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.LinearAlgebra.AffineSpace.AffineEquiv #align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2" open AffineMap AffineEquiv section variable (R : Type) {V V' P P' : Type} [Ring R] [Invertible (2 : R)] [AddCommGroup V] [Modu...	Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean	133	137	theorem midpoint_vsub (p₁ p₂ p : P) : midpoint R p₁ p₂ -ᵥ p = (⅟ 2 : R) • (p₁ -ᵥ p) + (⅟ 2 : R) • (p₂ -ᵥ p) := by	rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg, neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, ← vadd_vsub_assoc, midpoint_comm, midpoint, lineMap_apply]	[ " (pointReflection R (midpoint R x y)) x = y", " (pointReflection (midpoint R x y)) x = y", " midpoint R x y = midpoint R y x", " (pointReflection R (midpoint R x y)) y = x", " (pointReflection (midpoint R x y)) y = x", " midpoint R p₁ p₂ -ᵥ p₂ = ⅟2 • (p₁ -ᵥ p₂)", " p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ ...	[ " (pointReflection R (midpoint R x y)) x = y", " (pointReflection (midpoint R x y)) x = y", " midpoint R x y = midpoint R y x", " (pointReflection R (midpoint R x y)) y = x", " (pointReflection (midpoint R x y)) y = x", " midpoint R p₁ p₂ -ᵥ p₂ = ⅟2 • (p₁ -ᵥ p₂)", " p₂ -ᵥ midpoint R p₁ p₂ = ⅟2 • (p₂ -ᵥ ...
import Mathlib.Algebra.BigOperators.Group.Multiset import Mathlib.Data.Multiset.Dedup #align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e" assert_not_exists MonoidWithZero assert_not_exists MulAction universe v variable {α : Type*} {β : Type v} {γ δ : Ty...	Mathlib/Data/Multiset/Bind.lean	115	117	theorem coe_bind (l : List α) (f : α → List β) : (@bind α β l fun a => f a) = l.bind f := by	rw [List.bind, ← coe_join, List.map_map] rfl	[ " (↑(List.map ofList (l :: L))).join = ↑(l :: L).join", " a ∈ join 0 ↔ ∃ s ∈ 0, a ∈ s", " ∀ (a_1 : Multiset α) (s : Multiset (Multiset α)),\n (a ∈ s.join ↔ ∃ s_1 ∈ s, a ∈ s_1) → (a ∈ (a_1 ::ₘ s).join ↔ ∃ s_1 ∈ a_1 ::ₘ s, a ∈ s_1)", " card (join 0) = (map (⇑card) 0).sum", " ∀ (a : Multiset α) (s : Multise...	[ " (↑(List.map ofList (l :: L))).join = ↑(l :: L).join", " a ∈ join 0 ↔ ∃ s ∈ 0, a ∈ s", " ∀ (a_1 : Multiset α) (s : Multiset (Multiset α)),\n (a ∈ s.join ↔ ∃ s_1 ∈ s, a ∈ s_1) → (a ∈ (a_1 ::ₘ s).join ↔ ∃ s_1 ∈ a_1 ::ₘ s, a ∈ s_1)", " card (join 0) = (map (⇑card) 0).sum", " ∀ (a : Multiset α) (s : Multise...
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 powersetCard variable {n} {s t : Fi...	Mathlib/Data/Finset/Powerset.lean	220	225	theorem powersetCard_zero (s : Finset α) : s.powersetCard 0 = {∅} := by	ext; rw [mem_powersetCard, mem_singleton, card_eq_zero] refine ⟨fun h => h.2, fun h => by rw [h] exact ⟨empty_subset s, rfl⟩⟩	[ " s ∈ powersetCard n t ↔ s ⊆ t ∧ s.card = n", " { val := val✝, nodup := nodup✝ } ∈ powersetCard n t ↔\n { val := val✝, nodup := nodup✝ } ⊆ t ∧ { val := val✝, nodup := nodup✝ }.card = n", " powersetCard 0 s = {∅}", " a✝ ∈ powersetCard 0 s ↔ a✝ ∈ {∅}", " a✝ ⊆ s ∧ a✝ = ∅ ↔ a✝ = ∅", " a✝ ⊆ s ∧ a✝ = ∅", "...	[ " s ∈ powersetCard n t ↔ s ⊆ t ∧ s.card = n", " { val := val✝, nodup := nodup✝ } ∈ powersetCard n t ↔\n { val := val✝, nodup := nodup✝ } ⊆ t ∧ { val := val✝, nodup := nodup✝ }.card = n" ]
import Mathlib.Topology.Connected.Basic open Set Topology universe u v variable {α : Type u} {β : Type v} {ι : Type} {π : ι → Type} [TopologicalSpace α] {s t u v : Set α} section LocallyConnectedSpace class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where open_connected_basis : ∀ x,...	Mathlib/Topology/Connected/LocallyConnected.lean	125	132	theorem locallyConnectedSpace_of_connected_bases {ι : Type*} (b : α → ι → Set α) (p : α → ι → Prop) (hbasis : ∀ x, (𝓝 x).HasBasis (p x) (b x)) (hconnected : ∀ x i, p x i → IsPreconnected (b x i)) : LocallyConnectedSpace α := by	rw [locallyConnectedSpace_iff_connected_basis] exact fun x => (hbasis x).to_hasBasis (fun i hi => ⟨b x i, ⟨(hbasis x).mem_of_mem hi, hconnected x i hi⟩, subset_rfl⟩) fun s hs => ⟨(hbasis x).index s hs.1, ⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩	[ " LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V", " (∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id) ↔\n ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V", " (𝓝 x✝).HasBasis (fun s => IsOpen s ∧ x✝ ∈ s ∧ IsConnected s) i...	[ " LocallyConnectedSpace α ↔ ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V", " (∀ (x : α), (𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id) ↔\n ∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ⊆ U, IsOpen V ∧ x ∈ V ∧ IsConnected V", " (𝓝 x✝).HasBasis (fun s => IsOpen s ∧ x✝ ∈ s ∧ IsConnected s) i...
import Mathlib.Analysis.PSeries import Mathlib.Data.Real.Pi.Wallis import Mathlib.Tactic.AdaptationNote #align_import analysis.special_functions.stirling from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" open scoped Topology Real Nat Asymptotics open Finset Filter Nat Real namespace...	Mathlib/Analysis/SpecialFunctions/Stirling.lean	77	93	theorem log_stirlingSeq_diff_hasSum (m : ℕ) : HasSum (fun k : ℕ => (1 : ℝ) / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ ↑(k + 1)) (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by	let f (k : ℕ) := (1 : ℝ) / (2 * k + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ k change HasSum (fun k => f (k + 1)) _ rw [hasSum_nat_add_iff] convert (hasSum_log_one_add_inv m.cast_add_one_pos).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1 · ext k dsimp only [f] rw [← pow_mul, pow_add] push_cast field...	[ " stirlingSeq 0 = 0", " stirlingSeq 1 = rexp 1 / √2", " (stirlingSeq n).log = (↑n !).log - 1 / 2 * (2 * ↑n).log - ↑n * (↑n / rexp 1).log", " (stirlingSeq 0).log = (↑0!).log - 1 / 2 * (2 * ↑0).log - ↑0 * (↑0 / rexp 1).log", " (stirlingSeq (n✝ + 1)).log = (↑(n✝ + 1)!).log - 1 / 2 * (2 * ↑(n✝ + 1)).log - ↑(n✝ ...	[ " stirlingSeq 0 = 0", " stirlingSeq 1 = rexp 1 / √2", " (stirlingSeq n).log = (↑n !).log - 1 / 2 * (2 * ↑n).log - ↑n * (↑n / rexp 1).log", " (stirlingSeq 0).log = (↑0!).log - 1 / 2 * (2 * ↑0).log - ↑0 * (↑0 / rexp 1).log", " (stirlingSeq (n✝ + 1)).log = (↑(n✝ + 1)!).log - 1 / 2 * (2 * ↑(n✝ + 1)).log - ↑(n✝ ...
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic #align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a" variable {R M N N' : Type*} variable [CommRing R] [AddCommGroup M] [AddCo...	Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean	125	135	theorem liftAlternating_comp (g : N →ₗ[R] N') (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) : (liftAlternating (R := R) (M := M) (N := N') fun i => g.compAlternatingMap (f i)) = g ∘ₗ liftAlternating (R := R) (M := M) (N := N) f := by	ext v rw [LinearMap.comp_apply] induction' v using CliffordAlgebra.left_induction with r x y hx hy x m hx generalizing f · rw [liftAlternating_algebraMap, liftAlternating_algebraMap, map_smul, LinearMap.compAlternatingMap_apply] · rw [map_add, map_add, map_add, hx, hy] · rw [liftAlternating_ι_mul, li...	[ " Module R (M [⋀^ι]→ₗ[R] N)", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] N", " M [⋀^Fin 0]→ₗ[R] N ≃ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] (i : ℕ) → M [⋀^Fin i]→ₗ[R] N", " M →ₗ[R] ((i : ℕ) → M [⋀^Fin i]→ₗ[R]...	[ " Module R (M [⋀^ι]→ₗ[R] N)", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] N", " M [⋀^Fin 0]→ₗ[R] N ≃ₗ[R] N", " ((i : ℕ) → M [⋀^Fin i]→ₗ[R] N) →ₗ[R] ExteriorAlgebra R M →ₗ[R] (i : ℕ) → M [⋀^Fin i]→ₗ[R] N", " M →ₗ[R] ((i : ℕ) → M [⋀^Fin i]→ₗ[R]...
import Mathlib.Data.Matrix.Basic import Mathlib.Data.PEquiv #align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1" namespace PEquiv open Matrix universe u v variable {k l m n : Type*} variable {α : Type v} open Matrix def toMatrix [DecidableEq n] [Zer...	Mathlib/Data/Matrix/PEquiv.lean	96	99	theorem toPEquiv_mul_matrix [Fintype m] [DecidableEq m] [Semiring α] (f : m ≃ m) (M : Matrix m n α) : f.toPEquiv.toMatrix * M = M.submatrix f id := by	ext i j rw [mul_matrix_apply, Equiv.toPEquiv_apply, submatrix_apply, id]	[ " (f.toMatrix * M) i j = Option.casesOn (f i) 0 fun fi => M fi j", " ∑ j_1 : m, (if j_1 ∈ f i then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) (f i)", " ∑ j_1 : m, (if j_1 ∈ none then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) none", " ∑ j_1 : m, (if j_1 ∈ some fi then 1 else 0) * M j...	[ " (f.toMatrix * M) i j = Option.casesOn (f i) 0 fun fi => M fi j", " ∑ j_1 : m, (if j_1 ∈ f i then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) (f i)", " ∑ j_1 : m, (if j_1 ∈ none then 1 else 0) * M j_1 j = Option.rec 0 (fun val => M val j) none", " ∑ j_1 : m, (if j_1 ∈ some fi then 1 else 0) * M j...
import Mathlib.Data.ZMod.Basic import Mathlib.GroupTheory.Coxeter.Basic namespace CoxeterSystem open List Matrix Function Classical variable {B : Type} variable {W : Type} [Group W] variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W) local prefix:100 "s" => cs.simple local prefix:100 "π" => cs.wordProd ...	Mathlib/GroupTheory/Coxeter/Length.lean	91	98	theorem length_inv (w : W) : ℓ (w⁻¹) = ℓ w := by	apply Nat.le_antisymm · rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩ have := cs.length_wordProd_le (List.reverse ω) rwa [wordProd_reverse, length_reverse, hω] at this · rcases cs.exists_reduced_word w⁻¹ with ⟨ω, hω, h'ω⟩ have := cs.length_wordProd_le (List.reverse ω) rwa [wordProd_reverse, l...	[ " ∃ n ω, ω.length = n ∧ cs.wordProd ω = w", " ∃ n ω_1, ω_1.length = n ∧ cs.wordProd ω_1 = cs.wordProd ω", " ∃ ω, ω.length = cs.length w ∧ w = cs.wordProd ω", " ω.length = ω.length ∧ cs.wordProd ω = cs.wordProd ω", " cs.length w = 0 ↔ w = 1", " cs.length w = 0 → w = 1", " w = 1", " cs.wordProd ω = 1", ...	[ " ∃ n ω, ω.length = n ∧ cs.wordProd ω = w", " ∃ n ω_1, ω_1.length = n ∧ cs.wordProd ω_1 = cs.wordProd ω", " ∃ ω, ω.length = cs.length w ∧ w = cs.wordProd ω", " ω.length = ω.length ∧ cs.wordProd ω = cs.wordProd ω", " cs.length w = 0 ↔ w = 1", " cs.length w = 0 → w = 1", " w = 1", " cs.wordProd ω = 1", ...
import Mathlib.Analysis.Convex.Between import Mathlib.Analysis.Convex.Normed import Mathlib.Analysis.Normed.Group.AddTorsor #align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f" variable {R V V' P P' : Type*} open AffineEquiv AffineMap namespace Affine...	Mathlib/Analysis/Convex/Side.lean	109	119	theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P} {f : P →ᵃ[R] P'} (hf : Function.Injective f) : (s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y := by	refine ⟨fun h => ?_, fun h => h.map _⟩ rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩ rw [mem_map] at hfp₁ hfp₂ rcases hfp₁ with ⟨p₁, hp₁, rfl⟩ rcases hfp₂ with ⟨p₂, hp₂, rfl⟩ refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩ simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h exact h	[ " (AffineSubspace.map f s).WSameSide (f x) (f y)", " SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)", " SameRay R (f.linear (x -ᵥ p₁)) (f.linear (y -ᵥ p₂))", " (map f s).WSameSide (f x) (f y) ↔ s.WSameSide x y", " s.WSameSide x y", " SameRay R (x -ᵥ p₁) (y -ᵥ p₂)", " (map f s).SSameSide (f x) (f y) ↔ s.SSameSide...	[ " (AffineSubspace.map f s).WSameSide (f x) (f y)", " SameRay R (f x -ᵥ f p₁) (f y -ᵥ f p₂)", " SameRay R (f.linear (x -ᵥ p₁)) (f.linear (y -ᵥ p₂))", " (map f s).WSameSide (f x) (f y) ↔ s.WSameSide x y", " s.WSameSide x y", " SameRay R (x -ᵥ p₁) (y -ᵥ p₂)", " (map f s).SSameSide (f x) (f y) ↔ s.SSameSide...
import Mathlib.Control.Functor import Mathlib.Tactic.Common #align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a" universe u₀ u₁ u₂ v₀ v₁ v₂ open Function class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) where bimap : ∀ {α α' β β'}, (α → α') → (β → β'...	Mathlib/Control/Bifunctor.lean	86	87	theorem comp_fst {α₀ α₁ α₂ β} (f : α₀ → α₁) (f' : α₁ → α₂) (x : F α₀ β) : fst f' (fst f x) = fst (f' ∘ f) x := by	simp [fst, bimap_bimap]	[ " fst f' (fst f x) = fst (f' ∘ f) x" ]	[]
import Mathlib.Algebra.MvPolynomial.Rename #align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee" namespace MvPolynomial variable {σ : Type} {τ : Type} {υ : Type} {R : Type} [CommSemiring R] noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M...	Mathlib/Algebra/MvPolynomial/Comap.lean	48	50	theorem comap_id_apply (x : σ → R) : comap (AlgHom.id R (MvPolynomial σ R)) x = x := by	funext i simp only [comap, AlgHom.id_apply, id, aeval_X]	[ " comap (AlgHom.id R (MvPolynomial σ R)) x = x", " comap (AlgHom.id R (MvPolynomial σ R)) x i = x i" ]	[]
import Mathlib.Algebra.Quaternion import Mathlib.Analysis.InnerProductSpace.Basic import Mathlib.Analysis.InnerProductSpace.PiL2 import Mathlib.Topology.Algebra.Algebra #align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566" @[inherit_doc] scoped[Quaternion...	Mathlib/Analysis/Quaternion.lean	65	66	theorem normSq_eq_norm_mul_self (a : ℍ) : normSq a = ‖a‖ * ‖a‖ := by	rw [← inner_self, real_inner_self_eq_norm_mul_norm]	[ " (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ", " ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝ", " ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝ", " normSq a = ‖a‖ * ‖a‖" ]	[ " (starRingEnd ℝ) ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ", " ⟪x + y, z⟫_ℝ = ⟪x, z⟫_ℝ + ⟪y, z⟫_ℝ", " ⟪r • x, y⟫_ℝ = (starRingEnd ℝ) r * ⟪x, y⟫_ℝ" ]
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	63	64	theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by	rw [← exp_mul_I, abs_mul_exp_arg_mul_I]	[ " 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...	[ " 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.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	113	116	theorem N₁Γ₀_inv_app_f_f (K : ChainComplex C ℕ) (n : ℕ) : (N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n := by	rw [N₁Γ₀_inv_app] apply id_comp	[ " ∀ (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.CategoryTheory.CofilteredSystem import Mathlib.Combinatorics.SimpleGraph.Connectivity import Mathlib.Data.Finite.Set #align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4" universe u variable {V : Type u} (G : SimpleGraph V...	Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean	44	49	theorem ComponentCompl.supp_injective : Function.Injective (ComponentCompl.supp : G.ComponentCompl K → Set V) := by	refine ConnectedComponent.ind₂ ?_ rintro ⟨v, hv⟩ ⟨w, hw⟩ h simp only [Set.ext_iff, ConnectedComponent.eq, Set.mem_setOf_eq, ComponentCompl.supp] at h ⊢ exact ((h v).mp ⟨hv, Reachable.refl _⟩).choose_spec	[ " Function.Injective supp", " ∀ (v w : ↑Kᶜ),\n supp ((induce Kᶜ G).connectedComponentMk v) = supp ((induce Kᶜ G).connectedComponentMk w) →\n (induce Kᶜ G).connectedComponentMk v = (induce Kᶜ G).connectedComponentMk w", " (induce Kᶜ G).connectedComponentMk ⟨v, hv⟩ = (induce Kᶜ G).connectedComponentMk ⟨w,...	[]
import Mathlib.Algebra.Group.Support import Mathlib.Algebra.Order.Monoid.WithTop import Mathlib.Data.Nat.Cast.Field #align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829" open Function Set section AddMonoidWithOne variable {α M : Type*} [AddMonoidWith...	Mathlib/Algebra/CharZero/Lemmas.lean	88	89	theorem add_self_eq_zero {a : R} : a + a = 0 ↔ a = 0 := by	simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]	[ " 2 ≠ 0", " a + a = 0 ↔ a = 0" ]	[ " 2 ≠ 0" ]
import Mathlib.Data.Fintype.Order import Mathlib.Data.Set.Finite import Mathlib.Order.Category.FinPartOrd import Mathlib.Order.Category.LinOrd import Mathlib.CategoryTheory.Limits.Shapes.Images import Mathlib.CategoryTheory.Limits.Shapes.RegularMono import Mathlib.Data.Set.Subsingleton #align_import order.category.No...	Mathlib/Order/Category/NonemptyFinLinOrd.lean	150	163	theorem mono_iff_injective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) : Mono f ↔ Function.Injective f := by	refine ⟨?_, ConcreteCategory.mono_of_injective f⟩ intro intro a₁ a₂ h let X := NonemptyFinLinOrd.of (ULift (Fin 1)) let g₁ : X ⟶ A := ⟨fun _ => a₁, fun _ _ _ => by rfl⟩ let g₂ : X ⟶ A := ⟨fun _ => a₂, fun _ _ _ => by rfl⟩ change g₁ (ULift.up (0 : Fin 1)) = g₂ (ULift.up (0 : Fin 1)) have eq : g₁ ≫ f = g...	[ " ↑e ≫ ↑e.symm = 𝟙 α", " (↑e ≫ ↑e.symm) x = (𝟙 α) x", " ↑e.symm ≫ ↑e = 𝟙 β", " (↑e.symm ≫ ↑e) x = (𝟙 β) x", " x✝¹ = x✝", " x✝¹ x = x✝ x", " Mono f ↔ Function.Injective ⇑f", " Mono f → Function.Injective ⇑f", " Function.Injective ⇑f", " a₁ = a₂", " (fun x => a₁) x✝² ≤ (fun x => a₁) x✝¹", " ...	[ " ↑e ≫ ↑e.symm = 𝟙 α", " (↑e ≫ ↑e.symm) x = (𝟙 α) x", " ↑e.symm ≫ ↑e = 𝟙 β", " (↑e.symm ≫ ↑e) x = (𝟙 β) x", " x✝¹ = x✝", " x✝¹ x = x✝ x" ]
import Mathlib.Algebra.Algebra.Tower import Mathlib.Algebra.MvPolynomial.Basic #align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496" variable (R A B : Type) {σ : Type} namespace MvPolynomial section CommSemiring variable [CommSemiring R] ...	Mathlib/RingTheory/MvPolynomial/Tower.lean	48	53	theorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) := by	rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ← IsScalarTower.algebraMap_eq] -- Porting note: added simp only [Function.comp]	[ " (aeval (⇑(algebraMap A B) ∘ x)) p = (algebraMap A B) ((aeval x) p)", " (eval₂Hom (algebraMap R B) (⇑(algebraMap A B) ∘ x)) p = (eval₂Hom (algebraMap R B) fun i => (algebraMap A B) (x i)) p" ]	[]
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree import Mathlib.Algebra.Polynomial.EraseLead import Mathlib.Algebra.Polynomial.Eval #align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd" namespace Polynomial open Polynomial Finsupp Finset open...	Mathlib/Algebra/Polynomial/Reverse.lean	82	88	theorem revAt_add {N O n o : ℕ} (hn : n ≤ N) (ho : o ≤ O) : revAt (N + O) (n + o) = revAt N n + revAt O o := by	rcases Nat.le.dest hn with ⟨n', rfl⟩ rcases Nat.le.dest ho with ⟨o', rfl⟩ repeat' rw [revAt_le (le_add_right rfl.le)] rw [add_assoc, add_left_comm n' o, ← add_assoc, revAt_le (le_add_right rfl.le)] repeat' rw [add_tsub_cancel_left]	[ " revAtFun N (revAtFun N i) = i", " (if (if i ≤ N then N - i else i) ≤ N then N - if i ≤ N then N - i else i else if i ≤ N then N - i else i) = i", " N - (N - i) = i", " N - i = i", " False", " N - i ≤ N", " i = i", " Function.Injective (revAtFun N)", " a = b", " (revAt N) i = i", " (revAt (N + ...	[ " revAtFun N (revAtFun N i) = i", " (if (if i ≤ N then N - i else i) ≤ N then N - if i ≤ N then N - i else i else if i ≤ N then N - i else i) = i", " N - (N - i) = i", " N - i = i", " False", " N - i ≤ N", " i = i", " Function.Injective (revAtFun N)", " a = b", " (revAt N) i = i" ]
import Mathlib.Data.Real.Basic import Mathlib.Combinatorics.Pigeonhole import Mathlib.Algebra.Order.EuclideanAbsoluteValue #align_import number_theory.class_number.admissible_absolute_value from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c" local infixl:50 " ≺ " => EuclideanDomain.r na...	Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean	117	123	theorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0) (h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) : ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by	let e := Fintype.equivFin ι obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y) refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩ convert h (e k) <;> simp only [e.symm_apply_apply]	[ " ∃ t, ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε", " ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε", " i₁ = e.symm (e i₁)", " i₀ = e.symm (e i₀)", " ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁,...	[ " ∃ t, ∀ (i₀ i₁ : ι), t i₀ = t i₁ → ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε", " ↑(abv (A i₁ % b - A i₀ % b)) < abv b • ε", " i₁ = e.symm (e i₁)", " i₀ = e.symm (e i₀)", " ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁,...
import Mathlib.Analysis.Calculus.FDeriv.Basic #align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee" open Filter Asymptotics ContinuousLinearMap Set Metric open scoped Classical open Topology NNReal Filter Asymptotics ENNReal noncom...	Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean	99	102	theorem hasFDerivAt_of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivAt f g' x) (H : f'.restrictScalars 𝕜 = g') : HasFDerivAt f f' x := by	rw [← H] at h exact .of_isLittleO h.1	[ " HasFDerivWithinAt f f' s x", " HasFDerivAt f f' x" ]	[ " HasFDerivWithinAt f f' s x" ]
import Mathlib.LinearAlgebra.TensorProduct.Tower import Mathlib.Algebra.DirectSum.Module #align_import linear_algebra.direct_sum.tensor_product from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d" suppress_compilation universe u v₁ v₂ w₁ w₁' w₂ w₂' section Ring namespace TensorProduct ...	Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean	150	153	theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) : TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) = DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by	simp [TensorProduct.directSum]	[ " ((⨁ (i₁ : ι₁), M₁ i₁) ⊗[R] ⨁ (i₂ : ι₂), M₂ i₂) ≃ₗ[S] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " ((⨁ (i₁ : ι₁), M₁ i₁) ⊗[R] ⨁ (i₂ : ι₂), M₂ i₂) →ₗ[S] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " (⨁ (i₁ : ι₁), M₁ i₁) →ₗ[S] (⨁ (i₂ : ι₂), M₂ i₂) →ₗ[R] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " M₁ i₁ →ₗ[S] (⨁ (i₂ : ι₂), M₂...	[ " ((⨁ (i₁ : ι₁), M₁ i₁) ⊗[R] ⨁ (i₂ : ι₂), M₂ i₂) ≃ₗ[S] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " ((⨁ (i₁ : ι₁), M₁ i₁) ⊗[R] ⨁ (i₂ : ι₂), M₂ i₂) →ₗ[S] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " (⨁ (i₁ : ι₁), M₁ i₁) →ₗ[S] (⨁ (i₂ : ι₂), M₂ i₂) →ₗ[R] ⨁ (i : ι₁ × ι₂), M₁ i.1 ⊗[R] M₂ i.2", " M₁ i₁ →ₗ[S] (⨁ (i₂ : ι₂), M₂...
import Mathlib.Analysis.BoxIntegral.Partition.Additive import Mathlib.MeasureTheory.Measure.Lebesgue.Basic #align_import analysis.box_integral.partition.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" open Set noncomputable section open scoped ENNReal Classical BoxIntegral...	Mathlib/Analysis/BoxIntegral/Partition/Measure.lean	74	76	theorem coe_ae_eq_Icc : (I : Set (ι → ℝ)) =ᵐ[volume] Box.Icc I := by	rw [coe_eq_pi] exact Measure.univ_pi_Ioc_ae_eq_Icc	[ " ↑I =ᶠ[ae volume] Box.Icc I", " (univ.pi fun i => Ioc (I.lower i) (I.upper i)) =ᶠ[ae volume] Box.Icc I" ]	[]
import Mathlib.Algebra.CharP.Invertible import Mathlib.Algebra.Order.Invertible import Mathlib.Algebra.Order.Module.OrderedSMul import Mathlib.Algebra.Order.Group.Instances import Mathlib.LinearAlgebra.AffineSpace.Slope import Mathlib.LinearAlgebra.AffineSpace.Midpoint import Mathlib.Tactic.FieldSimp #align_import li...	Mathlib/LinearAlgebra/AffineSpace/Ordered.lean	57	59	theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r := by	simp only [lineMap_apply_module] exact add_lt_add_right (smul_lt_smul_of_pos_left ha (sub_pos.2 hr)) _	[ " (lineMap a b) r ≤ (lineMap a' b) r", " (1 - r) • a + r • b ≤ (1 - r) • a' + r • b", " (lineMap a b) r < (lineMap a' b) r", " (1 - r) • a + r • b < (1 - r) • a' + r • b" ]	[ " (lineMap a b) r ≤ (lineMap a' b) r", " (1 - r) • a + r • b ≤ (1 - r) • a' + r • b" ]
import Mathlib.Algebra.Category.GroupCat.Abelian import Mathlib.CategoryTheory.Limits.Shapes.Images #align_import algebra.category.Group.images from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" open CategoryTheory open CategoryTheory.Limits universe u namespace AddCommGroupCat set...	Mathlib/Algebra/Category/GroupCat/Images.lean	87	91	theorem image.lift_fac (F' : MonoFactorisation f) : image.lift F' ≫ F'.m = image.ι f := by	ext x change (F'.e ≫ F'.m) _ = _ rw [F'.fac, (Classical.indefiniteDescription _ x.2).2] rfl	[ " factorThruImage f ≫ ι f = f", " (factorThruImage f ≫ ι f) x✝ = f x✝", " (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) 0 = 0", " F'.m ((fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) 0) = F'.m 0", " (F'.e ≫ F'.m) ↑(Classical.indefiniteDescription ...	[ " factorThruImage f ≫ ι f = f", " (factorThruImage f ≫ ι f) x✝ = f x✝", " (fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) 0 = 0", " F'.m ((fun x => F'.e ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) ⋯)) 0) = F'.m 0", " (F'.e ≫ F'.m) ↑(Classical.indefiniteDescription ...
import Mathlib.LinearAlgebra.Ray import Mathlib.Analysis.NormedSpace.Real #align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7" open Real variable {E : Type} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type} [NormedAddCommGroup F] [NormedSp...	Mathlib/Analysis/NormedSpace/Ray.lean	59	65	theorem norm_injOn_ray_left (hx : x ≠ 0) : { y \| SameRay ℝ x y }.InjOn norm := by	rintro y hy z hz h rcases hy.exists_nonneg_left hx with ⟨r, hr, rfl⟩ rcases hz.exists_nonneg_left hx with ⟨s, hs, rfl⟩ rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr, norm_of_nonneg hs] at h rw [h]	[ " Set.InjOn Norm.norm {y \| SameRay ℝ x y}", " y = z", " r • x = z", " r • x = s • x" ]	[]
import Mathlib.Order.Interval.Set.OrdConnectedComponent import Mathlib.Topology.Order.Basic #align_import topology.algebra.order.t5 from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514" open Filter Set Function OrderDual Topology Interval variable {X : Type*} [LinearOrder X] [Topological...	Mathlib/Topology/Order/T5.lean	27	30	theorem ordConnectedComponent_mem_nhds : ordConnectedComponent s a ∈ 𝓝 a ↔ s ∈ 𝓝 a := by	refine ⟨fun h => mem_of_superset h ordConnectedComponent_subset, fun h => ?_⟩ rcases exists_Icc_mem_subset_of_mem_nhds h with ⟨b, c, ha, ha', hs⟩ exact mem_of_superset ha' (subset_ordConnectedComponent ha hs)	[ " s.ordConnectedComponent a ∈ 𝓝 a ↔ s ∈ 𝓝 a", " s.ordConnectedComponent a ∈ 𝓝 a" ]	[]
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation import Mathlib.LinearAlgebra.TensorProduct.Opposite import Mathlib.RingTheory.TensorProduct.Basic variable {R A V : Type*} variable [CommRing R] [CommRing A] [AddCommGroup V] variable [Algebra R A] [Mod...	Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean	124	137	theorem toBaseChange_comp_reverseOp (Q : QuadraticForm R V) : (toBaseChange A Q).op.comp reverseOp = ((Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)).toAlgHom.comp <\| (Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))).comp (toBaseChange A...	ext v show op (toBaseChange A Q (reverse (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))) = Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q) (Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q)) (toBaseChange A Q (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))) rw [toBaseChange_ι, re...	[ " { f // ∀ (m : V), f m * f m = (algebraMap R (CliffordAlgebra (QuadraticForm.baseChange A Q))) (Q m) }", " (↑R (ι (QuadraticForm.baseChange A Q)) ∘ₗ (TensorProduct.mk R A V) 1) v *\n (↑R (ι (QuadraticForm.baseChange A Q)) ∘ₗ (TensorProduct.mk R A V) 1) v =\n (algebraMap R (CliffordAlgebra (QuadraticForm....	[ " { f // ∀ (m : V), f m * f m = (algebraMap R (CliffordAlgebra (QuadraticForm.baseChange A Q))) (Q m) }", " (↑R (ι (QuadraticForm.baseChange A Q)) ∘ₗ (TensorProduct.mk R A V) 1) v *\n (↑R (ι (QuadraticForm.baseChange A Q)) ∘ₗ (TensorProduct.mk R A V) 1) v =\n (algebraMap R (CliffordAlgebra (QuadraticForm....
import Mathlib.SetTheory.Game.Basic import Mathlib.Tactic.NthRewrite #align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748" universe u namespace SetTheory open scoped PGame namespace PGame def ImpartialAux : PGame → Prop \| G => (G ≈ -G) ∧ (∀ i...	Mathlib/SetTheory/Game/Impartial.lean	137	142	theorem equiv_or_fuzzy_zero : (G ≈ 0) ∨ G ‖ 0 := by	rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h \| h \| h \| h) · exact ((nonneg G) h).elim · exact Or.inl h · exact ((nonpos G) h).elim · exact Or.inr h	[ " G.ImpartialAux ↔\n G ≈ -G ∧ (∀ (i : G.LeftMoves), (G.moveLeft i).ImpartialAux) ∧ ∀ (j : G.RightMoves), (G.moveRight j).ImpartialAux", " G.Impartial ↔\n G ≈ -G ∧ (∀ (i : G.LeftMoves), (G.moveLeft i).Impartial) ∧ ∀ (j : G.RightMoves), (G.moveRight j).Impartial", " Impartial 0", " 0 ≈ -0 ∧ (∀ (i : LeftMo...	[ " G.ImpartialAux ↔\n G ≈ -G ∧ (∀ (i : G.LeftMoves), (G.moveLeft i).ImpartialAux) ∧ ∀ (j : G.RightMoves), (G.moveRight j).ImpartialAux", " G.Impartial ↔\n G ≈ -G ∧ (∀ (i : G.LeftMoves), (G.moveLeft i).Impartial) ∧ ∀ (j : G.RightMoves), (G.moveRight j).Impartial", " Impartial 0", " 0 ≈ -0 ∧ (∀ (i : LeftMo...
import Mathlib.Data.Int.Bitwise import Mathlib.Data.Int.Order.Lemmas import Mathlib.Data.Set.Function import Mathlib.Order.Interval.Set.Basic #align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f" open Nat namespace Int theorem le_natCast_sub (m n : ℕ) : (m ...	Mathlib/Data/Int/Lemmas.lean	82	86	theorem natAbs_coe_sub_coe_le_of_le {a b n : ℕ} (a_le_n : a ≤ n) (b_le_n : b ≤ n) : natAbs (a - b : ℤ) ≤ n := by	rw [← Nat.cast_le (α := ℤ), natCast_natAbs] exact abs_sub_le_of_nonneg_of_le (ofNat_nonneg a) (ofNat_le.mpr a_le_n) (ofNat_nonneg b) (ofNat_le.mpr b_le_n)	[ " ↑m - ↑n ≤ ↑(m - n)", " 0 ≤ ↑n", " a.natAbs = b.natAbs ↔ a ^ 2 = b ^ 2", " a.natAbs = b.natAbs ↔ a * a = b * b", " a.natAbs < b.natAbs ↔ a ^ 2 < b ^ 2", " a.natAbs < b.natAbs ↔ a * a < b * b", " a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2", " a.natAbs ≤ b.natAbs ↔ a * a ≤ b * b", " a.natAbs = b.natAbs ↔ a ...	[ " ↑m - ↑n ≤ ↑(m - n)", " 0 ≤ ↑n", " a.natAbs = b.natAbs ↔ a ^ 2 = b ^ 2", " a.natAbs = b.natAbs ↔ a * a = b * b", " a.natAbs < b.natAbs ↔ a ^ 2 < b ^ 2", " a.natAbs < b.natAbs ↔ a * a < b * b", " a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2", " a.natAbs ≤ b.natAbs ↔ a * a ≤ b * b", " a.natAbs = b.natAbs ↔ a ...
import Mathlib.Order.Filter.AtTopBot import Mathlib.Order.Filter.Pi #align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1" open Set Function variable {ι α β : Type*} {l : Filter α} namespace Filter def cofinite : Filter α := comk Set.Finite finite_e...	Mathlib/Order/Filter/Cofinite.lean	101	104	theorem le_cofinite_iff_compl_singleton_mem : l ≤ cofinite ↔ ∀ x, {x}ᶜ ∈ l := by	refine ⟨fun h x => h (finite_singleton x).compl_mem_cofinite, fun h s (hs : sᶜ.Finite) => ?_⟩ rw [← compl_compl s, ← biUnion_of_singleton sᶜ, compl_iUnion₂, Filter.biInter_mem hs] exact fun x _ => h x	[ " cofinite = ⊥ ↔ Finite α", " (∃ᶠ (x : α) in cofinite, p x) ↔ {x \| p x}.Infinite", " l ≤ cofinite ↔ ∀ (x : α), {x}ᶜ ∈ l", " s ∈ l", " ∀ i ∈ sᶜ, {i}ᶜ ∈ l" ]	[ " cofinite = ⊥ ↔ Finite α", " (∃ᶠ (x : α) in cofinite, p x) ↔ {x \| p x}.Infinite" ]
import Mathlib.Data.Finset.Lattice import Mathlib.Order.Hom.Basic import Mathlib.Data.Set.Finite import Mathlib.Order.ConditionallyCompleteLattice.Basic #align_import order.partial_sups from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce" variable {α : Type*} section SemilatticeSup var...	Mathlib/Order/PartialSups.lean	97	101	theorem Monotone.partialSups_eq {f : ℕ → α} (hf : Monotone f) : (partialSups f : ℕ → α) = f := by	ext n induction' n with n ih · rfl · rw [partialSups_succ, ih, sup_eq_right.2 (hf (Nat.le_succ _))]	[ " p ((partialSups f) 0) ↔ ∀ k ≤ 0, p (f k)", " p ((partialSups f) (n + 1)) ↔ ∀ k ≤ n + 1, p (f k)", " upperBounds (Set.range ⇑(partialSups f)) = upperBounds (Set.range f)", " a ∈ upperBounds (Set.range ⇑(partialSups f)) ↔ a ∈ upperBounds (Set.range f)", " (∀ (i k : ℕ), k ≤ i → f k ≤ a) ↔ ∀ (i : ℕ), f i ≤ a"...	[ " p ((partialSups f) 0) ↔ ∀ k ≤ 0, p (f k)", " p ((partialSups f) (n + 1)) ↔ ∀ k ≤ n + 1, p (f k)", " upperBounds (Set.range ⇑(partialSups f)) = upperBounds (Set.range f)", " a ∈ upperBounds (Set.range ⇑(partialSups f)) ↔ a ∈ upperBounds (Set.range f)", " (∀ (i k : ℕ), k ≤ i → f k ≤ a) ↔ ∀ (i : ℕ), f i ≤ a"...
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	146	147	theorem degree_eq_iff_natDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) : p.degree = n ↔ p.natDegree = n := by	rw [degree_eq_natDegree hp]; exact WithBot.coe_eq_coe	[ " 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))", " AddMonoidAlgebra.supDegree id p.toFinsupp = p.natDegree", " AddMonoidAlgebra.supDegree id (toFinsupp 0) = natDegree 0", ...
import Mathlib.Algebra.GroupPower.IterateHom import Mathlib.Algebra.Module.Defs import Mathlib.Algebra.Order.Archimedean import Mathlib.Algebra.Order.Group.Instances import Mathlib.GroupTheory.GroupAction.Pi open Function Set structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where protected...	Mathlib/Algebra/AddConstMap/Basic.lean	78	79	theorem map_add_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b] (f : F) (x : G) (n : ℕ) : f (x + n) = f x + n • b := by	simp [← map_add_nsmul]	[ " f (x + n • a) = f x + n • b", " f (x + ↑n) = f x + n • b" ]	[ " f (x + n • a) = f x + n • b" ]
import Mathlib.Analysis.Convex.Between import Mathlib.Analysis.Convex.Jensen import Mathlib.Analysis.Convex.Topology import Mathlib.Analysis.Normed.Group.Pointwise import Mathlib.Analysis.NormedSpace.AddTorsor #align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052...	Mathlib/Analysis/Convex/Normed.lean	92	97	theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s) (hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by	rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩ rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩ use x', hx', y', hy' exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')	[ " ‖a • x‖ + ‖b • y‖ = a * ‖x‖ + b * ‖y‖", " ConvexOn ℝ s fun z' => dist z' z", " Convex ℝ (ball a r)", " Convex ℝ (closedBall a r)", " Convex ℝ (Metric.thickening δ s)", " Convex ℝ (s + ball 0 δ)", " Convex ℝ (Metric.cthickening δ s)", " Convex ℝ (⋂ ε, ⋂ (_ : δ < ε), Metric.thickening ε s)", " Conve...	[ " ‖a • x‖ + ‖b • y‖ = a * ‖x‖ + b * ‖y‖", " ConvexOn ℝ s fun z' => dist z' z", " Convex ℝ (ball a r)", " Convex ℝ (closedBall a r)", " Convex ℝ (Metric.thickening δ s)", " Convex ℝ (s + ball 0 δ)", " Convex ℝ (Metric.cthickening δ s)", " Convex ℝ (⋂ ε, ⋂ (_ : δ < ε), Metric.thickening ε s)", " Conve...
import Mathlib.Data.Countable.Basic import Mathlib.Data.Fin.VecNotation import Mathlib.Order.Disjointed import Mathlib.MeasureTheory.OuterMeasure.Defs #align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55" noncomputable section open Set F...	Mathlib/MeasureTheory/OuterMeasure/Basic.lean	129	129	theorem measure_union_null (hs : μ s = 0) (ht : μ t = 0) : μ (s ∪ t) = 0 := by	simp [*]	[ " μ (⋃ i, s i) ≤ ∑' (i : ι), μ (s i)", " (fun x x_1 => x ≤ x_1) (μ (⨆ i, t i)) (∑' (i : ℕ), μ (t i))", " μ (⋃ i, t i) = μ (⋃ i, disjointed t i)", " ∑' (i : ℕ), μ (disjointed t i) ≤ ∑' (i : ℕ), μ (t i)", " disjointed t a✝ ⊆ t a✝", " μ (⋃ i ∈ I, s i) ≤ ∑' (i : ↑I), μ (s ↑i)", " μ (⋃ x, s ↑x) ≤ ∑' (i : ↑I)...	[ " μ (⋃ i, s i) ≤ ∑' (i : ι), μ (s i)", " (fun x x_1 => x ≤ x_1) (μ (⨆ i, t i)) (∑' (i : ℕ), μ (t i))", " μ (⋃ i, t i) = μ (⋃ i, disjointed t i)", " ∑' (i : ℕ), μ (disjointed t i) ≤ ∑' (i : ℕ), μ (t i)", " disjointed t a✝ ⊆ t a✝", " μ (⋃ i ∈ I, s i) ≤ ∑' (i : ↑I), μ (s ↑i)", " μ (⋃ x, s ↑x) ≤ ∑' (i : ↑I)...
import Mathlib.Algebra.BigOperators.Group.Multiset import Mathlib.Data.PNat.Prime import Mathlib.Data.Nat.Factors import Mathlib.Data.Multiset.Sort #align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d" -- Porting note: `deriving` contained Inhabited, Canonic...	Mathlib/Data/PNat/Factors.lean	121	123	theorem coePNat_prime (v : PrimeMultiset) (p : ℕ+) (h : p ∈ (v : Multiset ℕ+)) : p.Prime := by	rcases Multiset.mem_map.mp h with ⟨⟨_, hp'⟩, ⟨_, h_eq⟩⟩ exact h_eq ▸ hp'	[ " ∀ (a : PrimeMultiset), ⊥ ≤ a", " Repr PrimeMultiset", " Repr (Multiset Nat.Primes)", " p.Prime" ]	[ " ∀ (a : PrimeMultiset), ⊥ ≤ a", " Repr PrimeMultiset", " Repr (Multiset Nat.Primes)", " p.Prime" ]
import Mathlib.RepresentationTheory.FdRep import Mathlib.LinearAlgebra.Trace import Mathlib.RepresentationTheory.Invariants #align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9" noncomputable section universe u open CategoryTheory LinearMap ...	Mathlib/RepresentationTheory/Character.lean	54	55	theorem char_mul_comm (V : FdRep k G) (g : G) (h : G) : V.character (h * g) = V.character (g * h) := by	simp only [trace_mul_comm, character, map_mul]	[ " V.character (h * g) = V.character (g * h)" ]	[]
import Mathlib.Algebra.BigOperators.Finsupp import Mathlib.Algebra.Module.Basic import Mathlib.Algebra.Regular.SMul import Mathlib.Data.Finset.Preimage import Mathlib.Data.Rat.BigOperators import Mathlib.GroupTheory.GroupAction.Hom import Mathlib.Data.Set.Subsingleton #align_import data.finsupp.basic from "leanprover...	Mathlib/Data/Finsupp/Basic.lean	78	80	theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by	cases c exact mk_mem_graph_iff	[ " (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0", " (∃ a_1, f a_1 ≠ 0 ∧ { toFun := fun a => (a, f a), inj' := ⋯ } a_1 = (a, m)) ↔ f a = m ∧ m ≠ 0", " (∃ a_1, f a_1 ≠ 0 ∧ { toFun := fun a => (a, f a), inj' := ⋯ } a_1 = (a, m)) → f a = m ∧ m ≠ 0", " f a = f a ∧ f a ≠ 0", " f a = m ∧ m ≠ 0 → ∃ a_2, f a_2 ≠ 0 ∧ { toFun :=...	[ " (a, m) ∈ f.graph ↔ f a = m ∧ m ≠ 0", " (∃ a_1, f a_1 ≠ 0 ∧ { toFun := fun a => (a, f a), inj' := ⋯ } a_1 = (a, m)) ↔ f a = m ∧ m ≠ 0", " (∃ a_1, f a_1 ≠ 0 ∧ { toFun := fun a => (a, f a), inj' := ⋯ } a_1 = (a, m)) → f a = m ∧ m ≠ 0", " f a = f a ∧ f a ≠ 0", " f a = m ∧ m ≠ 0 → ∃ a_2, f a_2 ≠ 0 ∧ { toFun :=...
import Mathlib.Algebra.MvPolynomial.Rename #align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee" namespace MvPolynomial variable {σ : Type} {τ : Type} {υ : Type} {R : Type} [CommSemiring R] noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M...	Mathlib/Algebra/MvPolynomial/Comap.lean	83	87	theorem comap_eq_id_of_eq_id (f : MvPolynomial σ R →ₐ[R] MvPolynomial σ R) (hf : ∀ φ, f φ = φ) (x : σ → R) : comap f x = x := by	convert comap_id_apply x ext1 φ simp [hf, AlgHom.id_apply]	[ " comap (AlgHom.id R (MvPolynomial σ R)) x = x", " comap (AlgHom.id R (MvPolynomial σ R)) x i = x i", " comap (AlgHom.id R (MvPolynomial σ R)) = id", " comap (AlgHom.id R (MvPolynomial σ R)) x = id x", " comap (g.comp f) x = comap f (comap g x)", " comap (g.comp f) x i = comap f (comap g x) i", " comap ...	[ " comap (AlgHom.id R (MvPolynomial σ R)) x = x", " comap (AlgHom.id R (MvPolynomial σ R)) x i = x i", " comap (AlgHom.id R (MvPolynomial σ R)) = id", " comap (AlgHom.id R (MvPolynomial σ R)) x = id x", " comap (g.comp f) x = comap f (comap g x)", " comap (g.comp f) x i = comap f (comap g x) i", " comap ...
import Mathlib.MeasureTheory.Covering.DensityTheorem #align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655" open Set Filter Metric MeasureTheory TopologicalSpace open scoped NNReal ENNReal Topology variable {α : Type*} [MetricSpace α] [...	Mathlib/MeasureTheory/Covering/LiminfLimsup.lean	41	150	theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α} (hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁) {M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) : (blimsup (fun i => cthickening (r₁ i) (s i)) atTop...	/- Sketch of proof: Assume that `p` is identically true for simplicity. Let `Y₁ i = cthickening (r₁ i) (s i)`, define `Y₂` similarly except using `r₂`, and let `(Z i) = ⋃_{j ≥ i} (Y₂ j)`. Our goal is equivalent to showing that `μ ((limsup Y₁) \ (Z i)) = 0` for all `i`. Assume for contradiction that `μ ((li...	[ " blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p", " blimsup Y₁ atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p", " blimsup Y₁ atTop p ≤ᶠ[ae μ] blimsup Y₂ atTop p", " ∀ (i : ℕ), μ (blimsup Y₁ atTop p \\ Z i) = 0", " μ (bl...	[]
import Mathlib.Order.Antichain import Mathlib.Order.UpperLower.Basic import Mathlib.Order.Interval.Set.Basic import Mathlib.Order.RelIso.Set #align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf" open Function Set variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s...	Mathlib/Order/Minimal.lean	113	115	theorem mem_minimals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] : x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt y x → y ∉ s := by	simp [minimals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]	[ " ∀ ⦃b : α⦄, b ∈ {a} → r a b → r b a", " r b b → r b b", " x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y", " x ∈ s → ((∀ ⦃b : α⦄, b ∈ s → r x b → r b x) ↔ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y)", " r y x", " y = x", " x = y", " x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, rlt y x → y ∉ s" ]	[ " ∀ ⦃b : α⦄, b ∈ {a} → r a b → r b a", " r b b → r b b", " x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y", " x ∈ s → ((∀ ⦃b : α⦄, b ∈ s → r x b → r b x) ↔ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y)", " r y x", " y = x", " x = y" ]
import Mathlib.Algebra.BigOperators.Pi import Mathlib.Algebra.BigOperators.Ring import Mathlib.Algebra.Order.BigOperators.Ring.Finset import Mathlib.Algebra.BigOperators.Fin import Mathlib.Algebra.Group.Submonoid.Membership import Mathlib.Data.Finsupp.Fin import Mathlib.Data.Finsupp.Indicator #align_import algebra.bi...	Mathlib/Algebra/BigOperators/Finsupp.lean	124	127	theorem sum_ite_self_eq_aux [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) : (if a ∈ f.support then f a else 0) = f a := by	simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not] exact fun h ↦ h.symm	[ " f.prod g = ∏ x ∈ s, g x (f x)", " f x = 0", " ∏ x ∈ {a}, h x ((single a b) x) = h a b", " h x✝¹ ((mapRange f hf g) x✝¹) = 1", " (f.prod fun x v => if a = x then b x v else 1) = if a ∈ f.support then b a (f a) else 1", " (∏ a_1 ∈ f.support, if a = a_1 then b a_1 (f a_1) else 1) = if a ∈ f.support then b ...	[ " f.prod g = ∏ x ∈ s, g x (f x)", " f x = 0", " ∏ x ∈ {a}, h x ((single a b) x) = h a b", " h x✝¹ ((mapRange f hf g) x✝¹) = 1", " (f.prod fun x v => if a = x then b x v else 1) = if a ∈ f.support then b a (f a) else 1", " (∏ a_1 ∈ f.support, if a = a_1 then b a_1 (f a_1) else 1) = if a ∈ f.support then b ...
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic import Mathlib.LinearAlgebra.GeneralLinearGroup import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup import Mathlib.Topology.Instances.Matrix import Mathlib.Topology.Algebra.Module.FiniteDimension #align_import number_theory.modular from "leanprover-community/mat...	Mathlib/NumberTheory/Modular.lean	85	89	theorem bottom_row_coprime {R : Type*} [CommRing R] (g : SL(2, R)) : IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) R) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) R) 1 1) := by	use -(↑g : Matrix (Fin 2) (Fin 2) R) 0 1, (↑g : Matrix (Fin 2) (Fin 2) R) 0 0 rw [add_comm, neg_mul, ← sub_eq_add_neg, ← det_fin_two] exact g.det_coe	[ " IsCoprime (↑g 1 0) (↑g 1 1)", " -↑g 0 1 * ↑g 1 0 + ↑g 0 0 * ↑g 1 1 = 1", " (↑g).det = 1" ]	[]
import Mathlib.Analysis.NormedSpace.Basic import Mathlib.Analysis.Normed.Group.Hom import Mathlib.Data.Real.Sqrt import Mathlib.RingTheory.Ideal.QuotientOperations import Mathlib.Topology.MetricSpace.HausdorffDistance #align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008...	Mathlib/Analysis/Normed/Group/Quotient.lean	147	148	theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ := by	rw [← neg_sub, quotient_norm_neg]	[ " ‖x‖ = infDist 0 {m \| ↑m = x}", " ‖↑x‖ = infDist x ↑S", " infDist x (⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m \| ↑m = ↑x}) = infDist x ↑S", " y ∈ ⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m \| ↑m = ↑x} ↔ y ∈ ↑S", " ‖-x‖ = ‖x‖", " sInf (norm '' {m \| ↑m = -x}) = sInf (norm '' {m \| ↑m = x})", " r ∈ norm '' {m \| ↑m...	[ " ‖x‖ = infDist 0 {m \| ↑m = x}", " ‖↑x‖ = infDist x ↑S", " infDist x (⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m \| ↑m = ↑x}) = infDist x ↑S", " y ∈ ⇑(IsometryEquiv.subLeft x).symm ⁻¹' {m \| ↑m = ↑x} ↔ y ∈ ↑S", " ‖-x‖ = ‖x‖", " sInf (norm '' {m \| ↑m = -x}) = sInf (norm '' {m \| ↑m = x})", " r ∈ norm '' {m \| ↑m...
import Mathlib.Algebra.CharP.Defs import Mathlib.RingTheory.Multiplicity import Mathlib.RingTheory.PowerSeries.Basic #align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60" noncomputable section open Polynomial open Finset (antidiagonal mem_anti...	Mathlib/RingTheory/PowerSeries/Order.lean	99	101	theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by	contrapose! h exact order_le _ h	[ " (∃ n, (coeff R n) φ ≠ 0) ↔ φ ≠ 0", " (¬∃ n, (coeff R n) φ ≠ 0) ↔ ¬φ ≠ 0", " (∀ (n : ℕ), (coeff R n) φ = 0) ↔ φ = 0", " φ.order.Dom ↔ φ ≠ 0", " (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom ↔ φ ≠ 0", " (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom → φ ≠ 0", " ⊤.Dom → φ ≠ 0", " (↑(Nat.find ⋯)).Dom → φ ≠...	[ " (∃ n, (coeff R n) φ ≠ 0) ↔ φ ≠ 0", " (¬∃ n, (coeff R n) φ ≠ 0) ↔ ¬φ ≠ 0", " (∀ (n : ℕ), (coeff R n) φ = 0) ↔ φ = 0", " φ.order.Dom ↔ φ ≠ 0", " (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom ↔ φ ≠ 0", " (if h : φ = 0 then ⊤ else ↑(Nat.find ⋯)).Dom → φ ≠ 0", " ⊤.Dom → φ ≠ 0", " (↑(Nat.find ⋯)).Dom → φ ≠...
import Mathlib.MeasureTheory.Constructions.Prod.Basic import Mathlib.MeasureTheory.Group.Measure #align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844" noncomputable section open Set hiding prod_eq open Function MeasureTheory open Filter hiding ma...	Mathlib/MeasureTheory/Group/Prod.lean	108	116	theorem measurable_measure_mul_right (hs : MeasurableSet s) : Measurable fun x => μ ((fun y => y * x) ⁻¹' s) := by	suffices Measurable fun y => μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s)) by convert this using 1; ext1 x; congr 1 with y : 1; simp apply measurable_measure_prod_mk_right apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs) infer_instance...	[ " Measurable fun x => μ ((fun y => y * x) ⁻¹' s)", " (fun x => μ ((fun y => y * x) ⁻¹' s)) = fun y => μ ((fun x => (x, y)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s))", " μ ((fun y => y * x) ⁻¹' s) = μ ((fun x_1 => (x_1, x)) ⁻¹' ((fun z => (1, z.1 * z.2)) ⁻¹' univ ×ˢ s))", " y ∈ (fun y => y * x) ⁻¹' s ↔ y ...	[]
import Mathlib.Data.Finsupp.ToDFinsupp import Mathlib.LinearAlgebra.Finsupp import Mathlib.LinearAlgebra.LinearIndependent #align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64" variable {ι : Type} {R : Type} {S : Type} {M : ι → Type} {N : Type*} n...	Mathlib/LinearAlgebra/DFinsupp.lean	206	209	theorem mapRange.linearMap_id : (mapRange.linearMap fun i => (LinearMap.id : β₂ i →ₗ[R] _)) = LinearMap.id := by	ext simp [linearMap]	[ " mapRange f hf (r • g) = r • mapRange f hf g", " (mapRange f hf (r • g)) i✝ = (r • mapRange f hf g) i✝", " (linearMap fun i => LinearMap.id) = LinearMap.id", " (((linearMap fun i => LinearMap.id) ∘ₗ lsingle i✝¹) x✝) i✝ = ((LinearMap.id ∘ₗ lsingle i✝¹) x✝) i✝" ]	[ " mapRange f hf (r • g) = r • mapRange f hf g", " (mapRange f hf (r • g)) i✝ = (r • mapRange f hf g) i✝" ]
import Mathlib.Topology.Sets.Closeds #align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988" variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] namespace TopologicalSpace @[mk_iff] class NoetherianSpace : Prop where wellFounded_open...	Mathlib/Topology/NoetherianSpace.lean	53	56	theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by	rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact, CompleteLattice.isSupFiniteCompact_iff_all_elements_compact] exact forall_congr' Opens.isCompactElement_iff	[ " NoetherianSpace α ↔ ∀ (s : Opens α), IsCompact ↑s", " (∀ (k : Opens α), CompleteLattice.IsCompactElement k) ↔ ∀ (s : Opens α), IsCompact ↑s" ]	[]
import Mathlib.CategoryTheory.Subobject.Lattice #align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d" universe v u noncomputable section open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite var...	Mathlib/CategoryTheory/Subobject/Limits.lean	98	100	theorem kernelSubobject_arrow : (kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow := by	simp [kernelSubobjectIso]	[ " (kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow" ]	[]
import Mathlib.Data.Set.Pairwise.Basic import Mathlib.Data.Set.Lattice import Mathlib.Data.SetLike.Basic #align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0" open scoped Classical open Set variable {α β : Type*} section Chain variable (r : α → α → Prop) ...	Mathlib/Order/Chain.lean	107	110	theorem Monotone.isChain_range [LinearOrder α] [Preorder β] {f : α → β} (hf : Monotone f) : IsChain (· ≤ ·) (range f) := by	rw [← image_univ] exact (isChain_of_trichotomous _).image (· ≤ ·) _ _ hf	[ " IsChain r univ ↔ IsTrichotomous α r", " r a b ∨ a = b ∨ r b a", " ¬a = b → r a b ∨ r b a", " IsChain (fun x x_1 => x ≤ x_1) (range f)", " IsChain (fun x x_1 => x ≤ x_1) (f '' univ)" ]	[ " IsChain r univ ↔ IsTrichotomous α r", " r a b ∨ a = b ∨ r b a", " ¬a = b → r a b ∨ r b a" ]
import Mathlib.AlgebraicTopology.DoldKan.Normalized #align_import algebraic_topology.dold_kan.homotopy_equivalence from "leanprover-community/mathlib"@"f951e201d416fb50cc7826171d80aa510ec20747" open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditive Simplicial DoldKan nonco...	Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean	52	58	theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) : ((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) = (homotopyPToId X q).hom n (n + 1) := by	simp only [homotopyHσToZero, AlternatingFaceMapComplex.obj_X, Nat.add_eq, Homotopy.trans_hom, Homotopy.ofEq_hom, Pi.zero_apply, Homotopy.add_hom, Homotopy.compLeft_hom, add_zero, Homotopy.nullHomotopy'_hom, ComplexShape.down_Rel, hσ'_eq_zero hqn (c_mk (n + 1) n rfl), dite_eq_ite, ite_self, comp_zero, zer...	[ " Homotopy (P (q + 1)) (𝟙 K[X])", " P (q + 1) = P q + P q ≫ Hσ q", " 𝟙 K[X] + P q ≫ 0 = 𝟙 K[X]", " (homotopyPToId X (q + 1)).hom n (n + 1) = (homotopyPToId X q).hom n (n + 1)" ]	[ " Homotopy (P (q + 1)) (𝟙 K[X])", " P (q + 1) = P q + P q ≫ Hσ q", " 𝟙 K[X] + P q ≫ 0 = 𝟙 K[X]" ]
import Mathlib.Algebra.DirectSum.Module import Mathlib.Algebra.Module.Submodule.Basic #align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441" variable {ι R M σ : Type*} open DirectSum namespace DirectSum section AddCommMonoid variable [Deci...	Mathlib/Algebra/DirectSum/Decomposition.lean	145	147	theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) : i = j := by	contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]	[ " x = y", " { decompose' := x, left_inv := xl, right_inv := xr } = y", " { decompose' := x, left_inv := xl, right_inv := xr } = { decompose' := y, left_inv := yl, right_inv := yr }", " ∀ (m : M), p m", " (decompose ℳ) ↑x = (of (fun i => ↥(ℳ i)) i) x", " ↑(((decompose ℳ) x) i) = x", " ↑(((decompose ℳ) x)...	[ " x = y", " { decompose' := x, left_inv := xl, right_inv := xr } = y", " { decompose' := x, left_inv := xl, right_inv := xr } = { decompose' := y, left_inv := yl, right_inv := yr }", " ∀ (m : M), p m", " (decompose ℳ) ↑x = (of (fun i => ↥(ℳ i)) i) x", " ↑(((decompose ℳ) x) i) = x", " ↑(((decompose ℳ) x)...
import Mathlib.Topology.Sheaves.PUnit import Mathlib.Topology.Sheaves.Stalks import Mathlib.Topology.Sheaves.Functors #align_import topology.sheaves.skyscraper from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" noncomputable section open TopologicalSpace TopCat CategoryTheory CategoryT...	Mathlib/Topology/Sheaves/Skyscraper.lean	68	74	theorem skyscraperPresheaf_eq_pushforward [hd : ∀ U : Opens (TopCat.of PUnit.{u + 1}), Decidable (PUnit.unit ∈ U)] : skyscraperPresheaf p₀ A = ContinuousMap.const (TopCat.of PUnit) p₀ _* skyscraperPresheaf (X := TopCat.of PUnit) PUnit.unit A := by	convert_to @skyscraperPresheaf X p₀ (fun U => hd <\| (Opens.map <\| ContinuousMap.const _ p₀).obj U) C _ _ A = _ <;> congr	[ " (fun U => if p₀ ∈ U.unop then A else ⊤_ C) U = (fun U => if p₀ ∈ U.unop then A else ⊤_ C) V", " (if p₀ ∈ U.unop then A else ⊤_ C) = if p₀ ∈ V.unop then A else ⊤_ C", " { obj := fun U => if p₀ ∈ U.unop then A else ⊤_ C,\n map := fun {U V} i =>\n if h : p₀ ∈ V.unop then eqToHom ⋯\n ...	[ " (fun U => if p₀ ∈ U.unop then A else ⊤_ C) U = (fun U => if p₀ ∈ U.unop then A else ⊤_ C) V", " (if p₀ ∈ U.unop then A else ⊤_ C) = if p₀ ∈ V.unop then A else ⊤_ C", " { obj := fun U => if p₀ ∈ U.unop then A else ⊤_ C,\n map := fun {U V} i =>\n if h : p₀ ∈ V.unop then eqToHom ⋯\n ...
import Mathlib.MeasureTheory.Group.Measure assert_not_exists NormedSpace namespace MeasureTheory open Measure TopologicalSpace open scoped ENNReal variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G} section MeasurableMul variable [Group G] [MeasurableMul G] @[to_additive "Translating a fu...	Mathlib/MeasureTheory/Group/LIntegral.lean	46	49	theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) : (∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ := by	convert (lintegral_map_equiv f <\| MeasurableEquiv.mulRight g).symm using 1 simp [map_mul_right_eq_self μ g]	[ " ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ", " μ = map (⇑(MeasurableEquiv.mulLeft g)) μ", " ∫⁻ (x : G), f (x * g) ∂μ = ∫⁻ (x : G), f x ∂μ", " ∫⁻ (x : G), f x ∂μ = ∫⁻ (a : G), f a ∂map (⇑(MeasurableEquiv.mulRight g)) μ" ]	[ " ∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ", " μ = map (⇑(MeasurableEquiv.mulLeft g)) μ" ]
import Mathlib.Topology.Instances.ENNReal import Mathlib.MeasureTheory.Measure.Dirac #align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d" noncomputable section variable {α β γ : Type*} open scoped Classical open NNReal ENNReal M...	Mathlib/Probability/ProbabilityMassFunction/Basic.lean	136	138	theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by	refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p) split_ifs with h <;> simp only [h, zero_le', le_rfl]	[ " p a = 0 ↔ a ∉ p.support", " p a = 1 ↔ p.support = {a}", " False", " 1 < ∑' (a : α), p a", " (p a + ∑' (b : α), if b = a then 0 else p b) = (if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b", " ((if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b) =\n (∑' (b : α), if b = ...	[ " p a = 0 ↔ a ∉ p.support", " p a = 1 ↔ p.support = {a}", " False", " 1 < ∑' (a : α), p a", " (p a + ∑' (b : α), if b = a then 0 else p b) = (if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b", " ((if a = a then p a else 0) + ∑' (b : α), if b = a then 0 else p b) =\n (∑' (b : α), if b = ...
import Mathlib.Data.List.Nodup #align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e" variable {α : Type*} namespace List inductive Duplicate (x : α) : List α → Prop \| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l) \| cons_duplicate {y : α} {l ...	Mathlib/Data/List/Duplicate.lean	98	99	theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by	simpa [duplicate_cons_iff, hx.symm] using h	[ " x ∈ l", " x ∈ x :: l'", " x ∈ y :: l'", " l ≠ [y]", " x :: l' ≠ [y]", " z :: l' ≠ [y]", " x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l", " y = x ∧ x ∈ l ∨ x ∈+ l", " x = x ∧ x ∈ l ∨ x ∈+ l", " x ∈+ y :: l", " x ∈+ x :: l", " x ∈+ l" ]	[ " x ∈ l", " x ∈ x :: l'", " x ∈ y :: l'", " l ≠ [y]", " x :: l' ≠ [y]", " z :: l' ≠ [y]", " x ∈+ y :: l ↔ y = x ∧ x ∈ l ∨ x ∈+ l", " y = x ∧ x ∈ l ∨ x ∈+ l", " x = x ∧ x ∈ l ∨ x ∈+ l", " x ∈+ y :: l", " x ∈+ x :: l" ]
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	94	97	theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot) (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by	have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg simpa [(· ∘ ·)] 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.Algebra.BigOperators.Ring.List import Mathlib.Data.Nat.ModEq import Mathlib.Data.Nat.GCD.BigOperators namespace Nat variable {ι : Type*} lemma modEq_list_prod_iff {a b} {l : List ℕ} (co : l.Pairwise Coprime) : a ≡ b [MOD l.prod] ↔ ∀ i, a ≡ b [MOD l.get i] := by induction' l with m l ih · si...	Mathlib/Data/Nat/ChineseRemainder.lean	93	105	theorem chineseRemainderOfList_modEq_unique (l : List ι) (co : l.Pairwise (Coprime on s)) {z} (hz : ∀ i ∈ l, z ≡ a i [MOD s i]) : z ≡ chineseRemainderOfList a s l co [MOD (l.map s).prod] := by	induction' l with i l ih · simp [modEq_one] · simp only [List.map_cons, List.prod_cons, chineseRemainderOfList] have : Coprime (s i) (l.map s).prod := by simp only [coprime_list_prod_right_iff, List.mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] intro j hj exact (L...	[ " a ≡ b [MOD l.prod] ↔ ∀ (i : Fin l.length), a ≡ b [MOD l.get i]", " a ≡ b [MOD [].prod] ↔ ∀ (i : Fin [].length), a ≡ b [MOD [].get i]", " a ≡ b [MOD (m :: l).prod] ↔ ∀ (i : Fin (m :: l).length), a ≡ b [MOD (m :: l).get i]", " (a ≡ b [MOD m] ∧ ∀ (i : Fin l.length), a ≡ b [MOD l.get i]) ↔ ∀ (i : Fin l.length.s...	[ " a ≡ b [MOD l.prod] ↔ ∀ (i : Fin l.length), a ≡ b [MOD l.get i]", " a ≡ b [MOD [].prod] ↔ ∀ (i : Fin [].length), a ≡ b [MOD [].get i]", " a ≡ b [MOD (m :: l).prod] ↔ ∀ (i : Fin (m :: l).length), a ≡ b [MOD (m :: l).get i]", " (a ≡ b [MOD m] ∧ ∀ (i : Fin l.length), a ≡ b [MOD l.get i]) ↔ ∀ (i : Fin l.length.s...
import Mathlib.Algebra.Group.Equiv.TypeTags import Mathlib.GroupTheory.FreeAbelianGroup import Mathlib.GroupTheory.FreeGroup.IsFreeGroup import Mathlib.LinearAlgebra.Dimension.StrongRankCondition #align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600...	Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean	54	59	theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup : toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by	ext x y; simp only [AddMonoidHom.id_comp] rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom] simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]	[ " toFreeAbelianGroup.comp (singleAddHom x) = (smulAddHom ℤ (FreeAbelianGroup X)).flip (of x)", " (toFreeAbelianGroup.comp (singleAddHom x)) 1 = ((smulAddHom ℤ (FreeAbelianGroup X)).flip (of x)) 1", " toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ)", " (((toFinsupp.comp toFreeAbelianGroup).comp (s...	[ " toFreeAbelianGroup.comp (singleAddHom x) = (smulAddHom ℤ (FreeAbelianGroup X)).flip (of x)", " (toFreeAbelianGroup.comp (singleAddHom x)) 1 = ((smulAddHom ℤ (FreeAbelianGroup X)).flip (of x)) 1" ]
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax import Mathlib.Algebra.Order.Monoid.WithTop import Mathlib.Data.Finset.Image import Mathlib.Data.Multiset.Fold #align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" -- TODO: -- assert_not_exists OrderedComm...	Mathlib/Data/Finset/Fold.lean	50	52	theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f := by	dsimp only [fold] rw [cons_val, Multiset.map_cons, fold_cons_left]	[ " fold op b f (cons a s h) = op (f a) (fold op b f s)", " Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))" ]	[]
import Mathlib.Algebra.GradedMonoid import Mathlib.Algebra.Order.Monoid.Canonical.Defs import Mathlib.Algebra.MvPolynomial.Basic #align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4" noncomputable section open Set Function Fins...	Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean	168	173	theorem weightedHomogeneousSubmodule_eq_finsupp_supported (w : σ → M) (m : M) : weightedHomogeneousSubmodule R w m = Finsupp.supported R R { d \| weightedDegree w d = m } := by	ext x rw [mem_supported, Set.subset_def] simp only [Finsupp.mem_support_iff, mem_coe] rfl	[ " (weightedDegree w) f = f.sum fun i c => c • w i", " weightedTotalDegree' w p = ⊥ ↔ p = 0", " (∀ (s : σ →₀ ℕ), coeff s p ≠ 0 → False) ↔ ∀ (d : σ →₀ ℕ), coeff d p = 0", " weightedTotalDegree' w 0 = ⊥", " (weightedDegree w) c = m", " coeff c a ≠ 0 ∨ coeff c b ≠ 0", " coeff c a + coeff c b = 0", " weigh...	[ " (weightedDegree w) f = f.sum fun i c => c • w i", " weightedTotalDegree' w p = ⊥ ↔ p = 0", " (∀ (s : σ →₀ ℕ), coeff s p ≠ 0 → False) ↔ ∀ (d : σ →₀ ℕ), coeff d p = 0", " weightedTotalDegree' w 0 = ⊥", " (weightedDegree w) c = m", " coeff c a ≠ 0 ∨ coeff c b ≠ 0", " coeff c a + coeff c b = 0" ]
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity #align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3" section Jacobi open Nat ZMod -- Since we need the fact that the factors are prime, we use `List.pmap`. def ...	Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean	331	337	theorem value_at (a : ℤ) {R : Type} [CommSemiring R] (χ : R → ℤ) (hp : ∀ (p : ℕ) (pp : p.Prime), p ≠ 2 → @legendreSym p ⟨pp⟩ a = χ p) {b : ℕ} (hb : Odd b) : J(a \| b) = χ b := by	conv_rhs => rw [← prod_factors hb.pos.ne', cast_list_prod, map_list_prod χ] rw [jacobiSym, List.map_map, ← List.pmap_eq_map Nat.Prime _ _ fun _ => prime_of_mem_factors] congr 1; apply List.pmap_congr exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <\| dvd_of_mem_factors h)	[ " J(a \| b) = χ ↑b", "a : ℤ\nR : Type u_1\ninst✝ : CommSemiring R\nχ : R →* ℤ\nhp : ∀ (p : ℕ) (pp : p.Prime), p ≠ 2 → legendreSym p a = χ ↑p\nb : ℕ\nhb : Odd b\n\| χ ↑b", " J(a \| b) = (List.map (⇑χ) (List.map Nat.cast b.factors)).prod", " (List.pmap (fun p pp => legendreSym p a) b.factors ⋯).prod =\n (List.p...	[]
import Mathlib.LinearAlgebra.Dimension.Free import Mathlib.Algebra.Module.Torsion #align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5" noncomputable section universe u v v' u₁' w w' variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}...	Mathlib/LinearAlgebra/Dimension/Constructions.lean	538	541	theorem subalgebra_top_rank_eq_submodule_top_rank : Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by	rw [← Algebra.top_toSubmodule] rfl	[ " Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} (Module.rank R M)", " Module.rank R (ι →₀ M) = #ι * Module.rank R M", " Module.rank R (ι →₀ R) = lift.{u, w} #ι", " Module.rank R (ι →₀ R) = #ι", " Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)", " Module.rank R (Matrix m n R) = l...	[ " Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} (Module.rank R M)", " Module.rank R (ι →₀ M) = #ι * Module.rank R M", " Module.rank R (ι →₀ R) = lift.{u, w} #ι", " Module.rank R (ι →₀ R) = #ι", " Module.rank R (⨁ (i : ι), M i) = sum fun i => Module.rank R (M i)", " Module.rank R (Matrix m n R) = l...
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	137	143	theorem naturality {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G] [CommApplicative H] (eta : ApplicativeTransformation G H) {α β : Type _} (f : α → G β) (x : Multiset α) : eta (traverse f x) = traverse (@eta _ ∘ f) x := by	refine Quotient.inductionOn x ?_ intro simp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply, ApplicativeTransformation.preserves_map, LawfulTraversable.naturality]	[ " ∀ {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.Analysis.NormedSpace.IndicatorFunction import Mathlib.MeasureTheory.Function.EssSup import Mathlib.MeasureTheory.Function.AEEqFun import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic #align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27...	Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean	117	120	theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by	rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one] exact (ne_of_lt hq0_lt).symm	[ " snorm f p μ = snorm' f p.toReal μ", " snorm f p μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ (1 / p.toReal)", " snorm f 1 μ = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ", " snorm f ⊤ μ = snormEssSup f μ", " ∫⁻ (a : α), ↑‖f a‖₊ ^ q ∂μ = snorm' f q μ ^ q", " q ≠ 0" ]	[ " snorm f p μ = snorm' f p.toReal μ", " snorm f p μ = (∫⁻ (x : α), ↑‖f x‖₊ ^ p.toReal ∂μ) ^ (1 / p.toReal)", " snorm f 1 μ = ∫⁻ (x : α), ↑‖f x‖₊ ∂μ", " snorm f ⊤ μ = snormEssSup f μ" ]
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	53	62	theorem mem_splitCenterBox {s : Set ι} {y : ι → ℝ} : y ∈ I.splitCenterBox s ↔ y ∈ I ∧ ∀ i, (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s := by	simp only [splitCenterBox, mem_def, ← forall_and] refine forall_congr' fun i ↦ ?_ dsimp only [Set.piecewise] split_ifs with hs <;> simp only [hs, iff_true_iff, iff_false_iff, not_lt] exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩, fun H ↦ ⟨H.2, H.1.2⟩⟩, ⟨fun H...	[ " 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.CategoryTheory.Preadditive.ProjectiveResolution import Mathlib.Algebra.Homology.HomotopyCategory import Mathlib.Tactic.SuppressCompilation suppress_compilation noncomputable section universe v u namespace CategoryTheory variable {C : Type u} [Category.{v} C] open Category Limits Projective set_...	Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean	99	102	theorem lift_commutes {Y Z : C} (f : Y ⟶ Z) (P : ProjectiveResolution Y) (Q : ProjectiveResolution Z) : lift f P Q ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f := by	ext simp [lift, liftFZero, liftFOne]	[ " (P.complex.d 1 0 ≫ liftFZero f P Q) ≫ (ShortComplex.mk (Q.complex.d 1 0) (Q.π.f 0) ⋯).g = 0", " liftFOne f P Q ≫ Q.complex.d 1 0 = P.complex.d 1 0 ≫ liftFZero f P Q", " (P.complex.d (n + 2) (n + 1) ≫ g') ≫ (ShortComplex.mk (Q.complex.d (n + 2) (n + 1)) (Q.complex.d (n + 1) n) ⋯).g = 0", " lift f P Q ≫ Q.π =...	[ " (P.complex.d 1 0 ≫ liftFZero f P Q) ≫ (ShortComplex.mk (Q.complex.d 1 0) (Q.π.f 0) ⋯).g = 0", " liftFOne f P Q ≫ Q.complex.d 1 0 = P.complex.d 1 0 ≫ liftFZero f P Q", " (P.complex.d (n + 2) (n + 1) ≫ g') ≫ (ShortComplex.mk (Q.complex.d (n + 2) (n + 1)) (Q.complex.d (n + 1) n) ⋯).g = 0" ]
import Mathlib.Algebra.Polynomial.AlgebraMap import Mathlib.Algebra.Polynomial.BigOperators import Mathlib.Algebra.Polynomial.Degree.Lemmas import Mathlib.Algebra.Polynomial.Div #align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8" noncomputable ...	Mathlib/Algebra/Polynomial/RingDivision.lean	401	407	theorem nmem_nonZeroDivisors_iff {P : R[X]} : P ∉ R[X]⁰ ↔ ∃ a : R, a ≠ 0 ∧ a • P = 0 := by	refine ⟨fun hP ↦ ?_, fun ⟨a, ha, h⟩ h1 ↦ ha <\| C_eq_zero.1 <\| (h1 _) <\| smul_eq_C_mul a ▸ h⟩ by_contra! h obtain ⟨Q, hQ⟩ := _root_.nmem_nonZeroDivisors_iff.1 hP refine hQ.2 (eq_zero_of_mul_eq_zero_of_smul P (fun a ha ↦ ?_) Q (mul_comm P _ ▸ hQ.1)) contrapose! ha exact h a ha	[ " IsUnit a", " IsUnit (C (a.coeff 0))", " ∀ (Q : R[X]), P * Q = 0 → Q = 0", " Q = 0", " Q.leadingCoeff = 0", " Q.leadingCoeff • P = 0", " ∀ (i : ℕ), P.coeff i • Q = 0", " ∃ n, ∀ m > n, P.coeff m • Q = 0", " ∀ m > P.natDegree, P.coeff m • Q = 0", " P.coeff i • Q = 0", " ∀ (n : ℕ), (∀ m > n, P.coe...	[ " IsUnit a", " IsUnit (C (a.coeff 0))", " ∀ (Q : R[X]), P * Q = 0 → Q = 0", " Q = 0", " Q.leadingCoeff = 0", " Q.leadingCoeff • P = 0", " ∀ (i : ℕ), P.coeff i • Q = 0", " ∃ n, ∀ m > n, P.coeff m • Q = 0", " ∀ m > P.natDegree, P.coeff m • Q = 0", " P.coeff i • Q = 0", " ∀ (n : ℕ), (∀ m > n, P.coe...
import Mathlib.Order.Filter.AtTopBot #align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1" variable {α β M E : Type*} open Set Filter @[to_additive]	Mathlib/Order/Filter/IndicatorFunction.lean	63	66	theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α) (hs : Monotone s) (f : α → β) (a : α) : (fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a := by	classical exact hs.piecewise_eventually_eq_iUnion f 1 a	[ " (fun i => (s i).mulIndicator f a) =ᶠ[atTop] fun x => (⋃ i, s i).mulIndicator f a" ]	[]
import Mathlib.Topology.Constructions #align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494" open Set Filter Function Topology Filter variable {α : Type} {β : Type} {γ : Type} {δ : Type} variable [TopologicalSpace α] @[simp] theorem nhds_bind_nhdsW...	Mathlib/Topology/ContinuousOn.lean	75	76	theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by	rw [nhdsWithin, principal_univ, inf_top_eq]	[ " (∃ᶠ (x : α) in 𝓝 z, x ∈ s ∧ p x) ↔ ∃ᶠ (x : α) in 𝓝 z, p x ∧ x ∈ s", " z ∈ closure (s \\ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s", " (∀ᶠ (y : α) in 𝓝[s] a, ∀ᶠ (x : α) in 𝓝[s] y, p x) ↔ ∀ᶠ (x : α) in 𝓝[s] a, p x", " ∀ᶠ (x : α) in 𝓝[s] a, p x", " ∀ᶠ (x : α) in 𝓝 a, x ∈ s → p x", " 𝓝[univ] a = 𝓝 a" ]	[ " (∃ᶠ (x : α) in 𝓝 z, x ∈ s ∧ p x) ↔ ∃ᶠ (x : α) in 𝓝 z, p x ∧ x ∈ s", " z ∈ closure (s \\ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s", " (∀ᶠ (y : α) in 𝓝[s] a, ∀ᶠ (x : α) in 𝓝[s] y, p x) ↔ ∀ᶠ (x : α) in 𝓝[s] a, p x", " ∀ᶠ (x : α) in 𝓝[s] a, p x", " ∀ᶠ (x : α) in 𝓝 a, x ∈ s → p x" ]
import Mathlib.Order.Filter.Basic import Mathlib.Order.Filter.CountableInter import Mathlib.SetTheory.Cardinal.Ordinal import Mathlib.SetTheory.Cardinal.Cofinality open Set Filter Cardinal universe u variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}} class CardinalInterFilter (l : Filter α) (c : Cardinal.{...	Mathlib/Order/Filter/CardinalInter.lean	102	105	theorem eventually_cardinal_forall {p : α → ι → Prop} (hic : #ι < c) : (∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by	simp only [Filter.Eventually, setOf_forall] exact cardinal_iInter_mem hic	[ " ∀ (S : Set (Set α)), #↑S < ℵ₀ → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l", " ⋂ i, s i ∈ l ↔ ∀ (i : ι), s i ∈ l", " (⋂₀ range fun i => s i) ∈ l ↔ ∀ (i : ι), s i ∈ l", " (∀ s_1 ∈ range fun i => s i, s_1 ∈ l) ↔ ∀ (i : ι), s i ∈ l", " ⋂ i, ⋂ (hi : i ∈ S), s i hi ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l", " ⋂ x, s ↑x ⋯ ∈...	[ " ∀ (S : Set (Set α)), #↑S < ℵ₀ → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l", " ⋂ i, s i ∈ l ↔ ∀ (i : ι), s i ∈ l", " (⋂₀ range fun i => s i) ∈ l ↔ ∀ (i : ι), s i ∈ l", " (∀ s_1 ∈ range fun i => s i, s_1 ∈ l) ↔ ∀ (i : ι), s i ∈ l", " ⋂ i, ⋂ (hi : i ∈ S), s i hi ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), s i hi ∈ l", " ⋂ x, s ↑x ⋯ ∈...
import Mathlib.SetTheory.Cardinal.Ordinal #align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925" namespace Cardinal universe u v open Cardinal def continuum : Cardinal.{u} := 2 ^ ℵ₀ #align cardinal.continuum Cardinal.continuum scoped notat...	Mathlib/SetTheory/Cardinal/Continuum.lean	52	54	theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 := by	-- Porting note: added explicit universes rw [← lift_continuum.{u,v}, lift_le]	[ " lift.{v, u_1} 𝔠 = 𝔠", " 𝔠 ≤ lift.{v, u} c ↔ 𝔠 ≤ c", " lift.{v, u} c ≤ 𝔠 ↔ c ≤ 𝔠" ]	[ " lift.{v, u_1} 𝔠 = 𝔠", " 𝔠 ≤ lift.{v, u} c ↔ 𝔠 ≤ c" ]
import Mathlib.CategoryTheory.Adjunction.Whiskering import Mathlib.CategoryTheory.Sites.PreservesSheafification #align_import category_theory.sites.adjunction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a" namespace CategoryTheory open GrothendieckTopology CategoryTheory Limits Op...	Mathlib/CategoryTheory/Sites/Adjunction.lean	148	160	theorem adjunctionToTypes_counit_app_val {G : Type max v u ⥤ D} (adj : G ⊣ forget D) (X : Sheaf J D) : ((adjunctionToTypes J adj).counit.app X).val = sheafifyLift J ((Functor.associator _ _ _).hom ≫ (adj.whiskerRight _).counit.app _) X.2 := by	apply sheafifyLift_unique dsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app, instCategorySheaf_comp_val, instCategorySheaf_id_val] rw [adjunction_counit_app_val] erw [Category.id_comp, sheafifyMap_sheafifyLift, toSheafify_sheafifyLift] ext dsimp [sheafEquivSheafOfTypes, Equivalence.symm...	[ " Function.LeftInverse\n (fun γ =>\n { val := sheafifyLift J ((A.homEquiv ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map γ)) ⋯ })\n fun η => { val := (A.homEquiv X.val Y.val) (toSheafify J (((whiskeringRight Cᵒᵖ E D).obj G).obj X.val) ≫ η.val) }", " (fun γ =>\n {\n ...	[ " Function.LeftInverse\n (fun γ =>\n { val := sheafifyLift J ((A.homEquiv ((sheafToPresheaf J E).obj X) Y.val).symm ((sheafToPresheaf J E).map γ)) ⋯ })\n fun η => { val := (A.homEquiv X.val Y.val) (toSheafify J (((whiskeringRight Cᵒᵖ E D).obj G).obj X.val) ≫ η.val) }", " (fun γ =>\n {\n ...
import Mathlib.Algebra.Order.Ring.Basic import Mathlib.Algebra.Order.Ring.Int import Mathlib.Algebra.Ring.Divisibility.Basic import Mathlib.Data.Nat.Cast.Order #align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1" #align_import data.nat.parity from "leanpr...	Mathlib/Algebra/Order/Ring/Abs.lean	192	193	theorem abs_dvd (a b : α) : \|a\| ∣ b ↔ a ∣ b := by	cases' abs_choice a with h h <;> simp only [h, neg_dvd]	[ " Odd \|a\| ↔ Odd a", " \|a\| ∣ b ↔ a ∣ b" ]	[ " Odd \|a\| ↔ Odd a" ]
import Mathlib.Algebra.IsPrimePow import Mathlib.Data.Nat.Factorization.Basic #align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f" variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ) theorem IsPrimePow.minFac_pow_factorization_eq ...	Mathlib/Data/Nat/Factorization/PrimePow.lean	63	73	theorem IsPrimePow.exists_ord_compl_eq_one {n : ℕ} (h : IsPrimePow n) : ∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by	rcases eq_or_ne n 0 with (rfl \| hn0); · cases not_isPrimePow_zero h rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩ rcases em' p.Prime with (pp \| pp) · refine absurd ?_ hk0.ne' simp [← Nat.factorization_eq_zero_of_non_prime n pp, h1] refine ⟨p, pp, ?_⟩ refine Nat.eq_of_factoriza...	[ " n.minFac ^ n.factorization n.minFac = n", " (p ^ k).minFac ^ (p ^ k).factorization (p ^ k).minFac = p ^ k", " IsPrimePow n", " IsPrimePow 0", " 0 < n.factorization n.minFac", " IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k", " (∃ p k, p.Prime ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ ...	[ " n.minFac ^ n.factorization n.minFac = n", " (p ^ k).minFac ^ (p ^ k).factorization (p ^ k).minFac = p ^ k", " IsPrimePow n", " IsPrimePow 0", " 0 < n.factorization n.minFac", " IsPrimePow n ↔ ∃ p k, 0 < k ∧ n.factorization = Finsupp.single p k", " (∃ p k, p.Prime ∧ 0 < k ∧ p ^ k = n) ↔ ∃ p k, 0 < k ∧ ...
import Mathlib.GroupTheory.OrderOfElement import Mathlib.Data.Finset.NoncommProd import Mathlib.Data.Fintype.BigOperators import Mathlib.Data.Nat.GCD.BigOperators import Mathlib.Order.SupIndep #align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8" ...	Mathlib/GroupTheory/NoncommPiCoprod.lean	125	137	theorem noncommPiCoprod_mulSingle (i : ι) (y : N i) : noncommPiCoprod ϕ hcomm (Pi.mulSingle i y) = ϕ i y := by	change Finset.univ.noncommProd (fun j => ϕ j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _) = ϕ i y rw [← Finset.insert_erase (Finset.mem_univ i)] rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ (Finset.not_mem_erase i _)] rw [Pi.mulSingle_eq_same] rw [Finset.noncommProd_eq_pow_card] · rw [one_p...	[ " (fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯) 1 = 1", " ∀ x ∈ Finset.univ, (ϕ x) (1 x) = 1", " { toFun := fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯, map_one' := ⋯ }.toFun (f * g) =\n { toFun := fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯, map_one' := ⋯ }.toFun f ...	[ " (fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯) 1 = 1", " ∀ x ∈ Finset.univ, (ϕ x) (1 x) = 1", " { toFun := fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯, map_one' := ⋯ }.toFun (f * g) =\n { toFun := fun f => Finset.univ.noncommProd (fun i => (ϕ i) (f i)) ⋯, map_one' := ⋯ }.toFun f ...
import Mathlib.NumberTheory.ZetaValues import Mathlib.NumberTheory.LSeries.RiemannZeta open Complex Real Set open scoped Nat namespace HurwitzZeta variable {k : ℕ} {x : ℝ} theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) : cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! * ...	Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean	113	124	theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) : sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) * ((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by	rw [sinZeta_two_mul_nat_add_one hk hx] congr 1 have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)), Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul, ← Nat.cast_add_one, ← Nat.cast_m...	[ " cosZeta (↑x) (2 * ↑k) =\n (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! \n Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 k)))", " 1 < (2 * ↑k).re", " ∑' (b : ℕ), ↑(2 * π * x * ↑b).cos / ↑b ^ (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k) * (2 * π * ↑b * x).cos)", ...	[ " cosZeta (↑x) (2 * ↑k) =\n (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! \n Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 k)))", " 1 < (2 * ↑k).re", " ∑' (b : ℕ), ↑(2 * π * x * ↑b).cos / ↑b ^ (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k) * (2 * π * ↑b * x).cos)", ...
import Mathlib.Data.Fin.VecNotation import Mathlib.GroupTheory.Abelianization import Mathlib.GroupTheory.Perm.ViaEmbedding import Mathlib.GroupTheory.Subgroup.Simple import Mathlib.SetTheory.Cardinal.Basic #align_import group_theory.solvable from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298...	Mathlib/GroupTheory/Solvable.lean	56	59	theorem derivedSeries_normal (n : ℕ) : (derivedSeries G n).Normal := by	induction' n with n ih · exact (⊤ : Subgroup G).normal_of_characteristic · exact @Subgroup.commutator_normal G _ (derivedSeries G n) (derivedSeries G n) ih ih	[ " (derivedSeries G n).Normal", " (derivedSeries G 0).Normal", " (derivedSeries G (n + 1)).Normal" ]	[]
import Mathlib.Algebra.Order.Field.Basic import Mathlib.Data.Nat.Cast.Order import Mathlib.Data.Nat.Choose.Basic import Mathlib.Data.Nat.Cast.Order #align_import data.nat.choose.bounds from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11" open Nat variable {α : Type*} [LinearOrderedSemif...	Mathlib/Data/Nat/Choose/Bounds.lean	32	37	theorem choose_le_pow (r n : ℕ) : (n.choose r : α) ≤ (n ^ r : α) / r ! := by	rw [le_div_iff'] · norm_cast rw [← Nat.descFactorial_eq_factorial_mul_choose] exact n.descFactorial_le_pow r exact mod_cast r.factorial_pos	[ " ↑(n.choose r) ≤ ↑n ^ r / ↑r !", " ↑r ! * ↑(n.choose r) ≤ ↑n ^ r", " r ! * n.choose r ≤ n ^ r", " n.descFactorial r ≤ n ^ r", " 0 < ↑r !" ]	[]
import Mathlib.GroupTheory.Solvable import Mathlib.FieldTheory.PolynomialGaloisGroup import Mathlib.RingTheory.RootsOfUnity.Basic #align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a" noncomputable section open scoped Classical Polynomial Intermedi...	Mathlib/FieldTheory/AbelRuffini.lean	49	49	theorem gal_X_isSolvable : IsSolvable (X : F[X]).Gal := by	infer_instance	[ " IsSolvable (Gal 0)", " IsSolvable (Gal 1)", " IsSolvable (C x).Gal", " IsSolvable X.Gal" ]	[ " IsSolvable (Gal 0)", " IsSolvable (Gal 1)", " IsSolvable (C x).Gal" ]
import Mathlib.Data.List.Infix #align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2" -- Make sure we don't import algebra assert_not_exists Monoid variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ) namespace List def rdrop : List α := l.take (l.leng...	Mathlib/Data/List/DropRight.lean	179	181	theorem dropWhile_idempotent : dropWhile p (dropWhile p l) = dropWhile p l := by	simp only [dropWhile_eq_self_iff] exact fun h => dropWhile_nthLe_zero_not p l h	[ " [].rdrop n = []", " l.rdrop 0 = l", " l.rdrop n = (drop n l.reverse).reverse", " take (l.length - n) l = (drop n l.reverse).reverse", " take ([].length - n) [] = (drop n [].reverse).reverse", " take ((xs ++ [x]).length - n) (xs ++ [x]) = (drop n (xs ++ [x]).reverse).reverse", " take ((xs ++ [x]).lengt...	[ " [].rdrop n = []", " l.rdrop 0 = l", " l.rdrop n = (drop n l.reverse).reverse", " take (l.length - n) l = (drop n l.reverse).reverse", " take ([].length - n) [] = (drop n [].reverse).reverse", " take ((xs ++ [x]).length - n) (xs ++ [x]) = (drop n (xs ++ [x]).reverse).reverse", " take ((xs ++ [x]).lengt...
import Mathlib.Algebra.Polynomial.FieldDivision import Mathlib.FieldTheory.Minpoly.Basic import Mathlib.RingTheory.Algebraic #align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5" open scoped Classical open Polynomial Set Function minpoly namespace...	Mathlib/FieldTheory/Minpoly/Field.lean	86	90	theorem dvd_map_of_isScalarTower (A K : Type) {R : Type} [CommRing A] [Field K] [CommRing R] [Algebra A K] [Algebra A R] [Algebra K R] [IsScalarTower A K R] (x : R) : minpoly K x ∣ (minpoly A x).map (algebraMap A K) := by	refine minpoly.dvd K x ?_ rw [aeval_map_algebraMap, minpoly.aeval]	[ " (Polynomial.aeval x) (p * C p.leadingCoeff⁻¹) = 0", " p = minpoly A x", " minpoly A x = p", " minpoly A x - p = 0", " False", " (Polynomial.aeval x) (minpoly A x - p) = 0", " (minpoly A x - p).degree < (minpoly A x).degree", " (minpoly A x).leadingCoeff = p.leadingCoeff", " (minpoly A x).degree = ...	[ " (Polynomial.aeval x) (p * C p.leadingCoeff⁻¹) = 0", " p = minpoly A x", " minpoly A x = p", " minpoly A x - p = 0", " False", " (Polynomial.aeval x) (minpoly A x - p) = 0", " (minpoly A x - p).degree < (minpoly A x).degree", " (minpoly A x).leadingCoeff = p.leadingCoeff", " (minpoly A x).degree = ...
import Mathlib.SetTheory.Game.Basic import Mathlib.Tactic.NthRewrite #align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748" universe u namespace SetTheory open scoped PGame namespace PGame def ImpartialAux : PGame → Prop \| G => (G ≈ -G) ∧ (∀ i...	Mathlib/SetTheory/Game/Impartial.lean	35	38	theorem impartialAux_def {G : PGame} : G.ImpartialAux ↔ (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j) := by	rw [ImpartialAux]	[ " G.ImpartialAux ↔\n G ≈ -G ∧ (∀ (i : G.LeftMoves), (G.moveLeft i).ImpartialAux) ∧ ∀ (j : G.RightMoves), (G.moveRight j).ImpartialAux" ]	[]
import Mathlib.Data.Finset.Pointwise #align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259" open MulOpposite open Pointwise variable {α : Type*} [DecidableEq α] namespace Finset section Group variable [Group α] (e : α) (x : Finset...	Mathlib/Combinatorics/Additive/ETransform.lean	142	145	theorem mulETransformLeft.fst_mul_snd_subset : (mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2 := by	refine inter_mul_union_subset_union.trans (union_subset Subset.rfl ?_) rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]	[ " mulETransformLeft 1 x = x", " mulETransformRight 1 x = x", " (mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2", " op e • x.1 * e⁻¹ • x.2 ⊆ x.1 * x.2" ]	[ " mulETransformLeft 1 x = x", " mulETransformRight 1 x = x" ]
import Mathlib.Topology.ContinuousFunction.ZeroAtInfty open Topology Filter variable {E F 𝓕 : Type*} variable [SeminormedAddGroup E] [SeminormedAddCommGroup F] variable [FunLike 𝓕 E F] [ZeroAtInftyContinuousMapClass 𝓕 E F]	Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean	24	34	theorem ZeroAtInftyContinuousMapClass.norm_le (f : 𝓕) (ε : ℝ) (hε : 0 < ε) : ∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε := by	have h := zero_at_infty f rw [tendsto_zero_iff_norm_tendsto_zero, tendsto_def] at h specialize h (Metric.ball 0 ε) (Metric.ball_mem_nhds 0 hε) rcases Metric.closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩ use r intro x hr' suffices x ∈ (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε by aesop apply hr a...	[ " ∃ r, ∀ (x : E), r < ‖x‖ → ‖f x‖ < ε", " ∀ (x : E), r < ‖x‖ → ‖f x‖ < ε", " ‖f x‖ < ε", " x ∈ (fun x => ‖f x‖) ⁻¹' Metric.ball 0 ε", " x ∈ (Metric.closedBall 0 r)ᶜ" ]	[]
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax import Mathlib.Algebra.Order.Monoid.WithTop import Mathlib.Data.Finset.Image import Mathlib.Data.Multiset.Fold #align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" -- TODO: -- assert_not_exists OrderedComm...	Mathlib/Data/Finset/Fold.lean	83	85	theorem fold_op_distrib {f g : α → β} {b₁ b₂ : β} : (s.fold op (b₁ * b₂) fun x => f x * g x) = s.fold op b₁ f * s.fold op b₂ g := by	simp only [fold, fold_distrib]	[ " fold op b f (cons a s h) = op (f a) (fold op b f s)", " Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))", " fold op b f (insert a s) = op (f a) (fold op b f s)", " Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b ...	[ " fold op b f (cons a s h) = op (f a) (fold op b f s)", " Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))", " fold op b f (insert a s) = op (f a) (fold op b f s)", " Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b ...
import Mathlib.CategoryTheory.Sites.Sheaf import Mathlib.CategoryTheory.Sites.CoverLifting import Mathlib.CategoryTheory.Adjunction.FullyFaithful #align_import category_theory.sites.dense_subsite from "leanprover-community/mathlib"@"1d650c2e131f500f3c17f33b4d19d2ea15987f2c" universe w v u namespace CategoryTheory...	Mathlib/CategoryTheory/Sites/DenseSubsite.lean	133	141	theorem functorPullback_pushforward_covering [Full G] {X : C} (T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) := by	refine K.superset_covering ?_ (K.bind_covering T.property fun Y f _ => G.is_cover_of_isCoverDense K Y) rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', ⟨rfl⟩⟩, rfl⟩ use W; use G.preimage (f' ≫ f); use g constructor · simpa using T.val.downward_closed hf f' · simp	[ " (g ≫ f₁) ≫ f₂ = g ≫ f✝", " 𝟙 (G.obj Y) ≫ f = f", " Sieve.coverByImage G U ∈ K.sieves U", " Sieve.coverByImage G B ∈ K.sieves B", " Sieve.generate (Presieve.singleton f) ≤ Sieve.coverByImage G B", " (Sieve.coverByImage G B).arrows f", " s = t", " ∀ ⦃Y : D⦄ ⦃f : Y ⟶ X⦄, (Sieve.coverByImage G X).arrow...	[ " (g ≫ f₁) ≫ f₂ = g ≫ f✝", " 𝟙 (G.obj Y) ≫ f = f", " Sieve.coverByImage G U ∈ K.sieves U", " Sieve.coverByImage G B ∈ K.sieves B", " Sieve.generate (Presieve.singleton f) ≤ Sieve.coverByImage G B", " (Sieve.coverByImage G B).arrows f", " s = t", " ∀ ⦃Y : D⦄ ⦃f : Y ⟶ X⦄, (Sieve.coverByImage G X).arrow...
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order #align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce" open Set Filter MeasureTheory MeasurableSpace open scoped Classical Topology NNReal ENNReal MeasureTheory univers...	Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean	56	66	theorem borel_eq_generateFrom_Ioi_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ioi (a : ℝ)}) := by	rw [borel_eq_generateFrom_Ioi] refine le_antisymm (generateFrom_le ?_) (generateFrom_mono <\| iUnion_subset fun q ↦ singleton_subset_iff.mpr <\| mem_range_self _) rintro _ ⟨a, rfl⟩ have : IsGLB (range ((↑) : ℚ → ℝ) ∩ Ioi a) a := by simp [isGLB_iff_le_iff, mem_lowerBounds, ← le_iff_forall_lt_rat_imp_l...	[ " borel ℝ = generateFrom (⋃ a, {Iio ↑a})", " generateFrom (range Iio) = generateFrom (⋃ a, {Iio ↑a})", " ∀ t ∈ range Iio, MeasurableSet t", " MeasurableSet (Iio a)", " IsLUB (range Rat.cast ∩ Iio a) a", " MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Iio a, Iio ↑y)", " Iio ↑b ∈ ⋃ a, {Iio ↑a}", " borel ℝ = generat...	[ " borel ℝ = generateFrom (⋃ a, {Iio ↑a})", " generateFrom (range Iio) = generateFrom (⋃ a, {Iio ↑a})", " ∀ t ∈ range Iio, MeasurableSet t", " MeasurableSet (Iio a)", " IsLUB (range Rat.cast ∩ Iio a) a", " MeasurableSet (⋃ y ∈ Rat.cast ⁻¹' Iio a, Iio ↑y)", " Iio ↑b ∈ ⋃ a, {Iio ↑a}" ]
import Mathlib.SetTheory.Cardinal.Finite #align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8" noncomputable section open scoped Classical variable {α β γ : Type} def Finite.equivFin (α : Type) [Finite α] : α ≃ Fin (Nat.card α) := by have := (Finite....	Mathlib/Data/Finite/Card.lean	83	85	theorem one_lt_card_iff_nontrivial [Finite α] : 1 < Nat.card α ↔ Nontrivial α := by	haveI := Fintype.ofFinite α simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]	[ " α ≃ Fin (Nat.card α)", " α ≃ Fin n", " Nat.card α = if h : Finite α then Fintype.card α else 0", " 0 < Nat.card α ↔ Nonempty α", " Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)", " Nat.card α ≤ 1 ↔ Subsingleton α", " 1 < Nat.card α ↔ Nontrivial α" ]	[ " α ≃ Fin (Nat.card α)", " α ≃ Fin n", " Nat.card α = if h : Finite α then Fintype.card α else 0", " 0 < Nat.card α ↔ Nonempty α", " Nat.card α = Nat.card β ↔ Nonempty (α ≃ β)", " Nat.card α ≤ 1 ↔ Subsingleton α" ]
import Mathlib.Algebra.Algebra.Bilinear import Mathlib.RingTheory.Localization.Basic #align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86" section IsLocalizedModule universe u v variable {R : Type*} [CommSemiring R] (S : Submonoid R) variabl...	Mathlib/Algebra/Module/LocalizedModule.lean	574	588	theorem IsLocalizedModule.of_linearEquiv (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f] : IsLocalizedModule S (e ∘ₗ f : M →ₗ[R] M'') where map_units s := by	rw [show algebraMap R (Module.End R M'') s = e ∘ₗ (algebraMap R (Module.End R M') s) ∘ₗ e.symm by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp, LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp] exact (Module.End_isUnit_iff _).m...	[ " f x₁ = f x₂", " IsUnit ((algebraMap R (Module.End R M'')) ↑s)", " (algebraMap R (Module.End R M'')) ↑s = ↑e ∘ₗ (algebraMap R (Module.End R M')) ↑s ∘ₗ ↑e.symm", " ((algebraMap R (Module.End R M'')) ↑s) x✝ = (↑e ∘ₗ (algebraMap R (Module.End R M')) ↑s ∘ₗ ↑e.symm) x✝", " Function.Bijective ⇑((algebraMap R (Mo...	[ " f x₁ = f x₂" ]
import Mathlib.Algebra.Field.Basic import Mathlib.Algebra.GroupWithZero.Units.Equiv import Mathlib.Algebra.Order.Field.Defs import Mathlib.Algebra.Order.Ring.Abs import Mathlib.Order.Bounds.OrderIso import Mathlib.Tactic.Positivity.Core #align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477...	Mathlib/Algebra/Order/Field/Basic.lean	86	86	theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b := by	rw [mul_comm, lt_div_iff hc]	[ " a ≤ b / c ↔ c * a ≤ b", " a = a / b * b", " c * b / b = c", " a / b ≤ c ↔ a ≤ b * c", " a / b ≤ c ↔ a / c ≤ b", " a < b / c ↔ c * a < b" ]	[ " a ≤ b / c ↔ c * a ≤ b", " a = a / b * b", " c * b / b = c", " a / b ≤ c ↔ a ≤ b * c", " a / b ≤ c ↔ a / c ≤ b" ]
import Mathlib.CategoryTheory.Limits.Types import Mathlib.CategoryTheory.IsConnected import Mathlib.CategoryTheory.Limits.Final import Mathlib.CategoryTheory.Conj universe w v u namespace CategoryTheory.Limits.Types variable (C : Type u) [Category.{v} C] def constPUnitFunctor : C ⥤ Type w := (Functor.const C).o...	Mathlib/CategoryTheory/Limits/IsConnected.lean	87	93	theorem zigzag_of_eqvGen_quot_rel (F : C ⥤ Type w) (c d : Σ j, F.obj j) (h : EqvGen (Quot.Rel F) c d) : Zigzag c.1 d.1 := by	induction h with \| rel _ _ h => exact Zigzag.of_hom <\| Exists.choose h \| refl _ => exact Zigzag.refl _ \| symm _ _ _ ih => exact zigzag_symmetric ih \| trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂	[ " (pUnitCocone C).ι.app j ≫ (fun s => s.ι.app Classical.ofNonempty) s = s.ι.app j", " ((pUnitCocone C).ι.app j ≫ (fun s => s.ι.app Classical.ofNonempty) s) PUnit.unit = s.ι.app j PUnit.unit", " ∀ (j₁ j₂ : C), (j₁ ⟶ j₂) → s.ι.app j₁ PUnit.unit = s.ι.app j₂ PUnit.unit", " s.ι.app X PUnit.unit = s.ι.app Y PUnit....	[ " (pUnitCocone C).ι.app j ≫ (fun s => s.ι.app Classical.ofNonempty) s = s.ι.app j", " ((pUnitCocone C).ι.app j ≫ (fun s => s.ι.app Classical.ofNonempty) s) PUnit.unit = s.ι.app j PUnit.unit", " ∀ (j₁ j₂ : C), (j₁ ⟶ j₂) → s.ι.app j₁ PUnit.unit = s.ι.app j₂ PUnit.unit", " s.ι.app X PUnit.unit = s.ι.app Y PUnit....
import Mathlib.Combinatorics.SetFamily.Shadow #align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1" open Finset variable {α : Type*}	Mathlib/Combinatorics/SetFamily/Compression/UV.lean	57	64	theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) : { x \| Disjoint u x ∧ v ≤ x }.InjOn fun x => (x ⊔ u) \ v := by	rintro a ha b hb hab have h : ((a ⊔ u) \ v) \ u ⊔ v = ((b ⊔ u) \ v) \ u ⊔ v := by dsimp at hab rw [hab] rwa [sdiff_sdiff_comm, ha.1.symm.sup_sdiff_cancel_right, sdiff_sdiff_comm, hb.1.symm.sup_sdiff_cancel_right, sdiff_sup_cancel ha.2, sdiff_sup_cancel hb.2] at h	[ " Set.InjOn (fun x => (x ⊔ u) \\ v) {x \| Disjoint u x ∧ v ≤ x}", " a = b", " ((a ⊔ u) \\ v) \\ u ⊔ v = ((b ⊔ u) \\ v) \\ u ⊔ v" ]	[]
import Mathlib.Algebra.Order.Ring.Int import Mathlib.Data.Nat.SuccPred #align_import data.int.succ_pred from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853" open Function Order namespace Int -- so that Lean reads `Int.succ` through `SuccOrder.succ` @[instance] abbrev instSuccOrder : Su...	Mathlib/Data/Int/SuccPred.lean	79	79	theorem sub_one_covBy (z : ℤ) : z - 1 ⋖ z := by	rw [Int.covBy_iff_succ_eq, sub_add_cancel]	[ " succ^[n + 1] a = a + ↑(n + 1)", " (succ ∘ succ^[n]) a = a + ↑n + 1", " pred^[n + 1] a = a - ↑(n + 1)", " (pred ∘ pred^[n]) a = a - ↑n - 1", " Order.succ^[(b - a).toNat] a = b", " Order.pred^[(b - a).toNat] b = a", " z - 1 ⋖ z" ]	[ " succ^[n + 1] a = a + ↑(n + 1)", " (succ ∘ succ^[n]) a = a + ↑n + 1", " pred^[n + 1] a = a - ↑(n + 1)", " (pred ∘ pred^[n]) a = a - ↑n - 1", " Order.succ^[(b - a).toNat] a = b", " Order.pred^[(b - a).toNat] b = a" ]
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	99	105	theorem t_inv (i j : D.J) : D.t i j ≫ D.t j i = 𝟙 _ := by	have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd ≫ inv pullback.fst := by simp have := D.cocycle i j i rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this rw [← IsIso.eq_inv_comp, ← Category.assoc, IsIso.comp_inv_e...	[ " 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.MeasureTheory.PiSystem import Mathlib.Order.OmegaCompletePartialOrder import Mathlib.Topology.Constructions import Mathlib.MeasureTheory.MeasurableSpace.Basic open Set namespace MeasureTheory variable {ι : Type _} {α : ι → Type _} section cylinder def cylinder (s : Finset ι) (S : Set (∀ i : s, α...	Mathlib/MeasureTheory/Constructions/Cylinders.lean	161	162	theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ := by	rw [cylinder, preimage_empty]	[ " cylinder s ∅ = ∅" ]	[]
import Mathlib.CategoryTheory.Closed.Cartesian import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts import Mathlib.CategoryTheory.Adjunction.FullyFaithful #align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184" noncomputable secti...	Mathlib/CategoryTheory/Closed/Functor.lean	166	168	theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C) [i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) := by	rw [← frobeniusMorphism_mate F h]; infer_instance	[ " IsIso ((frobeniusMorphism F h A).app B)", " IsIso (prodComparison L (F.obj A) B ≫ prod.map (h.counit.app A) (𝟙 (L.obj B)))", " prod.map (𝟙 (F.obj A)) ((expComparison F A).app B) ≫ (exp.ev (F.obj A)).app (F.obj B) =\n inv (prodComparison F A (A ⟹ B)) ≫ F.map ((exp.ev A).app B)", " inv (prodComparison F ...	[ " IsIso ((frobeniusMorphism F h A).app B)", " IsIso (prodComparison L (F.obj A) B ≫ prod.map (h.counit.app A) (𝟙 (L.obj B)))", " prod.map (𝟙 (F.obj A)) ((expComparison F A).app B) ≫ (exp.ev (F.obj A)).app (F.obj B) =\n inv (prodComparison F A (A ⟹ B)) ≫ F.map ((exp.ev A).app B)", " inv (prodComparison F ...
import Mathlib.Algebra.Group.Subgroup.Pointwise import Mathlib.Data.ZMod.Basic import Mathlib.GroupTheory.GroupAction.ConjAct import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup #align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f...	Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean	73	75	theorem Gamma_one_top : Gamma 1 = ⊤ := by	ext simp [eq_iff_true_of_subsingleton]	[ " γ ∈ Gamma N ↔ ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(↑γ 1 0) = 0 ∧ ↑(↑γ 1 1) = 1", " (SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ = 1 ↔\n ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(↑γ 1 0) = 0 ∧ ↑(↑γ 1 1) = 1", " (SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ = 1 →\n ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(...	[ " γ ∈ Gamma N ↔ ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(↑γ 1 0) = 0 ∧ ↑(↑γ 1 1) = 1", " (SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ = 1 ↔\n ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(↑γ 1 0) = 0 ∧ ↑(↑γ 1 1) = 1", " (SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ = 1 →\n ↑(↑γ 0 0) = 1 ∧ ↑(↑γ 0 1) = 0 ∧ ↑(...
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	125	133	theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) : (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal := by	constructor <;> rintro ⟨⟨x, hx⟩⟩ · have x_mem : x ∈ coeSubmodule S I := hx.symm ▸ Submodule.mem_span_singleton_self x obtain ⟨x, _, rfl⟩ := (mem_coeSubmodule _ _).mp x_mem refine ⟨⟨x, coeSubmodule_injective S h ?_⟩⟩ rw [Ideal.submodule_span_eq, hx, coeSubmodule_span_singleton] · refine ⟨⟨algebraMap R...	[ " 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 (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x}", " IsN...	[ " 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 (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x}", " IsN...

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