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a" namespace PosNum def minFacAux (n : PosNum) : ℕ → PosNum → PosNum \| 0, _ => n \| fuel + 1, k => if n < k.bit1...	Mathlib/Data/Num/Prime.lean	44	54	theorem minFacAux_to_nat {fuel : ℕ} {n k : PosNum} (h : Nat.sqrt n < fuel + k.bit1) : (minFacAux n fuel k : ℕ) = Nat.minFacAux n k.bit1 := by	induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux] · rw [Nat.zero_add, Nat.sqrt_lt] at h simp only [h, ite_true] simp_rw [← mul_to_nat] simp only [cast_lt, dvd_to_nat] split_ifs <;> try rfl rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;> simp only [_root_.bit...	[ " ↑(n.minFacAux fuel k) = (↑n).minFacAux ↑k.bit1", " ↑(n.minFacAux 0 k) = (↑n).minFacAux ↑k.bit1", " ↑(n.minFacAux (fuel + 1) k) = (↑n).minFacAux ↑k.bit1", " ↑n = if ↑n < ↑k.bit1 * ↑k.bit1 then ↑n else if ↑k.bit1 ∣ ↑n then ↑k.bit1 else (↑n).minFacAux (↑k.bit1 + 2)", " ↑(if n < k.bit1 * k.bit1 then n else if...	[ " ↑(n.minFacAux fuel k) = (↑n).minFacAux ↑k.bit1" ]	[ " ↑(n.minFacAux 0 k) = (↑n).minFacAux ↑k.bit1", " ↑(n.minFacAux (fuel + 1) k) = (↑n).minFacAux ↑k.bit1", " ↑n = if ↑n < ↑k.bit1 * ↑k.bit1 then ↑n else if ↑k.bit1 ∣ ↑n then ↑k.bit1 else (↑n).minFacAux (↑k.bit1 + 2)", " ↑(if n < k.bit1 * k.bit1 then n else if k.bit1 ∣ n then k.bit1 else n.minFacAux fuel k.succ)...
import Mathlib.Analysis.Convex.Side import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine #align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" noncomputable section open ...	Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean	85	86	theorem left_ne_right_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₃ := by	rw [← (vsub_left_injective p₂).ne_iff]; exact o.ne_of_oangle_ne_zero h	[ " ContinuousAt (fun y => ∡ y.1 y.2.1 y.2.2) x", " (f x).1 ≠ 0", " (f x).2 ≠ 0", " ∡ p₁ p₁ p₂ = 0", " ∡ p₁ p₂ p₂ = 0", " p₁ ≠ p₂", " p₁ -ᵥ p₂ ≠ 0", " p₃ ≠ p₂", " p₃ -ᵥ p₂ ≠ 0", " p₁ ≠ p₃", " p₁ -ᵥ p₂ ≠ p₃ -ᵥ p₂" ]	[ " ContinuousAt (fun y => ∡ y.1 y.2.1 y.2.2) x", " (f x).1 ≠ 0", " (f x).2 ≠ 0", " ∡ p₁ p₁ p₂ = 0", " ∡ p₁ p₂ p₂ = 0", " p₁ ≠ p₂", " p₁ -ᵥ p₂ ≠ 0", " p₃ ≠ p₂", " p₃ -ᵥ p₂ ≠ 0", " p₁ ≠ p₃" ]	[ " p₁ -ᵥ p₂ ≠ p₃ -ᵥ p₂" ]
import Mathlib.Topology.Algebra.Module.WeakDual import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed #align_import measure_theory.measure.finite_measure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable...	Mathlib/MeasureTheory/Measure/FiniteMeasure.lean	207	209	theorem mass_nonzero_iff (μ : FiniteMeasure Ω) : μ.mass ≠ 0 ↔ μ ≠ 0 := by	rw [not_iff_not] exact FiniteMeasure.mass_zero_iff μ	[ " ↑μ s = ↑ν s", " μ s₁ ≤ μ s₂", " (↑μ s₁).toNNReal ≤ (↑μ s₂).toNNReal", " μ s ≤ μ.mass", " μ.mass = 0 ↔ μ = 0", " μ.mass = 0", " μ = 0", " ↑μ = ↑0", " ↑μ univ = 0", " μ.mass ≠ 0 ↔ μ ≠ 0" ]	[ " ↑μ s = ↑ν s", " μ s₁ ≤ μ s₂", " (↑μ s₁).toNNReal ≤ (↑μ s₂).toNNReal", " μ s ≤ μ.mass", " μ.mass = 0 ↔ μ = 0", " μ.mass = 0", " μ = 0", " ↑μ = ↑0", " ↑μ univ = 0", " μ.mass ≠ 0 ↔ μ ≠ 0" ]	[]
import Batteries.Classes.Order namespace Batteries.PairingHeapImp inductive Heap (α : Type u) where \| nil : Heap α \| node (a : α) (child sibling : Heap α) : Heap α deriving Repr def Heap.size : Heap α → Nat \| .nil => 0 \| .node _ c s => c.size + 1 + s.size def Heap.singleton (a : α) : Heap α := ....	.lake/packages/batteries/Batteries/Data/PairingHeap.lean	95	101	theorem Heap.noSibling_combine (le) (s : Heap α) : (s.combine le).NoSibling := by	unfold combine; split · exact noSibling_merge _ _ _ · match s with \| nil \| node _ _ nil => constructor \| node _ _ (node _ _ s) => rename_i h; exact (h _ _ _ _ _ rfl).elim	[ " (merge le s₁ s₂).NoSibling", " (match s₁, s₂ with\n \| nil, nil => nil\n \| nil, node a₂ c₂ sibling => node a₂ c₂ nil\n \| node a₁ c₁ sibling, nil => node a₁ c₁ nil\n \| node a₁ c₁ sibling, node a₂ c₂ sibling_1 =>\n if le a₁ a₂ = true then node a₁ (node a₂ c₂ c₁) nil else node a₂ (node a₁ c₁ c₂) ni...	[ " (merge le s₁ s₂).NoSibling", " (match s₁, s₂ with\n \| nil, nil => nil\n \| nil, node a₂ c₂ sibling => node a₂ c₂ nil\n \| node a₁ c₁ sibling, nil => node a₁ c₁ nil\n \| node a₁ c₁ sibling, node a₂ c₂ sibling_1 =>\n if le a₁ a₂ = true then node a₁ (node a₂ c₂ c₁) nil else node a₂ (node a₁ c₁ c₂) ni...	[ " (match s with\n \| h₁@h_1:(node a child h₂@h:(node a_1 child_1 s)) => merge le (merge le h₁ h₂) (combine le s)\n \| h => h).NoSibling", " (merge le (merge le (node a✝¹ child✝¹ (node a✝ child✝ s✝)) (node a✝ child✝ s✝)) (combine le s✝)).NoSibling", " s.NoSibling", " (node a✝ child✝ nil).NoSibling", " (n...
import Mathlib.MeasureTheory.Constructions.Pi import Mathlib.MeasureTheory.Integral.Lebesgue open scoped Classical ENNReal open Set Function Equiv Finset noncomputable section namespace MeasureTheory section LMarginal variable {δ δ' : Type} {π : δ → Type} [∀ x, MeasurableSpace (π x)] variable {μ : ∀ i, Measu...	Mathlib/MeasureTheory/Integral/Marginal.lean	137	139	theorem lmarginal_union' (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f) {s t : Finset δ} (hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ := by	rw [Finset.union_comm, lmarginal_union μ f hf hst.symm]	[ " Measurable (∫⋯∫⁻_s, f ∂μ)", " Measurable (uncurry fun x y => f (updateFinset x s y))", " Measurable fun a => updateFinset a.1 s a.2", " ∀ (a : δ), Measurable fun x => updateFinset x.1 s x.2 a", " Measurable fun x => updateFinset x.1 s x.2 i", " Measurable fun x => x.2 ⟨i, ⋯⟩", " Measurable fun x => x....	[ " Measurable (∫⋯∫⁻_s, f ∂μ)", " Measurable (uncurry fun x y => f (updateFinset x s y))", " Measurable fun a => updateFinset a.1 s a.2", " ∀ (a : δ), Measurable fun x => updateFinset x.1 s x.2 a", " Measurable fun x => updateFinset x.1 s x.2 i", " Measurable fun x => x.2 ⟨i, ⋯⟩", " Measurable fun x => x....	[]
import Mathlib.Topology.Separation open Topology Filter Set TopologicalSpace section Basic variable {α : Type*} [TopologicalSpace α] {C : Set α}	Mathlib/Topology/Perfect.lean	62	68	theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) : AccPt x (𝓟 (U ∩ C)) := by	have : 𝓝[≠] x ≤ 𝓟 U := by rw [le_principal_iff] exact mem_nhdsWithin_of_mem_nhds hU rw [AccPt, ← inf_principal, ← inf_assoc, inf_of_le_left this] exact h_acc	[ " AccPt x (𝓟 (U ∩ C))", " 𝓝[≠] x ≤ 𝓟 U", " U ∈ 𝓝[≠] x", " (𝓝[≠] x ⊓ 𝓟 C).NeBot" ]	[ " AccPt x (𝓟 (U ∩ C))" ]	[ " 𝓝[≠] x ≤ 𝓟 U", " U ∈ 𝓝[≠] x", " (𝓝[≠] x ⊓ 𝓟 C).NeBot" ]
import Mathlib.FieldTheory.Galois #align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a" noncomputable section open scoped Polynomial open FiniteDimensional namespace Polynomial variable {F : Type} [Field F] (p q : F[X]) (E : Type) [...	Mathlib/FieldTheory/PolynomialGaloisGroup.lean	155	168	theorem mapRoots_bijective [h : Fact (p.Splits (algebraMap F E))] : Function.Bijective (mapRoots p E) := by	constructor · exact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h)) · intro y -- this is just an equality of two different ways to write the roots of `p` as an `E`-polynomial have key := roots_map (IsScalarTower.toAlgHom F p.SplittingField E : p.SplittingField →+* E) (...	[ " σ = τ", " (↑σ).equalizer ↑τ = ⊤", " f x = default x", " f ((algebraMap F p.SplittingField) y) = default ((algebraMap F p.SplittingField) y)", " Function.Bijective (mapRoots p E)", " Function.Injective (mapRoots p E)", " Function.Surjective (mapRoots p E)", " ∃ a, mapRoots p E a = y" ]	[ " σ = τ", " (↑σ).equalizer ↑τ = ⊤", " f x = default x", " f ((algebraMap F p.SplittingField) y) = default ((algebraMap F p.SplittingField) y)", " Function.Bijective (mapRoots p E)" ]	[ " Function.Injective (mapRoots p E)", " Function.Surjective (mapRoots p E)", " ∃ a, mapRoots p E a = y" ]
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots import Mathlib.NumberTheory.NumberField.Discriminant #align_import number_theory.cyclotomic.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1" universe u v open Algebra Polynomial Nat IsPrimitiveRoot PowerBasis open s...	Mathlib/NumberTheory/Cyclotomic/Discriminant.lean	37	48	theorem discr_zeta_eq_discr_zeta_sub_one (hζ : IsPrimitiveRoot ζ n) : discr ℚ (hζ.powerBasis ℚ).basis = discr ℚ (hζ.subOnePowerBasis ℚ).basis := by	haveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional {n} ℚ K) have H₁ : (aeval (hζ.powerBasis ℚ).gen) (X - 1 : ℤ[X]) = (hζ.subOnePowerBasis ℚ).gen := by simp have H₂ : (aeval (hζ.subOnePowerBasis ℚ).gen) (X + 1 : ℤ[X]) = (hζ.powerBasis ℚ).gen := by simp refine discr_eq_discr_...	[ " discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = discr ℚ ⇑(subOnePowerBasis ℚ hζ).basis", " (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen", " (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen", " IsIntegral ℤ (IsPrimitiveRoot.powerBas...	[ " discr ℚ ⇑(IsPrimitiveRoot.powerBasis ℚ hζ).basis = discr ℚ ⇑(subOnePowerBasis ℚ hζ).basis" ]	[ " (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen", " (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen", " IsIntegral ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen", " minpoly ℚ (IsPrimitiveRoot.powerBasis ℚ hζ).gen =\n Polynomial.map (algebr...
import Mathlib.Analysis.InnerProductSpace.Projection import Mathlib.Analysis.NormedSpace.PiLp import Mathlib.LinearAlgebra.FiniteDimensional import Mathlib.LinearAlgebra.UnitaryGroup #align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395" set_...	Mathlib/Analysis/InnerProductSpace/PiL2.lean	145	150	theorem EuclideanSpace.sphere_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) : Metric.sphere (0 : EuclideanSpace ℝ n) r = {x \| ∑ i, x i ^ 2 = r ^ 2} := by	ext x have : (0 : ℝ) ≤ ∑ i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _ simp_rw [mem_setOf, mem_sphere_zero_iff_norm, norm_eq, norm_eq_abs, sq_abs, Real.sqrt_eq_iff_sq_eq this hr, eq_comm]	[ " ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜", " ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜", " (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜", " (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)...	[ " ‖x‖ ^ 2 = re ⟪x, x⟫_𝕜", " ∀ (x y : PiLp 2 f), (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜", " (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜", " (starRingEnd 𝕜) ({ inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n { inner := fun x y => ∑ i : ι, InnerProductSpace.toInner.1 (x i) (y i)...	[ " x ∈ Metric.sphere 0 r ↔ x ∈ {x \| ∑ i : n, x i ^ 2 = r ^ 2}" ]
import Mathlib.Probability.ProbabilityMassFunction.Basic import Mathlib.Probability.ProbabilityMassFunction.Constructions import Mathlib.MeasureTheory.Integral.Bochner namespace PMF open MeasureTheory ENNReal TopologicalSpace section General variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α] v...	Mathlib/Probability/ProbabilityMassFunction/Integrals.lean	28	41	theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) : ∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc _ = ∫ a in p.support, f a ∂(p.toMeasure) := by	rw [restrict_toMeasure_support p] _ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by apply integral_countable f p.support_countable rwa [restrict_toMeasure_support p] _ = ∑' (a : support p), (p a).toReal • f a := by congr with x; congr 2 apply PMF.toMeasure_apply_singleton p x (Measur...	[ " ∫ (a : α), f a ∂p.toMeasure = ∫ (a : α) in p.support, f a ∂p.toMeasure", " ∫ (a : α) in p.support, f a ∂p.toMeasure = ∑' (a : ↑p.support), (p.toMeasure {↑a}).toReal • f ↑a", " Integrable f (p.toMeasure.restrict p.support)", " ∑' (a : ↑p.support), (p.toMeasure {↑a}).toReal • f ↑a = ∑' (a : ↑p.support), (p ↑a...	[ " ∫ (a : α), f a ∂p.toMeasure = ∫ (a : α) in p.support, f a ∂p.toMeasure" ]	[ " ∫ (a : α) in p.support, f a ∂p.toMeasure = ∑' (a : ↑p.support), (p.toMeasure {↑a}).toReal • f ↑a", " Integrable f (p.toMeasure.restrict p.support)", " ∑' (a : ↑p.support), (p.toMeasure {↑a}).toReal • f ↑a = ∑' (a : ↑p.support), (p ↑a).toReal • f ↑a", " (p.toMeasure {↑x}).toReal • f ↑x = (p ↑x).toReal • f ↑x...
import Mathlib.Data.List.Lattice import Mathlib.Data.List.Range import Mathlib.Data.Bool.Basic #align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213" open Nat namespace List def Ico (n m : ℕ) : List ℕ := range' n (m - n) #align list.Ico List.Ico names...	Mathlib/Data/List/Intervals.lean	95	100	theorem append_consecutive {n m l : ℕ} (hnm : n ≤ m) (hml : m ≤ l) : Ico n m ++ Ico m l = Ico n l := by	dsimp only [Ico] convert range'_append n (m-n) (l-m) 1 using 2 · rw [Nat.one_mul, Nat.add_sub_cancel' hnm] · rw [Nat.sub_add_sub_cancel hml hnm]	[ " Ico 0 n = range n", " (Ico n m).length = m - n", " (range' n (m - n)).length = m - n", " Pairwise (fun x x_1 => x < x_1) (Ico n m)", " Pairwise (fun x x_1 => x < x_1) (range' n (m - n))", " (Ico n m).Nodup", " (range' n (m - n)).Nodup", " l ∈ Ico n m ↔ n ≤ l ∧ l < m", " n ≤ l ∧ l < n + (m - n) ↔ n...	[ " Ico 0 n = range n", " (Ico n m).length = m - n", " (range' n (m - n)).length = m - n", " Pairwise (fun x x_1 => x < x_1) (Ico n m)", " Pairwise (fun x x_1 => x < x_1) (range' n (m - n))", " (Ico n m).Nodup", " (range' n (m - n)).Nodup", " l ∈ Ico n m ↔ n ≤ l ∧ l < m", " n ≤ l ∧ l < n + (m - n) ↔ n...	[ " range' n (m - n) ++ range' m (l - m) = range' n (l - n)", " range' m (l - m) = range' (n + 1 * (m - n)) (l - m)", " l - n = l - m + (m - n)" ]
import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Derivative import Mathlib.Data.Nat.Choose.Cast import Mathlib.Data.Nat.Choose.Vandermonde import Mathlib.Tactic.FieldSimp #align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358...	Mathlib/Algebra/Polynomial/HasseDeriv.lean	93	97	theorem hasseDeriv_eq_zero_of_lt_natDegree (p : R[X]) (n : ℕ) (h : p.natDegree < n) : hasseDeriv n p = 0 := by	rw [hasseDeriv_apply, sum_def] refine Finset.sum_eq_zero fun x hx => ?_ simp [Nat.choose_eq_zero_of_lt ((le_natDegree_of_mem_supp _ hx).trans_lt h)]	[ " (hasseDeriv k) f = f.sum fun i r => (monomial (i - k)) (↑(i.choose k) * r)", " (f.sum fun x x_1 => (monomial (x - k)) (x.choose k • x_1)) = f.sum fun i r => (monomial (i - k)) (↑(i.choose k) * r)", " (fun x x_1 => (monomial (x - k)) (x.choose k • x_1)) = fun i r => (monomial (i - k)) (↑(i.choose k) * r)", "...	[ " (hasseDeriv k) f = f.sum fun i r => (monomial (i - k)) (↑(i.choose k) * r)", " (f.sum fun x x_1 => (monomial (x - k)) (x.choose k • x_1)) = f.sum fun i r => (monomial (i - k)) (↑(i.choose k) * r)", " (fun x x_1 => (monomial (x - k)) (x.choose k • x_1)) = fun i r => (monomial (i - k)) (↑(i.choose k) * r)", "...	[ " ∑ n_1 ∈ p.support, (monomial (n_1 - n)) (↑(n_1.choose n) * p.coeff n_1) = 0", " (monomial (x - n)) (↑(x.choose n) * p.coeff x) = 0" ]
import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Fintype.Basic import Mathlib.Data.Int.GCD import Mathlib.RingTheory.Coprime.Basic #align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226" universe u v section IsCoprime variable {R : Type ...	Mathlib/RingTheory/Coprime/Lemmas.lean	69	70	theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by	simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)	[ " IsCoprime m n ↔ m.gcd n = 1", " IsCoprime m n → m.gcd n = 1", " m.gcd n = 1", " 1 = m * a + n * b", " m.gcd n = 1 → IsCoprime m n", " m.gcdA n * m + m.gcdB n * n = 1 → ∃ a b, a * m + b * n = 1", " ∃ a b, a * m + b * n = 1", " IsCoprime ↑m ↑n ↔ m.Coprime n", " IsCoprime ↑a ↑b", " IsCoprime ↑↑a ↑↑...	[ " IsCoprime m n ↔ m.gcd n = 1", " IsCoprime m n → m.gcd n = 1", " m.gcd n = 1", " 1 = m * a + n * b", " m.gcd n = 1 → IsCoprime m n", " m.gcdA n * m + m.gcdB n * n = 1 → ∃ a b, a * m + b * n = 1", " ∃ a b, a * m + b * n = 1", " IsCoprime ↑m ↑n ↔ m.Coprime n", " IsCoprime ↑a ↑b", " IsCoprime ↑↑a ↑↑...	[]
import Mathlib.Algebra.GroupPower.IterateHom import Mathlib.Algebra.Ring.Divisibility.Basic import Mathlib.Data.List.Cycle import Mathlib.Data.Nat.Prime import Mathlib.Data.PNat.Basic import Mathlib.Dynamics.FixedPoints.Basic import Mathlib.GroupTheory.GroupAction.Group #align_import dynamics.periodic_pts from "leanp...	Mathlib/Dynamics/PeriodicPts.lean	156	159	theorem left_of_comp {g : α → α} (hco : Commute f g) (hfg : IsPeriodicPt (f ∘ g) n x) (hg : IsPeriodicPt g n x) : IsPeriodicPt f n x := by	rw [IsPeriodicPt, hco.comp_iterate] at hfg exact hfg.left_of_comp hg	[ " IsPeriodicPt f (n + m) x", " IsFixedPt (f^[n] ∘ f^[m]) x", " IsPeriodicPt f n x", " IsPeriodicPt f m x", " IsPeriodicPt f (m - n) x", " IsPeriodicPt f (m - n + n) x", " IsPeriodicPt f 0 x", " IsPeriodicPt f (m * n) x", " IsPeriodicPt f (n * m) x", " IsPeriodicPt f^[m] n x", " IsFixedPt f^[n]^[...	[ " IsPeriodicPt f (n + m) x", " IsFixedPt (f^[n] ∘ f^[m]) x", " IsPeriodicPt f n x", " IsPeriodicPt f m x", " IsPeriodicPt f (m - n) x", " IsPeriodicPt f (m - n + n) x", " IsPeriodicPt f 0 x", " IsPeriodicPt f (m * n) x", " IsPeriodicPt f (n * m) x", " IsPeriodicPt f^[m] n x", " IsFixedPt f^[n]^[...	[]
import Mathlib.Algebra.Order.ToIntervalMod import Mathlib.Algebra.Ring.AddAut import Mathlib.Data.Nat.Totient import Mathlib.GroupTheory.Divisible import Mathlib.Topology.Connected.PathConnected import Mathlib.Topology.IsLocalHomeomorph #align_import topology.instances.add_circle from "leanprover-community/mathlib"@"...	Mathlib/Topology/Instances/AddCircle.lean	82	89	theorem continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x := by	rw [(funext fun y => Eq.trans (by rw [neg_neg]) <\| toIocMod_neg _ _ _ : toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)] -- Porting note: added have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg exact (continuous_sub_left _).continuousAt.comp_continuousWithinAt <\| (co...	[ " ContinuousWithinAt (toIcoMod hp a) (Ici x) x", " s ∈ Filter.map (toIcoMod hp a) (𝓝[≥] x)", " ∃ u ∈ 𝓝 x, u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s", " ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s", " ∃ u,\n (∃ l u_1, x ∈ Ioo l u_1 ∧ ∀ x ∈ Ioo l u_1, x ∈ u) ∧ ∀ (x_1 : 𝕜), x_1 ...	[ " ContinuousWithinAt (toIcoMod hp a) (Ici x) x", " s ∈ Filter.map (toIcoMod hp a) (𝓝[≥] x)", " ∃ u ∈ 𝓝 x, u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s", " ∃ u, (∃ l u_1, x ∈ Ioo l u_1 ∧ Ioo l u_1 ⊆ u) ∧ u ∩ Ici x ⊆ toIcoMod hp a ⁻¹' s", " ∃ u,\n (∃ l u_1, x ∈ Ioo l u_1 ∧ ∀ x ∈ Ioo l u_1, x ∈ u) ∧ ∀ (x_1 : 𝕜), x_1 ...	[ " toIocMod hp a y = toIocMod hp a (- -y)", " ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x" ]
import Mathlib.Analysis.Calculus.Deriv.AffineMap import Mathlib.Analysis.Calculus.Deriv.Slope import Mathlib.Analysis.Calculus.Deriv.Mul import Mathlib.Analysis.Calculus.Deriv.Comp import Mathlib.Analysis.Calculus.LocalExtr.Rolle import Mathlib.Analysis.Convex.Normed import Mathlib.Analysis.RCLike.Basic #align_import...	Mathlib/Analysis/Calculus/MeanValue.lean	92	124	theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ} (hf : ContinuousOn f (Icc a b)) -- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x` (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Ic...	change Icc a b ⊆ { x \| f x ≤ B x } set s := { x \| f x ≤ B x } ∩ Icc a b have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB have : IsClosed s := by simp only [s, inter_comm] exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' apply this.Icc_subset_of_fo...	[ " ∀ ⦃x : ℝ⦄, x ∈ Icc a b → f x ≤ B x", " Icc a b ⊆ {x \| f x ≤ B x}", " IsClosed s", " IsClosed (Icc a b ∩ {x \| f x ≤ B x})", " ∀ x ∈ {x \| f x ≤ B x} ∩ Ico a b, ∀ y ∈ Ioi x, ({x \| f x ≤ B x} ∩ Ioc x y).Nonempty", " ({x \| f x ≤ B x} ∩ Ioc x y).Nonempty", " {x \| f x ≤ B x} ∈ 𝓝[>] x", " z ∈ {x \| f x ≤ B ...	[ " ∀ ⦃x : ℝ⦄, x ∈ Icc a b → f x ≤ B x" ]	[ " Icc a b ⊆ {x \| f x ≤ B x}", " IsClosed s", " IsClosed (Icc a b ∩ {x \| f x ≤ B x})", " ∀ x ∈ {x \| f x ≤ B x} ∩ Ico a b, ∀ y ∈ Ioi x, ({x \| f x ≤ B x} ∩ Ioc x y).Nonempty", " ({x \| f x ≤ B x} ∩ Ioc x y).Nonempty", " {x \| f x ≤ B x} ∈ 𝓝[>] x", " z ∈ {x \| f x ≤ B x}" ]
import Mathlib.Topology.Separation import Mathlib.Algebra.Group.Defs #align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514" @[to_additive "Any nonempty compact Hausdorff additive semigroup where right-addition is continuous contains an ...	Mathlib/Topology/Algebra/Semigroup.lean	27	72	theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M] [TopologicalSpace M] [CompactSpace M] [T2Space M] (continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by	/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`. It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/ let S : Set (Set M) := { N \| IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N } rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul...	[ " ∃ m, m * m = m", " m * m = m", " (fun x => x * m) '' N = N", " (fun x => x * m) '' N ∈ S", " ∀ m_1 ∈ (fun x => x * m) '' N, ∀ m' ∈ (fun x => x * m) '' N, m_1 * m' ∈ (fun x => x * m) '' N", " (fun x => x * m) m'' * (fun x => x * m) m' ∈ (fun x => x * m) '' N", " (fun x => x * m) '' N ⊆ N", " (fun x =...	[ " ∃ m, m * m = m" ]	[ " m * m = m", " (fun x => x * m) '' N = N", " (fun x => x * m) '' N ∈ S", " ∀ m_1 ∈ (fun x => x * m) '' N, ∀ m' ∈ (fun x => x * m) '' N, m_1 * m' ∈ (fun x => x * m) '' N", " (fun x => x * m) m'' * (fun x => x * m) m' ∈ (fun x => x * m) '' N", " (fun x => x * m) '' N ⊆ N", " (fun x => x * m) m' ∈ N", "...
import Mathlib.MeasureTheory.Measure.Dirac set_option autoImplicit true open Set open scoped ENNReal Classical variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α} noncomputable section namespace MeasureTheory.Measure def count : Measure α := sum dirac #align measure_theory.measure.count MeasureTheo...	Mathlib/MeasureTheory/Measure/Count.lean	39	40	theorem count_apply (hs : MeasurableSet s) : count s = ∑' i : s, 1 := by	simp only [count, sum_apply, hs, dirac_apply', ← tsum_subtype s (1 : α → ℝ≥0∞), Pi.one_apply]	[ " count s = ∑' (i : ↑s), 1" ]	[ " count s = ∑' (i : ↑s), 1" ]	[]
import Mathlib.SetTheory.Game.State #align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225" namespace SetTheory namespace PGame namespace Domineering open Function @[simps!] def shiftUp : ℤ × ℤ ≃ ℤ × ℤ := (Equiv.refl ℤ).prodCongr (Equiv.addRig...	Mathlib/SetTheory/Game/Domineering.lean	117	122	theorem moveRight_card {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) : Finset.card (moveRight b m) + 2 = Finset.card b := by	dsimp [moveRight] rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)] rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)] exact tsub_add_cancel_of_le (card_of_mem_right h)	[ " (m.1 - 1, m.2) ∈ Finset.erase b m", " (m.1 - 1, m.2) ≠ m", " (m.1, m.2 - 1) ∈ Finset.erase b m", " (m.1, m.2 - 1) ≠ m", " 2 ≤ Finset.card b", " Finset.card (moveLeft b m) + 2 = Finset.card b", " ((Finset.erase b m).erase (m.1, m.2 - 1)).card + 2 = Finset.card b", " (Finset.erase b m).card - 1 + 2 = ...	[ " (m.1 - 1, m.2) ∈ Finset.erase b m", " (m.1 - 1, m.2) ≠ m", " (m.1, m.2 - 1) ∈ Finset.erase b m", " (m.1, m.2 - 1) ≠ m", " 2 ≤ Finset.card b", " Finset.card (moveLeft b m) + 2 = Finset.card b", " ((Finset.erase b m).erase (m.1, m.2 - 1)).card + 2 = Finset.card b", " (Finset.erase b m).card - 1 + 2 = ...	[ " ((Finset.erase b m).erase (m.1 - 1, m.2)).card + 2 = Finset.card b" ]
import Mathlib.Data.Complex.Basic import Mathlib.MeasureTheory.Integral.CircleIntegral #align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15" open Set MeasureTheory Metric Filter Function open scoped Interval Real noncomputable secti...	Mathlib/MeasureTheory/Integral/CircleTransform.lean	133	152	theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R) (hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧ ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B := by	obtain ⟨r, hr, hrx⟩ := exists_lt_mem_ball_of_mem_ball hx obtain ⟨ε', hε', H⟩ := exists_ball_subset_ball hrx obtain ⟨⟨⟨a, b⟩, ⟨ha, hb⟩⟩, hab⟩ := abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z let V : ℝ → ℂ → ℂ := fun θ w => circleTransformDeriv R z w (fun _ => 1) θ obtain ⟨X, -, HX2⟩...	[ " Periodic (circleTransformDeriv R z w f) (2 * π)", " ∀ (x : ℝ), circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x", " circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x", " (2 * ↑π * I)⁻¹ • deriv (circleMap z R) (x + 2 * π) • ((circleMap z R x - w) ^ 2)⁻¹ • f (c...	[ " Periodic (circleTransformDeriv R z w f) (2 * π)", " ∀ (x : ℝ), circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x", " circleTransformDeriv R z w f (x + 2 * π) = circleTransformDeriv R z w f x", " (2 * ↑π * I)⁻¹ • deriv (circleMap z R) (x + 2 * π) • ((circleMap z R x - w) ^ 2)⁻¹ • f (c...	[ " ‖circleTransformDeriv R z v f y‖ ≤ abs (V b a) * abs (f X)", " ‖circleTransformDeriv R z v f y1‖ ≤ abs (V b a) * abs (f X)" ]
import Mathlib.Data.Real.Irrational import Mathlib.Data.Nat.Fib.Basic import Mathlib.Data.Fin.VecNotation import Mathlib.Algebra.LinearRecurrence import Mathlib.Tactic.NormNum.NatFib import Mathlib.Tactic.NormNum.Prime #align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712...	Mathlib/Data/Real/GoldenRatio.lean	70	72	theorem gold_add_goldConj : φ + ψ = 1 := by	rw [goldenRatio, goldenConj] ring	[ " φ⁻¹ = -ψ", " 0 < 1", " 0 < 5", " 2 * 2 = 5 - 1", " ψ⁻¹ = -φ", " -ψ = φ⁻¹", " φ * ψ = -1", " (1 + √5) * (1 - √5) = -(2 * 2)", " 1 ^ 2 - √5 ^ 2 = -(2 * 2)", " ψ * φ = -1", " φ + ψ = 1", " (1 + √5) / 2 + (1 - √5) / 2 = 1" ]	[ " φ⁻¹ = -ψ", " 0 < 1", " 0 < 5", " 2 * 2 = 5 - 1", " ψ⁻¹ = -φ", " -ψ = φ⁻¹", " φ * ψ = -1", " (1 + √5) * (1 - √5) = -(2 * 2)", " 1 ^ 2 - √5 ^ 2 = -(2 * 2)", " ψ * φ = -1", " φ + ψ = 1" ]	[ " (1 + √5) / 2 + (1 - √5) / 2 = 1" ]
import Mathlib.Algebra.Polynomial.Mirror import Mathlib.Analysis.Complex.Polynomial #align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" namespace Polynomial open scoped Polynomial open Finset section Semiring variable {R : Type*} [Semirin...	Mathlib/Algebra/Polynomial/UnitTrinomial.lean	49	52	theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) : (trinomial k m n u v w).coeff n = w := by	rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]	[ " (trinomial k m n u v w).coeff n = w" ]	[ " (trinomial k m n u v w).coeff n = w" ]	[]
import Mathlib.Order.Lattice import Mathlib.Data.List.Sort import Mathlib.Logic.Equiv.Fin import Mathlib.Logic.Equiv.Functor import Mathlib.Data.Fintype.Card import Mathlib.Order.RelSeries #align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada" universe u ...	Mathlib/Order/JordanHolder.lean	116	117	theorem second_iso_of_eq {x y a b : X} (hm : IsMaximal x a) (ha : x ⊔ y = a) (hb : x ⊓ y = b) : Iso (x, a) (b, y) := by	substs a b; exact second_iso hm	[ " IsMaximal (x ⊓ y) y", " IsMaximal (y ⊓ x) y", " IsMaximal a y", " Iso (x, a) (b, y)", " Iso (x, x ⊔ y) (x ⊓ y, y)" ]	[ " IsMaximal (x ⊓ y) y", " IsMaximal (y ⊓ x) y", " IsMaximal a y", " Iso (x, a) (b, y)" ]	[ " Iso (x, x ⊔ y) (x ⊓ y, y)" ]
import Mathlib.RepresentationTheory.Rep import Mathlib.Algebra.Category.FGModuleCat.Limits import Mathlib.CategoryTheory.Preadditive.Schur import Mathlib.RepresentationTheory.Basic #align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b" suppress_comp...	Mathlib/RepresentationTheory/FdRep.lean	95	100	theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) : W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by	-- Porting note: Changed `rw` to `erw` erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply] rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)] exact (i.hom.comm g).symm	[ " Linear k (FdRep k G)", " AddCommGroup (CoeSort.coe V)", " AddCommGroup ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj", " Module k (CoeSort.coe V)", " Module k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj", " FiniteDimensional k (CoeSort.coe V)", " FiniteDimensional k ↑((forget₂ (FdRep k G) ...	[ " Linear k (FdRep k G)", " AddCommGroup (CoeSort.coe V)", " AddCommGroup ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj", " Module k (CoeSort.coe V)", " Module k ↑((forget₂ (FdRep k G) (FGModuleCat k)).obj V).obj", " FiniteDimensional k (CoeSort.coe V)", " FiniteDimensional k ↑((forget₂ (FdRep k G) ...	[ " W.ρ g =\n ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).inv ≫\n V.ρ g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom", " ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom ≫ W.ρ g =\n V.ρ g ≫ ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i).hom" ]
import Mathlib.Algebra.CharP.Basic import Mathlib.Algebra.CharP.Algebra import Mathlib.Data.Nat.Prime #align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" universe u variable (R : Type u) section Semiring variable [Semiring R] class inductive Ex...	Mathlib/Algebra/CharP/ExpChar.lean	120	125	theorem expChar_one_iff_char_zero (p q : ℕ) [CharP R p] [ExpChar R q] : q = 1 ↔ p = 0 := by	constructor · rintro rfl exact char_zero_of_expChar_one R p · rintro rfl exact expChar_one_of_char_zero R q	[ " ExpChar (R × S) p", " ExpChar (R × S) 1", " p = q", " 1 = q", " ringExpChar R = q", " ringExpChar R = 1", " max 0 1 = 1", " max q 1 = q", " q = 1", " 1 = 1", " p = q ↔ p.Prime", " p = 1 ↔ p.Prime", " 0 = 1 ↔ Nat.Prime 0", " p = 0", " CharZero R", " q = 1 ↔ p = 0", " q = 1 → p = 0",...	[ " ExpChar (R × S) p", " ExpChar (R × S) 1", " p = q", " 1 = q", " ringExpChar R = q", " ringExpChar R = 1", " max 0 1 = 1", " max q 1 = q", " q = 1", " 1 = 1", " p = q ↔ p.Prime", " p = 1 ↔ p.Prime", " 0 = 1 ↔ Nat.Prime 0", " p = 0", " CharZero R", " q = 1 ↔ p = 0" ]	[ " q = 1 → p = 0", " p = 0 → q = 1" ]
import Mathlib.MeasureTheory.Group.Measure import Mathlib.MeasureTheory.Integral.IntegrableOn import Mathlib.MeasureTheory.Function.LocallyIntegrable open Asymptotics MeasureTheory Set Filter variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {a b :...	Mathlib/MeasureTheory/Integral/Asymptotics.lean	81	85	theorem LocallyIntegrableOn.integrableOn_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))] (hf : LocallyIntegrableOn f (Iic a) μ) (ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot μ) : IntegrableOn f (Iic a) μ := by	refine integrableOn_Iic_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩ exact ⟨Iic a, Iic_mem_atBot a, hf.aestronglyMeasurable⟩	[ " IntegrableAtFilter f l μ", " ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖", " ‖f x‖ ≤ ‖C * ‖g x‖‖", " Integrable f μ", " IntegrableAtFilter f ⊤ μ", " StronglyMeasurableAtFilter f (cocompact α) μ", " StronglyMeasurableAtFilter f atTop μ", " StronglyMeasurableAtFilter f atBot μ", " IntegrableOn f (...	[ " IntegrableAtFilter f l μ", " ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖", " ‖f x‖ ≤ ‖C * ‖g x‖‖", " Integrable f μ", " IntegrableAtFilter f ⊤ μ", " StronglyMeasurableAtFilter f (cocompact α) μ", " StronglyMeasurableAtFilter f atTop μ", " StronglyMeasurableAtFilter f atBot μ", " IntegrableOn f (...	[]
import Mathlib.Analysis.SpecialFunctions.Pow.Real #align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8" noncomputable section open scoped Classical open Real NNReal ENNReal ComplexConjugate open Finset Function Set namespace NNReal var...	Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean	57	59	theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by	rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero] exact Real.rpow_eq_zero_iff_of_nonneg x.2	[ " x ^ y = 0 ↔ x = 0 ∧ y ≠ 0", " ↑x ^ y = ↑0 ↔ ↑x = 0 ∧ y ≠ 0" ]	[ " x ^ y = 0 ↔ x = 0 ∧ y ≠ 0" ]	[ " ↑x ^ y = ↑0 ↔ ↑x = 0 ∧ y ≠ 0" ]
import Mathlib.Analysis.Convex.Gauge import Mathlib.Analysis.Convex.Normed open Metric Bornology Filter Set open scoped NNReal Topology Pointwise noncomputable section section Module variable {E : Type*} [AddCommGroup E] [Module ℝ E] def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x the...	Mathlib/Analysis/Convex/GaugeRescale.lean	103	114	theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) : Continuous (gaugeRescale s t) := by	have hta : Absorbent ℝ t := absorbent_nhds_zero ht₀ refine continuous_iff_continuousAt.2 fun x ↦ ?_ rcases eq_or_ne x 0 with rfl \| hx · rw [ContinuousAt, gaugeRescale_zero] nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀] simp only [tendsto_comap_iff, (· ∘ ·), gauge_gaugeRescale _ hta htb] exact ten...	[ " gaugeRescale s t (c • x) = c • gaugeRescale s t x", " (c * gauge s x / (c * gauge t x) * c) • x = (c * (gauge s x / gauge t x)) • x", " gaugeRescale s s x = x", " gaugeRescale s s 0 = 0", " gauge s x ≠ 0", " gauge t (gaugeRescale s t x) = gauge s x", " gauge t (gaugeRescale s t 0) = gauge s 0", " ga...	[ " gaugeRescale s t (c • x) = c • gaugeRescale s t x", " (c * gauge s x / (c * gauge t x) * c) • x = (c * (gauge s x / gauge t x)) • x", " gaugeRescale s s x = x", " gaugeRescale s s 0 = 0", " gauge s x ≠ 0", " gauge t (gaugeRescale s t x) = gauge s x", " gauge t (gaugeRescale s t 0) = gauge s 0", " ga...	[ " ContinuousAt (gaugeRescale s t) x", " ContinuousAt (gaugeRescale s t) 0", " Tendsto (gaugeRescale s t) (𝓝 0) (𝓝 0)", " Tendsto (gaugeRescale s t) (𝓝 0) (comap (gauge t) (𝓝 0))", " Tendsto (fun x => gauge s x) (𝓝 0) (𝓝 0)" ]
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho import Mathlib.LinearAlgebra.Orientation #align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163" noncomputable section variable {E : Type*} [NormedAddCommGroup E] [InnerProduct...	Mathlib/Analysis/InnerProductSpace/Orientation.lean	91	96	theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) : e.toBasis.det = -f.toBasis.det := by	rw [e.toBasis.det.eq_smul_basis_det f.toBasis] -- Porting note: added `neg_one_smul` with explicit type simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h, neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]	[ " e.toBasis.det ⇑f = 1", " ¬e.toBasis.det ⇑f = -1", " 0 < e.toBasis.det ⇑f", " e.toBasis.det ⇑f = -1", " e.toBasis.orientation = f.toBasis.orientation", " e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation", " e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.o...	[ " e.toBasis.det ⇑f = 1", " ¬e.toBasis.det ⇑f = -1", " 0 < e.toBasis.det ⇑f", " e.toBasis.det ⇑f = -1", " e.toBasis.orientation = f.toBasis.orientation", " e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation", " e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.o...	[ " e.toBasis.det ⇑f.toBasis • f.toBasis.det = -f.toBasis.det" ]
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv #align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840" noncomputable section open Affine open Set section variable (k : Type) {V : Type} {P : Type*} [Ring k] [AddCommGroup V]...	Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean	128	132	theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan {s : Set P} {p : P} {v : V} (hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k s := by	rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩ rw [hv2p, vadd_vadd] exact ⟨p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl⟩	[ " vectorSpan k ∅ = ⊥", " vectorSpan k {p} = ⊥", " (spanPoints k s).Nonempty ↔ s.Nonempty", " (spanPoints k s).Nonempty → s.Nonempty", " ¬s.Nonempty → ¬(spanPoints k s).Nonempty", " s = ∅ → spanPoints k s = ∅", " spanPoints k s = ∅", " s.Nonempty → (spanPoints k s).Nonempty", " v +ᵥ p ∈ spanPoints k ...	[ " vectorSpan k ∅ = ⊥", " vectorSpan k {p} = ⊥", " (spanPoints k s).Nonempty ↔ s.Nonempty", " (spanPoints k s).Nonempty → s.Nonempty", " ¬s.Nonempty → ¬(spanPoints k s).Nonempty", " s = ∅ → spanPoints k s = ∅", " spanPoints k s = ∅", " s.Nonempty → (spanPoints k s).Nonempty", " v +ᵥ p ∈ spanPoints k ...	[ " v + v2 +ᵥ p2 ∈ spanPoints k s" ]
import Mathlib.Data.Set.Pointwise.SMul import Mathlib.GroupTheory.GroupAction.Hom open Set Pointwise	Mathlib/GroupTheory/GroupAction/Pointwise.lean	33	41	theorem MulAction.smul_bijective_of_is_unit {M : Type} [Monoid M] {α : Type} [MulAction M α] {m : M} (hm : IsUnit m) : Function.Bijective (fun (a : α) ↦ m • a) := by	lift m to Mˣ using hm rw [Function.bijective_iff_has_inverse] use fun a ↦ m⁻¹ • a constructor · intro x; simp [← Units.smul_def] · intro x; simp [← Units.smul_def]	[ " Function.Bijective fun a => m • a", " Function.Bijective fun a => ↑m • a", " ∃ g, (Function.LeftInverse g fun a => ↑m • a) ∧ Function.RightInverse g fun a => ↑m • a", " (Function.LeftInverse (fun a => m⁻¹ • a) fun a => ↑m • a) ∧ Function.RightInverse (fun a => m⁻¹ • a) fun a => ↑m • a", " Function.LeftInv...	[ " Function.Bijective fun a => m • a" ]	[ " Function.Bijective fun a => ↑m • a", " ∃ g, (Function.LeftInverse g fun a => ↑m • a) ∧ Function.RightInverse g fun a => ↑m • a", " (Function.LeftInverse (fun a => m⁻¹ • a) fun a => ↑m • a) ∧ Function.RightInverse (fun a => m⁻¹ • a) fun a => ↑m • a", " Function.LeftInverse (fun a => m⁻¹ • a) fun a => ↑m • a"...
import Mathlib.Data.List.Forall2 import Mathlib.Data.Set.Pairwise.Basic import Mathlib.Init.Data.Fin.Basic #align_import data.list.nodup from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0" universe u v open Nat Function variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α ...	Mathlib/Data/List/Nodup.lean	39	40	theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by	simp only [Nodup, pairwise_cons, forall_mem_ne]	[ " (a :: l).Nodup ↔ a ∉ l ∧ l.Nodup" ]	[ " (a :: l).Nodup ↔ a ∉ l ∧ l.Nodup" ]	[]
import Mathlib.Tactic.CategoryTheory.Coherence import Mathlib.CategoryTheory.Monoidal.Free.Coherence #align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe" open CategoryTheory Category Iso namespace CategoryTheory.MonoidalCategory v...	Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean	67	68	theorem unitors_inv_equal : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by	coherence	[ " (α_ (𝟙_ C) X Y).hom ≫ (λ_ (X ⊗ Y)).hom = (λ_ X).hom ⊗ 𝟙 Y", " (λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y)", " (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom", " 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ (X ⊗ Y)).inv ≫ (α_ X Y (𝟙_ C)).hom", " (λ_ X).inv ⊗ 𝟙 Y = (λ_ (X ⊗ Y)).inv ≫ (α_ (𝟙...	[ " (α_ (𝟙_ C) X Y).hom ≫ (λ_ (X ⊗ Y)).hom = (λ_ X).hom ⊗ 𝟙 Y", " (λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y)", " (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom", " 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ (X ⊗ Y)).inv ≫ (α_ X Y (𝟙_ C)).hom", " (λ_ X).inv ⊗ 𝟙 Y = (λ_ (X ⊗ Y)).inv ≫ (α_ (𝟙...	[]
import Mathlib.Algebra.Polynomial.Degree.Definitions import Mathlib.Data.ENat.Basic #align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836" noncomputable section open Function Polynomial Finsupp Finset open scoped Polynomial namespace ...	Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean	111	114	theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) : p.trailingDegree = n ↔ p.natTrailingDegree = n := by	rw [trailingDegree_eq_natTrailingDegree hp] exact WithTop.coe_eq_coe	[ " p.trailingDegree = ⊤", " p.trailingDegree = ↑p.natTrailingDegree", " ↑n = ↑(Option.getD (↑n) 0)", " p.trailingDegree = ↑n ↔ p.natTrailingDegree = n", " ↑p.natTrailingDegree = ↑n ↔ p.natTrailingDegree = n" ]	[ " p.trailingDegree = ⊤", " p.trailingDegree = ↑p.natTrailingDegree", " ↑n = ↑(Option.getD (↑n) 0)", " p.trailingDegree = ↑n ↔ p.natTrailingDegree = n" ]	[ " ↑p.natTrailingDegree = ↑n ↔ p.natTrailingDegree = n" ]
import Mathlib.Topology.Bases import Mathlib.Topology.DenseEmbedding #align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977" noncomputable section open Filter Set open Topology universe u v section Ultrafilter def ultrafilterBasis (α : Type u) : Set ...	Mathlib/Topology/StoneCech.lean	67	77	theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} : ↑u ≤ 𝓝 x ↔ x = joinM u := by	rw [eq_comm, ← Ultrafilter.coe_le_coe] change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α \| s ∈ v } ∈ u simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff, mem_setOf_eq] constructor · intro h a ha exact h _ ⟨ha, a, rfl⟩ · rintro h a ⟨xi, a, rfl⟩ exact h ...	[ " ∀ t₁ ∈ ultrafilterBasis α, ∀ t₂ ∈ ultrafilterBasis α, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ ultrafilterBasis α, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂", " ∃ t₃ ∈ ultrafilterBasis α, u ∈ t₃ ∧ t₃ ⊆ (fun s => {u \| s ∈ u}) a ∩ (fun s => {u \| s ∈ u}) b", " v ∈ (fun s => {u \| s ∈ u}) a", " v ∈ (fun s => {u \| s ∈ u}) b", " a ∩ b ⊆ a", " a ∩ b...	[ " ∀ t₁ ∈ ultrafilterBasis α, ∀ t₂ ∈ ultrafilterBasis α, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ ultrafilterBasis α, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂", " ∃ t₃ ∈ ultrafilterBasis α, u ∈ t₃ ∧ t₃ ⊆ (fun s => {u \| s ∈ u}) a ∩ (fun s => {u \| s ∈ u}) b", " v ∈ (fun s => {u \| s ∈ u}) a", " v ∈ (fun s => {u \| s ∈ u}) b", " a ∩ b ⊆ a", " a ∩ b...	[ " ↑u ≤ 𝓝 x ↔ ↑(joinM u) ≤ ↑x", " ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, {v \| s ∈ v} ∈ u", " (∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s => {u \| s ∈ u}) → i ∈ ↑u) ↔ ∀ s ∈ x, {v \| s ∈ v} ∈ u", " (∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s => {u \| s ∈ u}) → i ∈ ↑u) → ∀ s ∈ x, {v \| s ∈ v} ∈ u", " {v \| a ∈ ...
import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex #align_import measure_theory.function.special_functions.inner from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf" variable {α : Type} {𝕜 : Type} {E : Type*} variable [RCLike ...	Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean	41	47	theorem AEMeasurable.inner {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E] [SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f g : α → E} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : AEMeasurable (fun x => ⟪f x, g x⟫) μ := by	refine ⟨fun x => ⟪hf.mk f x, hg.mk g x⟫, hf.measurable_mk.inner hg.measurable_mk, ?_⟩ refine hf.ae_eq_mk.mp (hg.ae_eq_mk.mono fun x hxg hxf => ?_) dsimp only congr	[ " AEMeasurable (fun x => ⟪f x, g x⟫_𝕜) μ", " (fun x => ⟪f x, g x⟫_𝕜) =ᵐ[μ] fun x => ⟪mk f hf x, mk g hg x⟫_𝕜", " (fun x => ⟪f x, g x⟫_𝕜) x = (fun x => ⟪mk f hf x, mk g hg x⟫_𝕜) x", " ⟪f x, g x⟫_𝕜 = ⟪mk f hf x, mk g hg x⟫_𝕜" ]	[ " AEMeasurable (fun x => ⟪f x, g x⟫_𝕜) μ" ]	[ " (fun x => ⟪f x, g x⟫_𝕜) =ᵐ[μ] fun x => ⟪mk f hf x, mk g hg x⟫_𝕜", " (fun x => ⟪f x, g x⟫_𝕜) x = (fun x => ⟪mk f hf x, mk g hg x⟫_𝕜) x", " ⟪f x, g x⟫_𝕜 = ⟪mk f hf x, mk g hg x⟫_𝕜" ]
import Mathlib.Algebra.Order.Ring.Abs import Mathlib.Tactic.Ring #align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c" def hyperoperation : ℕ → ℕ → ℕ → ℕ \| 0, _, k => k + 1 \| 1, m, 0 => m \| 2, _, 0 => 0 \| _ + 3, _, 0 => 1 \| n + 1, m, k + 1 ...	Mathlib/Data/Nat/Hyperoperation.lean	69	78	theorem hyperoperation_two : hyperoperation 2 = (· * ·) := by	ext m k induction' k with bn bih · rw [hyperoperation] exact (Nat.mul_zero m).symm · rw [hyperoperation_recursion, hyperoperation_one, bih] -- Porting note: was `ring` dsimp only nth_rewrite 1 [← mul_one m] rw [← mul_add, add_comm]	[ " hyperoperation 0 m k = k.succ", " hyperoperation (n + 3) m 0 = 1", " hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)", " hyperoperation 1 = fun x x_1 => x + x_1", " hyperoperation 1 m k = m + k", " hyperoperation 1 m 0 = m + 0", " hyperoperation 1 m (bn + 1) = m + (b...	[ " hyperoperation 0 m k = k.succ", " hyperoperation (n + 3) m 0 = 1", " hyperoperation (n + 1) m (k + 1) = hyperoperation n m (hyperoperation (n + 1) m k)", " hyperoperation 1 = fun x x_1 => x + x_1", " hyperoperation 1 m k = m + k", " hyperoperation 1 m 0 = m + 0", " hyperoperation 1 m (bn + 1) = m + (b...	[ " hyperoperation 2 m k = m * k", " hyperoperation 2 m 0 = m * 0", " 0 = m * 0", " hyperoperation 2 m (bn + 1) = m * (bn + 1)", " (fun x x_1 => x + x_1) m (m * bn) = m * (bn + 1)", " m + m * bn = m * (bn + 1)", " m * 1 + m * bn = m * (bn + 1)" ]
import Mathlib.Data.Fintype.Basic import Mathlib.ModelTheory.Substructures #align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15" open FirstOrder namespace FirstOrder namespace Language open Structure variable (L : Language) (M : Type*) (N : T...	Mathlib/ModelTheory/ElementaryMaps.lean	78	94	theorem map_boundedFormula (f : M ↪ₑ[L] N) {α : Type*} {n : ℕ} (φ : L.BoundedFormula α n) (v : α → M) (xs : Fin n → M) : φ.Realize (f ∘ v) (f ∘ xs) ↔ φ.Realize v xs := by	classical rw [← BoundedFormula.realize_restrictFreeVar Set.Subset.rfl, Set.inclusion_eq_id, iff_eq_eq] have h := f.map_formula' ((φ.restrictFreeVar id).toFormula.relabel (Fintype.equivFin _)) (Sum.elim (v ∘ (↑)) xs ∘ (Fintype.equivFin _).symm) simp only [Formula.realize_relabel, BoundedForm...	[ " f = g", " { toFun := toFun✝, map_formula' := map_formula'✝ } = g", " { toFun := toFun✝¹, map_formula' := map_formula'✝¹ } = { toFun := toFun✝, map_formula' := map_formula'✝ }", " toFun✝¹ = toFun✝", " toFun✝¹ x = toFun✝ x", " φ.Realize (⇑f ∘ v) (⇑f ∘ xs) ↔ φ.Realize v xs", " (φ.restrictFreeVar id).Real...	[ " f = g", " { toFun := toFun✝, map_formula' := map_formula'✝ } = g", " { toFun := toFun✝¹, map_formula' := map_formula'✝¹ } = { toFun := toFun✝, map_formula' := map_formula'✝ }", " toFun✝¹ = toFun✝", " toFun✝¹ x = toFun✝ x", " φ.Realize (⇑f ∘ v) (⇑f ∘ xs) ↔ φ.Realize v xs" ]	[ " (φ.restrictFreeVar id).Realize ((⇑f ∘ v) ∘ Subtype.val) (⇑f ∘ xs) = φ.Realize v xs", " (φ.restrictFreeVar id).Realize\n (((Sum.elim (v ∘ Subtype.val) xs ∘ ⇑(Fintype.equivFin ({ x // x ∈ φ.freeVarFinset } ⊕ Fin n)).symm) ∘\n ⇑(Fintype.equivFin ({ x // x ∈ φ.freeVarFinset } ⊕ Fin n))) ∘\n Sum...
import Mathlib.Algebra.CharP.Invertible import Mathlib.Algebra.Order.Interval.Set.Group import Mathlib.Analysis.Convex.Segment import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional import Mathlib.Tactic.FieldSimp #align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c...	Mathlib/Analysis/Convex/Between.lean	80	83	theorem affineSegment_image (f : P →ᵃ[R] P') (x y : P) : f '' affineSegment R x y = affineSegment R (f x) (f y) := by	rw [affineSegment, affineSegment, Set.image_image, ← comp_lineMap] rfl	[ " affineSegment R x y = segment R x y", " affineSegment R x y = affineSegment R y x", " z ∈ affineSegment R x y ↔ z ∈ affineSegment R y x", " z ∈ affineSegment R x y → z ∈ affineSegment R y x", " z ∈ affineSegment R y x", " 1 - t ∈ Set.Icc 0 1", " (lineMap y x) (1 - t) = z", " z ∈ affineSegment R y x ...	[ " affineSegment R x y = segment R x y", " affineSegment R x y = affineSegment R y x", " z ∈ affineSegment R x y ↔ z ∈ affineSegment R y x", " z ∈ affineSegment R x y → z ∈ affineSegment R y x", " z ∈ affineSegment R y x", " 1 - t ∈ Set.Icc 0 1", " (lineMap y x) (1 - t) = z", " z ∈ affineSegment R y x ...	[ " (fun x_1 => f ((lineMap x y) x_1)) '' Set.Icc 0 1 = ⇑(f.comp (lineMap x y)) '' Set.Icc 0 1" ]
import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.BigOperators.NatAntidiagonal import Mathlib.Algebra.CharZero.Lemmas import Mathlib.Data.Finset.NatAntidiagonal import Mathlib.Data.Nat.Choose.Central import Mathlib.Data.Tree.Basic import Mathlib.Tactic.FieldSimp import Mathlib.Tactic.GCongr import Mathlib...	Mathlib/Combinatorics/Enumerative/Catalan.lean	148	149	theorem catalan_three : catalan 3 = 5 := by	norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]	[ " catalan 0 = 1", " catalan (n + 1) = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)", " catalan (n + 1) = ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2", " catalan 1 = 1", " gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n ↑i.centralBinom / (↑i + 1) * ↑(n - i).centralBinom / (↑n - ↑i + ...	[ " catalan 0 = 1", " catalan (n + 1) = ∑ i : Fin n.succ, catalan ↑i * catalan (n - ↑i)", " catalan (n + 1) = ∑ ij ∈ antidiagonal n, catalan ij.1 * catalan ij.2", " catalan 1 = 1", " gosperCatalan (n + 1) (i + 1) - gosperCatalan (n + 1) i =\n ↑i.centralBinom / (↑i + 1) * ↑(n - i).centralBinom / (↑n - ↑i + ...	[]
import Mathlib.Algebra.FreeMonoid.Basic import Mathlib.Algebra.Group.Submonoid.MulOpposite import Mathlib.Algebra.Group.Submonoid.Operations import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Data.Finset.NoncommProd import Mathlib.Data.Int.Order.Lemmas #align_import group_theory.submonoid.membership fro...	Mathlib/Algebra/Group/Submonoid/Membership.lean	262	265	theorem mem_sSup_of_mem {S : Set (Submonoid M)} {s : Submonoid M} (hs : s ∈ S) : ∀ {x : M}, x ∈ s → x ∈ sSup S := by	rw [← SetLike.le_def] exact le_sSup hs	[ " x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i", " x ∈ ⨆ i, S i → ∃ i, x ∈ S i", " x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i", " ∃ i, 1 ∈ S i", " ∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i", " ∃ i, x * y ∈ S i", " x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i)", " x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s", " x ∈ ↑(sSup S)...	[ " x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i", " x ∈ ⨆ i, S i → ∃ i, x ∈ S i", " x ∈ closure (⋃ i, ↑(S i)) → ∃ i, x ∈ S i", " ∃ i, 1 ∈ S i", " ∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i", " ∃ i, x * y ∈ S i", " x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i)", " x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s", " x ∈ ↑(sSup S)...	[ " s ≤ sSup S" ]
import Mathlib.Data.List.Cycle import Mathlib.GroupTheory.Perm.Cycle.Type import Mathlib.GroupTheory.Perm.List #align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a" open Equiv Equiv.Perm List variable {α : Type*} namespace Equiv.Perm secti...	Mathlib/GroupTheory/Perm/Cycle/Concrete.lean	312	320	theorem next_toList_eq_apply (p : Perm α) (x y : α) (hy : y ∈ toList p x) : next (toList p x) y hy = p y := by	rw [mem_toList_iff] at hy obtain ⟨k, hk, hk'⟩ := hy.left.exists_pow_eq_of_mem_support hy.right rw [← nthLe_toList p x k (by simpa using hk)] at hk' simp_rw [← hk'] rw [next_nthLe _ (nodup_toList _ _), nthLe_toList, nthLe_toList, ← mul_apply, ← pow_succ', length_toList, ← pow_mod_orderOf_cycleOf_apply p (...	[ " toList 1 x = []", " p.toList x = [] ↔ x ∉ p.support", " (p.toList x).length = (p.cycleOf x).support.card", " p.toList x ≠ [y]", " False", " 2 ≤ (p.toList x).length ↔ x ∈ p.support", " (p.toList x).get ⟨n, hn⟩ = (p ^ n) x", " (p.toList x).get ⟨0, ⋯⟩ = x", " (p.toList x).nthLe n hn = (p ^ n) x", "...	[ " toList 1 x = []", " p.toList x = [] ↔ x ∉ p.support", " (p.toList x).length = (p.cycleOf x).support.card", " p.toList x ≠ [y]", " False", " 2 ≤ (p.toList x).length ↔ x ∈ p.support", " (p.toList x).get ⟨n, hn⟩ = (p ^ n) x", " (p.toList x).get ⟨0, ⋯⟩ = x", " (p.toList x).nthLe n hn = (p ^ n) x", "...	[ " (p.toList x).next y hy✝ = p y", " k < (p.toList x).length", " (p.toList x).next ((p.toList x).nthLe k ⋯) ⋯ = p ((p.toList x).nthLe k ⋯)", " (p.cycleOf x).IsCycle" ]
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex #align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section namespace Real open Set Filter open scoped Topology Real theorem tan_add {x y : ℝ} ...	Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean	47	49	theorem tan_two_mul {x : ℝ} : tan (2 * x) = 2 * tan x / (1 - tan x ^ 2) := by	have := @Complex.tan_two_mul x norm_cast at *	[ " (x + y).tan = (x.tan + y.tan) / (1 - x.tan * y.tan)", " ((∀ (k : ℤ), ↑x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), ↑y ≠ (2 * ↑l + 1) * ↑π / 2) ∨\n (∃ k, ↑x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, ↑y = (2 * ↑l + 1) * ↑π / 2", " ↑x = (2 * ↑a✝ + 1) * ↑π / 2 ↔ x = (2 * ↑a✝ + 1) * π / 2", " ↑y = (2 * ↑a✝ + 1) * ↑π / 2 ↔ y...	[ " (x + y).tan = (x.tan + y.tan) / (1 - x.tan * y.tan)", " ((∀ (k : ℤ), ↑x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), ↑y ≠ (2 * ↑l + 1) * ↑π / 2) ∨\n (∃ k, ↑x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, ↑y = (2 * ↑l + 1) * ↑π / 2", " ↑x = (2 * ↑a✝ + 1) * ↑π / 2 ↔ x = (2 * ↑a✝ + 1) * π / 2", " ↑y = (2 * ↑a✝ + 1) * ↑π / 2 ↔ y...	[]
import Mathlib.Algebra.Order.Floor import Mathlib.Algebra.ContinuedFractions.Basic #align_import algebra.continued_fractions.computation.basic from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad" namespace GeneralizedContinuedFraction -- Fix a carrier `K`. variable (K : Type*) structu...	Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean	159	161	theorem stream_isSeq (v : K) : (IntFractPair.stream v).IsSeq := by	intro _ hyp simp [IntFractPair.stream, hyp]	[ " (IntFractPair.stream v).IsSeq", " IntFractPair.stream v (n✝ + 1) = none" ]	[ " (IntFractPair.stream v).IsSeq" ]	[ " IntFractPair.stream v (n✝ + 1) = none" ]
import Mathlib.Data.Finsupp.Multiset import Mathlib.Order.Bounded import Mathlib.SetTheory.Cardinal.PartENat import Mathlib.SetTheory.Ordinal.Principal import Mathlib.Tactic.Linarith #align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f" noncomputa...	Mathlib/SetTheory/Cardinal/Ordinal.lean	500	543	theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c := by	refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c) -- the only nontrivial part is `c * c ≤ c`. We prove it inductively. refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h -- consider the minimal well-order `r` on `α` (...	[ " c.ord.IsLimit", " ℵ₀ = 0", " c.ord ≤ a", " c ≤ a.card", " ℵ₀ ≤ a.card", " ℵ₀ ≤ (succ a).card", " ℵ₀.ord.IsLimit", " ω.IsLimit", " c * c = c", " c ≤ c * c", " c * c ≤ c", " ⟦α⟧ * ⟦α⟧ ≤ ⟦α⟧", " type s ≤ type r", " o < type r", " typein s p < type r", " (typein s p).card < #α", " (typ...	[ " c.ord.IsLimit", " ℵ₀ = 0", " c.ord ≤ a", " c ≤ a.card", " ℵ₀ ≤ a.card", " ℵ₀ ≤ (succ a).card", " ℵ₀.ord.IsLimit", " ω.IsLimit", " c * c = c" ]	[ " c ≤ c * c", " c * c ≤ c", " ⟦α⟧ * ⟦α⟧ ≤ ⟦α⟧", " type s ≤ type r", " o < type r", " typein s p < type r", " (typein s p).card < #α", " (typein s p).card ≤\n (succ (typein (fun x x_1 => x < x_1) (g p))).card * (succ (typein (fun x x_1 => x < x_1) (g p))).card", " {q \| s q p} ⊆ insert (g p) {x \| x...
import Mathlib.Algebra.Group.Support import Mathlib.Algebra.Order.Monoid.WithTop import Mathlib.Data.Nat.Cast.Field #align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829" open Function Set section AddMonoidWithOne variable {α M : Type*} [AddMonoidWith...	Mathlib/Algebra/CharZero/Lemmas.lean	188	188	theorem half_sub (a : R) : a / 2 - a = -(a / 2) := by	rw [← neg_sub, sub_half]	[ " 2 ≠ 0", " a + a = 0 ↔ a = 0", " 0 = bit0 a ↔ a = 0", " bit0 a = 0 ↔ a = 0", " n = 0 ∨ a = b", " a = b", " ↑2 * a = ↑2 * b", " bit1 a = 1 ↔ a = 0", " 1 = bit1 0", " 1 = bit1 a ↔ a = 0", " (a + a) / 2 = a", " a / 2 + a / 2 = a", " a - a / 2 = a / 2", " a / 2 - a = -(a / 2)" ]	[ " 2 ≠ 0", " a + a = 0 ↔ a = 0", " 0 = bit0 a ↔ a = 0", " bit0 a = 0 ↔ a = 0", " n = 0 ∨ a = b", " a = b", " ↑2 * a = ↑2 * b", " bit1 a = 1 ↔ a = 0", " 1 = bit1 0", " 1 = bit1 a ↔ a = 0", " (a + a) / 2 = a", " a / 2 + a / 2 = a", " a - a / 2 = a / 2", " a / 2 - a = -(a / 2)" ]	[]
import Mathlib.LinearAlgebra.Quotient #align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d" universe u v variable {R M M₂ M₃ : Type*} variable [Ring R] [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃] variable [Module R M] [Module R M₂] [Modul...	Mathlib/LinearAlgebra/Isomorphisms.lean	81	85	theorem quotientInfEquivSupQuotient_injective (p p' : Submodule R M) : Function.Injective (quotientInfToSupQuotient p p') := by	rw [← ker_eq_bot, quotientInfToSupQuotient, ker_liftQ_eq_bot] rw [ker_comp, ker_mkQ] exact fun ⟨x, hx1⟩ hx2 => ⟨hx1, hx2⟩	[ " comap p.subtype (p ⊓ p') ≤ ker (subToSupQuotient p p')", " comap p.subtype (p ⊓ p') ≤ comap p.subtype ((p ⊔ p') ⊓ p')", " Function.Injective ⇑(quotientInfToSupQuotient p p')", " ker (subToSupQuotient p p') ≤ comap p.subtype (p ⊓ p')", " comap (inclusion ⋯) (comap (p ⊔ p').subtype p') ≤ comap p.subtype (p ...	[ " comap p.subtype (p ⊓ p') ≤ ker (subToSupQuotient p p')", " comap p.subtype (p ⊓ p') ≤ comap p.subtype ((p ⊔ p') ⊓ p')", " Function.Injective ⇑(quotientInfToSupQuotient p p')" ]	[ " ker (subToSupQuotient p p') ≤ comap p.subtype (p ⊓ p')", " comap (inclusion ⋯) (comap (p ⊔ p').subtype p') ≤ comap p.subtype (p ⊓ p')" ]
import Mathlib.Order.Antichain import Mathlib.Order.UpperLower.Basic import Mathlib.Order.Interval.Set.Basic import Mathlib.Order.RelIso.Set #align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf" open Function Set variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s...	Mathlib/Order/Minimal.lean	121	128	theorem minimals_eq_minimals_of_subset_of_forall [IsTrans α r] (hts : t ⊆ s) (h : ∀ x ∈ s, ∃ y ∈ t, r y x) : minimals r s = minimals r t := by	refine Set.ext fun a ↦ ⟨fun ⟨has, hmin⟩ ↦ ⟨?_,fun b hbt ↦ hmin (hts hbt)⟩, fun ⟨hat, hmin⟩ ↦ ⟨hts hat, fun b hbs hba ↦ ?_⟩⟩ · obtain ⟨a', ha', haa'⟩ := h _ has rwa [antisymm (hmin (hts ha') haa') haa'] obtain ⟨b', hb't, hb'b⟩ := h b hbs rwa [antisymm (hmin hb't (Trans.trans hb'b hba)) (Trans.trans hb'b...	[ " ∀ ⦃b : α⦄, b ∈ {a} → r a b → r b a", " r b b → r b b", " x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y", " x ∈ s → ((∀ ⦃b : α⦄, b ∈ s → r x b → r b x) ↔ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y)", " r y x", " y = x", " x = y", " x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, rlt y x → y ∉ s", " x ∈ m...	[ " ∀ ⦃b : α⦄, b ∈ {a} → r a b → r b a", " r b b → r b b", " x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y", " x ∈ s → ((∀ ⦃b : α⦄, b ∈ s → r x b → r b x) ↔ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y)", " r y x", " y = x", " x = y", " x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, rlt y x → y ∉ s", " x ∈ m...	[ " a ∈ t", " r a b" ]
import Mathlib.Topology.PartialHomeomorph import Mathlib.Analysis.Normed.Group.AddTorsor import Mathlib.Analysis.NormedSpace.Pointwise import Mathlib.Data.Real.Sqrt #align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156" open Set Metric Pointwise var...	Mathlib/Analysis/NormedSpace/HomeomorphBall.lean	130	131	theorem univBall_target (c : P) {r : ℝ} (hr : 0 < r) : (univBall c r).target = ball c r := by	rw [univBall, dif_pos hr]; rfl	[ " (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x ∈ ball 0 1", " 0 < 1 + ‖x‖ ^ 2", " ‖x‖ ^ 2 < 1 + ‖x‖ ^ 2", " (fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) ((fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x) = x", " (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) ((fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) y) = y", " 0 < 1 - ‖y‖ ^ 2", " ContinuousOn\n ↑{ toFu...	[ " (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x ∈ ball 0 1", " 0 < 1 + ‖x‖ ^ 2", " ‖x‖ ^ 2 < 1 + ‖x‖ ^ 2", " (fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) ((fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) x) = x", " (fun x => (√(1 + ‖x‖ ^ 2))⁻¹ • x) ((fun y => (√(1 - ‖y‖ ^ 2))⁻¹ • y) y) = y", " 0 < 1 - ‖y‖ ^ 2", " ContinuousOn\n ↑{ toFu...	[ " (univUnitBall.trans' (unitBallBall c r hr) ⋯).target = ball c r" ]
import Mathlib.Logic.Encodable.Lattice import Mathlib.MeasureTheory.MeasurableSpace.Defs #align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90" open MeasurableSpace Set open scoped Classical open MeasureTheory def IsPiSystem {α} (C : Set (Set α)) :...	Mathlib/MeasureTheory/PiSystem.lean	256	261	theorem generatePiSystem_measurableSet {α} [M : MeasurableSpace α] {S : Set (Set α)} (h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S) : MeasurableSet t := by	induction' h_in_pi with s h_s s u _ _ _ h_s h_u · apply h_meas_S _ h_s · apply MeasurableSet.inter h_s h_u	[ " IsPiSystem {S}", " s ∩ t ∈ {S}", " IsPiSystem (insert ∅ S)", " s ∩ t ∈ insert ∅ S", " IsPiSystem (insert univ S)", " s ∩ t ∈ insert univ S", " IsPiSystem {s \| ∃ t ∈ S, f ⁻¹' t = s}", " f ⁻¹' s ∩ f ⁻¹' t ∈ {s \| ∃ t ∈ S, f ⁻¹' t = s}", " f ⁻¹' (s ∩ t) ∈ {s \| ∃ t ∈ S, f ⁻¹' t = s}", " IsPiSystem (⋃...	[ " IsPiSystem {S}", " s ∩ t ∈ {S}", " IsPiSystem (insert ∅ S)", " s ∩ t ∈ insert ∅ S", " IsPiSystem (insert univ S)", " s ∩ t ∈ insert univ S", " IsPiSystem {s \| ∃ t ∈ S, f ⁻¹' t = s}", " f ⁻¹' s ∩ f ⁻¹' t ∈ {s \| ∃ t ∈ S, f ⁻¹' t = s}", " f ⁻¹' (s ∩ t) ∈ {s \| ∃ t ∈ S, f ⁻¹' t = s}", " IsPiSystem (⋃...	[ " MeasurableSet s", " MeasurableSet (s ∩ u)" ]
import Mathlib.Analysis.SpecificLimits.Basic import Mathlib.Order.Interval.Set.IsoIoo import Mathlib.Topology.Order.MonotoneContinuity import Mathlib.Topology.UrysohnsBounded #align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" variable {X Y : T...	Mathlib/Topology/TietzeExtension.lean	220	262	theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f := by	/- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference between the previous approximation and `f`. -/ choose F hF_norm hF_dist using fun f : X →ᵇ ℝ => tietze_extension_step f e he set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0 have g0 :...	[ " ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖", " 0 < 3", " 0 < 2 / 3", " ∃ g, ‖g‖ ≤ ‖0‖ / 3 ∧ dist (g.compContinuous e) 0 ≤ 2 / 3 * ‖0‖", " ‖0‖ ≤ ‖0‖ / 3 ∧ dist (compContinuous 0 e) 0 ≤ 2 / 3 * ‖0‖", " Disjoint (⇑e '' (⇑f ⁻¹' Iic (-‖f‖ / 3))) (⇑e '' (⇑f ⁻¹' Ici (‖f‖ / 3)))", " Disjoi...	[ " ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖", " 0 < 3", " 0 < 2 / 3", " ∃ g, ‖g‖ ≤ ‖0‖ / 3 ∧ dist (g.compContinuous e) 0 ≤ 2 / 3 * ‖0‖", " ‖0‖ ≤ ‖0‖ / 3 ∧ dist (compContinuous 0 e) 0 ≤ 2 / 3 * ‖0‖", " Disjoint (⇑e '' (⇑f ⁻¹' Iic (-‖f‖ / 3))) (⇑e '' (⇑f ⁻¹' Ici (‖f‖ / 3)))", " Disjoi...	[ " ∀ (n : ℕ), dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖", " dist ((g n).compContinuous e) f ≤ (2 / 3) ^ n * ‖f‖", " dist ((g 0).compContinuous e) f ≤ (2 / 3) ^ 0 * ‖f‖", " dist ((g (n + 1)).compContinuous e) f ≤ (2 / 3) ^ (n + 1) * ‖f‖", " dist ((F (f - (g n).compContinuous e)).compContinuous e) (f...
import Mathlib.Analysis.RCLike.Lemmas import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex #align_import measure_theory.function.special_functions.is_R_or_C from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad" noncomputable section open NNReal ENNReal namespace RCLike variabl...	Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean	80	84	theorem aemeasurable_of_re_im (hre : AEMeasurable (fun x => RCLike.re (f x)) μ) (him : AEMeasurable (fun x => RCLike.im (f x)) μ) : AEMeasurable f μ := by	convert AEMeasurable.add (M := 𝕜) (RCLike.measurable_ofReal.comp_aemeasurable hre) ((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I) exact (RCLike.re_add_im _).symm	[ " Measurable f", " f x✝ = (RCLike.ofReal ∘ fun x => RCLike.re (f x)) x✝ + (RCLike.ofReal ∘ fun x => RCLike.im (f x)) x✝ * RCLike.I", " AEMeasurable f μ" ]	[ " Measurable f", " f x✝ = (RCLike.ofReal ∘ fun x => RCLike.re (f x)) x✝ + (RCLike.ofReal ∘ fun x => RCLike.im (f x)) x✝ * RCLike.I", " AEMeasurable f μ" ]	[]
import Mathlib.RingTheory.GradedAlgebra.Basic import Mathlib.Algebra.GradedMulAction import Mathlib.Algebra.DirectSum.Decomposition import Mathlib.Algebra.Module.BigOperators #align_import algebra.module.graded_module from "leanprover-community/mathlib"@"59cdeb0da2480abbc235b7e611ccd9a7e5603d7c" section open Dir...	Mathlib/Algebra/Module/GradedModule.lean	99	102	theorem smulAddMonoidHom_apply_of_of [DecidableEq ιA] [DecidableEq ιB] [GMonoid A] [Gmodule A M] {i j} (x : A i) (y : M j) : smulAddMonoidHom A M (DirectSum.of A i x) (of M j y) = of M (i +ᵥ j) (GSMul.smul x y) := by	simp [smulAddMonoidHom]	[ " ((smulAddMonoidHom A M) ((of A i) x)) ((of M j) y) = (of M (i +ᵥ j)) (GSMul.smul x y)" ]	[ " ((smulAddMonoidHom A M) ((of A i) x)) ((of M j) y) = (of M (i +ᵥ j)) (GSMul.smul x y)" ]	[]
import Mathlib.Order.BooleanAlgebra import Mathlib.Logic.Equiv.Basic #align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904" open Function OrderDual variable {ι α β : Type} {π : ι → Type} def symmDiff [Sup α] [SDiff α] (a b : α) : α := a \ b ⊔ b \ a #ali...	Mathlib/Order/SymmDiff.lean	125	125	theorem symmDiff_bot : a ∆ ⊥ = a := by	rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]	[ " ∀ (p q : Bool), p ∆ q = xor p q", " a ∆ b = b ∆ a", " a ∆ a = ⊥", " a ∆ ⊥ = a" ]	[ " ∀ (p q : Bool), p ∆ q = xor p q", " a ∆ b = b ∆ a", " a ∆ a = ⊥", " a ∆ ⊥ = a" ]	[]
import Mathlib.Order.BooleanAlgebra import Mathlib.Logic.Equiv.Basic #align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904" open Function OrderDual variable {ι α β : Type} {π : ι → Type} def symmDiff [Sup α] [SDiff α] (a b : α) : α := a \ b ⊔ b \ a #ali...	Mathlib/Order/SymmDiff.lean	113	113	theorem symmDiff_comm : a ∆ b = b ∆ a := by	simp only [symmDiff, sup_comm]	[ " ∀ (p q : Bool), p ∆ q = xor p q", " a ∆ b = b ∆ a" ]	[ " ∀ (p q : Bool), p ∆ q = xor p q", " a ∆ b = b ∆ a" ]	[]
import Mathlib.Algebra.Order.BigOperators.Group.Finset import Mathlib.Data.Nat.Factors import Mathlib.Order.Interval.Finset.Nat #align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" open scoped Classical open Finset namespace Nat variable (n : ℕ) d...	Mathlib/NumberTheory/Divisors.lean	95	99	theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by	rcases eq_or_ne m 0 with (rfl \| hm); · simp [divisors] simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter, mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff] exact le_of_dvd hm.bot_lt	[ " filter (fun x => x ∣ n) (range n.succ) = n.divisors", " a✝ ∈ filter (fun x => x ∣ n) (range n.succ) ↔ a✝ ∈ n.divisors", " a✝ ∣ n → a✝ < n.succ → 1 ≤ a✝", " filter (fun x => x ∣ n) (range n) = n.properDivisors", " a✝ ∈ filter (fun x => x ∣ n) (range n) ↔ a✝ ∈ n.properDivisors", " a✝ ∣ n → a✝ < n → 1 ≤ a✝...	[ " filter (fun x => x ∣ n) (range n.succ) = n.divisors", " a✝ ∈ filter (fun x => x ∣ n) (range n.succ) ↔ a✝ ∈ n.divisors", " a✝ ∣ n → a✝ < n.succ → 1 ≤ a✝", " filter (fun x => x ∣ n) (range n) = n.properDivisors", " a✝ ∈ filter (fun x => x ∣ n) (range n) ↔ a✝ ∈ n.properDivisors", " a✝ ∣ n → a✝ < n → 1 ≤ a✝...	[ " n ∈ divisors 0 ↔ n ∣ 0 ∧ 0 ≠ 0", " n ∣ m → n ≤ m" ]
import Mathlib.CategoryTheory.Action import Mathlib.Combinatorics.Quiver.Arborescence import Mathlib.Combinatorics.Quiver.ConnectedComponent import Mathlib.GroupTheory.FreeGroup.IsFreeGroup #align_import group_theory.nielsen_schreier from "leanprover-community/mathlib"@"1bda4fc53de6ade5ab9da36f2192e24e2084a2ce" n...	Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean	275	288	theorem path_nonempty_of_hom {G} [Groupoid.{u, u} G] [IsFreeGroupoid G] {a b : G} : Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b)) := by	rintro ⟨p⟩ rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ← mul_inv_eq_one] let X := FreeGroup (WeaklyConnectedComponent <\| Generators G) let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g) let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type...	[ " ∀ {X : Type u} [inst : Group X] (f : Labelling (Generators (ActionCategory G A)) X),\n ∃! F, ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ((fun {a b} e => ⟨IsFreeGroup.of ↑e, ⋯⟩) g) = f g", " ∃! F, ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ((fun {a b} e => ⟨IsFreeGroup.of ↑...	[ " ∀ {X : Type u} [inst : Group X] (f : Labelling (Generators (ActionCategory G A)) X),\n ∃! F, ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ((fun {a b} e => ⟨IsFreeGroup.of ↑e, ⋯⟩) g) = f g", " ∃! F, ∀ (a b : Generators (ActionCategory G A)) (g : a ⟶ b), F.map ((fun {a b} e => ⟨IsFreeGroup.of ↑...	[ " Nonempty (Path (IsFreeGroupoid.symgen a) (IsFreeGroupoid.symgen b))", " FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen b)) *\n (FreeGroup.of (WeaklyConnectedComponent.mk (IsFreeGroupoid.symgen a)))⁻¹ =\n 1", " F.map p = ((Functor.const G).obj ()).map p", " F = (Functor.const G).obj...
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots import Mathlib.Tactic.ByContra import Mathlib.Topology.Algebra.Polynomial import Mathlib.NumberTheory.Padics.PadicVal import Mathlib.Analysis.Complex.Arg #align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16...	Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean	29	32	theorem eval_one_cyclotomic_prime {R : Type*} [CommRing R] {p : ℕ} [hn : Fact p.Prime] : eval 1 (cyclotomic p R) = p := by	simp only [cyclotomic_prime, eval_X, one_pow, Finset.sum_const, eval_pow, eval_finset_sum, Finset.card_range, smul_one_eq_cast]	[ " eval 1 (cyclotomic p R) = ↑p" ]	[ " eval 1 (cyclotomic p R) = ↑p" ]	[]
import Mathlib.Analysis.Normed.Group.Basic #align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3" section HammingDistNorm open Finset Function variable {α ι : Type} {β : ι → Type} [Fintype ι] [∀ i, DecidableEq (β i)] variable {γ : ι → Type*} [∀ ...	Mathlib/InformationTheory/Hamming.lean	78	81	theorem hammingDist_triangle_right (x y z : ∀ i, β i) : hammingDist x y ≤ hammingDist x z + hammingDist y z := by	rw [hammingDist_comm y] exact hammingDist_triangle _ _ _	[ " hammingDist x x = 0", " ∀ ⦃x_1 : ι⦄, x_1 ∈ univ → ¬x x_1 ≠ x x_1", " hammingDist x y = hammingDist y x", " hammingDist x z ≤ hammingDist x y + hammingDist y z", " (filter (fun i => x i ≠ z i) univ).card ≤\n (filter (fun i => x i ≠ y i) univ).card + (filter (fun i => y i ≠ z i) univ).card", " filter (...	[ " hammingDist x x = 0", " ∀ ⦃x_1 : ι⦄, x_1 ∈ univ → ¬x x_1 ≠ x x_1", " hammingDist x y = hammingDist y x", " hammingDist x z ≤ hammingDist x y + hammingDist y z", " (filter (fun i => x i ≠ z i) univ).card ≤\n (filter (fun i => x i ≠ y i) univ).card + (filter (fun i => y i ≠ z i) univ).card", " filter (...	[]
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals import Mathlib.Analysis.Calculus.ParametricIntegral import Mathlib.MeasureTheory.Measure.Haar.NormedSpace #align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb" open MeasureTheory Set Filter A...	Mathlib/Analysis/MellinTransform.lean	47	50	theorem MellinConvergent.const_smul {f : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) : MellinConvergent (fun t => c • f t) s := by	simpa only [MellinConvergent, smul_comm] using hf.smul c	[ " MellinConvergent (fun t => c • f t) s" ]	[ " MellinConvergent (fun t => c • f t) s" ]	[]
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic #align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap open s...	Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean	75	112	theorem condexp_indicator (hf_int : Integrable f μ) (hs : MeasurableSet[m] s) : μ[s.indicator f\|m] =ᵐ[μ] s.indicator (μ[f\|m]) := by	by_cases hm : m ≤ m0 swap; · simp_rw [condexp_of_not_le hm, Set.indicator_zero']; rfl by_cases hμm : SigmaFinite (μ.trim hm) swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm, Set.indicator_zero']; rfl haveI : SigmaFinite (μ.trim hm) := hμm -- use `have` to perform what should be the first calc step becau...	[ " μ[f\|m] =ᶠ[ae (μ.restrict s)] 0", " 0 =ᶠ[ae (μ.restrict s)] 0", " SigmaFinite ((μ.restrict s).trim hm)", " SigmaFinite ((μ.trim hm).restrict s)", " ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s) s_1 < ⊤ → IntegrableOn (μ[f\|m]) s_1 (μ.restrict s)", " ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s...	[ " μ[f\|m] =ᶠ[ae (μ.restrict s)] 0", " 0 =ᶠ[ae (μ.restrict s)] 0", " SigmaFinite ((μ.restrict s).trim hm)", " SigmaFinite ((μ.trim hm).restrict s)", " ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s) s_1 < ⊤ → IntegrableOn (μ[f\|m]) s_1 (μ.restrict s)", " ∀ (s_1 : Set α), MeasurableSet s_1 → (μ.restrict s...	[ " s.indicator (μ[f\|m]) =ᶠ[ae μ] s.indicator (μ[s.indicator f + sᶜ.indicator f\|m])", " s.indicator (μ[s.indicator f + sᶜ.indicator f\|m]) =ᶠ[ae μ] μ[s.indicator f\|m]", " s.indicator (μ[s.indicator f + sᶜ.indicator f\|m]) =ᶠ[ae μ] s.indicator (μ[s.indicator f\|m] + μ[sᶜ.indicator f\|m])", " s.indicator (μ[s.indicat...
import Mathlib.Algebra.Module.Card import Mathlib.SetTheory.Cardinal.CountableCover import Mathlib.SetTheory.Cardinal.Continuum import Mathlib.Analysis.SpecificLimits.Normed import Mathlib.Topology.MetricSpace.Perfect universe u v open Filter Pointwise Set Function Cardinal open scoped Cardinal Topology theorem c...	Mathlib/Topology/Algebra/Module/Cardinality.lean	110	115	theorem cardinal_eq_of_isOpen {E : Type} (𝕜 : Type) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E} (hs : IsOpen s) (h's : s.Nonempty) : #s = #E := by	rcases h's with ⟨x, hx⟩ exact cardinal_eq_of_mem_nhds 𝕜 (hs.mem_nhds hx)	[ " 𝔠 ≤ #𝕜", " ∃ f, range f ⊆ Set.univ ∧ Continuous f ∧ Injective f", " Perfect Set.univ", " ∃ y ∈ U ∩ Set.univ, y ≠ x", " x + c ^ n ∈ U ∩ Set.univ", " x + c ^ n ≠ x", " ¬c ^ n = 0", " c ≠ 0", " 𝔠 ≤ #E", " lift.{v, u} 𝔠 ≤ lift.{v, u} #𝕜", " #↑s = #E", " ∀ (n : ℕ), c ^ n ≠ 0", " c ^ n ≠ 0"...	[ " 𝔠 ≤ #𝕜", " ∃ f, range f ⊆ Set.univ ∧ Continuous f ∧ Injective f", " Perfect Set.univ", " ∃ y ∈ U ∩ Set.univ, y ≠ x", " x + c ^ n ∈ U ∩ Set.univ", " x + c ^ n ≠ x", " ¬c ^ n = 0", " c ≠ 0", " 𝔠 ≤ #E", " lift.{v, u} 𝔠 ≤ lift.{v, u} #𝕜", " #↑s = #E", " ∀ (n : ℕ), c ^ n ≠ 0", " c ^ n ≠ 0"...	[]
import Mathlib.Data.Set.Pointwise.SMul #align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" class AddTorsor (G : outParam Type) (P : Type) [AddGroup G] extends AddAction G P, VSub G P where [nonempty : Nonempty P] vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ...	Mathlib/Algebra/AddTorsor.lean	165	167	theorem vsub_vadd_eq_vsub_sub (p₁ p₂ : P) (g : G) : p₁ -ᵥ (g +ᵥ p₂) = p₁ -ᵥ p₂ - g := by	rw [← add_right_inj (p₂ -ᵥ p₁ : G), vsub_add_vsub_cancel, ← neg_vsub_eq_vsub_rev, vadd_vsub, ← add_sub_assoc, ← neg_vsub_eq_vsub_rev, neg_add_self, zero_sub]	[ " g₁ = g₂", " g +ᵥ p₁ -ᵥ p₂ = g + (p₁ -ᵥ p₂)", " g +ᵥ p₁ -ᵥ p₂ +ᵥ p₂ = g + (p₁ -ᵥ p₂) +ᵥ p₂", " p -ᵥ p = 0", " p₁ = p₂", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) = p₁ -ᵥ p₃", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) +ᵥ p₃ = p₁ -ᵥ p₃ +ᵥ p₃", " -(p₁ -ᵥ p₂) = p₂ -ᵥ p₁", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₁) +ᵥ p₁ = 0 +ᵥ p₁", " g +ᵥ p -ᵥ q = g - (q...	[ " g₁ = g₂", " g +ᵥ p₁ -ᵥ p₂ = g + (p₁ -ᵥ p₂)", " g +ᵥ p₁ -ᵥ p₂ +ᵥ p₂ = g + (p₁ -ᵥ p₂) +ᵥ p₂", " p -ᵥ p = 0", " p₁ = p₂", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) = p₁ -ᵥ p₃", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₃) +ᵥ p₃ = p₁ -ᵥ p₃ +ᵥ p₃", " -(p₁ -ᵥ p₂) = p₂ -ᵥ p₁", " p₁ -ᵥ p₂ + (p₂ -ᵥ p₁) +ᵥ p₁ = 0 +ᵥ p₁", " g +ᵥ p -ᵥ q = g - (q...	[]
import Mathlib.Topology.Algebra.GroupWithZero import Mathlib.Topology.Order.OrderClosed #align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064" open Topology Filter TopologicalSpace Filter Set Function namespace WithZeroTopology variable {α...	Mathlib/Topology/Algebra/WithZeroTopology.lean	101	101	theorem singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({↑γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by	simp	[ " 𝓝 = update pure 0 (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ))", " pure 0 ≤ ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)", " 𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)", " (𝓝 0).HasBasis (fun γ => γ ≠ 0) Iio", " (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)).HasBasis (fun γ => γ ≠ 0) Iio", " DirectedOn ((fun γ => Iio γ) ⁻¹'o fun x x_1 => x ≥ x_1...	[ " 𝓝 = update pure 0 (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ))", " pure 0 ≤ ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)", " 𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)", " (𝓝 0).HasBasis (fun γ => γ ≠ 0) Iio", " (⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Iio γ)).HasBasis (fun γ => γ ≠ 0) Iio", " DirectedOn ((fun γ => Iio γ) ⁻¹'o fun x x_1 => x ≥ x_1...	[]
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic #align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a" variable {R M N N' : Type*} variable [CommRing R] [AddCommGroup M] [AddCo...	Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean	96	99	theorem liftAlternating_algebraMap (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (r : R) : liftAlternating (R := R) (M := M) (N := N) f (algebraMap _ (ExteriorAlgebra R M) r) = r • f 0 0 := by	rw [Algebra.algebraMap_eq_smul_one, map_smul, liftAlternating_one]	[ " Module R (M [⋀^ι]→ₗ[R] N)", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] N", " M [⋀^Fin 0]→ₗ[R] N ≃ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] (i : ℕ) → M [⋀^Fin i]→ₗ[R] N", " M →ₗ[R] ((i : ℕ) → M [⋀^Fin i]→ₗ[R]...	[ " Module R (M [⋀^ι]→ₗ[R] N)", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] N", " M [⋀^Fin 0]→ₗ[R] N ≃ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] (i : ℕ) → M [⋀^Fin i]→ₗ[R] N", " M →ₗ[R] ((i : ℕ) → M [⋀^Fin i]→ₗ[R]...	[]
import Mathlib.MeasureTheory.Group.Measure import Mathlib.MeasureTheory.Integral.IntegrableOn import Mathlib.MeasureTheory.Function.LocallyIntegrable open Asymptotics MeasureTheory Set Filter variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F] {f : α → E} {g : α → F} {a b :...	Mathlib/MeasureTheory/Integral/Asymptotics.lean	97	101	theorem LocallyIntegrable.integrable_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))] [OrderTop α] (hf : LocallyIntegrable f μ) (ho : f =O[atBot] g) (hg : IntegrableAtFilter g atBot μ) : Integrable f μ := by	refine integrable_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩ exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter	[ " IntegrableAtFilter f l μ", " ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖", " ‖f x‖ ≤ ‖C * ‖g x‖‖", " Integrable f μ", " IntegrableAtFilter f ⊤ μ", " StronglyMeasurableAtFilter f (cocompact α) μ", " StronglyMeasurableAtFilter f atTop μ", " StronglyMeasurableAtFilter f atBot μ", " IntegrableOn f (...	[ " IntegrableAtFilter f l μ", " ∀ᵐ (a : α) ∂μ.restrict s, ‖f a‖ ≤ ‖C * ‖g a‖‖", " ‖f x‖ ≤ ‖C * ‖g x‖‖", " Integrable f μ", " IntegrableAtFilter f ⊤ μ", " StronglyMeasurableAtFilter f (cocompact α) μ", " StronglyMeasurableAtFilter f atTop μ", " StronglyMeasurableAtFilter f atBot μ", " IntegrableOn f (...	[]
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic import Mathlib.Data.ZMod.Basic import Mathlib.RingTheory.GradedAlgebra.Basic #align_import linear_algebra.clifford_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0" namespace CliffordAlgebra variable {R M : Type*} [Co...	Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean	58	65	theorem evenOdd_mul_le (i j : ZMod 2) : evenOdd Q i * evenOdd Q j ≤ evenOdd Q (i + j) := by	simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span] apply Submodule.span_mono simp_rw [Set.iUnion_mul, Set.mul_iUnion, Set.iUnion_subset_iff, Set.mul_subset_iff] rintro ⟨xi, rfl⟩ ⟨yi, rfl⟩ x hx y hy refine Set.mem_iUnion.mpr ⟨⟨xi + yi, Nat.cast_add _ _⟩, ?_⟩ simp only [Subtype.coe_mk, Nat.ca...	[ " 1 ≤ evenOdd Q 0", " 1 ≤ LinearMap.range (ι Q) ^ ↑⟨0, ⋯⟩", " LinearMap.range (ι Q) ≤ evenOdd Q 1", " LinearMap.range (ι Q) ≤ LinearMap.range (ι Q) ^ ↑⟨1, ⋯⟩", " (ι Q) m₁ * (ι Q) m₂ ∈ LinearMap.range (ι Q) ^ ↑⟨2, ⋯⟩", " (ι Q) m₁ * (ι Q) m₂ ∈ LinearMap.range (ι Q) * LinearMap.range (ι Q)", " evenOdd Q i ...	[ " 1 ≤ evenOdd Q 0", " 1 ≤ LinearMap.range (ι Q) ^ ↑⟨0, ⋯⟩", " LinearMap.range (ι Q) ≤ evenOdd Q 1", " LinearMap.range (ι Q) ≤ LinearMap.range (ι Q) ^ ↑⟨1, ⋯⟩", " (ι Q) m₁ * (ι Q) m₂ ∈ LinearMap.range (ι Q) ^ ↑⟨2, ⋯⟩", " (ι Q) m₁ * (ι Q) m₂ ∈ LinearMap.range (ι Q) * LinearMap.range (ι Q)", " evenOdd Q i ...	[ " Submodule.span R ((⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1)) * ⋃ i, ↑(LinearMap.range (ι Q) ^ ↑i)) ≤\n Submodule.span R (⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1))", " (⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1)) * ⋃ i, ↑(LinearMap.range (ι Q) ^ ↑i) ⊆ ⋃ i_1, ↑(LinearMap.range (ι Q) ^ ↑i_1)", " ∀ (i_1 : { n // ↑n = ...
import Mathlib.Analysis.NormedSpace.PiLp import Mathlib.Analysis.InnerProductSpace.PiL2 #align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" noncomputable section open scoped NNReal Matrix namespace Matrix variable {R l m n α β : Type*} [Fintype l] [Fintyp...	Mathlib/Analysis/Matrix.lean	116	118	theorem nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) : ‖A.map f‖₊ = ‖A‖₊ := by	simp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]	[ " ‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)", " ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r", " ‖A‖₊ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ ≤ r", " ‖A‖ < r ↔ ∀ (i : m) (j : n), ‖A i j‖ < r", " ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r", " ‖A.map f‖₊ = ‖A‖₊" ]	[ " ‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)", " ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r", " ‖A‖₊ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ ≤ r", " ‖A‖ < r ↔ ∀ (i : m) (j : n), ‖A i j‖ < r", " ‖A‖₊ < r ↔ ∀ (i : m) (j : n), ‖A i j‖₊ < r", " ‖A.map f‖₊ = ‖A‖₊" ]	[]
import Mathlib.RingTheory.Derivation.ToSquareZero import Mathlib.RingTheory.Ideal.Cotangent import Mathlib.RingTheory.IsTensorProduct import Mathlib.Algebra.Exact import Mathlib.Algebra.MvPolynomial.PDeriv import Mathlib.Algebra.Polynomial.Derivation #align_import ring_theory.kaehler from "leanprover-community/mathli...	Mathlib/RingTheory/Kaehler.lean	78	99	theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) : D.tensorProductTo (x * y) = TensorProduct.lmul' (S := S) R x • D.tensorProductTo y + TensorProduct.lmul' (S := S) R y • D.tensorProductTo x := by	refine TensorProduct.induction_on x ?_ ?_ ?_ · rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero] swap · intro x₁ y₁ h₁ h₂ rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm] intro x₁ x₂ refine TensorProduct.induction_on y ?_ ?_ ?_ · rw [mul_zero, map_...	[ " 1 ⊗ₜ[R] a - a ⊗ₜ[R] 1 ∈ ideal R S", " D.tensorProductTo (x * y) =\n (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x", " D.tensorProductTo (0 * y) =\n (TensorProduct.lmul' R) 0 • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo 0", " ...	[ " 1 ⊗ₜ[R] a - a ⊗ₜ[R] 1 ∈ ideal R S", " D.tensorProductTo (x * y) =\n (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x" ]	[ " D.tensorProductTo (0 * y) =\n (TensorProduct.lmul' R) 0 • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo 0", " ∀ (x y_1 : S ⊗[R] S),\n D.tensorProductTo (x * y) =\n (TensorProduct.lmul' R) x • D.tensorProductTo y + (TensorProduct.lmul' R) y • D.tensorProductTo x →\n D.tens...
import Mathlib.Logic.Function.Basic import Mathlib.Logic.Relator import Mathlib.Init.Data.Quot import Mathlib.Tactic.Cases import Mathlib.Tactic.Use import Mathlib.Tactic.MkIffOfInductiveProp import Mathlib.Tactic.SimpRw #align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a...	Mathlib/Logic/Relation.lean	312	316	theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by	intro x y h induction' h with z w _ b c · rfl · apply Relation.ReflTransGen.head (h b) c	[ " ReflTransGen r a c", " ReflTransGen r a b", " ReflTransGen r a c✝", " Symmetric (ReflTransGen r)", " ReflTransGen r y x", " ReflTransGen r x x", " ReflTransGen r w x" ]	[ " ReflTransGen r a c", " ReflTransGen r a b", " ReflTransGen r a c✝", " Symmetric (ReflTransGen r)" ]	[ " ReflTransGen r y x", " ReflTransGen r x x", " ReflTransGen r w x" ]

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