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 | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Batteries.Data.List.Lemmas
namespace List
universe u v
variable {α : Type u} {β : Type v}
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by s... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 43 | 47 | theorem eraseIdx_append_of_lt_length {l : List α} {k : Nat} (hk : k < length l) (l' : List α) :
eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by |
rw [eraseIdx_eq_take_drop_succ, take_append_of_le_length, drop_append_of_le_length,
eraseIdx_eq_take_drop_succ, append_assoc]
all_goals omega
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,821 |
import Batteries.Data.List.Lemmas
namespace List
universe u v
variable {α : Type u} {β : Type v}
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by s... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 49 | 55 | theorem eraseIdx_append_of_length_le {l : List α} {k : Nat} (hk : length l ≤ k) (l' : List α) :
eraseIdx (l ++ l') k = l ++ eraseIdx l' (k - length l) := by |
rw [eraseIdx_eq_take_drop_succ, eraseIdx_eq_take_drop_succ,
take_append_eq_append_take, drop_append_eq_append_drop,
take_all_of_le hk, drop_eq_nil_of_le (by omega), nil_append, append_assoc]
congr
omega
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,821 |
import Batteries.Data.List.Lemmas
namespace List
universe u v
variable {α : Type u} {β : Type v}
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by s... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 76 | 80 | theorem mem_eraseIdx_iff_get? {x : α} {l} {k} : x ∈ eraseIdx l k ↔ ∃ i ≠ k, l.get? i = x := by |
simp only [mem_eraseIdx_iff_get, Fin.exists_iff, exists_and_left, get_eq_iff, exists_prop]
refine exists_congr fun i => and_congr_right' <| and_iff_right_of_imp fun h => ?_
obtain ⟨h, -⟩ := get?_eq_some.1 h
exact h
| 4 | 54.59815 | 2 | 1.666667 | 3 | 1,821 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
| Mathlib/Topology/Perfect.lean | 62 | 68 | theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by |
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_principal_iff]
exact mem_nhdsWithin_of_mem_nhds hU
rw [AccPt, ← inf_principal, ← inf_assoc, inf_of_le_left this]
exact h_acc
| 5 | 148.413159 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 87 | 88 | theorem preperfect_iff_nhds : Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by |
simp only [Preperfect, accPt_iff_nhds]
| 1 | 2.718282 | 0 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 111 | 115 | theorem Preperfect.open_inter {U : Set α} (hC : Preperfect C) (hU : IsOpen U) :
Preperfect (U ∩ C) := by |
rintro x ⟨xU, xC⟩
apply (hC _ xC).nhds_inter
exact hU.mem_nhds xU
| 3 | 20.085537 | 1 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 120 | 128 | theorem Preperfect.perfect_closure (hC : Preperfect C) : Perfect (closure C) := by |
constructor; · exact isClosed_closure
intro x hx
by_cases h : x ∈ C <;> apply AccPt.mono _ (principal_mono.mpr subset_closure)
· exact hC _ h
have : {x}ᶜ ∩ C = C := by simp [h]
rw [AccPt, nhdsWithin, inf_assoc, inf_principal, this]
rw [closure_eq_cluster_pts] at hx
exact hx
| 8 | 2,980.957987 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 132 | 144 | theorem preperfect_iff_perfect_closure [T1Space α] : Preperfect C ↔ Perfect (closure C) := by |
constructor <;> intro h
· exact h.perfect_closure
intro x xC
have H : AccPt x (𝓟 (closure C)) := h.acc _ (subset_closure xC)
rw [accPt_iff_frequently] at *
have : ∀ y, y ≠ x ∧ y ∈ closure C → ∃ᶠ z in 𝓝 y, z ≠ x ∧ z ∈ C := by
rintro y ⟨hyx, yC⟩
simp only [← mem_compl_singleton_iff, and_comm, ← fre... | 12 | 162,754.791419 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 147 | 153 | theorem Perfect.closure_nhds_inter {U : Set α} (hC : Perfect C) (x : α) (xC : x ∈ C) (xU : x ∈ U)
(Uop : IsOpen U) : Perfect (closure (U ∩ C)) ∧ (closure (U ∩ C)).Nonempty := by |
constructor
· apply Preperfect.perfect_closure
exact hC.acc.open_inter Uop
apply Nonempty.closure
exact ⟨x, ⟨xU, xC⟩⟩
| 5 | 148.413159 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 158 | 177 | theorem Perfect.splitting [T25Space α] (hC : Perfect C) (hnonempty : C.Nonempty) :
∃ C₀ C₁ : Set α,
(Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C) ∧ Disjoint C₀ C₁ := by |
cases' hnonempty with y yC
obtain ⟨x, xC, hxy⟩ : ∃ x ∈ C, x ≠ y := by
have := hC.acc _ yC
rw [accPt_iff_nhds] at this
rcases this univ univ_mem with ⟨x, xC, hxy⟩
exact ⟨x, xC.2, hxy⟩
obtain ⟨U, xU, Uop, V, yV, Vop, hUV⟩ := exists_open_nhds_disjoint_closure hxy
use closure (U ∩ C), closure (V ∩ ... | 17 | 24,154,952.753575 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 186 | 218 | theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
(hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by |
obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
let v := { U ∈ b | (U ∩ C).Countable }
let V := ⋃ U ∈ v, U
let D := C \ V
have Vct : (V ∩ C).Countable := by
simp only [V, iUnion_inter, mem_sep_iff]
apply Countable.biUnion
· exact Countable.mono inter_subset_left bct
· ... | 31 | 29,048,849,665,247.426 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 222 | 233 | theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
(hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C := by |
rcases exists_countable_union_perfect_of_isClosed hclosed with ⟨V, D, Vct, Dperf, VD⟩
refine ⟨D, ⟨Dperf, ?_⟩⟩
constructor
· rw [nonempty_iff_ne_empty]
by_contra h
rw [h, union_empty] at VD
rw [VD] at hunc
contradiction
rw [VD]
exact subset_union_right
| 10 | 22,026.465795 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 113 | 116 | theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by |
rw [← seqClosure_eq_closure]
rfl
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,823 |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 125 | 134 | theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := by |
refine
⟨fun hf u hu => hf.comp hu, fun h =>
((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 ?_⟩
rintro s ⟨hbs, hsc⟩
refine ⟨closure (f ⁻¹' s), ⟨mt ?_ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩
rw [← seqClosure_eq_closure]
rintro ⟨u, hus, hu⟩
exact hsc.mem_of_te... | 8 | 2,980.957987 | 2 | 1.666667 | 3 | 1,823 |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 139 | 151 | theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto
(h : ∀ (f : X → Prop) (a : X),
(∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a) :
FrechetUrysohnSpace X := by |
refine ⟨fun s x hcx => ?_⟩
by_cases hx : x ∈ s;
· exact subset_seqClosure hx
· obtain ⟨u, hux, hus⟩ : ∃ u : ℕ → X, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ x in atTop, u x ∈ s := by
simpa only [ContinuousAt, hx, tendsto_nhds_true, (· ∘ ·), ← not_frequently, exists_prop,
← mem_closure_iff_frequently, hcx, imp_... | 9 | 8,103.083928 | 2 | 1.666667 | 3 | 1,823 |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 23 | 28 | theorem range_sigmaMk (i : ι) : range (Sigma.mk i : α i → Sigma α) = Sigma.fst ⁻¹' {i} := by |
apply Subset.antisymm
· rintro _ ⟨b, rfl⟩
simp
· rintro ⟨x, y⟩ (rfl | _)
exact mem_range_self y
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,824 |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 31 | 34 | theorem preimage_image_sigmaMk_of_ne (h : i ≠ j) (s : Set (α j)) :
Sigma.mk i ⁻¹' (Sigma.mk j '' s) = ∅ := by |
ext x
simp [h.symm]
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,824 |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 43 | 50 | theorem image_sigmaMk_preimage_sigmaMap {β : ι' → Type*} {f : ι → ι'} (hf : Function.Injective f)
(g : ∀ i, α i → β (f i)) (i : ι) (s : Set (β (f i))) :
Sigma.mk i '' (g i ⁻¹' s) = Sigma.map f g ⁻¹' (Sigma.mk (f i) '' s) := by |
refine (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm ?_
rintro ⟨j, x⟩ ⟨y, hys, hxy⟩
simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy
rcases hxy with ⟨rfl, hxy⟩; rw [heq_iff_eq] at hxy; subst y
exact ⟨x, hys, rfl⟩
| 5 | 148.413159 | 2 | 1.666667 | 3 | 1,824 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 42 | 53 | theorem dense_of_not_isolated_zero (S : AddSubgroup G) (hS : ∀ ε > 0, ∃ g ∈ S, g ∈ Ioo 0 ε) :
Dense (S : Set G) := by |
cases subsingleton_or_nontrivial G
· refine fun x => _root_.subset_closure ?_
rw [Subsingleton.elim x 0]
exact zero_mem S
refine dense_of_exists_between fun a b hlt => ?_
rcases hS (b - a) (sub_pos.2 hlt) with ⟨g, hgS, hg0, hg⟩
rcases (existsUnique_add_zsmul_mem_Ioc hg0 0 a).exists with ⟨m, hm⟩
rw ... | 10 | 22,026.465795 | 2 | 1.666667 | 3 | 1,825 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 58 | 62 | theorem dense_of_no_min (S : AddSubgroup G) (hbot : S ≠ ⊥)
(H : ¬∃ a : G, IsLeast { g : G | g ∈ S ∧ 0 < g } a) : Dense (S : Set G) := by |
refine S.dense_of_not_isolated_zero fun ε ε0 => ?_
contrapose! H
exact exists_isLeast_pos hbot ε0 (disjoint_left.2 H)
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,825 |
import Mathlib.GroupTheory.Archimedean
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.archimedean from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set
theorem Rat.denseRange_cast {𝕜} [LinearOrderedField 𝕜] [TopologicalSpace 𝕜] [OrderTopology 𝕜]
... | Mathlib/Topology/Algebra/Order/Archimedean.lean | 67 | 71 | theorem dense_or_cyclic (S : AddSubgroup G) : Dense (S : Set G) ∨ ∃ a : G, S = closure {a} := by |
refine (em _).imp (dense_of_not_isolated_zero S) fun h => ?_
push_neg at h
rcases h with ⟨ε, ε0, hε⟩
exact cyclic_of_isolated_zero ε0 (disjoint_left.2 hε)
| 4 | 54.59815 | 2 | 1.666667 | 3 | 1,825 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 44 | 48 | theorem coeff_zero_mem_comap_of_root_mem_of_eval_mem {r : S} (hr : r ∈ I) {p : R[X]}
(hp : p.eval₂ f r ∈ I) : p.coeff 0 ∈ I.comap f := by |
rw [← p.divX_mul_X_add, eval₂_add, eval₂_C, eval₂_mul, eval₂_X] at hp
refine mem_comap.mpr ((I.add_mem_iff_right ?_).mp hp)
exact I.mul_mem_left _ hr
| 3 | 20.085537 | 1 | 1.666667 | 6 | 1,826 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 56 | 70 | theorem exists_coeff_ne_zero_mem_comap_of_non_zero_divisor_root_mem {r : S}
(r_non_zero_divisor : ∀ {x}, x * r = 0 → x = 0) (hr : r ∈ I) {p : R[X]} :
p ≠ 0 → p.eval₂ f r = 0 → ∃ i, p.coeff i ≠ 0 ∧ p.coeff i ∈ I.comap f := by |
refine p.recOnHorner ?_ ?_ ?_
· intro h
contradiction
· intro p a coeff_eq_zero a_ne_zero _ _ hp
refine ⟨0, ?_, coeff_zero_mem_comap_of_root_mem hr hp⟩
simp [coeff_eq_zero, a_ne_zero]
· intro p p_nonzero ih _ hp
rw [eval₂_mul, eval₂_X] at hp
obtain ⟨i, hi, mem⟩ := ih p_nonzero (r_non_zero_d... | 12 | 162,754.791419 | 2 | 1.666667 | 6 | 1,826 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 | theorem injective_quotient_le_comap_map (P : Ideal R[X]) :
Function.Injective <|
Ideal.quotientMap
(Ideal.map (Polynomial.mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(Polynomial.mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
le_comap_map := by |
refine quotientMap_injective' (le_of_eq ?_)
rw [comap_map_of_surjective (mapRingHom (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))))
(map_surjective (Ideal.Quotient.mk (P.comap (C : R →+* R[X]))) Ideal.Quotient.mk_surjective)]
refine le_antisymm (sup_le le_rfl ?_) (le_sup_of_le_left le_rfl)
refine fun p h... | 7 | 1,096.633158 | 2 | 1.666667 | 6 | 1,826 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 101 | 109 | theorem quotient_mk_maps_eq (P : Ideal R[X]) :
((Quotient.mk (map (mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)).comp C).comp
(Quotient.mk (P.comap (C : R →+* R[X]))) =
(Ideal.quotientMap (map (mapRingHom (Quotient.mk (P.comap (C : R →+* R[X])))) P)
(mapRingHom (Quotient.mk (P.c... |
refine RingHom.ext fun x => ?_
repeat' rw [RingHom.coe_comp, Function.comp_apply]
rw [quotientMap_mk, coe_mapRingHom, map_C]
| 3 | 20.085537 | 1 | 1.666667 | 6 | 1,826 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 116 | 126 | theorem exists_nonzero_mem_of_ne_bot {P : Ideal R[X]} (Pb : P ≠ ⊥) (hP : ∀ x : R, C x ∈ P → x = 0) :
∃ p : R[X], p ∈ P ∧ Polynomial.map (Quotient.mk (P.comap (C : R →+* R[X]))) p ≠ 0 := by |
obtain ⟨m, hm⟩ := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)
refine ⟨m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp ?_)⟩
refine
(injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk
(P.comap (C : R →+* R[X]))))).mp
?_ _ pp0
refine map_injective _ ... | 9 | 8,103.083928 | 2 | 1.666667 | 6 | 1,826 |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 139 | 149 | theorem comap_eq_of_scalar_tower_quotient [Algebra R S] [Algebra (R ⧸ p) (S ⧸ P)]
[IsScalarTower R (R ⧸ p) (S ⧸ P)] (h : Function.Injective (algebraMap (R ⧸ p) (S ⧸ P))) :
comap (algebraMap R S) P = p := by |
ext x
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,
IsScalarTower.algebraMap_apply R (R ⧸ p) (S ⧸ P), Quotient.algebraMap_eq]
constructor
· intro hx
exact (injective_iff_map_eq_zero (algebraMap (R ⧸ p) (S ⧸ P))).mp h _ hx
· intro hx
rw [hx, RingHom... | 8 | 2,980.957987 | 2 | 1.666667 | 6 | 1,826 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 83 | 86 | theorem map_mem (x : (inducedMap A).1) (n : ℕ) : (inducedMap A).2 x ∈ A (res x n) := by |
have := x.property.some_mem
rw [mem_iInter] at this
exact this n
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,827 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 99 | 115 | theorem Disjoint.map_injective (hA : CantorScheme.Disjoint A) : Injective (inducedMap A).2 := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
refine Subtype.coe_injective (res_injective ?_)
dsimp
ext n : 1
induction' n with n ih; · simp
simp only [res_succ, cons.injEq]
refine ⟨?_, ih⟩
contrapose hA
simp only [CantorScheme.Disjoint, _root_.Pairwise, Ne, not_forall, exists_prop]
refine ⟨res x n, _, _, hA, ?_⟩
r... | 16 | 8,886,110.520508 | 2 | 1.666667 | 3 | 1,827 |
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
namespace CantorScheme
open List Function Filter Set PiNat
open scoped Classical
open Topology
variable {β α : Type*} (A : List β → Set α)
... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 131 | 144 | theorem VanishingDiam.dist_lt (hA : VanishingDiam A) (ε : ℝ) (ε_pos : 0 < ε) (x : ℕ → β) :
∃ n : ℕ, ∀ (y) (_ : y ∈ A (res x n)) (z) (_ : z ∈ A (res x n)), dist y z < ε := by |
specialize hA x
rw [ENNReal.tendsto_atTop_zero] at hA
cases' hA (ENNReal.ofReal (ε / 2)) (by
simp only [gt_iff_lt, ENNReal.ofReal_pos]
linarith) with n hn
use n
intro y hy z hz
rw [← ENNReal.ofReal_lt_ofReal_iff ε_pos, ← edist_dist]
apply lt_of_le_of_lt (EMetric.edist_le_diam_of_mem hy hz)
appl... | 12 | 162,754.791419 | 2 | 1.666667 | 3 | 1,827 |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
| Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 45 | 56 | theorem iterateMapComap_le_succ (K : Submodule R N) (h : K.map f ≤ K.map i) (n : ℕ) :
f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K := by |
nth_rw 2 [iterateMapComap]
rw [iterate_succ', Function.comp_apply, ← iterateMapComap, ← map_le_iff_le_comap]
induction n with
| zero => exact h
| succ n ih =>
simp_rw [iterateMapComap, iterate_succ', Function.comp_apply]
calc
_ ≤ (f.iterateMapComap i n K).map i := map_comap_le _ _
_ ≤ (((... | 10 | 22,026.465795 | 2 | 1.666667 | 3 | 1,828 |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
theorem iterateMapComap... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 65 | 79 | theorem iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) (n : ℕ) :
f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K := by |
induction n with
| zero =>
contrapose! heq
induction m with
| zero => exact heq
| succ m ih =>
rw [iterateMapComap, iterateMapComap, iterate_succ', iterate_succ']
exact fun H ↦ ih (map_injective_of_injective hi (comap_injective_of_surjective hf H))
| succ n ih =>
rw [iterateMapCom... | 11 | 59,874.141715 | 2 | 1.666667 | 3 | 1,828 |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
theorem iterateMapComap... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 88 | 92 | theorem ker_le_of_iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) : LinearMap.ker f ≤ K := by |
rw [show K = _ from f.iterateMapComap_eq_succ i K m heq hf hi 0]
exact f.ker_le_comap
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,828 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 51 | 58 | theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i := by |
obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt
classical
have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn
obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩
refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩
exact of_not_not fun h ↦ hl ⟨k, mem_neLocus... | 6 | 403.428793 | 2 | 1.666667 | 3 | 1,829 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 61 | 64 | theorem lex_lt_of_lt [∀ i, PartialOrder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : Pi.Lex r (· < ·) x y := by |
simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,829 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 133 | 139 | theorem toLex_monotone : Monotone (@toLex (Π₀ i, α i)) := by |
intro a b h
refine le_of_lt_or_eq (or_iff_not_imp_right.2 fun hne ↦ ?_)
classical
exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ not_mem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_lt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩
| 6 | 403.428793 | 2 | 1.666667 | 3 | 1,829 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 77 | 79 | theorem mem_approxOrderOf_iff {A : Type*} [SeminormedGroup A] {n : ℕ} {δ : ℝ} {a : A} :
a ∈ approxOrderOf A n δ ↔ ∃ b : A, orderOf b = n ∧ a ∈ ball b δ := by |
simp only [approxOrderOf, thickening_eq_biUnion_ball, mem_iUnion₂, mem_setOf_eq, exists_prop]
| 1 | 2.718282 | 0 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 108 | 116 | theorem image_pow_subset_of_coprime (hm : 0 < m) (hmn : n.Coprime m) :
(fun (y : A) => y ^ m) '' approxOrderOf A n δ ⊆ approxOrderOf A n (m * δ) := by |
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb, hab⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {u : A | orderOf u = n} := by
rw [← hb] at hmn ⊢; exact hmn.orderOf_pow
apply ball_subset_thickening hb ((m : ℝ) • δ)
convert pow_mem_ball hm hab using 1
simp only [nsmul_eq_mul, Algebra.id.smul_eq_mul]
| 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 121 | 129 | theorem image_pow_subset (n : ℕ) (hm : 0 < m) :
(fun (y : A) => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (m * δ) := by |
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb : orderOf b = n * m, hab : a ∈ ball b δ⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {y : A | orderOf y = n} := by
rw [mem_setOf_eq, orderOf_pow' b hm.ne', hb, Nat.gcd_mul_left_left, n.mul_div_cancel hm]
apply ball_subset_thickening hb (m * δ)
convert pow_mem_b... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 134 | 142 | theorem smul_subset_of_coprime (han : (orderOf a).Coprime n) :
a • approxOrderOf A n δ ⊆ approxOrderOf A (orderOf a * n) δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
refine iUnion₂_subset_iff.mpr fun b hb c hc => ?_
simp only [mem_iUnion, exists_prop]
refine ⟨a * b, ?_, hc⟩
rw [← hb] at han ⊢
exact (Commute.all a b).orderOf_mul_eq_mul... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 147 | 166 | theorem smul_eq_of_mul_dvd (hn : 0 < n) (han : orderOf a ^ 2 ∣ n) :
a • approxOrderOf A n δ = approxOrderOf A n δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_fo... | 18 | 65,659,969.137331 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 174 | 180 | theorem mem_approxAddOrderOf_iff {δ : ℝ} {x : UnitAddCircle} {n : ℕ} (hn : 0 < n) :
x ∈ approxAddOrderOf UnitAddCircle n δ ↔ ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ := by |
simp only [mem_approx_add_orderOf_iff, mem_setOf_eq, ball, exists_prop, dist_eq_norm,
AddCircle.addOrderOf_eq_pos_iff hn, mul_one]
constructor
· rintro ⟨y, ⟨m, hm₁, hm₂, rfl⟩, hx⟩; exact ⟨m, hm₁, hm₂, hx⟩
· rintro ⟨m, hm₁, hm₂, hx⟩; exact ⟨↑((m : ℝ) / n), ⟨m, hm₁, hm₂, rfl⟩, hx⟩
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 183 | 191 | theorem mem_addWellApproximable_iff (δ : ℕ → ℝ) (x : UnitAddCircle) :
x ∈ addWellApproximable UnitAddCircle δ ↔
{n : ℕ | ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ n}.Infinite := by |
simp only [mem_add_wellApproximable_iff, ← Nat.cofinite_eq_atTop, cofinite.blimsup_set_eq,
mem_setOf_eq]
refine iff_of_eq (congr_arg Set.Infinite <| ext fun n => ⟨fun hn => ?_, fun hn => ?_⟩)
· exact (mem_approxAddOrderOf_iff hn.1).mp hn.2
· have h : 0 < n := by obtain ⟨m, hm₁, _, _⟩ := hn; exact pos_of_gt... | 6 | 403.428793 | 2 | 1.714286 | 7 | 1,835 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 97 | 99 | theorem int_valuation_ne_zero (x : R) (hx : x ≠ 0) : v.intValuationDef x ≠ 0 := by |
rw [intValuationDef, if_neg hx]
exact WithZero.coe_ne_zero
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 108 | 110 | theorem int_valuation_zero_le (x : nonZeroDivisors R) : 0 < v.intValuationDef x := by |
rw [v.intValuationDef_if_neg (nonZeroDivisors.coe_ne_zero x)]
exact WithZero.zero_lt_coe _
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 114 | 120 | theorem int_valuation_le_one (x : R) : v.intValuationDef x ≤ 1 := by |
rw [intValuationDef]
by_cases hx : x = 0
· rw [if_pos hx]; exact WithZero.zero_le 1
· rw [if_neg hx, ← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_le_coe, ofAdd_le,
Right.neg_nonpos_iff]
exact Int.natCast_nonneg _
| 6 | 403.428793 | 2 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 124 | 134 | theorem int_valuation_lt_one_iff_dvd (r : R) :
v.intValuationDef r < 1 ↔ v.asIdeal ∣ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
· simp [hr]
· rw [← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_lt_coe, ofAdd_lt, neg_lt_zero, ←
Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
exact h... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 139 | 147 | theorem int_valuation_le_pow_iff_dvd (r : R) (n : ℕ) :
v.intValuationDef r ≤ Multiplicative.ofAdd (-(n : ℤ)) ↔ v.asIdeal ^ n ∣ Ideal.span {r} := by |
rw [intValuationDef]
split_ifs with hr
· simp_rw [hr, Ideal.dvd_span_singleton, zero_le', Submodule.zero_mem]
· rw [WithZero.coe_le_coe, ofAdd_le, neg_le_neg_iff, Int.ofNat_le, Ideal.dvd_span_singleton, ←
Associates.le_singleton_iff,
Associates.prime_pow_dvd_iff_le (Associates.mk_ne_zero'.mpr hr)
... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 156 | 160 | theorem IntValuation.map_one' : v.intValuationDef 1 = 1 := by |
rw [v.intValuationDef_if_neg (zero_ne_one.symm : (1 : R) ≠ 0), Ideal.span_singleton_one, ←
Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
Associates.count_zero (by apply v.associates_irreducible), Int.ofNat_zero, neg_zero, ofAdd_zero,
WithZero.coe_one]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 164 | 175 | theorem IntValuation.map_mul' (x y : R) :
v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y := by |
simp only [intValuationDef]
by_cases hx : x = 0
· rw [hx, zero_mul, if_pos (Eq.refl _), zero_mul]
· by_cases hy : y = 0
· rw [hy, mul_zero, if_pos (Eq.refl _), mul_zero]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithZero.coe_mul, WithZero.coe_inj, ←
ofAdd_add, ← Ideal.span_singl... | 10 | 22,026.465795 | 2 | 1.714286 | 7 | 1,836 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 55 | 57 | theorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by |
erw [not_forall]
exact ⟨0, by simp⟩
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 60 | 63 | theorem Squarefree.ne_zero [MonoidWithZero R] [Nontrivial R] {m : R} (hm : Squarefree (m : R)) :
m ≠ 0 := by |
rintro rfl
exact not_squarefree_zero hm
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 67 | 72 | theorem Irreducible.squarefree [CommMonoid R] {x : R} (h : Irreducible x) : Squarefree x := by |
rintro y ⟨z, hz⟩
rw [mul_assoc] at hz
rcases h.isUnit_or_isUnit hz with (hu | hu)
· exact hu
· apply isUnit_of_mul_isUnit_left hu
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 92 | 98 | theorem Squarefree.eq_zero_or_one_of_pow_of_not_isUnit [CommMonoid R] {x : R} {n : ℕ}
(h : Squarefree (x ^ n)) (h' : ¬ IsUnit x) :
n = 0 ∨ n = 1 := by |
contrapose! h'
replace h' : 2 ≤ n := by omega
have : x * x ∣ x ^ n := by rw [← sq]; exact pow_dvd_pow x h'
exact h.squarefree_of_dvd this x (refl _)
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 120 | 126 | theorem squarefree_iff_multiplicity_le_one (r : R) :
Squarefree r ↔ ∀ x : R, multiplicity x r ≤ 1 ∨ IsUnit x := by |
refine forall_congr' fun a => ?_
rw [← sq, pow_dvd_iff_le_multiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]
norm_cast
rw [← one_add_one_eq_two]
simpa using PartENat.add_one_le_iff_lt (PartENat.natCast_ne_top 1)
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 147 | 152 | theorem squarefree_iff_no_irreducibles {x : R} (hx₀ : x ≠ 0) :
Squarefree x ↔ ∀ p, Irreducible p → ¬ (p * p ∣ x) := by |
refine ⟨fun h p hp hp' ↦ hp.not_unit (h p hp'), fun h d hd ↦ by_contra fun hdu ↦ ?_⟩
have hd₀ : d ≠ 0 := ne_zero_of_dvd_ne_zero (ne_zero_of_dvd_ne_zero hx₀ hd) (dvd_mul_left d d)
obtain ⟨p, irr, dvd⟩ := WfDvdMonoid.exists_irreducible_factor hdu hd₀
exact h p irr ((mul_dvd_mul dvd dvd).trans hd)
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 154 | 163 | theorem irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree (r : R) :
(∀ x : R, Irreducible x → ¬x * x ∣ r) ↔ (r = 0 ∧ ∀ x : R, ¬Irreducible x) ∨ Squarefree r := by |
refine ⟨fun h ↦ ?_, ?_⟩
· rcases eq_or_ne r 0 with (rfl | hr)
· exact .inl (by simpa using h)
· exact .inr ((squarefree_iff_no_irreducibles hr).mpr h)
· rintro (⟨rfl, h⟩ | h)
· simpa using h
intro x hx t
exact hx.not_unit (h x t)
| 8 | 2,980.957987 | 2 | 1.714286 | 7 | 1,837 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section Fermat
open GaussianInt
| Mathlib/NumberTheory/SumTwoSquares.lean | 33 | 36 | theorem Nat.Prime.sq_add_sq {p : ℕ} [Fact p.Prime] (hp : p % 4 ≠ 3) :
∃ a b : ℕ, a ^ 2 + b ^ 2 = p := by |
apply sq_add_sq_of_nat_prime_of_not_irreducible p
rwa [_root_.irreducible_iff_prime, prime_iff_mod_four_eq_three_of_nat_prime p]
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section General
theorem sq_add_sq_mul {R} [CommRing R] ... | Mathlib/NumberTheory/SumTwoSquares.lean | 56 | 61 | theorem Nat.sq_add_sq_mul {a b x y u v : ℕ} (ha : a = x ^ 2 + y ^ 2) (hb : b = u ^ 2 + v ^ 2) :
∃ r s : ℕ, a * b = r ^ 2 + s ^ 2 := by |
zify at ha hb ⊢
obtain ⟨r, s, h⟩ := _root_.sq_add_sq_mul ha hb
refine ⟨r.natAbs, s.natAbs, ?_⟩
simpa only [Int.natCast_natAbs, sq_abs]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 77 | 81 | theorem ZMod.isSquare_neg_one_of_dvd {m n : ℕ} (hd : m ∣ n) (hs : IsSquare (-1 : ZMod n)) :
IsSquare (-1 : ZMod m) := by |
let f : ZMod n →+* ZMod m := ZMod.castHom hd _
rw [← RingHom.map_one f, ← RingHom.map_neg]
exact hs.map f
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 86 | 94 | theorem ZMod.isSquare_neg_one_mul {m n : ℕ} (hc : m.Coprime n) (hm : IsSquare (-1 : ZMod m))
(hn : IsSquare (-1 : ZMod n)) : IsSquare (-1 : ZMod (m * n)) := by |
have : IsSquare (-1 : ZMod m × ZMod n) := by
rw [show (-1 : ZMod m × ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl]
obtain ⟨x, hx⟩ := hm
obtain ⟨y, hy⟩ := hn
rw [hx, hy]
exact ⟨(x, y), rfl⟩
simpa only [RingEquiv.map_neg_one] using this.map (ZMod.chineseRemainder hc).symm
| 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 98 | 103 | theorem Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one {p n : ℕ} (hpp : p.Prime) (hp : p ∣ n)
(hs : IsSquare (-1 : ZMod n)) : p % 4 ≠ 3 := by |
obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd hp hs
rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h
haveI : Fact p.Prime := ⟨hpp⟩
exact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero h
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 108 | 120 | theorem ZMod.isSquare_neg_one_iff {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q.Prime → q ∣ n → q % 4 ≠ 3 := by |
refine ⟨fun H q hqp hqd => hqp.mod_four_ne_three_of_dvd_isSquare_neg_one hqd H, fun H => ?_⟩
induction' n using induction_on_primes with p n hpp ih
· exact False.elim (hn.ne_zero rfl)
· exact ⟨0, by simp only [mul_zero, eq_iff_true_of_subsingleton]⟩
· haveI : Fact p.Prime := ⟨hpp⟩
have hcp : p.Coprime n ... | 11 | 59,874.141715 | 2 | 1.714286 | 7 | 1,838 |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 125 | 138 | theorem ZMod.isSquare_neg_one_iff' {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 := by |
have help : ∀ a b : ZMod 4, a ≠ 3 → b ≠ 3 → a * b ≠ 3 := by decide
rw [ZMod.isSquare_neg_one_iff hn]
refine ⟨?_, fun H q _ => H⟩
intro H
refine @induction_on_primes _ ?_ ?_ (fun p q hp hq hpq => ?_)
· exact fun _ => by norm_num
· exact fun _ => by norm_num
· replace hp := H hp (dvd_of_mul_right_dvd hpq... | 12 | 162,754.791419 | 2 | 1.714286 | 7 | 1,838 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 68 | 76 | theorem Ideal.finite_factors {I : Ideal R} (hI : I ≠ 0) :
{v : HeightOneSpectrum R | v.asIdeal ∣ I}.Finite := by |
rw [← Set.finite_coe_iff, Set.coe_setOf]
haveI h_fin := fintypeSubtypeDvd I hI
refine
Finite.of_injective (fun v => (⟨(v : HeightOneSpectrum R).asIdeal, v.2⟩ : { x // x ∣ I })) ?_
intro v w hvw
simp? at hvw says simp only [Subtype.mk.injEq] at hvw
exact Subtype.coe_injective ((HeightOneSpectrum.ext_iff... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 81 | 90 | theorem Associates.finite_factors {I : Ideal R} (hI : I ≠ 0) :
∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) = 0 := by |
have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count
(Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by
ext v
simp_rw [Int.natCast_eq_zero]
exact Associates.count_ne_zero_iff_dvd hI v.irreducible
rw [Filter.eventually_cofinite, h_supp]
ex... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 97 | 107 | theorem finite_mulSupport {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => v.maxPowDividing I).Finite :=
haveI h_subset : {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} ⊆
{v : HeightOneSpectrum R |
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) ≠ 0} := by |
intro v hv h_zero
have hv' : v.maxPowDividing I = 1 := by
rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
pow_zero _]
exact hv hv'
Finite.subset (Filter.eventually_cofinite.mp (Associates.finite_factors hI)) h_subset
| 6 | 403.428793 | 2 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 112 | 117 | theorem finite_mulSupport_coe {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)).Finite := by |
rw [mulSupport]
simp_rw [Ne, zpow_natCast, ← FractionalIdeal.coeIdeal_pow, FractionalIdeal.coeIdeal_eq_one]
exact finite_mulSupport hI
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 122 | 127 | theorem finite_mulSupport_inv {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal R⁰ K) ^
(-((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ))).Finite := by |
rw [mulSupport]
simp_rw [zpow_neg, Ne, inv_eq_one]
exact finite_mulSupport_coe hI
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 131 | 144 | theorem finprod_not_dvd (I : Ideal R) (hI : I ≠ 0) :
¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣
∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I := by |
have hf := finite_mulSupport hI
have h_ne_zero : v.maxPowDividing I ≠ 0 := pow_ne_zero _ v.ne_bot
rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]
intro h_contr
have hv_prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
obtain ⟨w, hw, hvw'⟩ :=
Prime.exists_mem... | 11 | 59,874.141715 | 2 | 1.714286 | 7 | 1,839 |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 149 | 156 | theorem Associates.finprod_ne_zero (I : Ideal R) :
Associates.mk (∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I) ≠ 0 := by |
rw [Associates.mk_ne_zero, finprod_def]
split_ifs
· rw [Finset.prod_ne_zero_iff]
intro v _
apply pow_ne_zero _ v.ne_bot
· exact one_ne_zero
| 6 | 403.428793 | 2 | 1.714286 | 7 | 1,839 |
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)
| Mathlib/Data/Nat/Factorization/PrimePow.lean | 20 | 24 | theorem IsPrimePow.minFac_pow_factorization_eq {n : ℕ} (hn : IsPrimePow n) :
n.minFac ^ n.factorization n.minFac = n := by |
obtain ⟨p, k, hp, hk, rfl⟩ := hn
rw [← Nat.prime_iff] at hp
rw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,840 |
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 | 27 | 33 | theorem isPrimePow_of_minFac_pow_factorization_eq {n : ℕ}
(h : n.minFac ^ n.factorization n.minFac = n) (hn : n ≠ 1) : IsPrimePow n := by |
rcases eq_or_ne n 0 with (rfl | hn')
· simp_all
refine ⟨_, _, (Nat.minFac_prime hn).prime, ?_, h⟩
simp [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.support_factorization, hn',
Nat.minFac_prime hn, Nat.minFac_dvd]
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,840 |
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 | 41 | 54 | theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k := by |
rw [isPrimePow_nat_iff]
refine exists₂_congr fun p k => ?_
constructor
· rintro ⟨hp, hk, hn⟩
exact ⟨hk, by rw [← hn, Nat.Prime.factorization_pow hp]⟩
· rintro ⟨hk, hn⟩
have hn0 : n ≠ 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw ... | 12 | 162,754.791419 | 2 | 1.714286 | 7 | 1,840 |
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 | 57 | 60 | theorem isPrimePow_iff_card_primeFactors_eq_one {n : ℕ} :
IsPrimePow n ↔ n.primeFactors.card = 1 := by |
simp_rw [isPrimePow_iff_factorization_eq_single, ← Nat.support_factorization,
Finsupp.card_support_eq_one', pos_iff_ne_zero]
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,840 |
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... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,840 |
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 | 76 | 84 | theorem exists_ord_compl_eq_one_iff_isPrimePow {n : ℕ} (hn : n ≠ 1) :
IsPrimePow n ↔ ∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by |
refine ⟨fun h => IsPrimePow.exists_ord_compl_eq_one h, fun h => ?_⟩
rcases h with ⟨p, pp, h⟩
rw [isPrimePow_nat_iff]
rw [← Nat.eq_of_dvd_of_div_eq_one (Nat.ord_proj_dvd n p) h] at hn ⊢
refine ⟨p, n.factorization p, pp, ?_, by simp⟩
contrapose! hn
simp [Nat.le_zero.1 hn]
| 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,840 |
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 | 89 | 108 | theorem isPrimePow_iff_unique_prime_dvd {n : ℕ} : IsPrimePow n ↔ ∃! p : ℕ, p.Prime ∧ p ∣ n := by |
rw [isPrimePow_nat_iff]
constructor
· rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ⟨hp, dvd_pow_self _ hk.ne'⟩, ?_⟩
rintro q ⟨hq, hq'⟩
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro ⟨p, ⟨hp, hn⟩, hq⟩
rcases eq_or_ne n 0 with (rfl | hn₀)
· cases (hq 2 ⟨Nat.prime_two, dvd_zero... | 19 | 178,482,300.963187 | 2 | 1.714286 | 7 | 1,840 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 74 | 79 | theorem taylorWithin_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithin f (n + 1) s x₀ = taylorWithin f n s x₀ +
PolynomialModule.comp (Polynomial.X - Polynomial.C x₀)
(PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n + 1) s x₀)) := by |
dsimp only [taylorWithin]
rw [Finset.sum_range_succ]
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 83 | 92 | theorem taylorWithinEval_succ (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f (n + 1) s x₀ x = taylorWithinEval f n s x₀ x +
(((n + 1 : ℝ) * n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ := by |
simp_rw [taylorWithinEval, taylorWithin_succ, LinearMap.map_add, PolynomialModule.comp_eval]
congr
simp only [Polynomial.eval_sub, Polynomial.eval_X, Polynomial.eval_C,
PolynomialModule.eval_single, mul_inv_rev]
dsimp only [taylorCoeffWithin]
rw [← mul_smul, mul_comm, Nat.factorial_succ, Nat.cast_mul, Na... | 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 97 | 102 | theorem taylor_within_zero_eval (f : ℝ → E) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f 0 s x₀ x = f x₀ := by |
dsimp only [taylorWithinEval]
dsimp only [taylorWithin]
dsimp only [taylorCoeffWithin]
simp
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 107 | 111 | theorem taylorWithinEval_self (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ : ℝ) :
taylorWithinEval f n s x₀ x₀ = f x₀ := by |
induction' n with k hk
· exact taylor_within_zero_eval _ _ _ _
simp [hk]
| 3 | 20.085537 | 1 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 114 | 120 | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k ∈ Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by |
induction' n with k hk
· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 125 | 136 | theorem continuousOn_taylorWithinEval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : Set ℝ}
(hs : UniqueDiffOn ℝ s) (hf : ContDiffOn ℝ n f s) :
ContinuousOn (fun t => taylorWithinEval f n s t x) s := by |
simp_rw [taylor_within_apply]
refine continuousOn_finset_sum (Finset.range (n + 1)) fun i hi => ?_
refine (continuousOn_const.mul ((continuousOn_const.sub continuousOn_id).pow _)).smul ?_
rw [contDiffOn_iff_continuousOn_differentiableOn_deriv hs] at hf
cases' hf with hf_left
specialize hf_left i
simp onl... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 141 | 146 | theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by |
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,841 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 29 | 34 | theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
Inducing (s.restrictPreimage f) := by |
simp_rw [← inducing_subtype_val.of_comp_iff, inducing_iff_nhds, restrictPreimage,
MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h ⊢
intro a
rw [← h, ← inducing_subtype_val.nhds_eq_comap]
| 4 | 54.59815 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 66 | 72 | theorem IsClosedMap.restrictPreimage (H : IsClosedMap f) (s : Set β) :
IsClosedMap (s.restrictPreimage f) := by |
intro t
suffices ∀ u, IsClosed u → Subtype.val ⁻¹' u = t →
∃ v, IsClosed v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
simpa [isClosed_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 78 | 84 | theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set β) :
IsOpenMap (s.restrictPreimage f) := by |
intro t
suffices ∀ u, IsOpen u → Subtype.val ⁻¹' u = t →
∃ v, IsOpen v ∧ Subtype.val ⁻¹' v = s.restrictPreimage f '' t by
simpa [isOpen_induced_iff]
exact fun u hu e => ⟨f '' u, H u hu, by simp [← e, image_restrictPreimage]⟩
| 5 | 148.413159 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 90 | 98 | theorem isOpen_iff_inter_of_iSup_eq_top (s : Set β) : IsOpen s ↔ ∀ i, IsOpen (s ∩ U i) := by |
constructor
· exact fun H i => H.inter (U i).2
· intro H
have : ⋃ i, (U i : Set β) = Set.univ := by
convert congr_arg (SetLike.coe) hU
simp
rw [← s.inter_univ, ← this, Set.inter_iUnion]
exact isOpen_iUnion H
| 8 | 2,980.957987 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 101 | 108 | theorem isOpen_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
IsOpen s ↔ ∀ i, IsOpen ((↑) ⁻¹' s : Set (U i)) := by |
-- Porting note: rewrote to avoid ´simp´ issues
rw [isOpen_iff_inter_of_iSup_eq_top hU s]
refine forall_congr' fun i => ?_
rw [(U _).2.openEmbedding_subtype_val.open_iff_image_open]
erw [Set.image_preimage_eq_inter_range]
rw [Subtype.range_coe, Opens.carrier_eq_coe]
| 6 | 403.428793 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 111 | 113 | theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by |
simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU sᶜ
| 1 | 2.718282 | 0 | 1.714286 | 7 | 1,842 |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 116 | 126 | theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by |
refine ⟨fun h i => h.restrictPreimage _, ?_⟩
rintro H s hs
rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
intro i
convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩
ext ⟨x, hx⟩
suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by
simpa [Set.restrictPreimage, ← Subtype.coe_inj]
exac... | 9 | 8,103.083928 | 2 | 1.714286 | 7 | 1,842 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 58 | 83 | theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
Cardinal.lift.{w'} #ι = Cardinal.lift.{w} #ι' := by |
classical
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite ι
· -- `v` is a finite basis, so by `basis_finite_of_finite_spans` so is `v'`.
-- haveI : Finite (range v) := Set.finite_range v
haveI := basis_finite_of_finite_spans _ (Set.finite_range v) v.span_eq v'
cases nonempty_f... | 24 | 26,489,122,129.84347 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 109 | 118 | theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M} [Fintype w]
(s : span R w = ⊤) : Fintype.card ι ≤ Fintype.card w := by |
-- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
-- by expressing a linear combination in `w` as a linear combination in `ι`.
fapply card_le_of_surjective' R
· exact b.repr.toLinearMap.comp (Finsupp.total w M R (↑))
· apply Surjective.comp (g := b.repr.toLinearMap)
· apply LinearEquiv.sur... | 8 | 2,980.957987 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 125 | 132 | theorem basis_le_span' {ι : Type*} (b : Basis ι R M) {w : Set M} [Fintype w] (s : span R w = ⊤) :
#ι ≤ Fintype.card w := by |
haveI := nontrivial_of_invariantBasisNumber R
haveI := basis_finite_of_finite_spans w (toFinite _) s b
cases nonempty_fintype ι
rw [Cardinal.mk_fintype ι]
simp only [Cardinal.natCast_le]
exact Basis.le_span'' b s
| 6 | 403.428793 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 140 | 164 | theorem Basis.le_span {J : Set M} (v : Basis ι R M) (hJ : span R J = ⊤) : #(range v) ≤ #J := by |
haveI := nontrivial_of_invariantBasisNumber R
cases fintypeOrInfinite J
· rw [← Cardinal.lift_le, Cardinal.mk_range_eq_of_injective v.injective, Cardinal.mk_fintype J]
convert Cardinal.lift_le.{v}.2 (basis_le_span' v hJ)
simp
· let S : J → Set ι := fun j => ↑(v.repr j).support
let S' : J → Set M :=... | 24 | 26,489,122,129.84347 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 177 | 191 | theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by |
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_o... | 12 | 162,754.791419 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 214 | 220 | theorem linearIndependent_le_span' {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : range v ≤ span R w) : #ι ≤ Fintype.card w := by |
haveI : Finite ι := i.finite_of_le_span_finite v w s
letI := Fintype.ofFinite ι
rw [Cardinal.mk_fintype]
simp only [Cardinal.natCast_le]
exact linearIndependent_le_span_aux' v i w s
| 5 | 148.413159 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 228 | 232 | theorem linearIndependent_le_span {ι : Type*} (v : ι → M) (i : LinearIndependent R v) (w : Set M)
[Fintype w] (s : span R w = ⊤) : #ι ≤ Fintype.card w := by |
apply linearIndependent_le_span' v i w
rw [s]
exact le_top
| 3 | 20.085537 | 1 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 236 | 238 | theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by |
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
| 1 | 2.718282 | 0 | 1.727273 | 11 | 1,843 |
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