Split (1)

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 0 224	goals_before listlengths 0 220
import Mathlib.Analysis.NormedSpace.AddTorsor import Mathlib.LinearAlgebra.AffineSpace.Ordered import Mathlib.Topology.ContinuousFunction.Basic import Mathlib.Topology.GDelta import Mathlib.Analysis.NormedSpace.FunctionSeries import Mathlib.Analysis.SpecificLimits.Basic #align_import topology.urysohns_lemma from "lea...	Mathlib/Topology/UrysohnsLemma.lean	185	192	theorem approx_le_one (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ 1 := by	induction' n with n ihn generalizing c · exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one · simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, ← div_eq_inv_mul] have := add_le_add (ihn (left c)) (ihn (right c)) set_option tactic.skipAssignedInstances false in norm_num...	[ " approx n c x = 0", " approx 0 c x = 0", " approx (n + 1) c x = 0", " midpoint ℝ (approx n c.left x) (approx n c.right x) = 0", " x ∈ c.left.C", " approx n c x = 1", " approx 0 c x = 1", " approx (n + 1) c x = 1", " midpoint ℝ (approx n c.left x) (approx n c.right x) = 1", " x ∉ c.left.U", " 0 ...	[ " approx n c x = 0", " approx 0 c x = 0", " approx (n + 1) c x = 0", " midpoint ℝ (approx n c.left x) (approx n c.right x) = 0", " x ∈ c.left.C", " approx n c x = 1", " approx 0 c x = 1", " approx (n + 1) c x = 1", " midpoint ℝ (approx n c.left x) (approx n c.right x) = 1", " x ∉ c.left.U", " 0 ...
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	121	122	theorem goldConj_neg : ψ < 0 := by	linarith [one_sub_goldConj, one_lt_gold]	[ " 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", " 1 - φ = ψ", " 1 - ψ = φ", " φ - ψ = √5", " φ ^ (n + 2) - φ ^ (...	[ " 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", " 1 - φ = ψ", " 1 - ψ = φ", " φ - ψ = √5", " φ ^ (n + 2) - φ ^ (...
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" open Filter Metric Set open scoped ComplexConjugate Real To...	Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean	33	37	theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by	unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]	[ " x.arg.sin = x.im / abs x", " (if 0 ≤ x.re then (x.im / abs x).arcsin\n else if 0 ≤ x.im then ((-x).im / abs x).arcsin + π else ((-x).im / abs x).arcsin - π).sin =\n x.im / abs x", " (x.im / abs x).arcsin.sin = x.im / abs x", " (((-x).im / abs x).arcsin + π).sin = x.im / abs x", " (((-x).im / abs x...	[]
import Mathlib.Geometry.Manifold.ChartedSpace #align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db" noncomputable section open scoped Classical open Manifold Topology open Set Filter TopologicalSpace variable {H M H' M' X : Typ...	Mathlib/Geometry/Manifold/LocalInvariantProperties.lean	648	666	theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H} [ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H} (hx : x ∈ f.source ∪ sᶜ) : G.IsLocalStructomorphWithinAt (⇑f) s x ↔ x ∈ s → ∃ e : PartialHomeomorph H H, e ∈ G ∧ e.source ⊆ f.sour...	constructor · intro hf h2x obtain ⟨e, he, hfe, hxe⟩ := hf h2x refine ⟨e.restr f.source, closedUnderRestriction' he f.open_source, ?_, ?_, hxe, ?_⟩ · simp_rw [PartialHomeomorph.restr_source] exact inter_subset_right.trans interior_subset · intro x' hx' exact hfe ⟨hx'.1, hx'.2.1⟩ · rw...	[ " ∀ {s : Set H} {x : H} {u : Set H} {f : H → H},\n IsOpen u → x ∈ u → (G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x)", " G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x", " G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt f...	[ " ∀ {s : Set H} {x : H} {u : Set H} {f : H → H},\n IsOpen u → x ∈ u → (G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x)", " G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x", " G.IsLocalStructomorphWithinAt f s x → G.IsLocalStructomorphWithinAt f...
import Mathlib.RingTheory.Localization.FractionRing import Mathlib.RingTheory.Localization.Ideal import Mathlib.RingTheory.Noetherian #align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177" variable {R : Type*} [CommRing R] (M : Submonoid R) ...	Mathlib/RingTheory/Localization/Submodule.lean	75	78	theorem coeSubmodule_span (s : Set R) : coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by	rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span] rfl	[ " coeSubmodule S ⊥ = ⊥", " coeSubmodule S ⊤ = 1", " coeSubmodule S (Ideal.span s) = Submodule.span R (⇑(algebraMap R S) '' s)", " Submodule.span R (⇑(Algebra.linearMap R S) '' s) = Submodule.span R (⇑(algebraMap R S) '' s)" ]	[ " coeSubmodule S ⊥ = ⊥", " coeSubmodule S ⊤ = 1" ]
import Mathlib.Combinatorics.Quiver.Basic import Mathlib.Combinatorics.Quiver.Path #align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e" universe v v₁ v₂ u u₁ u₂ variable {U : Type*} [Quiver.{u + 1} U] namespace Quiver def Hom.cast {u v u' v...	Mathlib/Combinatorics/Quiver/Cast.lean	87	90	theorem Path.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v) : p.cast hu hv = _root_.cast (by rw [hu, hv]) p := by	subst_vars rfl	[ " (u ⟶ v) = (u' ⟶ v')", " cast hu hv e = _root_.cast ⋯ e", " cast ⋯ ⋯ e = _root_.cast ⋯ e", " cast hu' hv' (cast hu hv e) = cast ⋯ ⋯ e", " cast ⋯ ⋯ (cast ⋯ ⋯ e) = cast ⋯ ⋯ e", " HEq (cast hu hv e) e", " HEq (cast ⋯ ⋯ e) e", " cast hu hv e = e' ↔ HEq e e'", " _root_.cast ⋯ e = e' ↔ HEq e e'", " e' ...	[ " (u ⟶ v) = (u' ⟶ v')", " cast hu hv e = _root_.cast ⋯ e", " cast ⋯ ⋯ e = _root_.cast ⋯ e", " cast hu' hv' (cast hu hv e) = cast ⋯ ⋯ e", " cast ⋯ ⋯ (cast ⋯ ⋯ e) = cast ⋯ ⋯ e", " HEq (cast hu hv e) e", " HEq (cast ⋯ ⋯ e) e", " cast hu hv e = e' ↔ HEq e e'", " _root_.cast ⋯ e = e' ↔ HEq e e'", " e' ...
import Mathlib.Analysis.RCLike.Lemmas import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner import Mathlib.MeasureTheory.Integral.SetIntegral #align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad" set_option linter.uppercaseLean3 false...	Mathlib/MeasureTheory/Function/L2Space.lean	154	167	theorem integral_inner_eq_sq_snorm (f : α →₂[μ] E) : ∫ a, ⟪f a, f a⟫ ∂μ = ENNReal.toReal (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ (2 : ℝ) ∂μ) := by	simp_rw [inner_self_eq_norm_sq_to_K] norm_cast rw [integral_eq_lintegral_of_nonneg_ae] rotate_left · exact Filter.eventually_of_forall fun x => sq_nonneg _ · exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable congr ext1 x have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by ...	[ " Integrable (fun x => f x ^ 2) μ", " Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ", " Memℒp f 2 μ ↔ Memℒp (fun x => ‖f x‖ ^ 2) 1 μ", " ‖f x✝‖ ^ 2 = ‖f x✝‖ ^ ENNReal.toReal 2", " 1 = 2 / 2", " Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ", " f x✝ ^ 2 = ‖f x✝‖ ^ 2", " snorm (fun x => ‖↑↑f x‖ ^ 2) 1...	[ " Integrable (fun x => f x ^ 2) μ", " Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ", " Memℒp f 2 μ ↔ Memℒp (fun x => ‖f x‖ ^ 2) 1 μ", " ‖f x✝‖ ^ 2 = ‖f x✝‖ ^ ENNReal.toReal 2", " 1 = 2 / 2", " Memℒp f 2 μ ↔ Integrable (fun x => f x ^ 2) μ", " f x✝ ^ 2 = ‖f x✝‖ ^ 2", " snorm (fun x => ‖↑↑f x‖ ^ 2) 1...
import Mathlib.Algebra.Lie.Nilpotent import Mathlib.Algebra.Lie.Normalizer #align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102" universe u v w w₁ w₂ variable {R : Type u} {L : Type v} variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie...	Mathlib/Algebra/Lie/CartanSubalgebra.lean	65	69	theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) : H.toLieSubmodule.ucs k = H.toLieSubmodule := by	induction' k with k ih · simp · simp [ih]	[ " H.toLieSubmodule.normalizer = H.toLieSubmodule", " LieSubmodule.ucs k H.toLieSubmodule = H.toLieSubmodule", " LieSubmodule.ucs 0 H.toLieSubmodule = H.toLieSubmodule", " LieSubmodule.ucs (k + 1) H.toLieSubmodule = H.toLieSubmodule" ]	[ " H.toLieSubmodule.normalizer = H.toLieSubmodule" ]
import Mathlib.RingTheory.Valuation.Basic import Mathlib.NumberTheory.Padics.PadicNorm import Mathlib.Analysis.Normed.Field.Basic #align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7" noncomputable section open scoped Classical open Nat m...	Mathlib/NumberTheory/Padics/PadicNumbers.lean	234	238	theorem val_eq_iff_norm_eq {f g : PadicSeq p} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : f.valuation = g.valuation ↔ f.norm = g.norm := by	rw [norm_eq_pow_val hf, norm_eq_pow_val hg, ← neg_inj, zpow_inj] · exact mod_cast (Fact.out : p.Prime).pos · exact mod_cast (Fact.out : p.Prime).ne_one	[ " padicNorm p (↑f m) = padicNorm p (↑f n)", " False", " f.norm = 0 ↔ f ≈ 0", " f.norm = 0 → f ≈ 0", " f ≈ 0", " ∃ i, ∀ j ≥ i, padicNorm p (↑(f - 0) j) < ε", " ∀ j ≥ stationaryPoint hf, padicNorm p (↑(f - 0) j) < ε", " padicNorm p (↑(f - 0) j) < ε", " f ≈ 0 → f.norm = 0", " f.norm = 0", " f.norm ...	[ " padicNorm p (↑f m) = padicNorm p (↑f n)", " False", " f.norm = 0 ↔ f ≈ 0", " f.norm = 0 → f ≈ 0", " f ≈ 0", " ∃ i, ∀ j ≥ i, padicNorm p (↑(f - 0) j) < ε", " ∀ j ≥ stationaryPoint hf, padicNorm p (↑(f - 0) j) < ε", " padicNorm p (↑(f - 0) j) < ε", " f ≈ 0 → f.norm = 0", " f.norm = 0", " f.norm ...
import Mathlib.MeasureTheory.Integral.IntervalIntegral #align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b" noncomputable section open scoped ENNReal MeasureTheory Topology open Set MeasureTheory Filter Measure namespace MeasureTheory se...	Mathlib/MeasureTheory/Integral/Layercake.lean	73	82	theorem countable_meas_le_ne_meas_lt (g : α → R) : {t : R \| μ {a : α \| t ≤ g a} ≠ μ {a : α \| t < g a}}.Countable := by	-- the target set is contained in the set of points where the function `t ↦ μ {a : α \| t ≤ g a}` -- jumps down on the right of `t`. This jump set is countable for any function. let F : R → ℝ≥0∞ := fun t ↦ μ {a : α \| t ≤ g a} apply (countable_image_gt_image_Ioi F).mono intro t ht have : μ {a \| t < g a} < μ ...	[ " {t \| μ {a \| t ≤ g a} ≠ μ {a \| t < g a}}.Countable", " {t \| μ {a \| t ≤ g a} ≠ μ {a \| t < g a}} ⊆ {x \| ∃ z < F x, ∀ (y : R), x < y → F y ≤ z}", " t ∈ {x \| ∃ z < F x, ∀ (y : R), x < y → F y ≤ z}" ]	[]
import Mathlib.LinearAlgebra.Span import Mathlib.LinearAlgebra.BilinearMap #align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940" universe uι u v open Set open Pointwise namespace Submodule variable {ι : Sort uι} {R M N P : Type*} variabl...	Mathlib/Algebra/Module/Submodule/Bilinear.lean	59	73	theorem map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : Set M) (t : Set N) : map₂ f (span R s) (span R t) = span R (Set.image2 (fun m n => f m n) s t) := by	apply le_antisymm · rw [map₂_le] apply @span_induction' R M _ _ _ s intro a ha apply @span_induction' R N _ _ _ t intro b hb exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩ all_goals intros; simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add, Linear...	[ " (f ↑⟨m, hm⟩) n = (f m) n", " map₂ f (span R s) (span R t) = span R (image2 (fun m n => (f m) n) s t)", " map₂ f (span R s) (span R t) ≤ span R (image2 (fun m n => (f m) n) s t)", " ∀ m ∈ span R s, ∀ n ∈ span R t, (f m) n ∈ span R (image2 (fun m n => (f m) n) s t)", " ∀ (a : R),\n ∀ x ∈ span R s,\n ...	[ " (f ↑⟨m, hm⟩) n = (f m) n" ]
import Mathlib.Algebra.Order.Monoid.Defs import Mathlib.Algebra.Order.Sub.Defs import Mathlib.Util.AssertExists #align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce" open Function universe u variable {α : Type u} class OrderedAddCommGroup (α : Ty...	Mathlib/Algebra/Order/Group/Defs.lean	178	179	theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by	rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]	[ " b ≤ c", " 1 < a⁻¹ ↔ a < 1", " a⁻¹ < 1 ↔ 1 < a", " b < a⁻¹ * c ↔ a * b < c", " a * b < a * (a⁻¹ * c) ↔ a * b < c", " b⁻¹ * a < c ↔ a < b * c" ]	[ " b ≤ c", " 1 < a⁻¹ ↔ a < 1", " a⁻¹ < 1 ↔ 1 < a", " b < a⁻¹ * c ↔ a * b < c", " a * b < a * (a⁻¹ * c) ↔ a * b < c" ]
import Mathlib.Algebra.PUnitInstances import Mathlib.Tactic.Abel import Mathlib.Tactic.Ring import Mathlib.Order.Hom.Lattice #align_import algebra.ring.boolean_ring from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3" open scoped symmDiff variable {α β γ : Type*} class BooleanRing (α) ...	Mathlib/Algebra/Ring/BooleanRing.lean	105	105	theorem mul_one_add_self : a * (1 + a) = 0 := by	rw [mul_add, mul_one, mul_self, add_self]	[ " a + a = 0", " a + a = (a + a) * (a + a)", " (a + a) * (a + a) = a * a + a * a + (a * a + a * a)", " a * a + a * a + (a * a + a * a) = a + a + (a + a)", " -a = -a + 0", " -a + 0 = -a + -a + a", " -a + -a + a = a", " a = -b ↔ a = b", " a * b + b * a = 0", " a + b = (a + b) * (a + b)", " (a + b) ...	[ " a + a = 0", " a + a = (a + a) * (a + a)", " (a + a) * (a + a) = a * a + a * a + (a * a + a * a)", " a * a + a * a + (a * a + a * a) = a + a + (a + a)", " -a = -a + 0", " -a + 0 = -a + -a + a", " -a + -a + a = a", " a = -b ↔ a = b", " a * b + b * a = 0", " a + b = (a + b) * (a + b)", " (a + b) ...
import Mathlib.Data.Fintype.Option import Mathlib.Topology.Separation import Mathlib.Topology.Sets.Opens #align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" open Set Filter Topology variable {X : Type} def OnePoint (X : Type) := Option X #ali...	Mathlib/Topology/Compactification/OnePoint.lean	144	145	theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by	induction x using OnePoint.rec <;> simp	[ " (some '' s)ᶜ = some '' sᶜ ∪ {∞}", " x ≠ ∞ ↔ ∃ y, ↑y = x", " ∞ ≠ ∞ ↔ ∃ y, ↑y = ∞", " ↑x✝ ≠ ∞ ↔ ∃ y, ↑y = ↑x✝" ]	[ " (some '' s)ᶜ = some '' sᶜ ∪ {∞}" ]
import Mathlib.Combinatorics.SimpleGraph.Finite import Mathlib.Combinatorics.SimpleGraph.Maps open Finset namespace SimpleGraph variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V) section ReplaceVertex def replaceVertex : SimpleGraph V where Adj v w := if v = t then if w = t then False else G...	Mathlib/Combinatorics/SimpleGraph/Operations.lean	82	86	theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet = (G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by	ext e; refine e.inductionOn ?_ simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff] intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]	[ " (fun v w => if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) v w →\n (fun v w => if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) w v", " (if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s el...	[ " (fun v w => if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) v w →\n (fun v w => if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s else G.Adj v w) w v", " (if v = t then if w = t then False else G.Adj s w else if w = t then G.Adj v s el...
import Mathlib.Algebra.ContinuedFractions.Basic import Mathlib.Algebra.GroupWithZero.Basic #align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad" namespace GeneralizedContinuedFraction section WithDivisionRing variable {K : Type*}...	Mathlib/Algebra/ContinuedFractions/Translations.lean	180	181	theorem convergents'Aux_succ_none {s : Stream'.Seq (Pair K)} (h : s.head = none) (n : ℕ) : convergents'Aux s (n + 1) = 0 := by	simp [convergents'Aux, h, convergents'Aux.match_1]	[ " ∃ conts, g.continuants n = conts ∧ conts.a = A", " ∃ conts, g.continuants n = conts ∧ conts.b = B", " g.convergents 0 = g.h", " g.continuantsAux 2 = { a := gp.b * g.h + gp.a, b := gp.b }", " g.continuants 1 = { a := gp.b * g.h + gp.a, b := gp.b }", " g.numerators 1 = gp.b * g.h + gp.a", " g.denominato...	[ " ∃ conts, g.continuants n = conts ∧ conts.a = A", " ∃ conts, g.continuants n = conts ∧ conts.b = B", " g.convergents 0 = g.h", " g.continuantsAux 2 = { a := gp.b * g.h + gp.a, b := gp.b }", " g.continuants 1 = { a := gp.b * g.h + gp.a, b := gp.b }", " g.numerators 1 = gp.b * g.h + gp.a", " g.denominato...
import Mathlib.Analysis.Asymptotics.Asymptotics import Mathlib.Analysis.Asymptotics.Theta import Mathlib.Analysis.Normed.Order.Basic #align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" namespace Asymptotics open Filter Function ...	Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean	128	130	theorem isEquivalent_zero_iff_eventually_zero : u ~[l] 0 ↔ u =ᶠ[l] 0 := by	rw [IsEquivalent, sub_zero] exact isLittleO_zero_right_iff	[ " v =O[l] u", " u x✝ = (u - v) x✝ + v x✝", " u ~[l] u", " 0 =o[l] u", " u ~[l] 0 ↔ u =ᶠ[l] 0", " u =o[l] 0 ↔ u =ᶠ[l] 0" ]	[ " v =O[l] u", " u x✝ = (u - v) x✝ + v x✝", " u ~[l] u", " 0 =o[l] u" ]
import Mathlib.Algebra.MvPolynomial.Equiv import Mathlib.Algebra.Polynomial.Eval #align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b89934139d3be96f9dab477f10c20f9f93da580" namespace MvPolynomial variable {R S σ : Type*}	Mathlib/Algebra/MvPolynomial/Polynomial.lean	19	28	theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring S] {x : S} (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R) : Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p := by	apply induction_on p · simp · intro p q hp hq simp [hp, hq] · intro p n hp simp [hp]	[ " Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p", " ∀ (a : R),\n Polynomial.eval x (eval₂ f g (C a)) =\n eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) (C a)", " ∀ (p q : MvPolynomial σ R),\n Polynomial.e...	[]
import Mathlib.Data.Matrix.Kronecker import Mathlib.LinearAlgebra.Matrix.ToLin import Mathlib.LinearAlgebra.TensorProduct.Basis #align_import linear_algebra.tensor_product.matrix from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081" variable {R : Type} {M N P M' N' : Type} {ι κ τ ι' κ' ...	Mathlib/LinearAlgebra/TensorProduct/Matrix.lean	49	53	theorem Matrix.toLin_kronecker (A : Matrix ι' ι R) (B : Matrix κ' κ R) : toLin (bM.tensorProduct bN) (bM'.tensorProduct bN') (A ⊗ₖ B) = TensorProduct.map (toLin bM bM' A) (toLin bN bN' B) := by	rw [← LinearEquiv.eq_symm_apply, toLin_symm, TensorProduct.toMatrix_map, toMatrix_toLin, toMatrix_toLin]	[ " (toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN')) (map f g) =\n kroneckerMap (fun x x_1 => x * x_1) ((toMatrix bM bM') f) ((toMatrix bN bN') g)", " (toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN')) (map f g) (i, j) (i', j') =\n kroneckerMap (fun x x_1 => x * x_1) ((toMatrix bM bM') f) ((toMa...	[ " (toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN')) (map f g) =\n kroneckerMap (fun x x_1 => x * x_1) ((toMatrix bM bM') f) ((toMatrix bN bN') g)", " (toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN')) (map f g) (i, j) (i', j') =\n kroneckerMap (fun x x_1 => x * x_1) ((toMatrix bM bM') f) ((toMa...
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	52	52	theorem powHalf_leftMoves (n) : (powHalf n).LeftMoves = PUnit := by	cases n <;> rfl	[ " (powHalf n).LeftMoves = PUnit.{u_1 + 1}", " (powHalf 0).LeftMoves = PUnit.{u_1 + 1}", " (powHalf (n✝ + 1)).LeftMoves = PUnit.{u_1 + 1}" ]	[]
import Mathlib.Algebra.Ring.Idempotents import Mathlib.Analysis.Normed.Group.Basic import Mathlib.Order.Basic import Mathlib.Tactic.NoncommRing #align_import analysis.normed_space.M_structure from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15" variable (X : Type*) [NormedAddCommGroup X] ...	Mathlib/Analysis/NormedSpace/MStructure.lean	105	144	theorem commute [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) : Commute P Q := by	have PR_eq_RPR : ∀ R : M, IsLprojection X R → P * R = R * P * R := fun R h₃ => by -- Porting note: Needed to fix function, which changes indent of following lines refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by rw [← norm_sub_eq_zero_iff] have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖(P * R) • x - (R * P...	[ " ‖x‖ = ‖(1 - P) • x‖ + ‖(1 - (1 - P)) • x‖", " ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖", " Commute P Q", " P * R = R * P * R", " (P * R) • x = (R * P * R) • x", " ‖(P * R) • x - (R * P * R) • x‖ = 0", " ‖R • x‖ = ‖R • P • R • x‖ + ‖(1 - R) • P • R • x‖ + (‖(R * R) • x - R • P • R • x‖ + ‖(1 - R) • (1 - P) • R • ...	[ " ‖x‖ = ‖(1 - P) • x‖ + ‖(1 - (1 - P)) • x‖", " ‖x‖ = ‖P • x‖ + ‖(1 - P) • x‖" ]
import Mathlib.Data.Vector.Basic import Mathlib.Data.Vector.Snoc set_option autoImplicit true namespace Vector section Fold section Unary variable (xs : Vector α n) (f₁ : β → σ₁ → σ₁ × γ) (f₂ : α → σ₂ → σ₂ × β) @[simp] theorem mapAccumr_mapAccumr : mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁ = let m := (...	Mathlib/Data/Vector/MapLemmas.lean	38	40	theorem mapAccumr_map (f₂ : α → β) : (mapAccumr f₁ (map f₂ xs) s) = (mapAccumr (fun x s => f₁ (f₂ x) s) xs s) := by	induction xs using Vector.revInductionOn generalizing s <;> simp_all	[ " mapAccumr f₁ (mapAccumr f₂ xs s₂).2 s₁ =\n let m :=\n mapAccumr\n (fun x s =>\n let r₂ := f₂ x s.2;\n let r₁ := f₁ r₂.2 s.1;\n ((r₁.1, r₂.1), r₁.2))\n xs (s₁, s₂);\n (m.1.1, m.2)", " mapAccumr f₁ (mapAccumr f₂ nil s₂).2 s₁ =\n let m :=\n mapAccumr\n ...	[ " mapAccumr f₁ (mapAccumr f₂ xs s₂).2 s₁ =\n let m :=\n mapAccumr\n (fun x s =>\n let r₂ := f₂ x s.2;\n let r₁ := f₁ r₂.2 s.1;\n ((r₁.1, r₂.1), r₁.2))\n xs (s₁, s₂);\n (m.1.1, m.2)", " mapAccumr f₁ (mapAccumr f₂ nil s₂).2 s₁ =\n let m :=\n mapAccumr\n ...
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.LatticeIntervals import Mathlib.Order.Interval.Set.OrdConnected #align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432" open scoped Classical open Set variable {ι : ...	Mathlib/Order/CompleteLatticeIntervals.lean	106	108	theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) : sInf t = default := by	simp [sInf, ht]	[ " sInf (Subtype.val '' t) = ↑(sInf t)", " sInf ∅ = default", " sInf t = default" ]	[ " sInf (Subtype.val '' t) = ↑(sInf t)", " sInf ∅ = default" ]
import Mathlib.CategoryTheory.Balanced import Mathlib.CategoryTheory.Limits.EssentiallySmall import Mathlib.CategoryTheory.Limits.Opposites import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms import Mathlib.CategoryTheory.Subobject.Lattice import Mathlib.CategoryTheory.Subobject.WellPowered import Mathlib.Data.S...	Mathlib/CategoryTheory/Generator.lean	117	126	theorem isDetecting_op_iff (𝒢 : Set C) : IsDetecting 𝒢.op ↔ IsCodetecting 𝒢 := by	refine ⟨fun h𝒢 X Y f hf => ?_, fun h𝒢 X Y f hf => ?_⟩ · refine (isIso_op_iff _).1 (h𝒢 _ fun G hG h => ?_) obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop exact ⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩ · refine (isIso_unop_iff _).1...	[ " IsSeparating 𝒢.op ↔ IsCoseparating 𝒢", " f = g", " (h ≫ f.op).unop = (h ≫ g.op).unop", " (f.unop ≫ h).op = (g.unop ≫ h).op", " IsCoseparating 𝒢.op ↔ IsSeparating 𝒢", " (f.op ≫ h).unop = (g.op ≫ h).unop", " (h ≫ f.unop).op = (h ≫ g.unop).op", " IsCoseparating 𝒢.unop ↔ IsSeparating 𝒢", " IsSep...	[ " IsSeparating 𝒢.op ↔ IsCoseparating 𝒢", " f = g", " (h ≫ f.op).unop = (h ≫ g.op).unop", " (f.unop ≫ h).op = (g.unop ≫ h).op", " IsCoseparating 𝒢.op ↔ IsSeparating 𝒢", " (f.op ≫ h).unop = (g.op ≫ h).unop", " (h ≫ f.unop).op = (h ≫ g.unop).op", " IsCoseparating 𝒢.unop ↔ IsSeparating 𝒢", " IsSep...
import Mathlib.Analysis.SpecialFunctions.Complex.Arg import Mathlib.Analysis.SpecialFunctions.Log.Basic #align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" noncomputable section namespace Complex open Set Filter Bornology open scop...	Mathlib/Analysis/SpecialFunctions/Complex/Log.lean	120	120	theorem log_neg_I : log (-I) = -(π / 2) * I := by	simp [log]	[ " x.log.re = (abs x).log", " x.log.im = x.arg", " -π < x.log.im", " x.log.im ≤ π", " cexp x.log = x", " x ∈ Set.range cexp → x ∈ {0}ᶜ", " cexp x ∈ {0}ᶜ", " (cexp x).log = x", " x = y", " (↑x.log).re = (↑x).log.re", " (↑x.log).im = (↑x).log.im", " (↑x).log.re = x.log", " (↑r * x).log = ↑r.log...	[ " x.log.re = (abs x).log", " x.log.im = x.arg", " -π < x.log.im", " x.log.im ≤ π", " cexp x.log = x", " x ∈ Set.range cexp → x ∈ {0}ᶜ", " cexp x ∈ {0}ᶜ", " (cexp x).log = x", " x = y", " (↑x.log).re = (↑x).log.re", " (↑x.log).im = (↑x).log.im", " (↑x).log.re = x.log", " (↑r * x).log = ↑r.log...
import Mathlib.Analysis.Asymptotics.Asymptotics import Mathlib.Analysis.Asymptotics.Theta import Mathlib.Analysis.Normed.Order.Basic #align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" namespace Asymptotics open Filter Function ...	Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean	151	154	theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) := by	rcases em <\| c = 0 with rfl \| h · exact (tendsto_congr' <\| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds · exact (isEquivalent_const_iff_tendsto h).mp hu	[ " v =O[l] u", " u x✝ = (u - v) x✝ + v x✝", " u ~[l] u", " 0 =o[l] u", " u ~[l] 0 ↔ u =ᶠ[l] 0", " u =o[l] 0 ↔ u =ᶠ[l] 0", " u ~[l] 0 ↔ u =O[l] 0", " u ~[l] 0", " ∃ s ∈ l, Set.EqOn u 0 s", " u ~[l] const α c ↔ Tendsto u l (𝓝 c)", " Tendsto (u - fun x => c) l (𝓝 0) ↔ Tendsto u l (𝓝 c)", " Tend...	[ " v =O[l] u", " u x✝ = (u - v) x✝ + v x✝", " u ~[l] u", " 0 =o[l] u", " u ~[l] 0 ↔ u =ᶠ[l] 0", " u =o[l] 0 ↔ u =ᶠ[l] 0", " u ~[l] 0 ↔ u =O[l] 0", " u ~[l] 0", " ∃ s ∈ l, Set.EqOn u 0 s", " u ~[l] const α c ↔ Tendsto u l (𝓝 c)", " Tendsto (u - fun x => c) l (𝓝 0) ↔ Tendsto u l (𝓝 c)", " Tend...
import Mathlib.Data.Matrix.Invertible import Mathlib.LinearAlgebra.Matrix.Adjugate import Mathlib.LinearAlgebra.FiniteDimensional #align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422" namespace Matrix universe u u' v variable {l : ...	Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean	120	129	theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 := suffices ∀ A B : Matrix n n α, A * B = 1 → B * A = 1 from ⟨this A B, this B A⟩ fun A B h => by letI : Invertible B.det := detInvertibleOfLeftInverse _ _ h letI : Invertible B := invertibleOfDetInvertible B calc B * A = B * A * (B * ⅟ B) := by	rw [mul_invOf_self, Matrix.mul_one] _ = B * (A * B * ⅟ B) := by simp only [Matrix.mul_assoc] _ = B * ⅟ B := by rw [h, Matrix.one_mul] _ = 1 := mul_invOf_self B	[ " ⅟A.det • A.adjugate * A = 1", " A * ⅟A.det • A.adjugate = 1", " ⅟A = ⅟A.det • A.adjugate", " B.det * A.det = 1", " A.det * B.det = 1", " (⅟A).det = ⅟A.det", " B * A = 1", " B * A = B * A * (B * ⅟B)", " B * A * (B * ⅟B) = B * (A * B * ⅟B)", " B * (A * B * ⅟B) = B * ⅟B" ]	[ " ⅟A.det • A.adjugate * A = 1", " A * ⅟A.det • A.adjugate = 1", " ⅟A = ⅟A.det • A.adjugate", " B.det * A.det = 1", " A.det * B.det = 1", " (⅟A).det = ⅟A.det" ]
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots import Mathlib.FieldTheory.Finite.Trace import Mathlib.Algebra.Group.AddChar import Mathlib.Data.ZMod.Units import Mathlib.Analysis.Complex.Polynomial #align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2...	Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean	189	192	theorem IsPrimitive.zmod_char_eq_one_iff (n : ℕ+) {ψ : AddChar (ZMod n) C} (hψ : IsPrimitive ψ) (a : ZMod n) : ψ a = 1 ↔ a = 0 := by	refine ⟨fun h => not_imp_comm.mp (hψ a) ?_, fun ha => by rw [ha, map_zero_eq_one]⟩ rw [zmod_char_isNontrivial_iff n (mulShift ψ a), mulShift_apply, mul_one, h, Classical.not_not]	[ " ⋯.unit ∈ rootsOfUnity (ringChar R).toPNat' R'", " (f.compAddChar φ).IsPrimitive", " ((f.compAddChar φ).mulShift a).IsNontrivial", " ∃ a_1, f (φ (a * a_1)) ≠ 1", " Function.Injective ψ.mulShift", " a = b", " ψ.IsPrimitive", " (ψ.mulShift a).IsNontrivial", " (ψ.mulShift a) (a⁻¹ * x) ≠ 1", " ¬(e.mu...	[ " ⋯.unit ∈ rootsOfUnity (ringChar R).toPNat' R'", " (f.compAddChar φ).IsPrimitive", " ((f.compAddChar φ).mulShift a).IsNontrivial", " ∃ a_1, f (φ (a * a_1)) ≠ 1", " Function.Injective ψ.mulShift", " a = b", " ψ.IsPrimitive", " (ψ.mulShift a).IsNontrivial", " (ψ.mulShift a) (a⁻¹ * x) ≠ 1", " ¬(e.mu...
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular import Mathlib.Topology.Category.CompHaus.EffectiveEpi import Mathlib.Topology.Category.Profinite.Limits import Mathlib.Topology.Category.Stonean.Basic universe u attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike open CategoryTh...	Mathlib/Topology/Category/Profinite/EffectiveEpi.lean	110	128	theorem effectiveEpiFamily_tfae {α : Type} [Finite α] {B : Profinite.{u}} (X : α → Profinite.{u}) (π : (a : α) → (X a ⟶ B)) : TFAE [ EffectiveEpiFamily X π , Epi (Sigma.desc π) , ∀ b : B, ∃ (a : α) (x : X a), π a x = b ] := by	tfae_have 2 → 1 · intro simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1] tfae_have 1 → 2 · intro; infer_instance tfae_have 3 ↔ 1 · erw [((CompHaus.effectiveEpiFamily_tfae (fun a ↦ profiniteToCompHaus.obj (X a)) (fun a ↦ profiniteToCompHaus.map (π a)))...	[ " { toFun := fun x => a, continuous_toFun := ⋯ } ≫ π = { toFun := fun x => b, continuous_toFun := ⋯ } ≫ π", " ({ toFun := fun x => a, continuous_toFun := ⋯ } ≫ π) x✝ = ({ toFun := fun x => b, continuous_toFun := ⋯ } ≫ π) x✝", " g = (fun {W} e h => ⋯.lift e ⋯) e ⋯", " g = ⋯.liftEquiv ⟨e, ⋯⟩", " ⋯.liftEquiv.s...	[ " { toFun := fun x => a, continuous_toFun := ⋯ } ≫ π = { toFun := fun x => b, continuous_toFun := ⋯ } ≫ π", " ({ toFun := fun x => a, continuous_toFun := ⋯ } ≫ π) x✝ = ({ toFun := fun x => b, continuous_toFun := ⋯ } ≫ π) x✝", " g = (fun {W} e h => ⋯.lift e ⋯) e ⋯", " g = ⋯.liftEquiv ⟨e, ⋯⟩", " ⋯.liftEquiv.s...
import Mathlib.Algebra.EuclideanDomain.Defs import Mathlib.Algebra.Ring.Divisibility.Basic import Mathlib.Algebra.Ring.Regular import Mathlib.Algebra.GroupWithZero.Divisibility import Mathlib.Algebra.Ring.Basic #align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0...	Mathlib/Algebra/EuclideanDomain/Basic.lean	84	85	theorem div_self {a : R} (a0 : a ≠ 0) : a / a = 1 := by	simpa only [one_mul] using mul_div_cancel_right₀ 1 a0	[ " a * b / b = a", " a - a * b / b = 0", " False", " b ∣ a", " b ∣ b * (a / b)", " a % b = 0", " b * c = b * (b * c / b)", " c ∣ a % b ↔ c ∣ a", " 0 / a = 0", " a / a = 1" ]	[ " a * b / b = a", " a - a * b / b = 0", " False", " b ∣ a", " b ∣ b * (a / b)", " a % b = 0", " b * c = b * (b * c / b)", " c ∣ a % b ↔ c ∣ a", " 0 / a = 0" ]
import Mathlib.Data.Set.Image import Mathlib.Data.List.GetD #align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4" open List variable {α β : Type*} (l : List α) namespace Set theorem range_list_map (f : α → β) : range (map f) = { l \| ∀ x ∈ l, x ∈ range f } :=...	Mathlib/Data/Set/List.lean	38	40	theorem range_list_get : range l.get = { x \| x ∈ l } := by	ext x rw [mem_setOf_eq, mem_iff_get, mem_range]	[ " range (map f) = {l \| ∀ x ∈ l, x ∈ range f}", " l ∈ range (map f)", " [] ∈ range (map f)", " a :: l ∈ range (map f)", " a :: map f l ∈ range (map f)", " f a :: map f l ∈ range (map f)", " range (map Subtype.val) = {l \| ∀ x ∈ l, x ∈ s}", " range l.get = {x \| x ∈ l}", " x ∈ range l.get ↔ x ∈ {x \| x ∈...	[ " range (map f) = {l \| ∀ x ∈ l, x ∈ range f}", " l ∈ range (map f)", " [] ∈ range (map f)", " a :: l ∈ range (map f)", " a :: map f l ∈ range (map f)", " f a :: map f l ∈ range (map f)", " range (map Subtype.val) = {l \| ∀ x ∈ l, x ∈ s}" ]
import Mathlib.Analysis.Normed.Group.Quotient import Mathlib.Topology.Instances.AddCircle #align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c" noncomputable section open Set open Int hiding mem_zmultiples_iff open AddSubgroup namespace A...	Mathlib/Analysis/Normed/Group/AddCircle.lean	44	68	theorem norm_coe_mul (x : ℝ) (t : ℝ) : ‖(↑(t * x) : AddCircle (t * p))‖ = \|t\| * ‖(x : AddCircle p)‖ := by	have aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b) := fun {a b c} h => by simp only [mem_zmultiples_iff] at h ⊢ obtain ⟨n, rfl⟩ := h exact ⟨n, (mul_smul_comm n c b).symm⟩ rcases eq_or_ne t 0 with (rfl \| ht); · simp have ht' : \|t\| ≠ 0 := (not_congr abs_eq_zero).mpr ht simp only ...	[ " ‖↑(t * x)‖ = \|t\| * ‖↑x‖", " c * a ∈ zmultiples (c * b)", " ∃ k, k • (c * b) = c * a", " ∃ k, k • (c * b) = c * n • b", " ‖↑(0 * x)‖ = \|0\| * ‖↑x‖", " sInf ((fun a => \|a\|) '' {m \| ↑m = ↑(t * x)}) = \|t\| * sInf ((fun a => \|a\|) '' {m \| ↑m = ↑x})", "p x t : ℝ\naux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈...	[]
import Mathlib.Data.Set.Function import Mathlib.Order.Interval.Set.OrdConnected #align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c" variable {α β : Type*} [LinearOrder α] open Function namespace Set def projIci (a x : α) : Ici a := ⟨max a x,...	Mathlib/Order/Interval/Set/ProjIcc.lean	99	99	theorem projIci_eq_self : projIci a x = ⟨a, le_rfl⟩ ↔ x ≤ a := by	simp [projIci, Subtype.ext_iff]	[ " projIcc a b h x = ⟨a, ⋯⟩", " projIcc a b h x = ⟨b, ⋯⟩", " projIci a x = ⟨a, ⋯⟩ ↔ x ≤ a" ]	[ " projIcc a b h x = ⟨a, ⋯⟩", " projIcc a b h x = ⟨b, ⋯⟩" ]
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	76	84	theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) = (dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) / (2 * √(a.im * c.im) * dist (b : ℂ) (conj ↑b)) := by	simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm] rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _), dist_comm (b : ℂ), Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im] congr 2 rw [sqrt_mul, sqrt_mul...	[ " (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))", " (dist z w / 2).cosh = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))", " (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))", " 2 ^ 2 * (z.im * w.im) + dist ...	[ " (dist z w / 2).sinh = dist ↑z ↑w / (2 * √(z.im * w.im))", " (dist z w / 2).cosh = dist (↑z) ((starRingEnd ℂ) ↑w) / (2 * √(z.im * w.im))", " (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))", " 2 ^ 2 * (z.im * w.im) + dist ...
import Mathlib.Analysis.Calculus.TangentCone import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics #align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01" open Filter Asymptotics ContinuousLinearMap Set Metric open scoped Classical open To...	Mathlib/Analysis/Calculus/FDeriv/Basic.lean	216	217	theorem fderivWithin_zero_of_isolated (h : 𝓝[s \ {x}] x = ⊥) : fderivWithin 𝕜 f s x = 0 := by	rw [fderivWithin, if_pos h]	[ " fderivWithin 𝕜 f s x = 0" ]	[]
import Mathlib.Algebra.Field.Defs import Mathlib.Algebra.GroupWithZero.Units.Lemmas import Mathlib.Algebra.Ring.Commute import Mathlib.Algebra.Ring.Invertible import Mathlib.Order.Synonym #align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102" open Function ...	Mathlib/Algebra/Field/Basic.lean	126	126	theorem neg_div_neg_eq (a b : K) : -a / -b = a / b := by	rw [div_neg_eq_neg_div, neg_div, neg_neg]	[ " -1 * -1 = 1", " 1 / -a = 1 / (-1 * a)", " 1 / (-1 * a) = 1 / a * (1 / -1)", " 1 / a * (1 / -1) = 1 / a * -1", " 1 / a * -1 = -(1 / a)", " b / -a = b * (1 / -a)", " b * (1 / -a) = b * -(1 / a)", " b * -(1 / a) = -(b * (1 / a))", " -(b * (1 / a)) = -(b / a)", " -b / a = -(b / a)", " -(b / a) = -...	[ " -1 * -1 = 1", " 1 / -a = 1 / (-1 * a)", " 1 / (-1 * a) = 1 / a * (1 / -1)", " 1 / a * (1 / -1) = 1 / a * -1", " 1 / a * -1 = -(1 / a)", " b / -a = b * (1 / -a)", " b * (1 / -a) = b * -(1 / a)", " b * -(1 / a) = -(b * (1 / a))", " -(b * (1 / a)) = -(b / a)", " -b / a = -(b / a)", " -(b / a) = -...
import Mathlib.Algebra.Polynomial.Monic #align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722" open Finset open Multiset open Polynomial universe u w variable {R : Type u} {ι : Type w} namespace Polynomial variable (s : Finset ι) sectio...	Mathlib/Algebra/Polynomial/BigOperators.lean	263	265	theorem prod_X_sub_C_nextCoeff {s : Finset ι} (f : ι → R) : nextCoeff (∏ i ∈ s, (X - C (f i))) = -∑ i ∈ s, f i := by	simpa using multiset_prod_X_sub_C_nextCoeff (s.1.map f)	[ " (Multiset.map (fun x => X - C x) t).prod.nextCoeff = -t.sum", " (Multiset.map (fun i => (X - C i).nextCoeff) t).sum = -t.sum", " (Multiset.map (fun x => -x) t).sum = -t.sum", " ∀ i ∈ t, (X - C i).Monic", " (X - C i✝).Monic", " (∏ i ∈ s, (X - C (f i))).nextCoeff = -∑ i ∈ s, f i" ]	[ " (Multiset.map (fun x => X - C x) t).prod.nextCoeff = -t.sum", " (Multiset.map (fun i => (X - C i).nextCoeff) t).sum = -t.sum", " (Multiset.map (fun x => -x) t).sum = -t.sum", " ∀ i ∈ t, (X - C i).Monic", " (X - C i✝).Monic" ]
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic import Mathlib.RingTheory.RootsOfUnity.Minpoly #align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f" namespace Polynomial variable {R : Type*} [CommRing R] {n : ℕ}	Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean	40	49	theorem isRoot_of_unity_of_root_cyclotomic {ζ : R} {i : ℕ} (hi : i ∈ n.divisors) (h : (cyclotomic i R).IsRoot ζ) : ζ ^ n = 1 := by	rcases n.eq_zero_or_pos with (rfl \| hn) · exact pow_zero _ have := congr_arg (eval ζ) (prod_cyclotomic_eq_X_pow_sub_one hn R).symm rw [eval_sub, eval_pow, eval_X, eval_one] at this convert eq_add_of_sub_eq' this convert (add_zero (M := R) _).symm apply eval_eq_zero_of_dvd_of_eval_eq_zero _ h exact Fins...	[ " ζ ^ n = 1", " ζ ^ 0 = 1", " 1 = 1 + eval ζ (∏ i ∈ n.divisors, cyclotomic i R)", " eval ζ (∏ i ∈ n.divisors, cyclotomic i R) = 0", " cyclotomic i R ∣ ∏ i ∈ n.divisors, cyclotomic i R" ]	[]
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow import Mathlib.Data.Finset.Fold import Mathlib.Data.Finset.Option import Mathlib.Data.Finset.Pi import Mathlib.Data.Finset.Prod import Mathlib.Data.Multiset.Lattice import Mathlib.Data.Set.Lattice import Mathlib.Order.Hom.Lattice import Mathlib.Order.Nat #align_import...	Mathlib/Data/Finset/Lattice.lean	90	93	theorem sup_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) : s₁.sup f = s₂.sup g := by	subst hs exact Finset.fold_congr hfg	[ " s.sup (f ⊔ g) = s.sup f ⊔ s.sup g", " ∅.sup (f ⊔ g) = ∅.sup f ⊔ ∅.sup g", " (cons a✝ s✝ h✝).sup (f ⊔ g) = (cons a✝ s✝ h✝).sup f ⊔ (cons a✝ s✝ h✝).sup g", " (f ⊔ g) a✝ ⊔ (s✝.sup f ⊔ s✝.sup g) = f a✝ ⊔ s✝.sup f ⊔ (g a✝ ⊔ s✝.sup g)", " s₁.sup f = s₂.sup g", " s₁.sup f = s₁.sup g" ]	[ " s.sup (f ⊔ g) = s.sup f ⊔ s.sup g", " ∅.sup (f ⊔ g) = ∅.sup f ⊔ ∅.sup g", " (cons a✝ s✝ h✝).sup (f ⊔ g) = (cons a✝ s✝ h✝).sup f ⊔ (cons a✝ s✝ h✝).sup g", " (f ⊔ g) a✝ ⊔ (s✝.sup f ⊔ s✝.sup g) = f a✝ ⊔ s✝.sup f ⊔ (g a✝ ⊔ s✝.sup g)" ]
import Mathlib.Data.Complex.Exponential import Mathlib.Analysis.SpecialFunctions.Log.Deriv #align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973" namespace Real open IsAbsoluteValue Finset CauSeq Complex theorem exp_one_near_10 : \|exp 1 - 224...	Mathlib/Data/Complex/ExponentialBounds.lean	51	55	theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by	rw [exp_neg, inv_lt (exp_pos _)] · refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2) norm_num · norm_num	[ " \|rexp 1 - 2244083 / 825552\| ≤ 1 / 10 ^ 10", " \|rexp 1 - expNear 0 1 (2244083 / 825552)\| ≤ \|1\| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)", " 0 + 1 = ?m.532", " 1 = ?m.532", " ↑1 = ?m.543", " \|rexp 1 - expNear 1 1 ((2244083 / 825552 - 1) * 1)\| ≤ \|1\| ^ 1 / ↑(Nat.factorial 1) * (1 / 10 ^ 10 * 1)", " 1 + 1 ...	[ " \|rexp 1 - 2244083 / 825552\| ≤ 1 / 10 ^ 10", " \|rexp 1 - expNear 0 1 (2244083 / 825552)\| ≤ \|1\| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)", " 0 + 1 = ?m.532", " 1 = ?m.532", " ↑1 = ?m.543", " \|rexp 1 - expNear 1 1 ((2244083 / 825552 - 1) * 1)\| ≤ \|1\| ^ 1 / ↑(Nat.factorial 1) * (1 / 10 ^ 10 * 1)", " 1 + 1 ...
import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Algebra.GCDMonoid.Nat #align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802" theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ...	Mathlib/RingTheory/Int/Basic.lean	147	152	theorem eq_pow_of_mul_eq_pow_bit1_left {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ} (h : a * b = c ^ bit1 k) : ∃ d, a = d ^ bit1 k := by	obtain ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow' hab h replace hd := hd.symm rw [associated_iff_natAbs, natAbs_eq_natAbs_iff, ← neg_pow_bit1] at hd obtain rfl \| rfl := hd <;> exact ⟨_, rfl⟩	[ " p ∣ m.natAbs ∨ p ∣ n.natAbs", " ↑p ∣ m ∨ ↑p ∣ n", " p ∣ n.natAbs", " ↑p ∣ n", " p = 2 ∨ p ∣ m.natAbs", " p = 2", " p ∣ m.natAbs", " ∃ p, Prime p ∧ p ∣ n", " Ideal.span {↑a.natAbs} = Ideal.span {a}", " Associated (↑a.natAbs) a", " ∃ d, a = d ^ bit1 k", " ∃ d_1, d ^ bit1 k = d_1 ^ bit1 k", "...	[ " p ∣ m.natAbs ∨ p ∣ n.natAbs", " ↑p ∣ m ∨ ↑p ∣ n", " p ∣ n.natAbs", " ↑p ∣ n", " p = 2 ∨ p ∣ m.natAbs", " p = 2", " p ∣ m.natAbs", " ∃ p, Prime p ∧ p ∣ n", " Ideal.span {↑a.natAbs} = Ideal.span {a}", " Associated (↑a.natAbs) a" ]
import Mathlib.Algebra.Order.Ring.Nat #align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b" namespace Nat def dist (n m : ℕ) := n - m + (m - n) #align nat.dist Nat.dist -- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr...	Mathlib/Data/Nat/Dist.lean	74	78	theorem dist_add_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m := calc dist (n + k) (m + k) = n + k - (m + k) + (m + k - (n + k)) := rfl _ = n - m + (m + k - (n + k)) := by	rw [@add_tsub_add_eq_tsub_right] _ = n - m + (m - n) := by rw [@add_tsub_add_eq_tsub_right]	[ " n.dist m = m.dist n", " n.dist n = 0", " n.dist m = 0", " n.dist m = m - n", " n.dist m = n - m", " m.dist n = n - m", " m ≤ n + n.dist m", " m ≤ n.dist m + n", " n ≤ n.dist m + m", " n ≤ m.dist n + m", " n ≤ m + n.dist m", " n ≤ m + m.dist n", " n + k - (m + k) + (m + k - (n + k)) = n - m...	[ " n.dist m = m.dist n", " n.dist n = 0", " n.dist m = 0", " n.dist m = m - n", " n.dist m = n - m", " m.dist n = n - m", " m ≤ n + n.dist m", " m ≤ n.dist m + n", " n ≤ n.dist m + m", " n ≤ m.dist n + m", " n ≤ m + n.dist m", " n ≤ m + m.dist n" ]
import Mathlib.ModelTheory.Quotients import Mathlib.Order.Filter.Germ import Mathlib.Order.Filter.Ultrafilter #align_import model_theory.ultraproducts from "leanprover-community/mathlib"@"f1ae620609496a37534c2ab3640b641d5be8b6f0" universe u v variable {α : Type} (M : α → Type) (u : Ultrafilter α) open FirstOr...	Mathlib/ModelTheory/Ultraproducts.lean	96	144	theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormula β n) (x : β → ∀ a, M a) (v : Fin n → ∀ a, M a) : (φ.Realize (fun i : β => (x i : (u : Filter α).Product M)) (fun i => (v i : (u : Filter α).Product M))) ↔ ∀ᶠ a : α in u, φ.Realize (fun i : β => x i a) fun i => v i a := b...	letI := (u : Filter α).productSetoid M induction' φ with _ _ _ _ _ _ _ _ m _ _ ih ih' k φ ih · simp only [BoundedFormula.Realize, eventually_const] · have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a := fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun ...	[ " funMap f x ≈ funMap f y", " a ∈ {x_1 \| (fun a => funMap f x a = funMap f y a) x_1}", " RelMap r x = RelMap r y", " RelMap r x ↔ RelMap r y", " RelMap r y", " {x_1 \| (fun a => RelMap r fun i => x i a) x_1} ∩ ⋂ i, {x_1 \| (fun a => x i a = y i a) x_1} ⊆\n {x \| (fun a => RelMap r fun i => y i a) x}", "...	[ " funMap f x ≈ funMap f y", " a ∈ {x_1 \| (fun a => funMap f x a = funMap f y a) x_1}", " RelMap r x = RelMap r y", " RelMap r x ↔ RelMap r y", " RelMap r y", " {x_1 \| (fun a => RelMap r fun i => x i a) x_1} ∩ ⋂ i, {x_1 \| (fun a => x i a = y i a) x_1} ⊆\n {x \| (fun a => RelMap r fun i => y i a) x}", "...
import Mathlib.Data.List.Basic #align_import data.list.palindrome from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec" variable {α β : Type*} namespace List inductive Palindrome : List α → Prop \| nil : Palindrome [] \| singleton : ∀ x, Palindrome [x] \| cons_concat : ∀ (x) {l}, Pa...	Mathlib/Data/List/Palindrome.lean	68	70	theorem append_reverse (l : List α) : Palindrome (l ++ reverse l) := by	apply of_reverse_eq rw [reverse_append, reverse_reverse]	[ " l.reverse = l", " [].reverse = []", " [x✝].reverse = [x✝]", " (x✝ :: (l✝ ++ [x✝])).reverse = x✝ :: (l✝ ++ [x✝])", " l.reverse = l → l.Palindrome", " ∀ (a : α) (l : List α) (b : α),\n (l.reverse = l → l.Palindrome) → (a :: (l ++ [b])).reverse = a :: (l ++ [b]) → (a :: (l ++ [b])).Palindrome", " (x :...	[ " l.reverse = l", " [].reverse = []", " [x✝].reverse = [x✝]", " (x✝ :: (l✝ ++ [x✝])).reverse = x✝ :: (l✝ ++ [x✝])", " l.reverse = l → l.Palindrome", " ∀ (a : α) (l : List α) (b : α),\n (l.reverse = l → l.Palindrome) → (a :: (l ++ [b])).reverse = a :: (l ++ [b]) → (a :: (l ++ [b])).Palindrome", " (x :...
import Mathlib.Data.Finset.Image #align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83" assert_not_exists MonoidWithZero -- TODO: After a lot more work, -- assert_not_exists OrderedCommMonoid open Function Multiset Nat variable {α β R : Type*} namespace Fin...	Mathlib/Data/Finset/Card.lean	114	118	theorem card_insert_le (a : α) (s : Finset α) : card (insert a s) ≤ s.card + 1 := by	by_cases h : a ∈ s · rw [insert_eq_of_mem h] exact Nat.le_succ _ · rw [card_insert_of_not_mem h]	[ " Monotone card", " ({a} ∩ s).card ≤ 1", " (insert a s).card = s.card + 1", " (insert a s).card = s.card", " (insert a s).card ≤ s.card + 1", " s.card ≤ s.card + 1" ]	[ " Monotone card", " ({a} ∩ s).card ≤ 1", " (insert a s).card = s.card + 1", " (insert a s).card = s.card" ]
import Mathlib.Data.Set.Lattice import Mathlib.Order.Directed #align_import data.set.Union_lift from "leanprover-community/mathlib"@"5a4ea8453f128345f73cc656e80a49de2a54f481" variable {α : Type*} {ι β : Sort _} namespace Set section UnionLift @[nolint unusedArguments] noncomputable def iUnionLift (S : ι → Set...	Mathlib/Data/Set/UnionLift.lean	75	76	theorem iUnionLift_of_mem (x : T) {i : ι} (hx : (x : α) ∈ S i) : iUnionLift S f hf T hT x = f i ⟨x, hx⟩ := by	cases' x with x hx; exact hf _ _ _ _ _	[ " iUnionLift S f hf T hT x = f i ⟨↑x, hx⟩", " iUnionLift S f hf T hT ⟨x, hx✝⟩ = f i ⟨↑⟨x, hx✝⟩, hx⟩" ]	[]
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject import Mathlib.CategoryTheory.Idempotents.HomologicalComplex #align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504" no...	Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean	86	91	theorem N₁Γ₀_hom_app (K : ChainComplex C ℕ) : N₁Γ₀.hom.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫ (toKaroubi _).map (Γ₀NondegComplexIso K).hom := by	change (N₁Γ₀.app K).hom = _ simp only [N₁Γ₀_app] rfl	[ " ∀ (i j : ℕ),\n (ComplexShape.down ℕ).Rel i j →\n ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) i).hom ≫ K.d i j =\n (Γ₀.splitting K).nondegComplex.d i j ≫ ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) j).hom", " ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) (n ...	[ " ∀ (i j : ℕ),\n (ComplexShape.down ℕ).Rel i j →\n ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) i).hom ≫ K.d i j =\n (Γ₀.splitting K).nondegComplex.d i j ≫ ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) j).hom", " ((fun n => Iso.refl ((Γ₀.splitting K).nondegComplex.X n)) (n ...
import Mathlib.Algebra.Polynomial.Eval import Mathlib.RingTheory.Ideal.Quotient #align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24" open Submodule open Polynomial variable {R : Type} [Ring R] variable {A : Type} [CommRing A] variable {M : Type*} [...	Mathlib/LinearAlgebra/SModEq.lean	102	102	theorem zero : x ≡ 0 [SMOD U] ↔ x ∈ U := by	rw [SModEq.def, Submodule.Quotient.eq, sub_zero]	[ " x ≡ y [SMOD U] ↔ x - y ∈ U", " x ≡ y [SMOD ⊥] ↔ x = y", " x₁ + x₂ ≡ y₁ + y₂ [SMOD U]", " Submodule.Quotient.mk (x₁ + x₂) = Submodule.Quotient.mk (y₁ + y₂)", " c • x ≡ c • y [SMOD U]", " Submodule.Quotient.mk (c • x) = Submodule.Quotient.mk (c • y)", " x₁ * x₂ ≡ y₁ * y₂ [SMOD I]", " (Ideal.Quotient.m...	[ " x ≡ y [SMOD U] ↔ x - y ∈ U", " x ≡ y [SMOD ⊥] ↔ x = y", " x₁ + x₂ ≡ y₁ + y₂ [SMOD U]", " Submodule.Quotient.mk (x₁ + x₂) = Submodule.Quotient.mk (y₁ + y₂)", " c • x ≡ c • y [SMOD U]", " Submodule.Quotient.mk (c • x) = Submodule.Quotient.mk (c • y)", " x₁ * x₂ ≡ y₁ * y₂ [SMOD I]", " (Ideal.Quotient.m...
import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.SumIntegralComparisons import Mathlib.NumberTheory.Harmonic.Defs theorem log_add_one_le_harmonic (n : ℕ) : Real.log ↑(n+1) ≤ harmonic n := by calc _ = ∫ x in (1:ℕ)..↑(n+1), x⁻¹ := ?_ _ ≤ ∑ d ∈ Finset.Icc 1 n, (d:ℝ)⁻¹ := ?_ ...	Mathlib/NumberTheory/Harmonic/Bounds.lean	64	69	theorem harmonic_floor_le_one_add_log (y : ℝ) (hy : 1 ≤ y) : harmonic ⌊y⌋₊ ≤ 1 + Real.log y := by	refine (harmonic_le_one_add_log _).trans ?_ gcongr · exact_mod_cast Nat.floor_pos.mpr hy · exact Nat.floor_le <\| zero_le_one.trans hy	[ " (↑(n + 1)).log ≤ ↑(harmonic n)", " (↑(n + 1)).log = ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹", " 0 ∉ Set.uIcc 1 ↑(n + 1)", " ¬1 ≤ 0", " ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹ ≤ ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹", " 0 < ↑1", " ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹ = ↑(harmonic n)", " ↑(harmonic n) ≤ 1 + (↑n).log", " ∑ x ∈ Finse...	[ " (↑(n + 1)).log ≤ ↑(harmonic n)", " (↑(n + 1)).log = ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹", " 0 ∉ Set.uIcc 1 ↑(n + 1)", " ¬1 ≤ 0", " ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹ ≤ ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹", " 0 < ↑1", " ∑ d ∈ Finset.Icc 1 n, (↑d)⁻¹ = ↑(harmonic n)", " ↑(harmonic n) ≤ 1 + (↑n).log", " ∑ x ∈ Finse...
import Mathlib.Analysis.BoxIntegral.Box.Basic import Mathlib.Analysis.SpecificLimits.Basic #align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982" open Set Finset Function Filter Metric Classical Topology Filter ENNReal noncomputable...	Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean	122	170	theorem subbox_induction_on' {p : Box ι → Prop} (I : Box ι) (H_ind : ∀ J ≤ I, (∀ s, p (splitCenterBox J s)) → p J) (H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ i, J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J) : p I := by	by_contra hpI -- First we use `H_ind` to construct a decreasing sequence of boxes such that `∀ m, ¬p (J m)`. replace H_ind := fun J hJ ↦ not_imp_not.2 (H_ind J hJ) simp only [exists_imp, not_forall] at H_ind choose! s hs using H_ind set J : ℕ → Box ι := fun m ↦ (fun J ↦ splitCenterBox J (s J))^[m] I have...	[ " s.piecewise (fun i => (I.lower i + I.upper i) / 2) I.lower i <\n s.piecewise I.upper (fun i => (I.lower i + I.upper i) / 2) i", " (if i ∈ s then (I.lower i + I.upper i) / 2 else I.lower i) < if i ∈ s then I.upper i else (I.lower i + I.upper i) / 2", " (I.lower i + I.upper i) / 2 < I.upper i", " I.lower i...	[ " s.piecewise (fun i => (I.lower i + I.upper i) / 2) I.lower i <\n s.piecewise I.upper (fun i => (I.lower i + I.upper i) / 2) i", " (if i ∈ s then (I.lower i + I.upper i) / 2 else I.lower i) < if i ∈ s then I.upper i else (I.lower i + I.upper i) / 2", " (I.lower i + I.upper i) / 2 < I.upper i", " I.lower i...
import Mathlib.Probability.Notation import Mathlib.Probability.Process.Stopping #align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca" open TopologicalSpace Filter open scoped NNReal ENNReal MeasureTheory ProbabilityTheory namespace MeasureTheor...	Mathlib/Probability/Martingale/Basic.lean	109	113	theorem setIntegral_eq [SigmaFiniteFiltration μ ℱ] (hf : Martingale f ℱ μ) {i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) : ∫ ω in s, f i ω ∂μ = ∫ ω in s, f j ω ∂μ := by	rw [← @setIntegral_condexp _ _ _ _ _ (ℱ i) m0 _ _ _ (ℱ.le i) _ (hf.integrable j) hs] refine setIntegral_congr_ae (ℱ.le i s hs) ?_ filter_upwards [hf.2 i j hij] with _ heq _ using heq.symm	[ " μ[(fun x_1 x_2 => x) j\|↑ℱ i] =ᶠ[ae μ] (fun x_1 x_2 => x) i", " Martingale (fun x => f) ℱ μ", " μ[(fun x => f) j\|↑ℱ i] =ᶠ[ae μ] (fun x => f) i", " μ[0 j\|↑ℱ i] =ᶠ[ae μ] 0 i", " 0 =ᶠ[ae μ] 0 i", " ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (ω : Ω) in s, f j ω ∂μ", " ∫ (ω : Ω) in s, f i ω ∂μ = ∫ (x : Ω) in s, (μ[f j\|↑ℱ...	[ " μ[(fun x_1 x_2 => x) j\|↑ℱ i] =ᶠ[ae μ] (fun x_1 x_2 => x) i", " Martingale (fun x => f) ℱ μ", " μ[(fun x => f) j\|↑ℱ i] =ᶠ[ae μ] (fun x => f) i", " μ[0 j\|↑ℱ i] =ᶠ[ae μ] 0 i", " 0 =ᶠ[ae μ] 0 i" ]
import Mathlib.Tactic.CategoryTheory.Elementwise import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer import Mathlib.CategoryTheory.Limits.Constructions.EpiMono import Mathlib.CategoryTheory.Limits.Preserves.Limits import Mathlib.CategoryTheory.Limits.Shapes.Types #align_import category_theory.glue_data from "l...	Mathlib/CategoryTheory/GlueData.lean	123	129	theorem t'_comp_eq_pullbackSymmetry (i j k : D.J) : D.t' j k i ≫ D.t' k i j = (pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom := by	trans inv (D.t' i j k) · exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _) · rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)] simp [t_fac, t_fac_assoc]	[ " D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom", " D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd", " D.t' j i i = (D.t' j i i ≫ pullback.snd) ≫ inv pullback.snd", " D.t' i j i = pullback.fst ≫ D.t i j ≫ inv pullback.snd", " D.t' i j i = (D.t' i j i ≫ pullback.snd) ≫ inv pullback.snd", ...	[ " D.t' i i j = (pullbackSymmetry (D.f i i) (D.f i j)).hom", " D.t' j i i = pullback.fst ≫ D.t j i ≫ inv pullback.snd", " D.t' j i i = (D.t' j i i ≫ pullback.snd) ≫ inv pullback.snd", " D.t' i j i = pullback.fst ≫ D.t i j ≫ inv pullback.snd", " D.t' i j i = (D.t' i j i ≫ pullback.snd) ≫ inv pullback.snd", ...
import Mathlib.Analysis.Complex.Circle import Mathlib.LinearAlgebra.Determinant import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup #align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5" noncomputable section open Complex open ComplexConjugate ...	Mathlib/Analysis/Complex/Isometry.lean	60	62	theorem rotation_trans (a b : circle) : (rotation a).trans (rotation b) = rotation (b * a) := by	ext1 simp	[ " Complex.abs (↑a * x) = Complex.abs x", " (rotation a).trans (rotation b) = rotation (b * a)", " ((rotation a).trans (rotation b)) x✝ = (rotation (b * a)) x✝" ]	[ " Complex.abs (↑a * x) = Complex.abs x" ]
import Mathlib.Algebra.Field.Defs import Mathlib.Algebra.GroupWithZero.Units.Lemmas import Mathlib.Algebra.Ring.Commute import Mathlib.Algebra.Ring.Invertible import Mathlib.Order.Synonym #align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102" open Function ...	Mathlib/Algebra/Field/Basic.lean	66	67	theorem add_div' (a b c : α) (hc : c ≠ 0) : b + a / c = (b * c + a) / c := by	rw [add_div, mul_div_cancel_right₀ _ hc]	[ " (a + b) / c = a / c + b / c", " (b + a) / b = 1 + a / b", " (a + b) / b = a / b + 1", " 1 / a * (a + b) * (1 / b) = 1 / a + 1 / b", " (a + b / c) * c = a * c + b", " b + a / c = (b * c + a) / c" ]	[ " (a + b) / c = a / c + b / c", " (b + a) / b = 1 + a / b", " (a + b) / b = a / b + 1", " 1 / a * (a + b) * (1 / b) = 1 / a + 1 / b", " (a + b / c) * c = a * c + b" ]
import Mathlib.Algebra.Group.Pi.Basic import Mathlib.CategoryTheory.Limits.Shapes.Products import Mathlib.CategoryTheory.Limits.Shapes.Images import Mathlib.CategoryTheory.IsomorphismClasses import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects #align_import category_theory.limits.shapes.zero_morphisms from "leanpr...	Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean	150	151	theorem eq_zero_of_image_eq_zero {X Y : C} {f : X ⟶ Y} [HasImage f] (w : image.ι f = 0) : f = 0 := by	rw [← image.fac f, w, HasZeroMorphisms.comp_zero]	[]	[]
import Mathlib.Analysis.Normed.Field.Basic #align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8" namespace Int	Mathlib/Analysis/NormedSpace/Int.lean	24	26	theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by	obtain rfl \| rfl := units_eq_one_or e <;> simp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]	[ " ‖↑e‖₊ = 1", " ‖↑1‖₊ = 1", " ‖↑(-1)‖₊ = 1" ]	[]
import Mathlib.Algebra.Order.BigOperators.Ring.Finset import Mathlib.Analysis.Convex.Hull import Mathlib.LinearAlgebra.AffineSpace.Basis #align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d" open Set Function open scoped Classical open Pointwise ...	Mathlib/Analysis/Convex/Combination.lean	87	88	theorem Finset.centerMass_smul : (t.centerMass w fun i => c • z i) = c • t.centerMass w z := by	simp only [Finset.centerMass, Finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]	[ " ∅.centerMass w z = 0", " {i, j}.centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j", " (insert i t).centerMass w z =\n (w i / (w i + ∑ j ∈ t, w j)) • z i + ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z", " (w i / (w i + ∑ i ∈ t, w i)) • z i + (w i + ∑ i ∈ t, w i)⁻¹ • ∑ i ...	[ " ∅.centerMass w z = 0", " {i, j}.centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j", " (insert i t).centerMass w z =\n (w i / (w i + ∑ j ∈ t, w j)) • z i + ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z", " (w i / (w i + ∑ i ∈ t, w i)) • z i + (w i + ∑ i ∈ t, w i)⁻¹ • ∑ i ...
import Mathlib.Logic.Pairwise import Mathlib.Order.CompleteBooleanAlgebra import Mathlib.Order.Directed import Mathlib.Order.GaloisConnection #align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd" open Function Set universe u variable {α β γ : Type*} {ι ι' ι...	Mathlib/Data/Set/Lattice.lean	207	211	theorem exists_set_mem_of_union_eq_top {ι : Type*} (t : Set ι) (s : ι → Set β) (w : ⋃ i ∈ t, s i = ⊤) (x : β) : ∃ i ∈ t, x ∈ s i := by	have p : x ∈ ⊤ := Set.mem_univ x rw [← w, Set.mem_iUnion] at p simpa using p	[ " x ∈ ⋃ i, ⋃ j, s i j ↔ ∃ i j, x ∈ s i j", " x ∈ ⋂ i, ⋂ j, s i j ↔ ∀ (i : ι) (j : κ i), x ∈ s i j", " ∀ {ι : Type u_1} {κ : ι → Type u_1} (f : (a : ι) → κ a → Set α), ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)", " ⨅ a, ⨆ b, f✝ a b = ⨆ g, ⨅ a, f✝ a (g a)", " x✝ ∈ ⨅ a, ⨆ b, f✝ a b ↔ x✝ ∈ ⨆ g, ⨅ a, f✝ a (g a)", "...	[ " x ∈ ⋃ i, ⋃ j, s i j ↔ ∃ i j, x ∈ s i j", " x ∈ ⋂ i, ⋂ j, s i j ↔ ∀ (i : ι) (j : κ i), x ∈ s i j", " ∀ {ι : Type u_1} {κ : ι → Type u_1} (f : (a : ι) → κ a → Set α), ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)", " ⨅ a, ⨆ b, f✝ a b = ⨆ g, ⨅ a, f✝ a (g a)", " x✝ ∈ ⨅ a, ⨆ b, f✝ a b ↔ x✝ ∈ ⨆ g, ⨅ a, f✝ a (g a)", "...
import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.Data.Nat.Lattice #align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2" namespace SimpleGraph variable {V : Type*} (G : SimpleGraph V) noncomputable def dist (u v : V)...	Mathlib/Combinatorics/SimpleGraph/Metric.lean	137	142	theorem dist_eq_one_iff_adj {u v : V} : G.dist u v = 1 ↔ G.Adj u v := by	refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩ · let ⟨w, hw⟩ := exists_walk_of_dist_ne_zero <\| ne_zero_of_eq_one h exact w.adj_of_length_eq_one <\| h ▸ hw · have : h.toWalk.length = 1 := Walk.length_cons _ _ exact ge_antisymm (h.reachable.pos_dist_of_ne h.ne) (this ▸ dist_le _)	[ " G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v", " G.dist v v = 0", " G.dist u v = 0 ↔ u = v", " G.dist u v ≠ 0", " False", " G.dist u v = 0", " Set.univ.Nonempty", " G.dist u w ≤ G.dist u v + G.dist v w", " G.dist u w ≤ (p.append q).length", " G.dist u v ≤ G.dist v u", " G.dist u v ≤ p.reverse.len...	[ " G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v", " G.dist v v = 0", " G.dist u v = 0 ↔ u = v", " G.dist u v ≠ 0", " False", " G.dist u v = 0", " Set.univ.Nonempty", " G.dist u w ≤ G.dist u v + G.dist v w", " G.dist u w ≤ (p.append q).length", " G.dist u v ≤ G.dist v u", " G.dist u v ≤ p.reverse.len...
import Mathlib.Combinatorics.SimpleGraph.Finite import Mathlib.Combinatorics.SimpleGraph.Maps open Finset namespace SimpleGraph variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V) section AddEdge def edge : SimpleGraph V := fromEdgeSet {s(s, t)} lemma edge_adj (v w : V) : (edge s t).Adj v w ↔ ...	Mathlib/Combinatorics/SimpleGraph/Operations.lean	171	175	theorem edgeFinset_sup_edge [Fintype (edgeSet (G ⊔ edge s t))] (hn : ¬G.Adj s t) (h : s ≠ t) : (G ⊔ edge s t).edgeFinset = G.edgeFinset.cons s(s, t) (by simp_all) := by	letI := Classical.decEq V rw [edgeFinset_sup, cons_eq_insert, insert_eq, union_comm] simp_rw [edgeFinset, edge_edgeSet_of_ne h]; rfl	[ " (edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w", " Decidable ((edge s t).Adj x✝¹ x✝)", " Decidable ((x✝¹ = s ∧ x✝ = t ∨ x✝¹ = t ∧ x✝ = s) ∧ x✝¹ ≠ x✝)", " edge s s = ⊥", " (edge s s).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝", " (x✝¹ = s ∧ x✝ = s ∨ x✝¹ = s ∧ x✝ = s) ∧ x✝¹ ≠ x✝ ↔ ⊥.Adj x✝¹ x✝", " G ⊔ edge...	[ " (edge s t).Adj v w ↔ (v = s ∧ w = t ∨ v = t ∧ w = s) ∧ v ≠ w", " Decidable ((edge s t).Adj x✝¹ x✝)", " Decidable ((x✝¹ = s ∧ x✝ = t ∨ x✝¹ = t ∧ x✝ = s) ∧ x✝¹ ≠ x✝)", " edge s s = ⊥", " (edge s s).Adj x✝¹ x✝ ↔ ⊥.Adj x✝¹ x✝", " (x✝¹ = s ∧ x✝ = s ∨ x✝¹ = s ∧ x✝ = s) ∧ x✝¹ ≠ x✝ ↔ ⊥.Adj x✝¹ x✝", " G ⊔ edge...
import Mathlib.Data.Set.Image import Mathlib.Order.Interval.Set.Basic #align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105" open Set variable {α : Type*} namespace WithTop @[simp] theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =...	Mathlib/Order/Interval/Set/WithBotTop.lean	67	67	theorem preimage_coe_Ioc : (some : α → WithTop α) ⁻¹' Ioc a b = Ioc a b := by	simp [← Ioi_inter_Iic]	[ " range some = Iio ⊤", " x ∈ range some ↔ x ∈ Iio ⊤", " some ⁻¹' Icc ↑a ↑b = Icc a b", " some ⁻¹' Ico ↑a ↑b = Ico a b", " some ⁻¹' Ioc ↑a ↑b = Ioc a b" ]	[ " range some = Iio ⊤", " x ∈ range some ↔ x ∈ Iio ⊤", " some ⁻¹' Icc ↑a ↑b = Icc a b", " some ⁻¹' Ico ↑a ↑b = Ico a b" ]
import Mathlib.Algebra.Group.Defs import Mathlib.Algebra.GroupWithZero.Defs import Mathlib.Data.Int.Cast.Defs import Mathlib.Tactic.Spread import Mathlib.Util.AssertExists #align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f" universe u v w x variable {α : ...	Mathlib/Algebra/Ring/Defs.lean	244	245	theorem mul_boole {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) : (a * if P then 1 else 0) = if P then a else 0 := by	simp	[]	[]
import Mathlib.Data.Fintype.Basic #align_import data.fintype.quotient from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf" def Quotient.finChoiceAux {ι : Type} [DecidableEq ι] {α : ι → Type} [S : ∀ i, Setoid (α i)] : ∀ l : List ι, (∀ i ∈ l, Quotient (S i)) → @Quotient (∀ i ∈ l, α ...	Mathlib/Data/Fintype/Quotient.lean	76	84	theorem Quotient.finChoice_eq {ι : Type} [DecidableEq ι] [Fintype ι] {α : ι → Type} [∀ i, Setoid (α i)] (f : ∀ i, α i) : (Quotient.finChoice fun i => ⟦f i⟧) = ⟦f⟧ := by	dsimp only [Quotient.finChoice] conv_lhs => enter [1] tactic => change _ = ⟦fun i _ => f i⟧ exact Quotient.inductionOn (@Finset.univ ι _).1 fun l => Quotient.finChoiceAux_eq _ _ rfl	[ " Setoid ((i : ι) → i ∈ l → α i)", " Quotient inferInstance", " α i → ((i : ι) → i ∈ l → α i) → Quotient inferInstance", " α j", " α i", " ∀ (a₁ : α i) (b₁ : (i : ι) → i ∈ l → α i) (a₂ : α i) (b₂ : (i : ι) → i ∈ l → α i),\n a₁ ≈ a₂ →\n b₁ ≈ b₂ →\n ⟦fun j h => if e : j = i then ⋯.mpr a₁ else...	[ " Setoid ((i : ι) → i ∈ l → α i)", " Quotient inferInstance", " α i → ((i : ι) → i ∈ l → α i) → Quotient inferInstance", " α j", " α i", " ∀ (a₁ : α i) (b₁ : (i : ι) → i ∈ l → α i) (a₂ : α i) (b₂ : (i : ι) → i ∈ l → α i),\n a₁ ≈ a₂ →\n b₁ ≈ b₂ →\n ⟦fun j h => if e : j = i then ⋯.mpr a₁ else...
import Mathlib.LinearAlgebra.Finsupp import Mathlib.Algebra.MonoidAlgebra.Support import Mathlib.Algebra.DirectSum.Internal import Mathlib.RingTheory.GradedAlgebra.Basic #align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3" noncomputable sectio...	Mathlib/Algebra/MonoidAlgebra/Grading.lean	72	78	theorem mem_grade_iff' (m : M) (a : R[M]) : a ∈ grade R m ↔ a ∈ (LinearMap.range (Finsupp.lsingle m : R →ₗ[R] M →₀ R) : Submodule R R[M]) := by	rw [mem_grade_iff, Finsupp.support_subset_singleton'] apply exists_congr intro r constructor <;> exact Eq.symm	[ " f m = i", " a ∈ gradeBy R f i ↔ ↑a.support ⊆ f ⁻¹' {i}", " a ∈ grade R m ↔ a.support ⊆ {m}", " a ∈ grade R m ↔ ↑a.support ⊆ {m}", " a ∈ grade R m ↔ a ∈ LinearMap.range (Finsupp.lsingle m)", " (∃ b, a = Finsupp.single m b) ↔ a ∈ LinearMap.range (Finsupp.lsingle m)", " ∀ (a_1 : R), a = Finsupp.single m ...	[ " f m = i", " a ∈ gradeBy R f i ↔ ↑a.support ⊆ f ⁻¹' {i}", " a ∈ grade R m ↔ a.support ⊆ {m}", " a ∈ grade R m ↔ ↑a.support ⊆ {m}" ]
import Mathlib.Algebra.CharP.ExpChar import Mathlib.GroupTheory.OrderOfElement #align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450" variable {R ι : Type*} namespace CharTwo section Semiring variable [Semiring R] [CharP R 2] theorem two_eq_zero : (2 : ...	Mathlib/Algebra/CharP/Two.lean	44	44	theorem bit0_apply_eq_zero (x : R) : (bit0 x : R) = 0 := by	simp	[ " 2 = 0", " x + x = 0", " bit0 = 0", " bit0 x✝ = 0 x✝", " bit0 x = 0" ]	[ " 2 = 0", " x + x = 0", " bit0 = 0", " bit0 x✝ = 0 x✝" ]
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries import Mathlib.Algebra.ContinuedFractions.Computation.Translations import Mathlib.Data.Real.Irrational import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.Tactic.Basic #align_import number_theory.diophantine_approximation from "leanpro...	Mathlib/NumberTheory/DiophantineApproximation.lean	139	144	theorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) : ∃ k : ℕ, 0 < k ∧ k ≤ n ∧ \|↑k * ξ - round (↑k * ξ)\| ≤ 1 / (n + 1) := by	obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos have hk := toNat_of_nonneg hk₀.le rw [← hk] at hk₀ hk₁ h exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩	[ " ∃ j k, 0 < k ∧ k ≤ ↑n ∧ \|↑k * ξ - ↑j\| ≤ 1 / (↑n + 1)", "ξ : ℝ\nn : ℕ\nn_pos : 0 < n\nf : ℤ → ℤ := fun m => ⌊fract (ξ * ↑m) * (↑n + 1)⌋\nhn : 0 < ↑n + 1\nhfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1\nj k : ℤ\n\| \|↑k * ξ - ↑j\| ≤ 1 / (↑n + 1)", " ∃ j k, 0 < k ∧ k ≤ ↑n ∧ \|(ξ * ↑k - ↑j) * (↑n + 1)\| ≤ 1", "...	[ " ∃ j k, 0 < k ∧ k ≤ ↑n ∧ \|↑k * ξ - ↑j\| ≤ 1 / (↑n + 1)", "ξ : ℝ\nn : ℕ\nn_pos : 0 < n\nf : ℤ → ℤ := fun m => ⌊fract (ξ * ↑m) * (↑n + 1)⌋\nhn : 0 < ↑n + 1\nhfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1\nj k : ℤ\n\| \|↑k * ξ - ↑j\| ≤ 1 / (↑n + 1)", " ∃ j k, 0 < k ∧ k ≤ ↑n ∧ \|(ξ * ↑k - ↑j) * (↑n + 1)\| ≤ 1", "...
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	77	78	theorem PartialHomeomorph.univUnitBall_apply_zero : univUnitBall (0 : E) = 0 := by	simp [PartialHomeomorph.univUnitBall_apply]	[ " (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...
import Mathlib.Data.Set.Pointwise.Interval import Mathlib.Topology.Algebra.Field import Mathlib.Topology.Algebra.Order.Group #align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd" open Set Filter TopologicalSpace Function open scoped Pointwise Top...	Mathlib/Topology/Algebra/Order/Field.lean	79	82	theorem Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by	have := hf.atTop_mul (neg_pos.2 hC) hg.neg simpa only [(· ∘ ·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this	[ " TopologicalRing R", " ∀ (f : R → R), ∀ c ≥ 0, (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)", " ∃ ia, 0 < ia ∧ ∀ x ∈ {x \| norm x < ia}, f x ∈ {x \| norm x < ε}", " c * norm x < ε", " ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)", " Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ �...	[ " TopologicalRing R", " ∀ (f : R → R), ∀ c ≥ 0, (∀ (x : R), norm (f x) ≤ c * norm x) → Tendsto f (𝓝 0) (𝓝 0)", " ∃ ia, 0 < ia ∧ ∀ x ∈ {x \| norm x < ia}, f x ∈ {x \| norm x < ε}", " c * norm x < ε", " ∀ (x₀ : R), Tendsto (fun x => x * x₀) (𝓝 0) (𝓝 0)", " Tendsto (uncurry fun x x_1 => x * x_1) (𝓝 0 ×ˢ �...
import Mathlib.Topology.Algebra.InfiniteSum.Order import Mathlib.Topology.Algebra.InfiniteSum.Ring import Mathlib.Topology.Instances.Real import Mathlib.Topology.MetricSpace.Isometry #align_import topology.instances.nnreal from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514" noncomputabl...	Mathlib/Topology/Instances/NNReal.lean	163	164	theorem hasSum_coe {f : α → ℝ≥0} {r : ℝ≥0} : HasSum (fun a => (f a : ℝ)) (r : ℝ) ↔ HasSum f r := by	simp only [HasSum, ← coe_sum, tendsto_coe]	[ " Tendsto Real.toNNReal atTop atTop", " Tendsto (fun a => ↑a.toNNReal) atTop atTop", " ⨅ l, ⨅ (_ : ⊥ < l), 𝓟 (Set.Iio l) = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Set.Iio a)", " ⨅ l, ⨅ (_ : l ≠ ⊥), 𝓟 (Set.Iio l) = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Set.Iio a)", " HasSum (fun a => ↑(f a)) ↑r ↔ HasSum f r" ]	[ " Tendsto Real.toNNReal atTop atTop", " Tendsto (fun a => ↑a.toNNReal) atTop atTop", " ⨅ l, ⨅ (_ : ⊥ < l), 𝓟 (Set.Iio l) = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Set.Iio a)", " ⨅ l, ⨅ (_ : l ≠ ⊥), 𝓟 (Set.Iio l) = ⨅ a, ⨅ (_ : a ≠ 0), 𝓟 (Set.Iio a)" ]
import Mathlib.Data.Set.Prod #align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654" open Function namespace Set variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ} variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v...	Mathlib/Data/Set/NAry.lean	107	108	theorem image2_union_right : image2 f s (t ∪ t') = image2 f s t ∪ image2 f s t' := by	rw [← image2_swap, image2_union_left, image2_swap f, image2_swap f]	[ " f a b ∈ image2 f s t → a ∈ s ∧ b ∈ t", " a ∈ s ∧ b ∈ t", " a' ∈ s ∧ b' ∈ t", " image2 f s t ⊆ image2 f s' t'", " f a b ∈ image2 f s' t'", " image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u", " image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u", " x✝ ∈ (fun x => f x.1 x.2) '' s ×ˢ t ↔ x✝ ∈ i...	[ " f a b ∈ image2 f s t → a ∈ s ∧ b ∈ t", " a ∈ s ∧ b ∈ t", " a' ∈ s ∧ b' ∈ t", " image2 f s t ⊆ image2 f s' t'", " f a b ∈ image2 f s' t'", " image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u", " image2 f s t ⊆ u ↔ ∀ b ∈ t, (fun a => f a b) '' s ⊆ u", " x✝ ∈ (fun x => f x.1 x.2) '' s ×ˢ t ↔ x✝ ∈ i...
import Mathlib.Algebra.CharP.Two import Mathlib.Algebra.CharP.Reduced import Mathlib.Algebra.NeZero import Mathlib.Algebra.Polynomial.RingDivision import Mathlib.GroupTheory.SpecificGroups.Cyclic import Mathlib.NumberTheory.Divisors import Mathlib.RingTheory.IntegralDomain import Mathlib.Tactic.Zify #align_import rin...	Mathlib/RingTheory/RootsOfUnity/Basic.lean	268	271	theorem mem_rootsOfUnity_prime_pow_mul_iff (p k : ℕ) (m : ℕ+) [ExpChar R p] {ζ : Rˣ} : ζ ∈ rootsOfUnity (⟨p, expChar_pos R p⟩ ^ k * m) R ↔ ζ ∈ rootsOfUnity m R := by	simp only [mem_rootsOfUnity', PNat.mul_coe, PNat.pow_coe, PNat.mk_coe, ExpChar.pow_prime_pow_mul_eq_one_iff]	[ " a✝ * b✝ ∈ {ζ \| ζ ^ ↑k = 1}", " x✝¹⁻¹ ∈ { carrier := {ζ \| ζ ^ ↑k = 1}, mul_mem' := ⋯, one_mem' := ⋯ }.carrier", " ζ ∈ rootsOfUnity k M ↔ ↑ζ ^ ↑k = 1", " ζ ^ ↑k = 1 ↔ ↑ζ ^ ↑k = 1", " rootsOfUnity 1 M = ⊥", " x✝ ∈ rootsOfUnity 1 M ↔ x✝ ∈ ⊥", " rootsOfUnity k M ≤ rootsOfUnity l M", " rootsOfUnity k M ≤ ...	[ " a✝ * b✝ ∈ {ζ \| ζ ^ ↑k = 1}", " x✝¹⁻¹ ∈ { carrier := {ζ \| ζ ^ ↑k = 1}, mul_mem' := ⋯, one_mem' := ⋯ }.carrier", " ζ ∈ rootsOfUnity k M ↔ ↑ζ ^ ↑k = 1", " ζ ^ ↑k = 1 ↔ ↑ζ ^ ↑k = 1", " rootsOfUnity 1 M = ⊥", " x✝ ∈ rootsOfUnity 1 M ↔ x✝ ∈ ⊥", " rootsOfUnity k M ≤ rootsOfUnity l M", " rootsOfUnity k M ≤ ...
import Mathlib.MeasureTheory.Decomposition.RadonNikodym import Mathlib.MeasureTheory.Measure.Haar.OfBasis import Mathlib.Probability.Independence.Basic #align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520" open scoped Classical MeasureTheory NNReal ENNRea...	Mathlib/Probability/Density.lean	82	86	theorem hasPDF_iff_of_aemeasurable {X : Ω → E} {ℙ : Measure Ω} {μ : Measure E} (hX : AEMeasurable X ℙ) : HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by	rw [hasPDF_iff] simp only [hX, true_and]	[ " HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ", " AEMeasurable X ℙ ∧ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ ↔\n (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ" ]	[]
import Mathlib.Probability.ProbabilityMassFunction.Constructions import Mathlib.Tactic.FinCases namespace PMF open ENNReal noncomputable def binomial (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1)) := .ofFintype (fun i => p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ)) (by convert (add_pow ...	Mathlib/Probability/ProbabilityMassFunction/Binomial.lean	40	42	theorem binomial_apply_zero (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : binomial p h n 0 = (1-p)^n := by	simp [binomial_apply]	[ " ∑ a : Fin (n + 1), (fun i => p ^ ↑i * (1 - p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)) a = 1", " ∑ a : Fin (n + 1), (fun i => p ^ ↑i * (1 - p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)) a =\n ∑ m ∈ Finset.range (n + 1), p ^ m * (1 - p) ^ (n - m) * ↑(n.choose m)", " (∑ i ∈ Finset.range (n + 1),\n if h : ...	[ " ∑ a : Fin (n + 1), (fun i => p ^ ↑i * (1 - p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)) a = 1", " ∑ a : Fin (n + 1), (fun i => p ^ ↑i * (1 - p) ^ (↑(Fin.last n) - ↑i) * ↑(n.choose ↑i)) a =\n ∑ m ∈ Finset.range (n + 1), p ^ m * (1 - p) ^ (n - m) * ↑(n.choose m)", " (∑ i ∈ Finset.range (n + 1),\n if h : ...
import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Analysis.SpecialFunctions.Sqrt import Mathlib.Analysis.NormedSpace.HomeomorphBall #align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88" noncomputable section open RCLike Real ...	Mathlib/Analysis/InnerProductSpace/Calculus.lean	115	118	theorem HasDerivAt.inner {f g : ℝ → E} {f' g' : E} {x : ℝ} : HasDerivAt f f' x → HasDerivAt g g' x → HasDerivAt (fun t => ⟪f t, g t⟫) (⟪f x, g'⟫ + ⟪f', g x⟫) x := by	simpa only [← hasDerivWithinAt_univ] using HasDerivWithinAt.inner 𝕜	[ " HasDerivWithinAt (fun t => ⟪f t, g t⟫_𝕜) (⟪f x, g'⟫_𝕜 + ⟪f', g x⟫_𝕜) s x", " HasDerivAt f f' x → HasDerivAt g g' x → HasDerivAt (fun t => ⟪f t, g t⟫_𝕜) (⟪f x, g'⟫_𝕜 + ⟪f', g x⟫_𝕜) x" ]	[ " HasDerivWithinAt (fun t => ⟪f t, g t⟫_𝕜) (⟪f x, g'⟫_𝕜 + ⟪f', g x⟫_𝕜) s x" ]
import Mathlib.Analysis.SpecificLimits.Basic import Mathlib.Data.Rat.Denumerable import Mathlib.Data.Set.Pointwise.Interval import Mathlib.SetTheory.Cardinal.Continuum #align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d" open Nat Set open Cardinal no...	Mathlib/Data/Real/Cardinality.lean	69	70	theorem cantorFunctionAux_false (h : f n = false) : cantorFunctionAux c f n = 0 := by	simp [cantorFunctionAux, h]	[ " cantorFunctionAux c f n = c ^ n", " cantorFunctionAux c f n = 0" ]	[ " cantorFunctionAux c f n = c ^ n" ]
import Mathlib.Analysis.Calculus.LineDeriv.Basic import Mathlib.Analysis.Calculus.FDeriv.Measurable open MeasureTheory variable {𝕜 : Type} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜] {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E] {F : Type*} [Norm...	Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean	47	52	theorem stronglyMeasurable_lineDeriv [SecondCountableTopologyEither E F] (hf : Continuous f) : StronglyMeasurable (fun x ↦ lineDeriv 𝕜 f x v) := by	borelize 𝕜 let g : E → 𝕜 → F := fun x t ↦ f (x + t • v) have hg : Continuous g.uncurry := by apply hf.comp; continuity exact (stronglyMeasurable_deriv_with_param hg).comp_measurable measurable_prod_mk_right	[ " MeasurableSet {x \| LineDifferentiableAt 𝕜 f x v}", " Continuous (Function.uncurry g)", " Continuous fun x => x.1 + x.2 • v", " Measurable fun x => lineDeriv 𝕜 f x v", " StronglyMeasurable fun x => lineDeriv 𝕜 f x v" ]	[ " MeasurableSet {x \| LineDifferentiableAt 𝕜 f x v}", " Continuous (Function.uncurry g)", " Continuous fun x => x.1 + x.2 • v", " Measurable fun x => lineDeriv 𝕜 f x v" ]
import Mathlib.Analysis.Calculus.Conformal.NormedSpace import Mathlib.Analysis.InnerProductSpace.ConformalLinearMap #align_import analysis.calculus.conformal.inner_product from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" noncomputable section variable {E F : Type*} variable [NormedA...	Mathlib/Analysis/Calculus/Conformal/InnerProduct.lean	36	38	theorem conformalAt_iff {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : HasFDerivAt f f' x) : ConformalAt f x ↔ ∃ c : ℝ, 0 < c ∧ ∀ u v : E, ⟪f' u, f' v⟫ = c * ⟪u, v⟫ := by	simp only [conformalAt_iff', h.fderiv]	[ " ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫_ℝ = c * ⟪u, v⟫_ℝ", " ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪f' u, f' v⟫_ℝ = c * ⟪u, v⟫_ℝ" ]	[ " ConformalAt f x ↔ ∃ c, 0 < c ∧ ∀ (u v : E), ⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫_ℝ = c * ⟪u, v⟫_ℝ" ]
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup import Mathlib.Tactic.AdaptationNote #align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45...	Mathlib/NumberTheory/ModularForms/SlashActions.lean	67	70	theorem SlashAction.smul_slash_of_tower {R β G α : Type} (γ : Type) [Group G] [AddGroup α] [Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ] (k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by	rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]	[ " (-a) ∣[k;γ] g + a ∣[k;γ] g = 0", " (r • a) ∣[k;γ] g = r • a ∣[k;γ] g" ]	[ " (-a) ∣[k;γ] g + a ∣[k;γ] g = 0" ]
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	136	143	theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints {s : Set P} {p1 p2 : P} (hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s) : p1 -ᵥ p2 ∈ vectorSpan k s := by	rcases hp1 with ⟨p1a, ⟨hp1a, ⟨v1, ⟨hv1, hv1p⟩⟩⟩⟩ rcases hp2 with ⟨p2a, ⟨hp2a, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩ rw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +ᵥ p1a), vadd_vsub_assoc, add_comm, add_sub_assoc] have hv1v2 : v1 - v2 ∈ vectorSpan k s := (vectorSpan k s).sub_mem hv1 hv2 refine (vectorSpan k s).add_mem ?_ hv1v2 e...	[ " 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 ...
import Mathlib.Data.ULift import Mathlib.Data.ZMod.Defs import Mathlib.SetTheory.Cardinal.PartENat #align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8" set_option autoImplicit true open Cardinal Function noncomputable section variable {α β : Typ...	Mathlib/SetTheory/Cardinal/Finite.lean	167	170	theorem card_sum [Finite α] [Finite β] : Nat.card (α ⊕ β) = Nat.card α + Nat.card β := by	have := Fintype.ofFinite α have := Fintype.ofFinite β simp_rw [Nat.card_eq_fintype_card, Fintype.card_sum]	[ " Nat.card { x // x ∈ s } = s.card", " Nat.card ↑s = s.toFinset.card", " Nat.card ↑s = hs.toFinset.card", " Nat.card α = 0", " Nat.card α = 0 ↔ IsEmpty α ∨ Infinite α", " Nat.card α ≠ 0 ↔ Nonempty α ∧ Finite α", " 0 < Nat.card α ↔ Nonempty α ∧ Finite α", " Nat.card α ≤ Nat.card β", " lift.{u, v} #β ...	[ " Nat.card { x // x ∈ s } = s.card", " Nat.card ↑s = s.toFinset.card", " Nat.card ↑s = hs.toFinset.card", " Nat.card α = 0", " Nat.card α = 0 ↔ IsEmpty α ∨ Infinite α", " Nat.card α ≠ 0 ↔ Nonempty α ∧ Finite α", " 0 < Nat.card α ↔ Nonempty α ∧ Finite α", " Nat.card α ≤ Nat.card β", " lift.{u, v} #β ...
import Mathlib.Algebra.MonoidAlgebra.Degree import Mathlib.Algebra.Polynomial.Coeff import Mathlib.Algebra.Polynomial.Monomial import Mathlib.Data.Fintype.BigOperators import Mathlib.Data.Nat.WithBot import Mathlib.Data.Nat.Cast.WithTop import Mathlib.Data.Nat.SuccPred #align_import data.polynomial.degree.definitions...	Mathlib/Algebra/Polynomial/Degree/Definitions.lean	138	144	theorem supDegree_eq_natDegree (p : R[X]) : p.toFinsupp.supDegree id = p.natDegree := by	obtain rfl\|h := eq_or_ne p 0 · simp apply WithBot.coe_injective rw [← AddMonoidAlgebra.supDegree_withBot_some_comp, Function.comp_id, supDegree_eq_degree, degree_eq_natDegree h, Nat.cast_withBot] rwa [support_toFinsupp, nonempty_iff_ne_empty, Ne, support_eq_empty]	[ " Decidable p.Monic", " Decidable (p.leadingCoeff = 1)", " p.degree = ⊥", " p.natDegree = 0", " p.degree = ↑p.natDegree", " Option.some n = ↑(WithBot.unbot' 0 (Option.some n))", " AddMonoidAlgebra.supDegree id p.toFinsupp = p.natDegree", " AddMonoidAlgebra.supDegree id (toFinsupp 0) = natDegree 0", ...	[ " Decidable p.Monic", " Decidable (p.leadingCoeff = 1)", " p.degree = ⊥", " p.natDegree = 0", " p.degree = ↑p.natDegree", " Option.some n = ↑(WithBot.unbot' 0 (Option.some n))" ]
import Mathlib.Algebra.Module.Submodule.EqLocus import Mathlib.Algebra.Module.Submodule.RestrictScalars import Mathlib.Algebra.Ring.Idempotents import Mathlib.Data.Set.Pointwise.SMul import Mathlib.LinearAlgebra.Basic import Mathlib.Order.CompactlyGenerated.Basic import Mathlib.Order.OmegaCompletePartialOrder #align_...	Mathlib/LinearAlgebra/Span.lean	154	157	theorem span_preimage_le (f : F) (s : Set M₂) : span R (f ⁻¹' s) ≤ (span R₂ s).comap f := by	rw [span_le, comap_coe] exact preimage_mono subset_span	[ " ↑(span R ↑s) = ↑s", " ↑(span R ↑s) ≤ ↑s", " span R (insert 0 s) = span R s", " span R (insert 0 s) ≤ span R s", " 0 ∈ ↑(span R s) ∧ s ⊆ ↑(span R s)", " 0 ∈ ↑(span R s)", " span R (⇑f ⁻¹' s) ≤ comap f (span R₂ s)", " ⇑f ⁻¹' s ⊆ ⇑f ⁻¹' ↑(span R₂ s)" ]	[ " ↑(span R ↑s) = ↑s", " ↑(span R ↑s) ≤ ↑s", " span R (insert 0 s) = span R s", " span R (insert 0 s) ≤ span R s", " 0 ∈ ↑(span R s) ∧ s ⊆ ↑(span R s)", " 0 ∈ ↑(span R s)" ]
import Mathlib.Topology.Order.IsLUB open Set Filter TopologicalSpace Topology Function open OrderDual (toDual ofDual) variable {α β γ : Type*} section DenselyOrdered variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α} {s : Set α} theorem closure_Ioi' {a : α} (h : (Io...	Mathlib/Topology/Order/DenselyOrdered.lean	136	143	theorem Ioc_subset_closure_interior (a b : α) : Ioc a b ⊆ closure (interior (Ioc a b)) := by	rcases eq_or_ne a b with (rfl \| h) · simp · calc Ioc a b ⊆ Icc a b := Ioc_subset_Icc_self _ = closure (Ioo a b) := (closure_Ioo h).symm _ ⊆ closure (interior (Ioc a b)) := closure_mono (interior_maximal Ioo_subset_Ioc_self isOpen_Ioo)	[ " closure (Ioi a) = Ici a", " closure (Ioi a) ⊆ Ici a", " Ici a ⊆ closure (Ioi a)", " a ∈ closure (Ioi a)", " closure (Ioo a b) = Icc a b", " closure (Ioo a b) ⊆ Icc a b", " Icc a b ⊆ closure (Ioo a b)", " {a, b} ⊆ closure (Ioo a b)", " a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)", " ∅ ⊆ closure...	[ " closure (Ioi a) = Ici a", " closure (Ioi a) ⊆ Ici a", " Ici a ⊆ closure (Ioi a)", " a ∈ closure (Ioi a)", " closure (Ioo a b) = Icc a b", " closure (Ioo a b) ⊆ Icc a b", " Icc a b ⊆ closure (Ioo a b)", " {a, b} ⊆ closure (Ioo a b)", " a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)", " ∅ ⊆ closure...
import Mathlib.Topology.Separation import Mathlib.Topology.UniformSpace.Basic import Mathlib.Topology.UniformSpace.Cauchy #align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9" noncomputable section open Topology Uniformity Filter S...	Mathlib/Topology/UniformSpace/UniformConvergence.lean	124	127	theorem tendstoUniformlyOn_iff_tendsto {F : ι → α → β} {f : α → β} {p : Filter ι} {s : Set α} : TendstoUniformlyOn F f p s ↔ Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β) := by	simp [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, tendstoUniformlyOnFilter_iff_tendsto]	[ " TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s)", " (∀ u ∈ 𝓤 β, ∀ᶠ (n : ι) in p, ∀ x ∈ s, (f x, F n x) ∈ u) ↔\n ∀ u ∈ 𝓤 β, ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, (f n.2, F n.1 n.2) ∈ u", " ∀ a ∈ 𝓤 β, (∀ᶠ (n : ι) in p, ∀ x ∈ s, (f x, F n x) ∈ a) ↔ ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, (f n.2, F n.1 n.2) ...	[ " TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s)", " (∀ u ∈ 𝓤 β, ∀ᶠ (n : ι) in p, ∀ x ∈ s, (f x, F n x) ∈ u) ↔\n ∀ u ∈ 𝓤 β, ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, (f n.2, F n.1 n.2) ∈ u", " ∀ a ∈ 𝓤 β, (∀ᶠ (n : ι) in p, ∀ x ∈ s, (f x, F n x) ∈ a) ↔ ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, (f n.2, F n.1 n.2) ...
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp import Mathlib.MeasureTheory.Integral.Bochner import Mathlib.Order.Filter.IndicatorFunction import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner import Mathlib.MeasureTheory.Function.LpSeminorm.Trim #align_import measure_theory.function.conditional...	Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean	62	64	theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) : AEStronglyMeasurable' m g μ := by	obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩	[ " AEStronglyMeasurable' m g μ" ]	[]
import Mathlib.GroupTheory.GroupAction.Prod import Mathlib.Algebra.Ring.Int import Mathlib.Data.Nat.Cast.Basic assert_not_exists DenselyOrdered variable {M : Type} class NatPowAssoc (M : Type) [MulOneClass M] [Pow M ℕ] : Prop where protected npow_add : ∀ (k n: ℕ) (x : M), x ^ (k + n) = x ^ k * x ^ n ...	Mathlib/Algebra/Group/NatPowAssoc.lean	69	70	theorem npow_mul_comm (m n : ℕ) (x : M) : x ^ m * x ^ n = x ^ n * x ^ m := by	simp only [← npow_add, add_comm]	[ " x ^ k * x ^ m * x ^ n = x ^ k * (x ^ m * x ^ n)", " x ^ m * x ^ n = x ^ n * x ^ m" ]	[ " x ^ k * x ^ m * x ^ n = x ^ k * (x ^ m * x ^ n)" ]
import Mathlib.Data.Fintype.Basic import Mathlib.Data.Set.Finite #align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce" open Finset universe u v namespace HallMarriageTheorem variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α} s...	Mathlib/Combinatorics/Hall/Finite.lean	125	133	theorem hall_cond_of_restrict {ι : Type u} {t : ι → Finset α} {s : Finset ι} (ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (s' : Finset (s : Set ι)) : s'.card ≤ (s'.biUnion fun a' => t a').card := by	classical rw [← card_image_of_injective s' Subtype.coe_injective] convert ht (s'.image fun z => z.1) using 1 apply congr_arg ext y simp	[ " s'.card ≤ (s'.biUnion fun x' => (t ↑x').erase a).card", " s'.card < (s'.biUnion fun x_1 => t ↑x_1).card", " False", " (s'.biUnion fun x_1 => t ↑x_1) = (image (fun z => ↑z) s').biUnion t", " (x ∈ s'.biUnion fun x => t ↑x) ↔ x ∈ (image (fun z => ↑z) s').biUnion t", " s'.card ≤ ((s'.biUnion fun x' => t ↑x'...	[ " s'.card ≤ (s'.biUnion fun x' => (t ↑x').erase a).card", " s'.card < (s'.biUnion fun x_1 => t ↑x_1).card", " False", " (s'.biUnion fun x_1 => t ↑x_1) = (image (fun z => ↑z) s').biUnion t", " (x ∈ s'.biUnion fun x => t ↑x) ↔ x ∈ (image (fun z => ↑z) s').biUnion t", " s'.card ≤ ((s'.biUnion fun x' => t ↑x'...
import Mathlib.AlgebraicTopology.SplitSimplicialObject import Mathlib.AlgebraicTopology.DoldKan.PInfty #align_import algebraic_topology.dold_kan.functor_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504" noncomputable section open CategoryTheory CategoryTheory.Category CategoryT...	Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean	112	117	theorem mapMono_δ₀' (i : Δ' ⟶ Δ) [Mono i] (hi : Isδ₀ i) : mapMono K i = K.d Δ.len Δ'.len := by	unfold mapMono suffices Δ ≠ Δ' by simp only [dif_neg this, dif_pos hi] rintro rfl simpa only [self_eq_add_right, Nat.one_ne_zero] using hi.1	[ " K.X Δ.len ⟶ K.X Δ'.len", " K.X Δ.len = K.X Δ'.len", " mapMono K (𝟙 Δ) = 𝟙 (K.X Δ.len)", " (if h : Δ = Δ then eqToHom ⋯ else if h : Isδ₀ (𝟙 Δ) then K.d Δ.len Δ.len else 0) = 𝟙 (K.X Δ.len)", " mapMono K i = K.d Δ.len Δ'.len", " (if h : Δ = Δ' then eqToHom ⋯ else if h : Isδ₀ i then K.d Δ.len Δ'.len els...	[ " K.X Δ.len ⟶ K.X Δ'.len", " K.X Δ.len = K.X Δ'.len", " mapMono K (𝟙 Δ) = 𝟙 (K.X Δ.len)", " (if h : Δ = Δ then eqToHom ⋯ else if h : Isδ₀ (𝟙 Δ) then K.d Δ.len Δ.len else 0) = 𝟙 (K.X Δ.len)" ]
import Mathlib.Algebra.MvPolynomial.Monad #align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6" namespace MvPolynomial variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S] noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno...	Mathlib/Algebra/MvPolynomial/Expand.lean	71	73	theorem expand_bind₁ (p : ℕ) (f : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : expand p (bind₁ f φ) = bind₁ (fun i ↦ expand p (f i)) φ := by	rw [← AlgHom.comp_apply, expand_comp_bind₁]	[ " (expand 1) f = f", " expand 1 = AlgHom.id R (MvPolynomial σ R)", " (expand 1) (X f) = (AlgHom.id R (MvPolynomial σ R)) (X f)", " (expand p).comp (bind₁ f) = bind₁ fun i => (expand p) (f i)", " ∀ (i : σ), ((expand p).comp (bind₁ f)) (X i) = (bind₁ fun i => (expand p) (f i)) (X i)", " ((expand p).comp (bi...	[ " (expand 1) f = f", " expand 1 = AlgHom.id R (MvPolynomial σ R)", " (expand 1) (X f) = (AlgHom.id R (MvPolynomial σ R)) (X f)", " (expand p).comp (bind₁ f) = bind₁ fun i => (expand p) (f i)", " ∀ (i : σ), ((expand p).comp (bind₁ f)) (X i) = (bind₁ fun i => (expand p) (f i)) (X i)", " ((expand p).comp (bi...
import Mathlib.Data.Multiset.Bind import Mathlib.Control.Traversable.Lemmas import Mathlib.Control.Traversable.Instances #align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58" universe u namespace Multiset open List instance functor : Functor Multiset...	Mathlib/Data/Multiset/Functor.lean	129	134	theorem traverse_map {G : Type* → Type _} [Applicative G] [CommApplicative G] {α β γ : Type _} (g : α → β) (h : β → G γ) (x : Multiset α) : traverse h (map g x) = traverse (h ∘ g) x := by	refine Quotient.inductionOn x ?_ intro simp only [traverse, quot_mk_to_coe, map_coe, lift_coe, Function.comp_apply] rw [← Traversable.traverse_map h g, List.map_eq_map]	[ " ∀ {α : Type ?u.133} (x : Multiset α), id <$> x = x", " ∀ {α β γ : Type ?u.133} (g : α → β) (h : β → γ) (x : Multiset α), (h ∘ g) <$> x = h <$> g <$> x", " Multiset α' → F (Multiset β')", " ∀ (a b : List α'),\n a ≈ b → (Functor.map Coe.coe ∘ Traversable.traverse f) a = (Functor.map Coe.coe ∘ Traversable.t...	[ " ∀ {α : Type ?u.133} (x : Multiset α), id <$> x = x", " ∀ {α β γ : Type ?u.133} (g : α → β) (h : β → γ) (x : Multiset α), (h ∘ g) <$> x = h <$> g <$> x", " Multiset α' → F (Multiset β')", " ∀ (a b : List α'),\n a ≈ b → (Functor.map Coe.coe ∘ Traversable.traverse f) a = (Functor.map Coe.coe ∘ Traversable.t...
import Mathlib.Algebra.Group.Commute.Basic import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.Dynamics.PeriodicPts import Mathlib.Data.Set.Pointwise.SMul namespace MulAction open Pointwise variable {α : Type} variable {G : Type} [Group G] [MulAction G α] variable {M : Type*} [Monoid M] [MulAction M α] ...	Mathlib/GroupTheory/GroupAction/FixedPoints.lean	195	199	theorem fixedBy_mem_fixedBy_of_commute {g h : G} (comm: Commute g h) : (fixedBy α g) ∈ fixedBy (Set α) h := by	ext x rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, ← mul_smul, comm.inv_right, mul_smul, smul_left_cancel_iff, mem_fixedBy]	[ " fixedBy α g ∈ fixedBy (Set α) h", " x ∈ h • fixedBy α g ↔ x ∈ fixedBy α g" ]	[]
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine #align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5" noncomputable section open scoped EuclideanGeometry ...	Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean	69	73	theorem angle_add_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) : angle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by	rw [angle, inner_add_right, h, add_zero, real_inner_self_eq_norm_mul_norm] by_cases hx : ‖x‖ = 0; · simp [hx] rw [div_mul_eq_div_div, mul_self_div_self]	[ " ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2", " ⟪x, y⟫_ℝ = 0 ↔ angle x y = π / 2", " ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2", " angle x (x + y) = (‖x‖ / ‖x + y‖).arccos", " (‖x‖ * ‖x‖ / (‖x‖ * ‖x + y‖)).arccos = (‖x‖ / ‖x + y‖).arccos" ]	[ " ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2", " ⟪x, y⟫_ℝ = 0 ↔ angle x y = π / 2", " ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2" ]
import Mathlib.Logic.Pairwise import Mathlib.Logic.Relation import Mathlib.Data.List.Basic #align_import data.list.pairwise from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e" open Nat Function namespace List variable {α β : Type*} {R S T : α → α → Prop} {a : α} {l : List α} mk_iff_o...	Mathlib/Data/List/Pairwise.lean	152	156	theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h : ∀ a ∈ l, ∀ b ∈ l, r a b) : l.Pairwise r := by	rw [pairwise_iff_forall_sublist] intro a b hab apply h <;> (apply hab.subset; simp)	[ " Pairwise (flip fun x => R x) l", " ∀ ⦃a : α⦄, a ∈ l → ∀ ⦃b : α⦄, b ∈ l → a ≠ b → R a b", " Symmetric fun x y => x ≠ y → R x y", " ∀ (x : α), x ∈ l → x ≠ x → R x x", " Pairwise (fun x y => x ≠ y → R x y) l", " R a b", " Pairwise R (map f []) ↔ Pairwise (fun a b => R (f a) (f b)) []", " Pairwise R (ma...	[ " Pairwise (flip fun x => R x) l", " ∀ ⦃a : α⦄, a ∈ l → ∀ ⦃b : α⦄, b ∈ l → a ≠ b → R a b", " Symmetric fun x y => x ≠ y → R x y", " ∀ (x : α), x ∈ l → x ≠ x → R x x", " Pairwise (fun x y => x ≠ y → R x y) l", " R a b", " Pairwise R (map f []) ↔ Pairwise (fun a b => R (f a) (f b)) []", " Pairwise R (ma...
import Mathlib.Geometry.Manifold.ContMDiff.Basic open Set Function Filter ChartedSpace SmoothManifoldWithCorners open scoped Topology Manifold variable {𝕜 : Type} [NontriviallyNormedField 𝕜] -- declare a smooth manifold `M` over the pair `(E, H)`. {E : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H ...	Mathlib/Geometry/Manifold/ContMDiff/Product.lean	66	70	theorem ContMDiffWithinAt.prod_mk_space {f : M → E'} {g : M → F'} (hf : ContMDiffWithinAt I 𝓘(𝕜, E') n f s x) (hg : ContMDiffWithinAt I 𝓘(𝕜, F') n g s x) : ContMDiffWithinAt I 𝓘(𝕜, E' × F') n (fun x => (f x, g x)) s x := by	rw [contMDiffWithinAt_iff] at * exact ⟨hf.1.prod hg.1, hf.2.prod hg.2⟩	[ " ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x", " ContinuousWithinAt (fun x => (f x, g x)) s x ∧\n ContDiffWithinAt 𝕜 n (↑(extChartAt (I'.prod J') (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)\n (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)", " ContMDi...	[ " ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x", " ContinuousWithinAt (fun x => (f x, g x)) s x ∧\n ContDiffWithinAt 𝕜 n (↑(extChartAt (I'.prod J') (f x, g x)) ∘ (fun x => (f x, g x)) ∘ ↑(extChartAt I x).symm)\n (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)" ]
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison import Mathlib.CategoryTheory.Sites.Equivalence namespace CategoryTheory variable {C : Type} [Category C] open GrothendieckTopology namespace Equivalence variable {D : Type} [Category D] variable (e : C ≌ D) section Regular variable [Preregular C] ...	Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean	101	106	theorem preregular_isSheaf_iff (F : Cᵒᵖ ⥤ A) : haveI := e.preregular IsSheaf (regularTopology C) F ↔ IsSheaf (regularTopology D) (e.inverse.op ⋙ F) := by	refine ⟨fun hF ↦ ((e.sheafCongrPreregular A).functor.obj ⟨F, hF⟩).cond, fun hF ↦ ?_⟩ rw [isSheaf_of_iso_iff (P' := e.functor.op ⋙ e.inverse.op ⋙ F)] · exact (e.sheafCongrPreregular A).inverse.obj ⟨e.inverse.op ⋙ F, hF⟩ \|>.cond · exact isoWhiskerRight e.op.unitIso F	[ " IsSheaf (regularTopology C) F ↔ IsSheaf (regularTopology D) (e.inverse.op ⋙ F)", " IsSheaf (regularTopology C) F", " IsSheaf (regularTopology C) (e.functor.op ⋙ e.inverse.op ⋙ F)", " F ≅ e.functor.op ⋙ e.inverse.op ⋙ F" ]	[]
import Mathlib.CategoryTheory.Sites.Sieves import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer import Mathlib.CategoryTheory.Category.Preorder import Mathlib.Order.Copy import Mathlib.Data.Set.Subsingleton #align_import category_theory.sites.grothendieck fr...	Mathlib/CategoryTheory/Sites/Grothendieck.lean	105	109	theorem ext {J₁ J₂ : GrothendieckTopology C} (h : (J₁ : ∀ X : C, Set (Sieve X)) = J₂) : J₁ = J₂ := by	cases J₁ cases J₂ congr	[ " J₁ = J₂", " { sieves := sieves✝, top_mem' := top_mem'✝, pullback_stable' := pullback_stable'✝, transitive' := transitive'✝ } = J₂", " { sieves := sieves✝¹, top_mem' := top_mem'✝¹, pullback_stable' := pullback_stable'✝¹, transitive' := transitive'✝¹ } =\n { sieves := sieves✝, top_mem' := top_mem'✝, pullback...	[]
import Mathlib.Algebra.Algebra.Basic import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue import Mathlib.Data.Real.Archimedean #align_import number_theory.class_number.admissible_abs from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b" namespace AbsoluteValue open Int	Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean	31	52	theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) : ∃ t : Fin n → Fin ⌈1 / ε⌉₊, ∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε := by	have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb) have hbε : 0 < abs b • ε := by rw [Algebra.smul_def] exact mul_pos hb' hε have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) := fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg.mpr (emod_nonneg _ hb)) hbε.le) refine ⟨fun ...	[ " ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑\|A i₁ % b - A i₀ % b\| < \|b\| • ε", " 0 < \|b\| • ε", " 0 < (algebraMap ℤ ℝ) \|b\| * ε", " ⌊↑(A i % b) / \|b\| • ε⌋.natAbs < ⌈1 / ε⌉₊", " ↑(A i % b) / \|b\| • ε < ↑↑⌈1 / ε⌉₊", " ↑(A i % b) / \|b\| • ε < 1 / ε", " A i % b < \|b\|", " ∀ (i₀ i₁ : Fin n),\n (fun i => ⟨⌊↑(A i ...	[]
import Mathlib.Data.Finset.Lattice import Mathlib.Data.Multiset.Powerset #align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" namespace Finset open Function Multiset variable {α : Type*} {s t : Finset α} section Powerset def powerset (s : Finset...	Mathlib/Data/Finset/Powerset.lean	93	96	theorem not_mem_of_mem_powerset_of_not_mem {s t : Finset α} {a : α} (ht : t ∈ s.powerset) (h : a ∉ s) : a ∉ t := by	apply mt _ h apply mem_powerset.1 ht	[ " s ∈ t.powerset ↔ s ⊆ t", " { val := val✝, nodup := nodup✝ } ∈ t.powerset ↔ { val := val✝, nodup := nodup✝ } ⊆ t", " ↑s.powerset = toSet ⁻¹' 𝒫↑s", " x✝ ∈ ↑s.powerset ↔ x✝ ∈ toSet ⁻¹' 𝒫↑s", " s.powerset = {∅} ↔ s = ∅", " a ∉ t", " a ∈ t → a ∈ s" ]	[ " s ∈ t.powerset ↔ s ⊆ t", " { val := val✝, nodup := nodup✝ } ∈ t.powerset ↔ { val := val✝, nodup := nodup✝ } ⊆ t", " ↑s.powerset = toSet ⁻¹' 𝒫↑s", " x✝ ∈ ↑s.powerset ↔ x✝ ∈ toSet ⁻¹' 𝒫↑s", " s.powerset = {∅} ↔ s = ∅" ]
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics #align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8" noncomputable section open scoped Classical open Real Topology NNReal ENNReal Filter ComplexConjugate open Filter Finset...	Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean	105	109	theorem Filter.Tendsto.const_cpow {l : Filter α} {f : α → ℂ} {a b : ℂ} (hf : Tendsto f l (𝓝 b)) (h : a ≠ 0 ∨ b ≠ 0) : Tendsto (fun x => a ^ f x) l (𝓝 (a ^ b)) := by	cases h with \| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf \| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf	[ " (fun x => 0 ^ x) =ᶠ[𝓝 b] 0", " (fun x => 0 ^ x) x = 0 x", " 0 ^ x = OfNat.ofNat 0 x", " ∀ᶠ (x : ℂ) in 𝓝 b, x ≠ 0", " (fun x => x ^ b) =ᶠ[𝓝 a] fun x => cexp (x.log * b)", " (fun x => x ^ b) x = (fun x => cexp (x.log * b)) x", " x ^ b = cexp (x.log * b)", " ∀ᶠ (x : ℂ) in 𝓝 a, x ≠ 0", " (fun x =>...	[ " (fun x => 0 ^ x) =ᶠ[𝓝 b] 0", " (fun x => 0 ^ x) x = 0 x", " 0 ^ x = OfNat.ofNat 0 x", " ∀ᶠ (x : ℂ) in 𝓝 b, x ≠ 0", " (fun x => x ^ b) =ᶠ[𝓝 a] fun x => cexp (x.log * b)", " (fun x => x ^ b) x = (fun x => cexp (x.log * b)) x", " x ^ b = cexp (x.log * b)", " ∀ᶠ (x : ℂ) in 𝓝 a, x ≠ 0", " (fun x =>...
import Mathlib.MeasureTheory.Group.GeometryOfNumbers import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic #align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30" ...	Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean	326	333	theorem norm_le_convexBodySumFun (x : E K) : ‖x‖ ≤ convexBodySumFun x := by	rw [norm_eq_sup'_normAtPlace] refine (Finset.sup'_le_iff _ _).mpr fun w _ ↦ ?_ rw [convexBodySumFun_apply, ← Finset.univ.add_sum_erase _ (Finset.mem_univ w)] refine le_add_of_le_of_nonneg ?_ ?_ · exact le_mul_of_one_le_left (normAtPlace_nonneg w x) one_le_mult · exact Finset.sum_nonneg (fun _ _ => mul_non...	[ " convexBodySumFun x = ∑ w : { w // w.IsReal }, ‖x.1 w‖ + 2 * ∑ w : { w // w.IsComplex }, ‖x.2 w‖", " ∑ x_1 ∈ Finset.subtype (fun x => x.IsReal) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x +\n ∑ x_1 ∈ Finset.subtype (fun x => x.IsComplex) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x =\n ∑ x_1 ∈ F...	[ " convexBodySumFun x = ∑ w : { w // w.IsReal }, ‖x.1 w‖ + 2 * ∑ w : { w // w.IsComplex }, ‖x.2 w‖", " ∑ x_1 ∈ Finset.subtype (fun x => x.IsReal) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x +\n ∑ x_1 ∈ Finset.subtype (fun x => x.IsComplex) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x =\n ∑ x_1 ∈ F...

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