name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
Std.Tactic.BVDecide.BVExpr.bitblast.blastUmod.denote_go_eq_divRec_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Umod.lean | theorem denote_go_eq_divRec_r (aig : AIG α) (assign : α → Bool) (curr : Nat) (lhs rhs rbv qbv : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d q r : AIG.RefVec aig w) (wn wr : Nat)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx ... | case succ
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs... | rw [go, BitVec.divRec_succ, BitVec.divSubtractShift] | case succ
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
assign : α → Bool
lhs rhs : BitVec w
curr : Nat
ih :
∀ (aig : AIG α) (rbv qbv : BitVec w) (falseRef trueRef : aig.Ref) (n d q r : aig.RefVec w) (wn wr : Nat),
(∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs... | 123e2e165ae62131 |
List.erase_filter | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem erase_filter [LawfulBEq α] (f : α → Bool) (l : List α) :
(filter f l).erase a = filter f (l.erase a) | case pos
α : Type u_1
inst✝¹ : BEq α
a : α
inst✝ : LawfulBEq α
f : α → Bool
x : α
xs : List α
ih : (filter f xs).erase a = filter f (xs.erase a)
h : a = x
⊢ (if f x = true then x :: filter f xs else filter f xs).erase x = filter f xs | split | case pos.isTrue
α : Type u_1
inst✝¹ : BEq α
a : α
inst✝ : LawfulBEq α
f : α → Bool
x : α
xs : List α
ih : (filter f xs).erase a = filter f (xs.erase a)
h : a = x
h✝ : f x = true
⊢ (x :: filter f xs).erase x = filter f xs
case pos.isFalse
α : Type u_1
inst✝¹ : BEq α
a : α
inst✝ : LawfulBEq α
f : α → Bool
x : α
xs : Lis... | 05d8083de9ecc8f1 |
iSupIndep.injOn | Mathlib/Order/SupIndep.lean | theorem iSupIndep.injOn (ht : iSupIndep t) : InjOn t {i | t i ≠ ⊥} | α : Type u_1
ι : Type u_3
inst✝ : CompleteLattice α
t : ι → α
ht : iSupIndep t
i : ι
a✝ : i ∈ {i | t i ≠ ⊥}
j : ι
hj : t j ≠ ⊥
h : t i = t j
contra : j ≠ i
⊢ t j ≤ ⨆ k, ⨆ (_ : k ≠ i), t k | exact le_iSup₂ (f := fun x _ ↦ t x) j contra | no goals | 8c90a91a85885220 |
continuousWithinAt_right_of_monotoneOn_of_exists_between | Mathlib/Topology/Order/MonotoneContinuity.lean | theorem continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
ContinuousWithinAt f (Ici a) a | α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[≥] a
hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b
⊢ ContinuousWithinAt f (Ici a) a | have ha : a ∈ Ici a := left_mem_Ici | α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : LinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
s : Set α
a : α
h_mono : MonotoneOn f s
hs : s ∈ 𝓝[≥] a
hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b
ha : a ∈ Ici a
⊢ ContinuousWithi... | 2a29b6e29ca65136 |
ciSup_subtype | Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean | theorem ciSup_subtype [Nonempty ι] {p : ι → Prop} [Nonempty (Subtype p)] {f : Subtype p → α}
(hf : BddAbove (Set.range f)) (hf' : sSup ∅ ≤ iSup f) :
iSup f = ⨆ (i) (h : p i), f ⟨i, h⟩ | case refine_1
α : Type u_1
ι : Sort u_4
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : Nonempty ι
p : ι → Prop
inst✝ : Nonempty (Subtype p)
f : Subtype p → α
hf : BddAbove (range f)
hf' : sSup ∅ ≤ iSup f
i : ι
h : p i
this : f ⟨i, h⟩ = (fun i => ⨆ (h : p i), f ⟨i, h⟩) i
⊢ (fun i => ⨆ (h : p i), f ⟨i, h⟩) i ≤ ⨆ i, ⨆ (... | refine le_ciSup (f := (fun i : ι ↦ ⨆ (h : p i), f ⟨i, h⟩)) ?_ i | case refine_1
α : Type u_1
ι : Sort u_4
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : Nonempty ι
p : ι → Prop
inst✝ : Nonempty (Subtype p)
f : Subtype p → α
hf : BddAbove (range f)
hf' : sSup ∅ ≤ iSup f
i : ι
h : p i
this : f ⟨i, h⟩ = (fun i => ⨆ (h : p i), f ⟨i, h⟩) i
⊢ BddAbove (range fun i => ⨆ (h : p i), f ⟨i, h... | 41d37decd86a07eb |
Matroid.cRk_mono | Mathlib/Data/Matroid/Rank/Cardinal.lean | theorem cRk_mono (M : Matroid α) : Monotone M.cRk | α : Type u
M : Matroid α
X Y : Set α
hXY : X ⊆ Y
I : Set α
hIX : M.IsBasis' I X
⊢ #↑I ≤ M.cRk Y | obtain ⟨J, hJ, hIJ⟩ := hIX.indep.subset_isBasis'_of_subset (hIX.subset.trans hXY) | case intro.intro
α : Type u
M : Matroid α
X Y : Set α
hXY : X ⊆ Y
I : Set α
hIX : M.IsBasis' I X
J : Set α
hJ : M.IsBasis' J Y
hIJ : I ⊆ J
⊢ #↑I ≤ M.cRk Y | e3dc976a5390a380 |
MeasureTheory.ae_eq_of_ae_le_of_lintegral_le | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem ae_eq_of_ae_le_of_lintegral_le {f g : α → ℝ≥0∞} (hfg : f ≤ᵐ[μ] g) (hf : ∫⁻ x, f x ∂μ ≠ ∞)
(hg : AEMeasurable g μ) (hgf : ∫⁻ x, g x ∂μ ≤ ∫⁻ x, f x ∂μ) : f =ᵐ[μ] g | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᶠ[ae μ] g
hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hg : AEMeasurable g μ
hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ
n : ℕ
⊢ ∀ᵐ (x : α) ∂μ, g x < f x + (↑n)⁻¹ | simp only [ae_iff, not_lt] | α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hfg : f ≤ᶠ[ae μ] g
hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤
hg : AEMeasurable g μ
hgf : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f x ∂μ
n : ℕ
⊢ μ {a | f a + (↑n)⁻¹ ≤ g a} = 0 | acae6282b7426a58 |
Multiset.lt_cons_self | Mathlib/Data/Multiset/ZeroCons.lean | theorem lt_cons_self (s : Multiset α) (a : α) : s < a ::ₘ s :=
Quot.inductionOn s fun l =>
suffices l <+~ a :: l ∧ ¬l ~ a :: l by simpa [lt_iff_le_and_ne]
⟨(sublist_cons_self _ _).subperm,
fun p => _root_.ne_of_lt (lt_succ_self (length l)) p.length_eq⟩
| α : Type u_1
s : Multiset α
a : α
l : List α
this : l <+~ a :: l ∧ ¬l ~ a :: l
⊢ Quot.mk (⇑(isSetoid α)) l < a ::ₘ Quot.mk (⇑(isSetoid α)) l | simpa [lt_iff_le_and_ne] | no goals | e5d23f1b5d1546c1 |
Cardinal.mk_option | Mathlib/SetTheory/Cardinal/Basic.lean | theorem mk_option {α : Type u} : #(Option α) = #α + 1 | α : Type u
⊢ #(Option α) = #α + 1 | rw [(Equiv.optionEquivSumPUnit.{u, u} α).cardinal_eq, mk_sum, mk_eq_one PUnit, lift_id, lift_id] | no goals | 43638101d0e4e4a1 |
MvPolynomial.comp_eval₂Hom | Mathlib/Algebra/MvPolynomial/Eval.lean | theorem comp_eval₂Hom [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₁) (φ : S₁ →+* S₂) :
φ.comp (eval₂Hom f g) = eval₂Hom (φ.comp f) fun i => φ (g i) | case hX
R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
inst✝ : CommSemiring S₂
f : R →+* S₁
g : σ → S₁
φ : S₁ →+* S₂
⊢ ∀ (i : σ), (φ.comp (eval₂Hom f g)) (X i) = (eval₂Hom (φ.comp f) fun i => φ (g i)) (X i) | intro i | case hX
R : Type u
S₁ : Type v
S₂ : Type w
σ : Type u_1
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
inst✝ : CommSemiring S₂
f : R →+* S₁
g : σ → S₁
φ : S₁ →+* S₂
i : σ
⊢ (φ.comp (eval₂Hom f g)) (X i) = (eval₂Hom (φ.comp f) fun i => φ (g i)) (X i) | 1db4c03b0dbe6e59 |
Isometry.hausdorffMeasure_image | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem hausdorffMeasure_image (hf : Isometry f) (hd : 0 ≤ d ∨ Surjective f) (s : Set X) :
μH[d] (f '' s) = μH[d] s | X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
d : ℝ
hf : Isometry f
hd : 0 ≤ d ∨ Surjective f
s : Set X
⊢ 0 ≤ d → Monotone fun r => r ^ d | exact ENNReal.monotone_rpow_of_nonneg | no goals | fa3ad9276810d865 |
Finset.pow_ssubset_pow_succ_of_pow_ne_closure | Mathlib/Geometry/Group/Growth/LinearLowerBound.lean | @[to_additive]
lemma pow_ssubset_pow_succ_of_pow_ne_closure (hX₁ : (1 : G) ∈ X) (hX : X.Nontrivial)
(hXclosure : (X ^ n : Set G) ≠ closure (X : Set G)) : X ^ n ⊂ X ^ (n + 1) | G✝ : Type u_1
inst✝³ : Group G✝
inst✝² : DecidableEq G✝
X✝ : Finset G✝
n : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : True
hXn : X = X ^ 2
x : G
hx : x ∈ { carrier := ↑X, mul_mem' := ⋯, one_mem' := hX₁ }.carrier
⊢ x⁻¹ ∈ { carrier := ↑X, mul_mem' := ⋯, one_mem' ... | dsimp at * | G✝ : Type u_1
inst✝³ : Group G✝
inst✝² : DecidableEq G✝
X✝ : Finset G✝
n : ℕ
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq G
X : Finset G
hX₁ : 1 ∈ X
hX : X.Nontrivial
hn : True
hXn : X = X ^ 2
x : G
hx : x ∈ ↑X
⊢ x⁻¹ ∈ ↑X | baf8f0ea21c560bf |
DyckWord.firstReturn_pos | Mathlib/Combinatorics/Enumerative/DyckWord.lean | lemma firstReturn_pos : 0 < p.firstReturn | p : DyckWord
h : p ≠ 0
f : findIdx (fun i => decide (count U (List.take (i + 1) ↑p) = count D (List.take (i + 1) ↑p))) (range (↑p).length) = 0
⊢ 0 < (range (↑p).length).length | rw [length_range, length_pos] | p : DyckWord
h : p ≠ 0
f : findIdx (fun i => decide (count U (List.take (i + 1) ↑p) = count D (List.take (i + 1) ↑p))) (range (↑p).length) = 0
⊢ ↑p ≠ [] | d21cfbbb2df312ba |
collapse_modular | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | lemma collapse_modular [ExistsAddOfLE β]
(hu : a ∉ u) (h₁ : 0 ≤ f₁) (h₂ : 0 ≤ f₂) (h₃ : 0 ≤ f₃) (h₄ : 0 ≤ f₄)
(h : ∀ ⦃s⦄, s ⊆ insert a u → ∀ ⦃t⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t))
(𝒜 ℬ : Finset (Finset α)) :
∀ ⦃s⦄, s ⊆ u → ∀ ⦃t⦄, t ⊆ u → collapse 𝒜 a f₁ s * collapse ℬ a f₂ t ≤
... | case pos
α : Type u_1
β : Type u_2
inst✝² : DecidableEq α
inst✝¹ : LinearOrderedCommSemiring β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ ... | exact mul_le_mul_of_nonneg_right (le_collapse_of_mem ‹_› h₃ rfl <| inter_mem_infs ‹_› ‹_›) <|
add_nonneg (h₄ _) <| h₄ _ | no goals | 4b3d90b1fc92ae23 |
Finset.noncommProd_union_of_disjoint | Mathlib/Data/Finset/NoncommProd.lean | theorem noncommProd_union_of_disjoint [DecidableEq α] {s t : Finset α} (h : Disjoint s t)
(f : α → β) (comm : { x | x ∈ s ∪ t }.Pairwise (Commute on f)) :
noncommProd (s ∪ t) f comm =
noncommProd s f (comm.mono <| coe_subset.2 subset_union_left) *
noncommProd t f (comm.mono <| coe_subset.2 subset_... | case h.e'_2
α : Type u_3
β : Type u_4
inst✝¹ : Monoid β
inst✝ : DecidableEq α
f : α → β
sl : List α
sl' : sl.Nodup
tl : List α
tl' : tl.Nodup
h : sl.Disjoint tl
comm : {x | x ∈ sl.toFinset ∪ tl.toFinset}.Pairwise (Commute on f)
⊢ ↑{ val := ↑(sl ++ tl), nodup := ⋯ } = {x | x ∈ sl.toFinset ∪ tl.toFinset} | simp [Set.ext_iff] | no goals | b1392de476fe3f45 |
List.forM_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem forM_append [Monad m] [LawfulMonad m] (l₁ l₂ : List α) (f : α → m PUnit) :
forM (l₁ ++ l₂) f = (do forM l₁ f; forM l₂ f) | m : Type u_1 → Type u_2
α : Type u_3
inst✝¹ : Monad m
inst✝ : LawfulMonad m
l₁ l₂ : List α
f : α → m PUnit
⊢ forM (l₁ ++ l₂) f = do
forM l₁ f
forM l₂ f | induction l₁ <;> simp [*] | no goals | 1efb461b1fe0cfe0 |
UpperHalfPlane.image_coe_closedBall | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) | case h.mpr
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ closedBall (↑(z.center r)) (z.im * Real.sinh r) → w ∈ UpperHalfPlane.coe '' closedBall z r | intro hw | case h.mpr
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ closedBall (↑(z.center r)) (z.im * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r | 9dc4a85b3fc2501b |
Surreal.Multiplication.ih₂₁ | Mathlib/SetTheory/Surreal/Multiplication.lean | lemma ih₂₁ (ih' : ∀ a, ArgsRel a (Args.P24 x₁ x₂ y) → P124 a) : IH24 x₂ x₁ y := ih₁₂ <| by
simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢
suffices {x₁, y, x₂} = {x₂, y, x₁} by rwa [← this]
dsimp only [Multiset.insert_eq_cons, ← Multiset.singleton_add] at ih' ⊢
abel
| x₁ x₂ y : PGame
ih' : ∀ (a : Args), ArgsRel a (Args.P24 x₁ x₂ y) → P124 a
⊢ ∀ (a : Args), ArgsRel a (Args.P24 x₂ x₁ y) → P124 a | simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' ⊢ | x₁ x₂ y : PGame
ih' :
∀ (a : Args),
TransGen (CutExpand IsOption)
(match a with
| Args.P1 x y => {y, x}
| Args.P24 x₁ x₂ y => {x₁, y, x₂})
{x₁, y, x₂} →
P124 a
⊢ ∀ (a : Args),
TransGen (CutExpand IsOption)
(match a with
| Args.P1 x y => {y, x}
| Ar... | d29f04d7bd0d5fb9 |
DirectSum.AddSubmonoidClass.IsHomogeneous.mem_iff | Mathlib/Algebra/DirectSum/Decomposition.lean | theorem AddSubmonoidClass.IsHomogeneous.mem_iff
{P : Type*} [SetLike P M] [AddSubmonoidClass P M] (p : P)
(hp : SetLike.IsHomogeneous ℳ p) {x} :
x ∈ p ↔ ∀ i, (decompose ℳ x i : M) ∈ p | ι : Type u_1
M : Type u_3
σ : Type u_4
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid M
inst✝⁴ : SetLike σ M
inst✝³ : AddSubmonoidClass σ M
ℳ : ι → σ
inst✝² : Decomposition ℳ
P : Type u_5
inst✝¹ : SetLike P M
inst✝ : AddSubmonoidClass P M
p : P
hp : SetLike.IsHomogeneous ℳ p
x : M
⊢ x ∈ p ↔ ∀ (i : ι), ↑(((decompose ℳ) x... | classical
refine ⟨fun hx i ↦ hp i hx, fun hx ↦ ?_⟩
rw [← DirectSum.sum_support_decompose ℳ x]
exact sum_mem (fun i _ ↦ hx i) | no goals | 5718da3f2b805fbd |
List.lookmap.go_append | Mathlib/Data/List/Lookmap.lean | theorem lookmap.go_append (l : List α) (acc : Array α) :
lookmap.go f l acc = acc.toListAppend (lookmap f l) | case cons
α : Type u_1
f : α → Option α
acc : Array α
hd : α
tl : List α
⊢ (match f hd with
| some b => acc.toListAppend (b :: tl)
| none => go f tl (acc.push hd)) =
acc.toListAppend
(match f hd with
| some b => #[].toListAppend (b :: tl)
| none => go f tl (#[].push hd)) | cases f hd with
| none =>
simp only [go_append tl _, Array.toListAppend_eq, append_assoc, Array.push_toList]
rfl
| some a => rfl | no goals | fd84e79e13d58a75 |
Nat.mod_two_add_add_odd_mod_two | Mathlib/Algebra/Ring/Parity.lean | lemma mod_two_add_add_odd_mod_two (m : ℕ) {n : ℕ} (hn : Odd n) : m % 2 + (m + n) % 2 = 1 :=
((even_or_odd m).elim fun hm ↦ by rw [even_iff.1 hm, odd_iff.1 (hm.add_odd hn)]) fun hm ↦ by
rw [odd_iff.1 hm, even_iff.1 (hm.add_odd hn)]
| m n : ℕ
hn : Odd n
hm : Odd m
⊢ m % 2 + (m + n) % 2 = 1 | rw [odd_iff.1 hm, even_iff.1 (hm.add_odd hn)] | no goals | 8665ec8b15bbd0cc |
Equiv.Perm.IsThreeCycle.isThreeCycle_sq | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem isThreeCycle_sq {g : Perm α} (ht : IsThreeCycle g) : IsThreeCycle (g * g) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
ht : g.IsThreeCycle
⊢ (g * g).IsThreeCycle | rw [← pow_two, ← card_support_eq_three_iff, support_pow_coprime, ht.card_support] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
g : Perm α
ht : g.IsThreeCycle
⊢ Nat.Coprime 2 (_root_.orderOf g) | bc88a8d9742619dc |
biSup_lt_eq_iSup | Mathlib/Order/CompleteLattice.lean | lemma biSup_lt_eq_iSup {ι : Type*} [LT ι] [NoMaxOrder ι] {f : ι → α} :
⨆ (i) (j < i), f j = ⨆ i, f i | case a
α : Type u_1
inst✝² : CompleteLattice α
ι : Type u_8
inst✝¹ : LT ι
inst✝ : NoMaxOrder ι
f : ι → α
⊢ ⨆ i, f i ≤ ⨆ i, ⨆ j, ⨆ (_ : j < i), f j | refine iSup_le fun j ↦ ?_ | case a
α : Type u_1
inst✝² : CompleteLattice α
ι : Type u_8
inst✝¹ : LT ι
inst✝ : NoMaxOrder ι
f : ι → α
j : ι
⊢ f j ≤ ⨆ i, ⨆ j, ⨆ (_ : j < i), f j | 8a880b742010296f |
Complex.logDeriv_sin | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | theorem Complex.logDeriv_sin : logDeriv (Complex.sin) = Complex.cot | case h
x✝ : ℂ
⊢ logDeriv sin x✝ = x✝.cot | rw [logDeriv, Complex.deriv_sin, Pi.div_apply, Complex.cot] | no goals | df53217c06fc5292 |
PNat.count_factorMultiset | Mathlib/Data/PNat/Factors.lean | theorem count_factorMultiset (m : ℕ+) (p : Nat.Primes) (k : ℕ) :
(p : ℕ+) ^ k ∣ m ↔ k ≤ m.factorMultiset.count p | case a.h.e'_3.h
m : ℕ+
p : Nat.Primes
k : ℕ
e_1✝ : PrimeMultiset = Multiset Nat.Primes
⊢ Multiset.card (k • PrimeMultiset.ofPrime p) = k ∧ ∀ b ∈ k • PrimeMultiset.ofPrime p, b = p | constructor | case a.h.e'_3.h.left
m : ℕ+
p : Nat.Primes
k : ℕ
e_1✝ : PrimeMultiset = Multiset Nat.Primes
⊢ Multiset.card (k • PrimeMultiset.ofPrime p) = k
case a.h.e'_3.h.right
m : ℕ+
p : Nat.Primes
k : ℕ
e_1✝ : PrimeMultiset = Multiset Nat.Primes
⊢ ∀ b ∈ k • PrimeMultiset.ofPrime p, b = p | e40b7dbbae73693a |
AffineIndependent.affineSpan_disjoint_of_disjoint | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | theorem AffineIndependent.affineSpan_disjoint_of_disjoint [Nontrivial k] {p : ι → P}
(ha : AffineIndependent k p) {s1 s2 : Set ι} (hd : Disjoint s1 s2) :
Disjoint (affineSpan k (p '' s1) : Set P) (affineSpan k (p '' s2)) | k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s1 s2 : Set ι
hd : Disjoint s1 s2
⊢ Disjoint ↑(affineSpan k (p '' s1)) ↑(affineSpan k (p '' s2)) | refine Set.disjoint_left.2 fun p0 hp0s1 hp0s2 => ?_ | k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : Ring k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
ι : Type u_4
inst✝ : Nontrivial k
p : ι → P
ha : AffineIndependent k p
s1 s2 : Set ι
hd : Disjoint s1 s2
p0 : P
hp0s1 : p0 ∈ ↑(affineSpan k (p '' s1))
hp0s2 : p0 ∈ ↑(affineSpan k (p '' s2))
⊢ Fals... | 4f622f10dcf2d681 |
Cardinal.powerlt_aleph0_le | Mathlib/SetTheory/Cardinal/Arithmetic.lean | theorem powerlt_aleph0_le (c : Cardinal) : c ^< ℵ₀ ≤ max c ℵ₀ | case inr
c : Cardinal.{u_1}
h : c < ℵ₀
⊢ ∀ x < ℵ₀, c ^ x ≤ c ⊔ ℵ₀ | exact fun c' hc' => (power_lt_aleph0 h hc').le.trans (le_max_right _ _) | no goals | 7ba0d3bf557c7250 |
WittVector.mul_pow_charP_coeff_zero | Mathlib/RingTheory/WittVector/Identities.lean | theorem mul_pow_charP_coeff_zero [CharP R p] (x : 𝕎 R) {m n : ℕ} (h : m < n) :
(x * p ^ n).coeff m = 0 | case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
x : 𝕎 R
n : ℕ
ih : ∀ {m : ℕ}, m < n → (x * ↑p ^ n).coeff m = 0
m : ℕ
h : m < n + 1
⊢ (x * ↑p ^ (n + 1)).coeff m = 0 | rw [pow_succ, ← mul_assoc] | case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
x : 𝕎 R
n : ℕ
ih : ∀ {m : ℕ}, m < n → (x * ↑p ^ n).coeff m = 0
m : ℕ
h : m < n + 1
⊢ (x * ↑p ^ n * ↑p).coeff m = 0 | 47c51437be6452d5 |
MeasureTheory.innerRegularWRT_isCompact_isClosed_iff_innerRegularWRT_isCompact_closure | Mathlib/MeasureTheory/Measure/RegularityCompacts.lean | lemma innerRegularWRT_isCompact_isClosed_iff_innerRegularWRT_isCompact_closure
[TopologicalSpace α] [R1Space α] :
μ.InnerRegularWRT (fun s ↦ IsCompact s ∧ IsClosed s) IsClosed
↔ μ.InnerRegularWRT (IsCompact ∘ closure) IsClosed | case mp.intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT (fun s => IsCompact s ∧ IsClosed s) IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
K : Set α
hK1 : K ⊆ A
hK4 : r < μ K
hK2 : IsCompact K
right✝ : IsClosed K... | refine ⟨K, hK1, ?_, hK4⟩ | case mp.intro.intro.intro.intro
α : Type u_1
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace α
inst✝ : R1Space α
h : μ.InnerRegularWRT (fun s => IsCompact s ∧ IsClosed s) IsClosed
A : Set α
hA : IsClosed A
r : ℝ≥0∞
hr : r < μ A
K : Set α
hK1 : K ⊆ A
hK4 : r < μ K
hK2 : IsCompact K
right✝ : IsClosed K... | 66f3b6035580f668 |
IntermediateField.isSplittingField_iff | Mathlib/FieldTheory/SplittingField/IsSplittingField.lean | theorem IntermediateField.isSplittingField_iff :
p.IsSplittingField K F ↔ p.Splits (algebraMap K F) ∧ F = adjoin K (p.rootSet L) | K : Type v
L : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
p : K[X]
F : IntermediateField K L
this : ?m.53096 → (Algebra.adjoin K (p.rootSet ↥F) = ⊤ ↔ F = adjoin K (p.rootSet L))
⊢ IsSplittingField K (↥F) p ↔ Splits (algebraMap K ↥F) p ∧ F = adjoin K (p.rootSet L) | exact ⟨fun h ↦ ⟨h.1, (this h.1).mp h.2⟩, fun h ↦ ⟨h.1, (this h.1).mpr h.2⟩⟩ | no goals | 98b3e50465dee2ba |
Path.trans_range | Mathlib/Topology/Path.lean | theorem trans_range {a b c : X} (γ₁ : Path a b) (γ₂ : Path b c) :
range (γ₁.trans γ₂) = range γ₁ ∪ range γ₂ | case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : ¬t = 0
⊢ { toFun := (fun t => if t ≤ 1 / 2 then γ₁.extend (2 * t) else γ₂.extend (2 * t - 1)) ∘ Subtype.val,
continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
... | replace h : t ≠ 0 := h | case h
X : Type u_1
inst✝ : TopologicalSpace X
a b c : X
γ₁ : Path a b
γ₂ : Path b c
x : X
t : ℝ
ht0 : 0 ≤ t
ht1 : t ≤ 1
hxt : γ₂ ⟨t, ⋯⟩ = x
h : t ≠ 0
⊢ { toFun := (fun t => if t ≤ 1 / 2 then γ₁.extend (2 * t) else γ₂.extend (2 * t - 1)) ∘ Subtype.val,
continuous_toFun := ⋯, source' := ⋯, target' := ⋯ }
⟨... | 4045b3b0d5c91c52 |
BoxIntegral.Prepartition.IsPartition.compl_eq_bot | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | theorem IsPartition.compl_eq_bot {π : Prepartition I} (h : IsPartition π) : π.compl = ⊥ | ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
h : π.IsPartition
⊢ π.compl = ⊥ | rw [← iUnion_eq_empty, iUnion_compl, h.iUnion_eq, diff_self] | no goals | 784fee7a9144e16b |
mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | /-- The composition of the derivative of the inverse of `extChartAt` with the derivative of
`extChartAt` gives the identity.
Version where the basepoint belongs to `(extChartAt I x).target`. -/
lemma mfderivWithin_extChartAt_symm_comp_mfderiv_extChartAt
{y : E} (hy : y ∈ (extChartAt I x).target) :
(mfderivWithi... | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).ta... | rw [this, ← mfderivWithin_comp_of_eq] | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
inst✝ : IsManifold I 1 M
x : M
y : E
hy : y ∈ (extChartAt I x).ta... | b3cda8e75032e62a |
OrdinalApprox.lfpApprox_monotone | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | theorem lfpApprox_monotone : Monotone (lfpApprox f x) | α : Type u
inst✝ : CompleteLattice α
f : α →o α
x : α
⊢ Monotone (lfpApprox f x) | intros a b h | α : Type u
inst✝ : CompleteLattice α
f : α →o α
x : α
a b : Ordinal.{u}
h : a ≤ b
⊢ lfpApprox f x a ≤ lfpApprox f x b | 4ebcb31c8da6c33d |
IntermediateField.exists_finset_of_mem_adjoin | Mathlib/FieldTheory/IntermediateField/Adjoin/Basic.lean | theorem exists_finset_of_mem_adjoin {S : Set E} {x : E} (hx : x ∈ adjoin F S) :
∃ T : Finset E, (T : Set E) ⊆ S ∧ x ∈ adjoin F (T : Set E) | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
hx : x ∈ adjoin F S
⊢ ∃ T, ↑T ⊆ S ∧ x ∈ adjoin F ↑T | simp_rw [← biSup_adjoin_simple S, ← iSup_subtype''] at hx | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
x : E
hx : x ∈ ⨆ i, F⟮↑i⟯
⊢ ∃ T, ↑T ⊆ S ∧ x ∈ adjoin F ↑T | 1166f60e91d3f2e9 |
Bool.le_and | Mathlib/Data/Bool/Basic.lean | theorem le_and : ∀ {x y z : Bool}, x ≤ y → x ≤ z → x ≤ (y && z) | ⊢ ∀ {x y z : Bool}, x ≤ y → x ≤ z → x ≤ (y && z) | decide | no goals | 432cc046d95e6e71 |
Set.prod_subset_prod_iff | Mathlib/Data/Set/Prod.lean | theorem prod_subset_prod_iff : s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ | case inr.refine_1
α : Type u_1
β : Type u_2
s s₁ : Set α
t t₁ : Set β
h : (s ×ˢ t).Nonempty
st : s.Nonempty ∧ t.Nonempty
H : s ×ˢ t ⊆ s₁ ×ˢ t₁
this : Prod.fst '' s ×ˢ t ⊆ Prod.fst '' s₁ ×ˢ t₁
⊢ s ⊆ s₁ | rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this | no goals | 1b0ab7383e10acaf |
MeasureTheory.lintegral_const_mul'' | Mathlib/MeasureTheory/Integral/Lebesgue.lean | theorem lintegral_const_mul'' (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
∫⁻ a, r * f a ∂μ = r * ∫⁻ a, f a ∂μ | α : Type u_1
m : MeasurableSpace α
μ : Measure α
r : ℝ≥0∞
f : α → ℝ≥0∞
hf : AEMeasurable f μ
A : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), AEMeasurable.mk f hf a ∂μ
B : ∫⁻ (a : α), r * f a ∂μ = ∫⁻ (a : α), r * AEMeasurable.mk f hf a ∂μ
⊢ ∫⁻ (a : α), r * f a ∂μ = r * ∫⁻ (a : α), f a ∂μ | rw [A, B, lintegral_const_mul _ hf.measurable_mk] | no goals | cf82783884c5cc86 |
List.sections_eq_sectionsTR | Mathlib/.lake/packages/batteries/Batteries/Data/List/Basic.lean | theorem sections_eq_sectionsTR : @sections = @sectionsTR | case h.h.false.cons
α : Type u_1
l : List α
L : List (List α)
IH : L.sections = (foldr sectionsTR.go #[[]] L).toList
⊢ flatMap (fun s => map (fun a => a :: s) l) (foldr sectionsTR.go #[[]] L).toList =
#[].toList ++ flatMap ?h.h.false.cons.G (foldr sectionsTR.go #[[]] L).toList
case h.h.false.cons.G
α : Type u_1
l ... | rfl | case h.h.false.cons.H
α : Type u_1
l : List α
L : List (List α)
IH : L.sections = (foldr sectionsTR.go #[[]] L).toList
⊢ ∀ (acc : Array (List α)) (a : List α),
(foldl (fun acc' a_1 => acc'.push (a_1 :: a)) acc l).toList = acc.toList ++ map (fun a_1 => a_1 :: a) l | a50d5431031fde51 |
PiTensorProduct.mapL_comp | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | theorem mapL_comp : mapL (fun (i : ι) ↦ g i ∘L f i) = mapL g ∘L mapL f | ι : Type uι
inst✝⁷ : Fintype ι
𝕜 : Type u𝕜
inst✝⁶ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
E' : ι → Type u_1
E'' : ι → Type u_2
inst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E' i)
inst✝¹ :... | apply ContinuousLinearMap.coe_injective | case a
ι : Type uι
inst✝⁷ : Fintype ι
𝕜 : Type u𝕜
inst✝⁶ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
E' : ι → Type u_1
E'' : ι → Type u_2
inst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E' i)
i... | beb9820a89fa1157 |
seminormFromConst_apply_c | Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean | theorem seminormFromConst_apply_c : seminormFromConst' hf1 hc hpm c = f c :=
have hlim : Tendsto (seminormFromConst_seq c f c) atTop (𝓝 (f c)) | R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
⊢ seminormFromConst_seq c f c = fun _n => f c | ext n | case h
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
n : ℕ
⊢ seminormFromConst_seq c f c n = f c | 63f75e8cc9f932ab |
IsLocalizedModule.of_exists_mul_mem | Mathlib/Algebra/Module/LocalizedModule/Basic.lean | lemma IsLocalizedModule.of_exists_mul_mem {N : Type*} [AddCommGroup N] [Module R N]
(S T : Submonoid R) (h : S ≤ T) (h' : ∀ x : T, ∃ m : R, m * x ∈ S)
(f : M →ₗ[R] N) [IsLocalizedModule S f] :
IsLocalizedModule T f where
map_units x | case intro
R : Type u_1
inst✝⁵ : CommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_6
inst✝² : AddCommGroup N
inst✝¹ : Module R N
S T : Submonoid R
h : S ≤ T
h' : ∀ (x : ↥T), ∃ m, m * ↑x ∈ S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule S f
x : ↥T
m : R
mx : m * ↑x ∈ S
⊢ IsUnit ((algebraMap R... | have := IsLocalizedModule.map_units f ⟨_, mx⟩ | case intro
R : Type u_1
inst✝⁵ : CommSemiring R
M : Type u_2
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
N : Type u_6
inst✝² : AddCommGroup N
inst✝¹ : Module R N
S T : Submonoid R
h : S ≤ T
h' : ∀ (x : ↥T), ∃ m, m * ↑x ∈ S
f : M →ₗ[R] N
inst✝ : IsLocalizedModule S f
x : ↥T
m : R
mx : m * ↑x ∈ S
this : IsUnit ((algebra... | 66196422735502be |
bne_self_eq_false' | Mathlib/.lake/packages/lean4/src/lean/Init/SimpLemmas.lean | theorem bne_self_eq_false' [DecidableEq α] (a : α) : (a != a) = false | α : Type u_1
inst✝ : DecidableEq α
a : α
⊢ (a != a) = false | simp | no goals | 0fc27ef842602ede |
Polynomial.Chebyshev.S_eq_U_comp_half_mul_X | Mathlib/RingTheory/Polynomial/Chebyshev.lean | theorem S_eq_U_comp_half_mul_X [Invertible (2 : R)] (n : ℤ) :
S R n = (U R n).comp (Polynomial.C ⅟2 * X) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible 2
n : ℤ
this : (fun x => x.comp (Polynomial.C ⅟2 * X)) ((S R n).comp (2 * X)) = (fun x => x.comp (Polynomial.C ⅟2 * X)) (U R n)
⊢ S R n = (U R n).comp (Polynomial.C ⅟2 * X) | simp_rw [comp_assoc, mul_comp, ofNat_comp, X_comp, ← mul_assoc, ← C_eq_natCast, ← C_mul,
Nat.cast_ofNat, mul_invOf_self', map_one, one_mul, comp_X] at this | R : Type u_1
inst✝¹ : CommRing R
inst✝ : Invertible 2
n : ℤ
this : S R n = (U R n).comp (Polynomial.C ⅟2 * X)
⊢ S R n = (U R n).comp (Polynomial.C ⅟2 * X) | a8649861e3e1957a |
ContinuousMap.tendsto_iff_tendstoUniformly | Mathlib/Topology/UniformSpace/CompactConvergence.lean | theorem tendsto_iff_tendstoUniformly :
Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => F i a) f p | α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : CompactSpace α
⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformly (fun i a => (F i) a) (⇑f) p | rw [tendsto_iff_forall_isCompact_tendstoUniformlyOn, ← tendstoUniformlyOn_univ] | α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : CompactSpace α
⊢ (∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K) ↔
TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p univ | 39812cab1e68af13 |
Equiv.Perm.card_fixedPoints | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem card_fixedPoints (σ : Equiv.Perm α) :
Fintype.card (Function.fixedPoints σ) = Fintype.card α - σ.cycleType.sum | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ Fintype.card ↑(Function.fixedPoints ⇑σ) = Fintype.card α - σ.cycleType.sum | rw [Equiv.Perm.sum_cycleType, ← Finset.card_compl, Fintype.card_ofFinset] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
⊢ (Finset.filter (Membership.mem (Function.fixedPoints ⇑σ)) Finset.univ).card = σ.supportᶜ.card | 868ccbc5bc12036b |
MvPowerSeries.mul_invOfUnit | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | theorem mul_invOfUnit (φ : MvPowerSeries σ R) (u : Rˣ) (h : constantCoeff σ R φ = u) :
φ * invOfUnit φ u = 1 :=
ext fun n =>
letI := Classical.decEq (σ →₀ ℕ)
if H : n = 0 then by
rw [H]
simp [coeff_mul, support_single_ne_zero, h]
else by
classical
have : ((0 : σ →₀ ℕ), n) ∈ ant... | case mk.intro.hc.bc.right.h
σ : Type u_1
R : Type u_2
inst✝ : Ring R
φ : MvPowerSeries σ R
u : Rˣ
h : (constantCoeff σ R) φ = ↑u
this✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)
i j : σ →₀ ℕ
H✝ : ¬(i, j).1 + (i, j).2 = 0
this : (0, (i, j).1 + (i, j).2) ∈ antidiagonal ((i, j).1 + (i, j).2)
h₁ : (i, j) ≠ (0, (i, ... | exact Nat.eq_zero_of_le_zero (H s) | no goals | 56896b65324500b8 |
Valuation.IsDiscrete.surj | Mathlib/RingTheory/Valuation/Discrete/Basic.lean | /-- A discrete valuation on a field `K` is surjective. -/
lemma IsDiscrete.surj (v : Valuation K ℤₘ₀) [hv : IsDiscrete v] :
Surjective v | K : Type u_2
inst✝ : Field K
v : Valuation K ℤₘ₀
hv : v.IsDiscrete
⊢ Surjective ⇑v | intro c | K : Type u_2
inst✝ : Field K
v : Valuation K ℤₘ₀
hv : v.IsDiscrete
c : ℤₘ₀
⊢ ∃ a, v a = c | c180e6b232cba5c8 |
zpow_induction_right | Mathlib/Algebra/Group/Basic.lean | /-- To show a property of all powers of `g` it suffices to show it is closed under multiplication
by `g` and `g⁻¹` on the right. For subgroups generated by more than one element, see
`Subgroup.closure_induction_right`. -/
@[to_additive "To show a property of all multiples of `g` it suffices to show it is closed under
a... | case hz
G : Type u_3
inst✝ : Group G
g : G
P : G → Prop
h_one : P 1
h_mul : ∀ (a : G), P a → P (a * g)
h_inv : ∀ (a : G), P a → P (a * g⁻¹)
⊢ P (g ^ 0) | rwa [zpow_zero] | no goals | ada9e9579b0aa862 |
MeasureTheory.Lp.cauchy_tendsto_of_tendsto | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem cauchy_tendsto_of_tendsto {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(f_lim : α → E) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N)
(h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
atTop.Tendsto (fun n ... | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n =... | have h_B : ∃ N : ℕ, B N ≤ ε := by
suffices h_tendsto_zero : ∃ N : ℕ, ∀ n : ℕ, N ≤ n → B n ≤ ε from
⟨h_tendsto_zero.choose, h_tendsto_zero.choose_spec _ le_rfl⟩
exact (ENNReal.tendsto_atTop_zero.mp (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)) ε hε | α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
f_lim : α → E
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ⊤
h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n =... | 4d90eb848aa97c8b |
Array.foldr_filterMap' | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem foldr_filterMap' (f : α → Option β) (g : β → γ → γ) (l : Array α) (init : γ)
(w : start = (l.filterMap f).size) :
(l.filterMap f).foldr g init start 0 = l.foldr (fun x y => match f x with | some b => g b y | none => y) init | α : Type u_1
β : Type u_2
γ : Type u_3
f : α → Option β
g : β → γ → γ
l : Array α
init : γ
⊢ foldr g init (filterMap f l) =
foldr
(fun x y =>
match f x with
| some b => g b y
| none => y)
init l | cases l | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
f : α → Option β
g : β → γ → γ
init : γ
toList✝ : List α
⊢ foldr g init (filterMap f { toList := toList✝ }) =
foldr
(fun x y =>
match f x with
| some b => g b y
| none => y)
init { toList := toList✝ } | ab7e8eac404b1c3c |
Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction | Mathlib/LinearAlgebra/Matrix/Transvection.lean | theorem exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction
(IH :
∀ M : Matrix (Fin r) (Fin r) 𝕜,
∃ (L₀ L₀' : List (TransvectionStruct (Fin r) 𝕜)) (D₀ : Fin r → 𝕜),
(L₀.map toMatrix).prod * M * (L₀'.map toMatrix).prod = diagonal D₀)
(M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Uni... | case intro.intro
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
∀ (M : Matrix (Fin r) (Fin r) 𝕜),
∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)
hM : ((List.map toMatrix L₁).prod... | rcases IH M'' with ⟨L₀, L₀', D₀, h₀⟩ | case intro.intro.intro.intro.intro
𝕜 : Type u_3
inst✝ : Field 𝕜
r : ℕ
IH :
∀ (M : Matrix (Fin r) (Fin r) 𝕜),
∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀
M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)
hM : ((List.map... | 1db945989babbcf8 |
IsVisible.of_convexHull_of_pos | Mathlib/Analysis/Convex/Visible.lean | /-- If a point `x` sees a convex combination of points of a set `s` through `convexHull ℝ s ∌ x`,
then it sees all terms of that combination.
Note that the converse does not hold. -/
lemma IsVisible.of_convexHull_of_pos {ι : Type*} {t : Finset ι} {a : ι → V} {w : ι → 𝕜}
(hw₀ : ∀ i ∈ t, 0 ≤ w i) (hw₁ : ∑ i ∈ t, w ... | case inr.intro.intro.intro.intro.refine_1.refine_2
𝕜 : Type u_1
V : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVi... | rw [← sum_div, sum_erase_eq_sub hi, hw₁, div_self hwi.ne'] | no goals | 7c7847924d1006dd |
FormalMultilinearSeries.norm_apply_eq_norm_coef | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | theorem norm_apply_eq_norm_coef : ‖p n‖ = ‖coeff p n‖ | 𝕜 : Type u
E : Type v
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
p : FormalMultilinearSeries 𝕜 𝕜 E
n : ℕ
⊢ ‖p n‖ = ‖p.coeff n‖ | rw [← mkPiRing_coeff_eq p, ContinuousMultilinearMap.norm_mkPiRing] | no goals | 86c28f40f560ee79 |
Polynomial.hilbertPoly_eq_zero_of_le_rootMultiplicity_one | Mathlib/RingTheory/Polynomial/HilbertPoly.lean | lemma hilbertPoly_eq_zero_of_le_rootMultiplicity_one
{p : F[X]} {d : ℕ} (hdp : d ≤ p.rootMultiplicity 1) :
hilbertPoly p d = 0 | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
p : F[X]
d : ℕ
hdp : d ≤ rootMultiplicity 1 p
hp : ¬p = 0
q : F[X]
hq1 : p = (X - C 1) ^ rootMultiplicity 1 p * q
hq2 : ¬X - C 1 ∣ q
⊢ p = q * (-1) ^ rootMultiplicity 1 p * (1 - X) ^ rootMultiplicity 1 p | simp only [mul_assoc, ← mul_pow, neg_mul, one_mul, neg_sub] | F : Type u_1
inst✝¹ : Field F
inst✝ : CharZero F
p : F[X]
d : ℕ
hdp : d ≤ rootMultiplicity 1 p
hp : ¬p = 0
q : F[X]
hq1 : p = (X - C 1) ^ rootMultiplicity 1 p * q
hq2 : ¬X - C 1 ∣ q
⊢ p = q * (X - 1) ^ rootMultiplicity 1 p | 1a90db0201b3b7f2 |
SimpContFract.determinant_aux | Mathlib/Algebra/ContinuedFractions/Determinant.lean | theorem determinant_aux (hyp : n = 0 ∨ ¬(↑s : GenContFract K).TerminatedAt (n - 1)) :
((↑s : GenContFract K).contsAux n).a * ((↑s : GenContFract K).contsAux (n + 1)).b -
((↑s : GenContFract K).contsAux n).b * ((↑s : GenContFract K).contsAux (n + 1)).a =
(-1) ^ n | case succ.intro
K : Type u_1
inst✝ : Field K
s : SimpContFract K
n✝ n : ℕ
hyp : n + 1 = 0 ∨ ¬(↑s).TerminatedAt (n + 1 - 1)
g : GenContFract K := ↑s
conts : Pair K := g.contsAux (n + 2)
pred_conts : Pair K := g.contsAux (n + 1)
pred_conts_eq : pred_conts = g.contsAux (n + 1)
ppred_conts : Pair K := g.contsAux n
IH : n =... | exact IH <| Or.inr <| mt (terminated_stable <| n.sub_le 1) not_terminated_at_n | no goals | 5478ed7427a774c4 |
Vector.foldr_map_hom | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem foldr_map_hom (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : Vector α n)
(h : ∀ x y, f' (g x) (g y) = g (f x y)) :
(l.map g).foldr f' (g a) = g (l.foldr f a) | α : Type u_1
β : Type u_2
n : Nat
g : α → β
f : α → α → α
f' : β → β → β
a : α
l : Vector α n
h : ∀ (x y : α), f' (g x) (g y) = g (f x y)
⊢ foldr f' (g a) (map g l) = g (foldr f a l) | rcases l with ⟨l, rfl⟩ | case mk
α : Type u_1
β : Type u_2
g : α → β
f : α → α → α
f' : β → β → β
a : α
h : ∀ (x y : α), f' (g x) (g y) = g (f x y)
l : Array α
⊢ foldr f' (g a) (map g { toArray := l, size_toArray := ⋯ }) = g (foldr f a { toArray := l, size_toArray := ⋯ }) | d84ece43bb68e6a7 |
Besicovitch.TauPackage.color_lt | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N | α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ k < i, k < p.lastStep → p.color k < N
hi : i < p.lastStep
A : Set ℕ :=
⋃ j,
⋃ (_ :
(closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.i... | rw [color] | no goals | 70d4d5d8aa8a7de2 |
PowerBasis.leftMulMatrix | Mathlib/RingTheory/PowerBasis.lean | theorem leftMulMatrix (pb : PowerBasis A S) : Algebra.leftMulMatrix pb.basis pb.gen =
@Matrix.of (Fin pb.dim) (Fin pb.dim) _ fun i j =>
if ↑j + 1 = pb.dim then -pb.minpolyGen.coeff ↑i else if (i : ℕ) = j + 1 then 1 else 0 | case pos
S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
h✝ : Nontrivial A
k : Fin pb.dim
h : ↑k + 1 = pb.dim
⊢ pb.gen ^ pb.dim + ∑ x : Fin pb.dim, pb.minpolyGen.coeff ↑x • pb.gen ^ ↑x = 0 | convert pb.aeval_minpolyGen | case h.e'_2
S : Type u_2
inst✝² : Ring S
A : Type u_4
inst✝¹ : CommRing A
inst✝ : Algebra A S
pb : PowerBasis A S
h✝ : Nontrivial A
k : Fin pb.dim
h : ↑k + 1 = pb.dim
⊢ pb.gen ^ pb.dim + ∑ x : Fin pb.dim, pb.minpolyGen.coeff ↑x • pb.gen ^ ↑x = (aeval pb.gen) pb.minpolyGen | ef0f48e9c36978b9 |
IsClosed.Icc_subset_of_forall_exists_gt | Mathlib/Topology/Order/IntermediateValue.lean | theorem IsClosed.Icc_subset_of_forall_exists_gt {a b : α} {s : Set α} (hs : IsClosed (s ∩ Icc a b))
(ha : a ∈ s) (hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty) : Icc a b ⊆ s | α : Type u
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty
y : α
hy : y ∈ Icc a b
⊢ s ∩ Icc a y = s ∩ (Icc a b ∩ Icc a y) | congr | case e_a
α : Type u
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed (s ∩ Icc a b)
ha : a ∈ s
hgt : ∀ x ∈ s ∩ Ico a b, ∀ y ∈ Ioi x, (s ∩ Ioc x y).Nonempty
y : α
hy : y ∈ Icc a b
⊢ Icc a y = Icc a b ∩ Icc a y | 4b8c631eef6c52ae |
AEMeasurable.sum_measure | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | theorem sum_measure [Countable ι] {μ : ι → Measure α} (h : ∀ i, AEMeasurable f (μ i)) :
AEMeasurable f (sum μ) | case refine_3
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) ... | refine measure_mono_null (fun x (hx : f x ≠ g x) => ?_) (hsμ i) | case refine_3
ι : Type u_1
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
f : α → β
inst✝ : Countable ι
μ : ι → Measure α
h : ∀ (i : ι), AEMeasurable f (μ i)
a✝ : Nontrivial β
inhabited_h : Inhabited β
s : ι → Set α := fun i => toMeasurable (μ i) {x | f x ≠ mk f ⋯ x}
hsμ : ∀ (i : ι), (μ i) ... | df9a789791001abe |
Associates.eq_factors_of_eq_counts | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem eq_factors_of_eq_counts {a b : Associates α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : ∀ p : Associates α, Irreducible p → p.count a.factors = p.count b.factors) :
a.factors = b.factors | case intro.intro.a.h.mk
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
a b : Associates α
ha : a ≠ 0
hb : b ≠ 0
sa : Multiset { p // Irreducible p }
h_sa : a.factors = ↑sa
sb : Multiset { p ... | rw [Multiset.toFinsupp_apply, Multiset.toFinsupp_apply, h_count p hp] | no goals | 92d336a2c52c647d |
AkraBazziRecurrence.eventually_atTop_sumTransform_le | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma eventually_atTop_sumTransform_le :
∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ i, sumTransform (p a b) g (r i n) n ≤ c * g n | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g... | intro i | case h
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g... | 9769bbce1ca34368 |
LieModule.lowerCentralSeries_one_inf_center_le_ker_traceForm | Mathlib/Algebra/Lie/TraceForm.lean | lemma lowerCentralSeries_one_inf_center_le_ker_traceForm [Module.Free R M] [Module.Finite R M] :
lowerCentralSeries R L L 1 ⊓ LieAlgebra.center R L ≤ LinearMap.ker (traceForm R L M) | R : Type u_1
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : Free R M
inst✝ : Module.Fini... | rw [← LieSubmodule.baseChange_top, ← LieSubmodule.lie_baseChange] | R : Type u_1
L : Type u_3
M : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
inst✝⁴ : LieRing.IsNilpotent L
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : Free R M
inst✝ : Module.Fini... | f82e668bfa6bbbba |
Fin.castSucc_succAbove_castSucc | Mathlib/Data/Fin/Basic.lean | /-- `castSucc` commutes with `succAbove`. -/
@[simp]
lemma castSucc_succAbove_castSucc {n : ℕ} {i : Fin (n + 1)} {j : Fin n} :
i.castSucc.succAbove j.castSucc = (i.succAbove j).castSucc | case inr
n : ℕ
i : Fin (n + 1)
j : Fin n
h : j.castSucc < i
⊢ i.castSucc.succAbove j.castSucc = (i.succAbove j).castSucc | rw [succAbove_of_castSucc_lt _ _ h, succAbove_castSucc_of_lt _ _ h] | no goals | fde3027e08d01812 |
Ordinal.principal_add_iff_zero_or_omega0_opow | Mathlib/SetTheory/Ordinal/Principal.lean | theorem principal_add_iff_zero_or_omega0_opow :
Principal (· + ·) o ↔ o = 0 ∨ o ∈ Set.range (ω ^ · : Ordinal → Ordinal) | case inr
o : Ordinal.{u}
ho : o ≠ 0
H : ∀ a < o, a + o = o
h : ω ^ log ω o < o
⊢ False | have := H _ h | case inr
o : Ordinal.{u}
ho : o ≠ 0
H : ∀ a < o, a + o = o
h : ω ^ log ω o < o
this : ω ^ log ω o + o = o
⊢ False | b95345052c5b9163 |
CategoryTheory.Functor.rightDerivedNatTrans_id | Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean | @[simp]
lemma rightDerivedNatTrans_id :
rightDerivedNatTrans RF RF α α W (𝟙 F) = 𝟙 RF :=
rightDerived_ext RF α W _ _ _ (by simp)
| C : Type u_5
D : Type u_1
H : Type u_3
inst✝⁴ : Category.{u_6, u_5} C
inst✝³ : Category.{u_4, u_1} D
inst✝² : Category.{u_2, u_3} H
RF : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : RF.IsRightDerivedFunctor α W
⊢ α ≫ whiskerLeft L (RF.rightDerivedNatTrans RF α α W ... | simp | no goals | 3e394d4714e931fd |
List.push_append_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem push_append_toArray {as : Array α} {a : α} {bs : List α} : as.push a ++ bs.toArray = as ++ (a ::bs).toArray | α : Type u_1
as : Array α
a : α
bs : List α
⊢ as.push a ++ bs.toArray = as ++ (a :: bs).toArray | cases as | case mk
α : Type u_1
a : α
bs toList✝ : List α
⊢ { toList := toList✝ }.push a ++ bs.toArray = { toList := toList✝ } ++ (a :: bs).toArray | 54ccf7ab80e7271c |
CategoryTheory.CosimplicialObject.δ_comp_σ_of_gt | Mathlib/AlgebraicTopology/SimplicialObject/Basic.lean | theorem δ_comp_σ_of_gt {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : Fin.castSucc j < i) :
X.δ i.succ ≫ X.σ (Fin.castSucc j) = X.σ j ≫ X.δ i | C : Type u
inst✝ : Category.{v, u} C
X : CosimplicialObject C
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
H : j.castSucc < i
⊢ X.map (SimplexCategory.δ i.succ) ≫ X.map (SimplexCategory.σ j.castSucc) =
X.map (SimplexCategory.σ j) ≫ X.map (SimplexCategory.δ i) | simp only [← X.map_comp, SimplexCategory.δ_comp_σ_of_gt H] | no goals | 92adab57d8d79fe0 |
Polynomial.Monic.of_mul_monic_left | Mathlib/Algebra/Polynomial/Monic.lean | theorem Monic.of_mul_monic_left (hp : p.Monic) (hpq : (p * q).Monic) : q.Monic | R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p.Monic
hpq : (p * q).Monic
⊢ q.Monic | contrapose! hpq | R : Type u
inst✝ : Semiring R
p q : R[X]
hp : p.Monic
hpq : ¬q.Monic
⊢ ¬(p * q).Monic | 0f1198985ec0278d |
MeasureTheory.tendsto_of_integral_tendsto_of_monotone | Mathlib/MeasureTheory/Integral/Bochner.lean | /-- If a monotone sequence of functions has an upper bound and the sequence of integrals of these
functions tends to the integral of the upper bound, then the sequence of functions converges
almost everywhere to the upper bound. -/
lemma tendsto_of_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α → ℝ} {F : α → ℝ... | case h
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i => ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i => f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ ... | exact (ENNReal.continuousAt_toReal ENNReal.ofReal_ne_top).tendsto.comp ha | no goals | a54f15f3a6cf0f39 |
CategoryTheory.OverPresheafAux.YonedaCollection.map₂_snd | Mathlib/CategoryTheory/Comma/Presheaf/Basic.lean | @[simp]
lemma map₂_snd {Y : C} (f : X ⟶ Y) (p : YonedaCollection F Y) :
(YonedaCollection.map₂ F f p).snd = F.map ((CostructuredArrow.mkPrecomp p.fst f).op ≫
eqToHom (by rw [YonedaCollection.map₂_fst f])) p.snd | C : Type u
inst✝ : Category.{v, u} C
A : Cᵒᵖ ⥤ Type v
F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v
X Y : C
f : X ⟶ Y
p : YonedaCollection F Y
⊢ op (CostructuredArrow.mk (yoneda.map f ≫ p.fst)) = op (CostructuredArrow.mk (map₂ F f p).fst) | rw [YonedaCollection.map₂_fst f] | no goals | c86bf83d3b4c8847 |
Ideal.ramificationIdx_spec | Mathlib/NumberTheory/RamificationInertia/Basic.lean | theorem ramificationIdx_spec {n : ℕ} (hle : map f p ≤ P ^ n) (hgt : ¬map f p ≤ P ^ (n + 1)) :
ramificationIdx f p P = n | R : Type u
inst✝¹ : CommRing R
S : Type v
inst✝ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
n : ℕ
hle : map f p ≤ P ^ n
hgt : ¬map f p ≤ P ^ (n + 1)
Q : ℕ → Prop := fun m => ∀ (k : ℕ), map f p ≤ P ^ k → k ≤ m
this : Q n
h : Nat.find ⋯ < n
this' : Q (Nat.find ⋯)
⊢ False | exact h.not_le (this' _ hle) | no goals | fee84ed3d55e0f56 |
Tree.traverse_eq_map_id | Mathlib/Data/Tree/Traversable.lean | lemma traverse_eq_map_id (f : α → β) (t : Tree α) :
t.traverse ((pure : β → Id β) ∘ f) = t.map f | α : Type u_1
β : Type u_2
f : α → β
t : Tree α
⊢ traverse (pure ∘ f) t = pure (map f t) | induction t with
| nil => rw [traverse, map]
| node v l r hl hr =>
rw [traverse, map, hl, hr, Function.comp_apply, map_pure, pure_seq, map_pure, pure_seq,
map_pure] | no goals | a52a2b0c225468f9 |
MeasureTheory.condExpL2_indicator_of_measurable | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | theorem condExpL2_indicator_of_measurable (hm : m ≤ m0) (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞)
(c : E) :
(condExpL2 E 𝕜 hm (indicatorConstLp 2 (hm s hs) hμs c) : α →₂[μ] E) =
indicatorConstLp 2 (hm s hs) hμs c | α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
c : E
this : Fact (m ≤ m0)
h_mem : indicatorConstLp 2 ⋯ hμs c ∈ lpMeas E 𝕜 m 2... | have h_coe_ind : (ind : α →₂[μ] E) = indicatorConstLp 2 (hm s hs) hμs c := rfl | α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ⊤
c : E
this : Fact (m ≤ m0)
h_mem : indicatorConstLp 2 ⋯ hμs c ∈ lpMeas E 𝕜 m 2... | 01e0fd6a9ac7548d |
AkraBazziRecurrence.exists_eventually_r_le_const_mul | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | lemma exists_eventually_r_le_const_mul :
∃ c ∈ Set.Ioo (0 : ℝ) 1, ∀ᶠ (n : ℕ) in atTop, ∀ i, r i n ≤ c * n | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
c : ℝ := b (max_bi b) + (1 - b (max_bi b)) / 2
h_max_bi_pos : 0 < b (max_bi b)
h_max_bi_lt_one : 0 < 1 - b (max_bi b)
hc_pos : 0 < c
h₁ : 0 < (1 - b (max_bi b)) / 2
⊢ 1 * (1 / 2) + 1 / 2 = ... | norm_num | no goals | 261123087b919b7c |
List.Chain'.rel_head? | Mathlib/Data/List/Chain.lean | theorem Chain'.rel_head? {x l} (h : Chain' R (x :: l)) ⦃y⦄ (hy : y ∈ head? l) : R x y | α : Type u
R : α → α → Prop
x : α
l : List α
h : Chain' R (x :: l)
y : α
hy : y ∈ l.head?
⊢ R x y | rw [← cons_head?_tail hy] at h | α : Type u
R : α → α → Prop
x : α
l : List α
y : α
h : Chain' R (x :: y :: l.tail)
hy : y ∈ l.head?
⊢ R x y | 79b824ef06f96f23 |
Nat.sum_four_squares_of_two_mul_sum_four_squares | Mathlib/NumberTheory/SumFourSquares.lean | theorem sum_four_squares_of_two_mul_sum_four_squares {m a b c d : ℤ}
(h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m) :
∃ w x y z : ℤ, w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = m | m a b c d : ℤ
h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m
this :
∀ (f : Fin 4 → ZMod 2),
f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →
∃ i, f i ^ 2 + f ((swap i 0) 1) ^ 2 = 0 ∧ f ((swap i 0) 2) ^ 2 + f ((swap i 0) 3) ^ 2 = 0
f : Fin 4 → ℤ := ![a, b, c, d]
⊢ (fun x => ↑(f x)) 0 ^ 2 + (fun x => ↑(f x)) 1 ^ 2 + (f... | simp only [Int.cast_add, Int.cast_pow] | m a b c d : ℤ
h : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m
this :
∀ (f : Fin 4 → ZMod 2),
f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →
∃ i, f i ^ 2 + f ((swap i 0) 1) ^ 2 = 0 ∧ f ((swap i 0) 2) ^ 2 + f ((swap i 0) 3) ^ 2 = 0
f : Fin 4 → ℤ := ![a, b, c, d]
⊢ ↑(f 0) ^ 2 + ↑(f 1) ^ 2 + ↑(f 2) ^ 2 + ↑(f 3) ^ 2 = ↑a... | f5981f23cd7bc522 |
num_isRoot_scaleRoots_of_aeval_eq_zero | Mathlib/RingTheory/Polynomial/RationalRoot.lean | theorem num_isRoot_scaleRoots_of_aeval_eq_zero [UniqueFactorizationMonoid A] {p : A[X]} {x : K}
(hr : aeval x p = 0) : IsRoot (scaleRoots p (den A x)) (num A x) | A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0
⊢ eval₂ (algebraMap A K) ((algebraMap A K) (num A x)) (p.scaleRoots ↑(den A x)) = 0 | refine scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero ?_ | A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0
⊢ (aeval (mk' K (num A x) (den A x))) p = 0 | 465a444fec6c83d7 |
NumberField.RingOfIntegers.HeightOneSpectrum.one_lt_absNorm | Mathlib/NumberTheory/NumberField/FinitePlaces.lean | /-- The norm of a maximal ideal is `> 1` -/
lemma one_lt_absNorm : 1 < absNorm v.asIdeal | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
v : HeightOneSpectrum (𝓞 K)
h : absNorm v.asIdeal ≤ 1
⊢ False | apply IsPrime.ne_top v.isPrime | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
v : HeightOneSpectrum (𝓞 K)
h : absNorm v.asIdeal ≤ 1
⊢ v.asIdeal = ⊤ | 849af9e987d28ab6 |
Cardinal.mul_lt_aleph0_iff | Mathlib/SetTheory/Cardinal/Basic.lean | theorem mul_lt_aleph0_iff {a b : Cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ | case neg.h.right
a b : Cardinal.{u_1}
h : a * b < ℵ₀
ha : 1 ≤ a
hb : 1 ≤ b
⊢ b < ℵ₀ | rw [← one_mul b] | case neg.h.right
a b : Cardinal.{u_1}
h : a * b < ℵ₀
ha : 1 ≤ a
hb : 1 ≤ b
⊢ 1 * b < ℵ₀ | afac76d49f436d4e |
direction_affineSpan | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace/Defs.lean | theorem direction_affineSpan (s : Set P) : (affineSpan k s).direction = vectorSpan k s | k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
⊢ (affineSpan k s).direction = vectorSpan k s | apply le_antisymm | case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
⊢ (affineSpan k s).direction ≤ vectorSpan k s
case a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s :... | 002080c61aac1fc7 |
CategoryTheory.IsPushout.inr_fst' | Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean | theorem inr_fst' {b : BinaryBicone X Y} (h : b.IsBilimit) :
IsPushout b.inr (0 : Y ⟶ 0) b.fst (0 : 0 ⟶ X) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y : C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
b : BinaryBicone X Y
h : b.IsBilimit
⊢ IsPushout 0 (0 ≫ 0) (b.inl ≫ b.fst) 0 | simp | no goals | 79952a470535d47b |
Continuous.strictMono_of_inj | Mathlib/Topology/Order/IntermediateValue.lean | theorem Continuous.strictMono_of_inj {f : α → δ}
(hf_c : Continuous f) (hf_i : Injective f) : StrictMono f ∨ StrictAnti f | case neg
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
H : ∀ {c d : α}, c < d → StrictMon... | haveI : Subsingleton α := ⟨fun c d => Trans.trans (h c).symm (h d)⟩ | case neg
α : Type u
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
f : α → δ
hf_c : Continuous f
hf_i : Injective f
H : ∀ {c d : α}, c < d → StrictMon... | a6f33dd3051e4aa2 |
PerfectPairing.exists_basis_basis_of_span_eq_top_of_mem_algebraMap | Mathlib/LinearAlgebra/PerfectPairing/Restrict.lean | /-- If a perfect pairing over a field `L` takes values in a subfield `K` along two `K`-subspaces
whose `L` span is full, then these subspaces induce a `K`-structure in the sense of
[*Algebra I*, Bourbaki : Chapter II, §8.1 Definition 1][bourbaki1989]. -/
lemma exists_basis_basis_of_span_eq_top_of_mem_algebraMap
(M'... | case intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairing L M N
M' : Subm... | obtain ⟨w, hw₁, hw₂, hw₃⟩ := exists_linearIndependent L (N' : Set N) | case intro.intro.intro.intro.intro.intro
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : AddCommGroup M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module L M
inst✝³ : Module L N
inst✝² : Module K M
inst✝¹ : Module K N
inst✝ : IsScalarTower K L M
p : PerfectPairi... | 28e714660261030d |
eq_star_of_eq_star | Mathlib/Algebra/Star/Basic.lean | theorem eq_star_of_eq_star [InvolutiveStar R] {r s : R} (h : r = star s) : s = star r | R : Type u
inst✝ : InvolutiveStar R
r s : R
h : r = star s
⊢ s = star r | simp [h] | no goals | 38b7054c9bb01fe2 |
isUniformEmbedding_translate_mul | Mathlib/Topology/Algebra/UniformGroup/Basic.lean | theorem isUniformEmbedding_translate_mul (a : α) : IsUniformEmbedding fun x : α => x * a :=
{ comap_uniformity | case mk.mk
α : Type u_1
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
a p₁ p₂ q₁ q₂ : α
⊢ (fun x => (x.1 * a, x.2 * a)) (p₁, p₂) = (fun x => (x.1 * a, x.2 * a)) (q₁, q₂) → (p₁, p₂) = (q₁, q₂) | simp only [Prod.mk.injEq, mul_left_inj, imp_self] | no goals | f36b458599dbd9ca |
Lagrange.basis_ne_zero | Mathlib/LinearAlgebra/Lagrange.lean | theorem basis_ne_zero (hvs : Set.InjOn v s) (hi : i ∈ s) : Lagrange.basis s v i ≠ 0 | case intro
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
hij : j ≠ i
hj : j ∈ s
⊢ ¬basisDivisor (v i) (v j) = 0 | rw [basisDivisor_eq_zero_iff, hvs.eq_iff hi hj] | case intro
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : DecidableEq ι
s : Finset ι
v : ι → F
i : ι
hvs : Set.InjOn v ↑s
hi : i ∈ s
j : ι
hij : j ≠ i
hj : j ∈ s
⊢ ¬i = j | 12bc6496d8081252 |
Set.isCompl_range_inl_range_inr | Mathlib/Data/Set/Image.lean | theorem isCompl_range_inl_range_inr : IsCompl (range <| @Sum.inl α β) (range Sum.inr) :=
IsCompl.of_le
(by
rintro y ⟨⟨x₁, rfl⟩, ⟨x₂, h⟩⟩
exact Sum.noConfusion h)
(by rintro (x | y) - <;> [left; right] <;> exact mem_range_self _)
| α : Type u_1
β : Type u_2
⊢ ⊤ ≤ range Sum.inl ⊔ range Sum.inr | rintro (x | y) - <;> [left; right] <;> exact mem_range_self _ | no goals | ac20dd38237740f9 |
isIntegral_of_smul_mem_submodule | Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean | theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
[IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
(x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x | case intro.intro
R : Type u_1
A : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : NoZeroSMulDivisors A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Suba... | haveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top] | case intro.intro
R : Type u_1
A : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : NoZeroSMulDivisors A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Suba... | f9f48486d67e3d31 |
Multiset.IsDershowitzMannaLT.trans | Mathlib/Data/Multiset/DershowitzManna.lean | /-- `IsDershowitzMannaLT` is transitive. -/
lemma IsDershowitzMannaLT.trans :
IsDershowitzMannaLT M N → IsDershowitzMannaLT N P → IsDershowitzMannaLT M P | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝ : Preorder α
X₁ Y₁ Z₁ : Multiset α
hYZ₁ : ∀ y ∈ Y₁, ∃ z ∈ Z₁, y < z
X₂ Y₂ Z₂ : Multiset α
hZ₂ : Z₂ ≠ ∅
hXZXY : Z₁ + X₁ = Y₂ + X₂
hYZ₂ : ∀ y ∈ Y₂, ∃ z ∈ Z₂, y < z
⊢ Z₂ + (Z₁ - Y₂) ≠ ∅ | simpa [-not_and, not_and_or] using .inl hZ₂ | no goals | a8b88d02b0e355c1 |
Filter.blimsup_sup_le_or | Mathlib/Order/LiminfLimsup.lean | theorem blimsup_sup_le_or : blimsup u f p ⊔ blimsup u f q ≤ blimsup u f fun x => p x ∨ q x :=
sup_le (blimsup_mono <| by tauto) (blimsup_mono <| by tauto)
| α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
f : Filter β
p q : β → Prop
u : β → α
⊢ ∀ (x : β), q x → p x ∨ q x | tauto | no goals | fc841e25a686af54 |
WellFounded.prod_lex_of_wellFoundedOn_fiber | Mathlib/Order/WellFoundedSet.lean | theorem WellFounded.prod_lex_of_wellFoundedOn_fiber (hα : WellFounded (rα on f))
(hβ : ∀ a, (f ⁻¹' {a}).WellFoundedOn (rβ on g)) :
WellFounded (Prod.Lex rα rβ on fun c => (f c, g c)) | case inl
α : Type u_2
β : Type u_3
γ : Type u_4
rα : α → α → Prop
rβ : β → β → Prop
f : γ → α
g : γ → β
hα : WellFounded (rα on f)
hβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)
c c' : γ
h : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'
h' : rα ((fun c => (f c, g c)) c).1 ((fun c => (f c, g c)) c').1
⊢ ((PSigma.Lex... | exact PSigma.Lex.left _ _ h' | no goals | 56092b86f3486016 |
List.foldlM_cons_eq_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Monadic.lean | theorem foldlM_cons_eq_append [Monad m] [LawfulMonad m] (f : α → m β) (as : List α) (b : β) (bs : List β) :
(as.foldlM (init := b :: bs) fun acc a => return ((← f a) :: acc)) =
(· ++ b :: bs) <$> as.foldlM (init := []) fun acc a => return ((← f a) :: acc) | case cons
m : Type u_1 → Type u_2
α : Type u_3
β : Type u_1
inst✝¹ : Monad m
inst✝ : LawfulMonad m
f : α → m β
a : α
as : List α
b : β
bs : List β
ih :
∀ (b : β) (bs : List β),
List.foldlM (fun acc a => (fun a => a :: acc) <$> f a) (b :: bs) as =
(fun x => x ++ b :: bs) <$> List.foldlM (fun acc a => (fun a ... | simp [ih, _root_.map_bind, Functor.map_map, Function.comp_def] | no goals | 7164fb3aab3cf5ef |
Measurable.mul' | Mathlib/MeasureTheory/Group/Arithmetic.lean | /-- Compositional version of `Measurable.mul` for use by `fun_prop`. -/
@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))
"Compositional version of `Measurable.add` for use by `fun_prop`."]
lemma Measurable.mul' [MeasurableMul₂ M] {f g : α → β → M} {h : α → β} (hf : Measurable ↿f)
(h... | M : Type u_2
α : Type u_3
β : Type u_4
inst✝² : MeasurableSpace M
inst✝¹ : Mul M
m : MeasurableSpace α
mβ : MeasurableSpace β
inst✝ : MeasurableMul₂ M
f g : α → β → M
h : α → β
hf : Measurable ↿f
hg : Measurable ↿g
hh : Measurable h
⊢ Measurable fun a => f a (h a) * g a (h a) | fun_prop | no goals | cda2abf530c3de3c |
Complex.betaIntegral_eval_nat_add_one_right | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem betaIntegral_eval_nat_add_one_right {u : ℂ} (hu : 0 < re u) (n : ℕ) :
betaIntegral u (n + 1) = n ! / ∏ j ∈ Finset.range (n + 1), (u + j) | case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < (↑n.succ).re | rw [← ofReal_natCast, ofReal_re] | case succ.refine_1
n : ℕ
IH : ∀ {u : ℂ}, 0 < u.re → u.betaIntegral (↑n + 1) = ↑n ! / ∏ j ∈ Finset.range (n + 1), (u + ↑j)
u : ℂ
hu : 0 < u.re
⊢ 0 < ↑n.succ | 9c2559214b6b9868 |
Std.Sat.AIG.RefVec.map.go_le_size | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem map.go_le_size {aig : AIG α} (idx : Nat) (hidx) (s : RefVec aig idx)
(input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
[LawfulOperator α Ref f] [LawfulMapOperator α f] :
aig.decls.size ≤ (go aig idx hidx s input f).aig.decls.size | α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
idx : Nat
hidx : idx ≤ len
s : aig.RefVec idx
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
⊢ aig.decls.size ≤
(if hidx : idx < len then
let res := ... | split | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
idx : Nat
hidx : idx ≤ len
s : aig.RefVec idx
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
h✝ : idx < len
⊢ aig.decls.size ≤
(let res := f aig (inp... | b78794ce350736da |
Part.Fix.approx_mono' | Mathlib/Control/LawfulFix.lean | theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i) | case succ
α : Type u_1
β : α → Type u_2
f : ((a : α) → Part (β a)) →o (a : α) → Part (β a)
n✝ : ℕ
i_ih : approx (⇑f) n✝ ≤ approx (⇑f) n✝.succ
⊢ approx (⇑f) (n✝ + 1) ≤ approx (⇑f) (n✝ + 1).succ | intro | case succ
α : Type u_1
β : α → Type u_2
f : ((a : α) → Part (β a)) →o (a : α) → Part (β a)
n✝ : ℕ
i_ih : approx (⇑f) n✝ ≤ approx (⇑f) n✝.succ
i✝ : α
⊢ approx (⇑f) (n✝ + 1) i✝ ≤ approx (⇑f) (n✝ + 1).succ i✝ | c85ff91073cb5b69 |
nilpotencyClass_eq_succ_iff | Mathlib/RingTheory/Nilpotent/Defs.lean | lemma nilpotencyClass_eq_succ_iff {k : ℕ} :
nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | R : Type u_1
x : R
inst✝ : MonoidWithZero R
k : ℕ
⊢ nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | let s : Set ℕ := {k | x ^ k = 0} | R : Type u_1
x : R
inst✝ : MonoidWithZero R
k : ℕ
s : Set ℕ := {k | x ^ k = 0}
⊢ nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 | 88cf1fbdc3a16b39 |
EMetric.diam_pos_iff | Mathlib/Topology/EMetricSpace/Diam.lean | theorem diam_pos_iff : 0 < diam s ↔ s.Nontrivial | β : Type u_2
inst✝ : EMetricSpace β
s : Set β
⊢ 0 < diam s ↔ s.Nontrivial | simp only [pos_iff_ne_zero, Ne, diam_eq_zero_iff, Set.not_subsingleton_iff] | no goals | d31ac5b75e5cc8f3 |
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