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 |
|---|---|---|---|---|---|---|
Nat.one_sub | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem one_sub : ∀ n, 1 - n = if n = 0 then 1 else 0
| 0 => rfl
| _+1 => by rw [if_neg (Nat.succ_ne_zero _), Nat.succ_sub_succ, Nat.zero_sub]
| n✝ : Nat
⊢ 1 - (n✝ + 1) = if n✝ + 1 = 0 then 1 else 0 | rw [if_neg (Nat.succ_ne_zero _), Nat.succ_sub_succ, Nat.zero_sub] | no goals | 9a7412d67d205851 |
List.suffix_cons_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Sublist.lean | theorem suffix_cons_iff : l₁ <:+ a :: l₂ ↔ l₁ = a :: l₂ ∨ l₁ <:+ l₂ | case mp.intro.cons
α✝ : Type u_1
l₁ : List α✝
a : α✝
l₂ : List α✝
head✝ : α✝
tail✝ : List α✝
hl₃ : head✝ :: tail✝ ++ l₁ = a :: l₂
⊢ l₁ = a :: l₂ ∨ l₁ <:+ l₂ | simp only [cons_append] at hl₃ | case mp.intro.cons
α✝ : Type u_1
l₁ : List α✝
a : α✝
l₂ : List α✝
head✝ : α✝
tail✝ : List α✝
hl₃ : head✝ :: (tail✝ ++ l₁) = a :: l₂
⊢ l₁ = a :: l₂ ∨ l₁ <:+ l₂ | 3985de2c4a53c38e |
Multiset.toEmbedding_coeEquiv_trans | Mathlib/Data/Multiset/Fintype.lean | theorem toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding | α : Type u_1
inst✝ : DecidableEq α
m : Multiset α
⊢ m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype fun x => x ∈ m.toEnumFinset) = m.coeEmbedding | ext <;> rfl | no goals | 894800d40a2b0a9a |
WeierstrassCurve.map_b₆ | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | @[simp]
lemma map_b₆ : (W.map φ).b₆ = φ W.b₆ | R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
A : Type v
inst✝ : CommRing A
φ : R →+* A
⊢ φ W.a₃ ^ 2 + 4 * φ W.a₆ = φ (W.a₃ ^ 2 + 4 * W.a₆) | map_simp | no goals | b870cc1b285dec7e |
Polynomial.valuation_of_mk | Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | theorem valuation_of_mk (f : Polynomial K) {g : Polynomial K} (hg : g ≠ 0) :
(Polynomial.idealX K).valuation _ (RatFunc.mk f g) =
(Polynomial.idealX K).intValuation f / (Polynomial.idealX K).intValuation g | K : Type u_1
inst✝ : Field K
f g : K[X]
hg : g ≠ 0
⊢ (valuation (RatFunc K) (idealX K)) (RatFunc.mk f g) = (idealX K).intValuation f / (idealX K).intValuation g | simp only [RatFunc.mk_eq_mk' _ hg, valuation_of_mk'] | no goals | ace054a4a9f8a2f2 |
Std.DHashMap.Internal.List.getValueCast!_alterKey | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getValueCast!_alterKey {k k' : α} [Inhabited (β k')] {f : Option (β k) → Option (β k)}
(l : List ((a : α) × β a)) (hl : DistinctKeys l) : getValueCast! k' (alterKey k f l) =
if heq : k == k' then
(f (getValueCast? k l)).map (cast (congrArg β <| eq_of_beq heq)) |>.get!
else
getVal... | α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : LawfulBEq α
k k' : α
inst✝ : Inhabited (β k')
f : Option (β k) → Option (β k)
l : List ((a : α) × β a)
hl : DistinctKeys l
⊢ (if h : (k == k') = true then cast ⋯ (f (getValueCast? k l)) else getValueCast? k' l).get! =
if heq : (k == k') = true then (Option.map (cast... | split | case isTrue
α : Type u
β : α → Type v
inst✝² : BEq α
inst✝¹ : LawfulBEq α
k k' : α
inst✝ : Inhabited (β k')
f : Option (β k) → Option (β k)
l : List ((a : α) × β a)
hl : DistinctKeys l
h✝ : (k == k') = true
⊢ (cast ⋯ (f (getValueCast? k l))).get! = (Option.map (cast ⋯) (f (getValueCast? k l))).get!
case isFalse
α : Ty... | 8eb57811416acb53 |
LieAlgebra.derivedSeries_of_bot_eq_bot | Mathlib/Algebra/Lie/Solvable.lean | theorem derivedSeries_of_bot_eq_bot (k : ℕ) : derivedSeriesOfIdeal R L k ⊥ = ⊥ | R : Type u
L : Type v
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
k : ℕ
⊢ D k ⊥ ≤ ⊥ | exact derivedSeriesOfIdeal_le_self ⊥ k | no goals | 2c82da78fd86ffef |
Finmap.erase_union_singleton | Mathlib/Data/Finmap.lean | theorem erase_union_singleton (a : α) (b : β a) (s : Finmap β) (h : s.lookup a = some b) :
s.erase a ∪ singleton a b = s :=
ext_lookup fun x => by
by_cases h' : x = a
· subst a
rw [lookup_union_right not_mem_erase_self, lookup_singleton_eq, h]
· have : x ∉ singleton a b | α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
b : β a
s : Finmap β
h : lookup a s = some b
x : α
⊢ lookup x (erase a s ∪ singleton a b) = lookup x s | by_cases h' : x = a | case pos
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
b : β a
s : Finmap β
h : lookup a s = some b
x : α
h' : x = a
⊢ lookup x (erase a s ∪ singleton a b) = lookup x s
case neg
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
b : β a
s : Finmap β
h : lookup a s = some b
x : α
h' : ¬x = a
⊢ lookup x (eras... | 1c1c576ab183c6a1 |
Finset.piAntidiag_cons | Mathlib/Algebra/Order/Antidiag/Pi.lean | lemma piAntidiag_cons (hi : i ∉ s) (n : μ) :
piAntidiag (cons i s hi) n = (antidiagonal n).disjiUnion (fun p : μ × μ ↦
(piAntidiag s p.snd).map (addRightEmbedding fun t ↦ if t = i then p.fst else 0))
(pairwiseDisjoint_piAntidiag_map_addRightEmbedding hi _) | case h.mpr.intro.intro.intro.intro.intro.intro
ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n a : μ
g : ι → μ
hg : ∀ (i : ι), ¬g i = 0 → i ∈ s
hn : a + s.sum g = n
⊢ (addRightEmbedding fun t => if t = i the... | have := hg i | case h.mpr.intro.intro.intro.intro.intro.intro
ι : Type u_1
μ : Type u_2
inst✝³ : DecidableEq ι
inst✝² : AddCancelCommMonoid μ
inst✝¹ : HasAntidiagonal μ
inst✝ : DecidableEq μ
i : ι
s : Finset ι
hi : i ∉ s
n a : μ
g : ι → μ
hg : ∀ (i : ι), ¬g i = 0 → i ∈ s
hn : a + s.sum g = n
this : ¬g i = 0 → i ∈ s
⊢ (addRightEmbeddi... | 19ca49fbd299904c |
Polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime | Mathlib/RingTheory/Polynomial/Basic.lean | theorem sup_ker_aeval_eq_ker_aeval_mul_of_coprime (f : M →ₗ[R] M) {p q : R[X]}
(hpq : IsCoprime p q) :
LinearMap.ker (aeval f p) ⊔ LinearMap.ker (aeval f q) = LinearMap.ker (aeval f (p * q)) | case intro.intro
R : Type u
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
p q : R[X]
v : M
hv : v ∈ LinearMap.ker ((aeval f) (p * q))
p' q' : R[X]
hpq' : p' * p + q' * q = 1
⊢ ∃ y ∈ LinearMap.ker ((aeval f) p), ∃ z ∈ LinearMap.ker ((aeval f) q), y + z = v | have h_eval₂_qpp' :=
calc
aeval f (q * (p * p')) v = aeval f (p' * (p * q)) v := by
rw [mul_comm, mul_assoc, mul_comm, mul_assoc, mul_comm q p]
_ = 0 := by rw [aeval_mul, LinearMap.mul_apply, LinearMap.mem_ker.1 hv, LinearMap.map_zero] | case intro.intro
R : Type u
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : M →ₗ[R] M
p q : R[X]
v : M
hv : v ∈ LinearMap.ker ((aeval f) (p * q))
p' q' : R[X]
hpq' : p' * p + q' * q = 1
h_eval₂_qpp' : ((aeval f) (q * (p * p'))) v = 0
⊢ ∃ y ∈ LinearMap.ker ((aeval f) p), ∃ z ∈ LinearMap.ker... | bb84be2e4bee1df0 |
List.Perm.eq_of_sorted | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Perm.lean | theorem Perm.eq_of_sorted : ∀ {l₁ l₂ : List α}
(_ : ∀ a b, a ∈ l₁ → b ∈ l₂ → le a b → le b a → a = b)
(_ : l₁.Pairwise le) (_ : l₂.Pairwise le) (_ : l₁ ~ l₂), l₁ = l₂
| [], [], _, _, _, _ => rfl
| [], b :: l₂, _, _, _, h => by simp_all
| a :: l₁, [], _, _, _, h => by simp_all
| a :: l₁, b :: l₂, w, h₁, ... | α : Type u_1
le : α → α → Prop
a : α
l₁ l₂ : List α
h₁ : Pairwise le (a :: l₁)
w : ∀ (a_1 b : α), a_1 ∈ a :: l₁ → b ∈ a :: l₂ → le a_1 b → le b a_1 → a_1 = b
h₂ : Pairwise le (a :: l₂)
h : a :: l₁ ~ a :: l₂
am : a ∈ a :: l₂
bm : a ∈ a :: l₁
⊢ a :: l₁ = a :: l₂ | simp only [perm_cons] at h | α : Type u_1
le : α → α → Prop
a : α
l₁ l₂ : List α
h₁ : Pairwise le (a :: l₁)
w : ∀ (a_1 b : α), a_1 ∈ a :: l₁ → b ∈ a :: l₂ → le a_1 b → le b a_1 → a_1 = b
h₂ : Pairwise le (a :: l₂)
am : a ∈ a :: l₂
bm : a ∈ a :: l₁
h : l₁ ~ l₂
⊢ a :: l₁ = a :: l₂ | ac7218468435b086 |
Real.Wallis.W_eq_factorial_ratio | Mathlib/Data/Real/Pi/Wallis.lean | theorem W_eq_factorial_ratio (n : ℕ) :
W n = 2 ^ (4 * n) * n ! ^ 4 / ((2 * n)! ^ 2 * (2 * n + 1)) | n : ℕ
⊢ W n = 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)! ^ 2 * (2 * ↑n + 1)) | induction' n with n IH | case zero
⊢ W 0 = 2 ^ (4 * 0) * ↑0! ^ 4 / (↑(2 * 0)! ^ 2 * (2 * ↑0 + 1))
case succ
n : ℕ
IH : W n = 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)! ^ 2 * (2 * ↑n + 1))
⊢ W (n + 1) = 2 ^ (4 * (n + 1)) * ↑(n + 1)! ^ 4 / (↑(2 * (n + 1))! ^ 2 * (2 * ↑(n + 1) + 1)) | 386b2268ddd44bec |
Nat.pow_lt_pow_succ | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem pow_lt_pow_succ (h : 1 < a) : a ^ n < a ^ (n + 1) | a n : Nat
h : 1 < a
⊢ a ^ n < a ^ (n + 1) | rw [← Nat.mul_one (a^n), Nat.pow_succ] | a n : Nat
h : 1 < a
⊢ a ^ n * 1 < a ^ n * a | 4954a059e63dd328 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastShiftRightConst.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/ShiftRight.lean | theorem go_get_aux (aig : AIG α) (distance : Nat) (input : AIG.RefVec aig w)
(curr : Nat) (hcurr : curr ≤ w) (s : AIG.RefVec aig curr) :
∀ (idx : Nat) (hidx : idx < curr) (hfoo),
(go aig input distance curr hcurr s).vec.get idx (by omega)
=
(s.get idx hidx).cast hfoo | case isFalse
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
aig : AIG α
distance curr : Nat
s : aig.RefVec curr
idx : Nat
hidx : idx < curr
input : aig.RefVec curr
hcurr : curr ≤ curr
res : RefVecEntry α curr
h✝ : ¬curr < curr
hgo : { aig := aig, vec := ⋯ ▸ s } = res
⊢ ∀ (hfoo : True), (⋯ ▸ s).get idx ⋯ = (s.get id... | simp | no goals | 411d6c36c2c282f7 |
Complex.hasFPowerSeriesOnBall_of_differentiable_off_countable | Mathlib/Analysis/Complex/CauchyIntegral.lean | theorem hasFPowerSeriesOnBall_of_differentiable_off_countable {R : ℝ≥0} {c : ℂ} {f : ℂ → E}
{s : Set ℂ} (hs : s.Countable) (hc : ContinuousOn f (closedBall c R))
(hd : ∀ z ∈ ball c R \ s, DifferentiableAt ℂ f z) (hR : 0 < R) :
HasFPowerSeriesOnBall f (cauchyPowerSeries f c R) c R where
r_le := le_radius_c... | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ≥0
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ z ∈ ball c ↑R \ s, DifferentiableAt ℂ f z
hR : 0 < R
w : ℂ
hw : w ∈ EMetric.ball 0 ↑R
hw' : c + w ∈ ball c ↑R
⊢ HasSum (fun n => (cauc... | rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
hs hw' hc hd] | E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ≥0
c : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ z ∈ ball c ↑R \ s, DifferentiableAt ℂ f z
hR : 0 < R
w : ℂ
hw : w ∈ EMetric.ball 0 ↑R
hw' : c + w ∈ ball c ↑R
⊢ HasSum (fun n => (cauc... | cdf5cba971469b1e |
Nat.AM_GM | Mathlib/Data/Nat/Sqrt.lean | private lemma AM_GM : {a b : ℕ} → (4 * a * b ≤ (a + b) * (a + b))
| 0, _ => by rw [Nat.mul_zero, Nat.zero_mul]; exact zero_le _
| _, 0 => by rw [Nat.mul_zero]; exact zero_le _
| a + 1, b + 1 => by
simpa only [Nat.mul_add, Nat.add_mul, show (4 : ℕ) = 1 + 1 + 1 + 1 from rfl, Nat.one_mul,
Nat.mul_one, Nat.... | a b : ℕ
⊢ 4 * (a + 1) * (b + 1) ≤ (a + 1 + (b + 1)) * (a + 1 + (b + 1)) | simpa only [Nat.mul_add, Nat.add_mul, show (4 : ℕ) = 1 + 1 + 1 + 1 from rfl, Nat.one_mul,
Nat.mul_one, Nat.add_assoc, Nat.add_left_comm, Nat.add_le_add_iff_left]
using Nat.add_le_add_right (@AM_GM a b) 4 | no goals | c5edabcb7b92faa4 |
ContDiffBump.tendsto_support_normed_smallSets | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | theorem tendsto_support_normed_smallSets {ι} {φ : ι → ContDiffBump c} {l : Filter ι}
(hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) :
Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets | E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
ι : Type u_2
φ : ι → ContDiffBump c
l : Filter ι
hφ : ∀ ε > 0, ∀... | rw [(φ i).support_normed_eq] | E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsOpenPosMeasure
ι : Type u_2
φ : ι → ContDiffBump c
l : Filter ι
hφ : ∀ ε > 0, ∀... | 3e51bd2d58daa661 |
Finset.prod_range_div_prod_range | Mathlib/Algebra/BigOperators/Intervals.lean | theorem prod_range_div_prod_range {α : Type*} [CommGroup α] {f : ℕ → α} {n m : ℕ} (hnm : n ≤ m) :
((∏ k ∈ range m, f k) / ∏ k ∈ range n, f k) =
∏ k ∈ (range m).filter fun k => n ≤ k, f k | α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
⊢ (∏ k ∈ range m, f k) / ∏ k ∈ range n, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k | rw [← prod_Ico_eq_div f hnm] | α : Type u_3
inst✝ : CommGroup α
f : ℕ → α
n m : ℕ
hnm : n ≤ m
⊢ ∏ k ∈ Ico n m, f k = ∏ k ∈ filter (fun k => n ≤ k) (range m), f k | d8c3c7f95763a0f2 |
Polynomial.Gal.ext | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | theorem ext {σ τ : p.Gal} (h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x) : σ = τ | F : Type u_1
inst✝ : Field F
p : F[X]
σ τ : p.Gal
h : ∀ x ∈ p.rootSet p.SplittingField, σ x = τ x
x : p.SplittingField
⊢ AlgHom.equalizer ↑σ ↑τ = ⊤ | rwa [eq_top_iff, ← SplittingField.adjoin_rootSet, Algebra.adjoin_le_iff] | no goals | fbf3c970f0153c29 |
Algebra.adjoin_induction₂ | Mathlib/RingTheory/Adjoin/Basic.lean | theorem adjoin_induction₂ {s : Set A} {p : (x y : A) → x ∈ adjoin R s → y ∈ adjoin R s → Prop}
(mem_mem : ∀ (x) (y) (hx : x ∈ s) (hy : y ∈ s), p x y (subset_adjoin hx) (subset_adjoin hy))
(algebraMap_both : ∀ r₁ r₂, p (algebraMap R A r₁) (algebraMap R A r₂) (algebraMap_mem _ r₁)
(algebraMap_mem _ r₂))
... | case mem.algebraMap
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
p : (x y : A) → x ∈ adjoin R s → y ∈ adjoin R s → Prop
mem_mem : ∀ (x y : A) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
algebraMap_both : ∀ (r₁ r₂ : R), p ((algebraMap R A) r₁) ((algebraMap R A) r₂) ⋯ ⋯
algeb... | exact algebraMap_left _ _ hz | no goals | a48d8c6c65956c35 |
CoxeterSystem.IsReduced.nodup_rightInvSeq | Mathlib/GroupTheory/Coxeter/Inversion.lean | theorem IsReduced.nodup_rightInvSeq {ω : List B} (rω : cs.IsReduced ω) : List.Nodup (ris ω) | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
dup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?
j'_lt_length : j' < ω.length
⊢ j < (cs.rightInvSeq ω).length | simp | B : Type u_1
W : Type u_2
inst✝ : Group W
M : CoxeterMatrix B
cs : CoxeterSystem M W
ω : List B
rω : cs.IsReduced ω
j j' : ℕ
j_lt_j' : j < j'
dup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?
j'_lt_length : j' < ω.length
⊢ j < ω.length | 4f092e79e4ea2c15 |
Finpartition.exists_mem | Mathlib/Order/Partition/Finpartition.lean | theorem exists_mem (ha : a ∈ s) : ∃ t ∈ P.parts, a ∈ t | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
ha : a ∈ s
⊢ ∃ t ∈ P.parts, a ∈ t | simp_rw [← P.sup_parts] at ha | α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
a : α
ha : a ∈ P.parts.sup id
⊢ ∃ t ∈ P.parts, a ∈ t | bc797f3d133f7276 |
Filter.isCoboundedUnder_ge_add | Mathlib/Order/LiminfLimsup.lean | lemma isCoboundedUnder_ge_add (hu : f.IsBoundedUnder (· ≤ ·) u)
(hv : f.IsCoboundedUnder (· ≥ ·) v) :
f.IsCoboundedUnder (· ≥ ·) (u + v) | case intro.intro
α : Type u_6
R : Type u_7
inst✝⁴ : LinearOrder R
inst✝³ : Add R
f : Filter α
inst✝² : f.NeBot
inst✝¹ : CovariantClass R R (fun a b => a + b) fun x1 x2 => x1 ≤ x2
inst✝ : CovariantClass R R (fun a b => b + a) fun x1 x2 => x1 ≤ x2
u v : α → R
hu : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
hv : IsCobounde... | apply IsCoboundedUnder.of_frequently_le (a := U + V) | case intro.intro
α : Type u_6
R : Type u_7
inst✝⁴ : LinearOrder R
inst✝³ : Add R
f : Filter α
inst✝² : f.NeBot
inst✝¹ : CovariantClass R R (fun a b => a + b) fun x1 x2 => x1 ≤ x2
inst✝ : CovariantClass R R (fun a b => b + a) fun x1 x2 => x1 ≤ x2
u v : α → R
hu : IsBoundedUnder (fun x1 x2 => x1 ≤ x2) f u
hv : IsCobounde... | 5e0281c3505ce3a2 |
BoxIntegral.unitPartition.tendsto_card_div_pow₂ | Mathlib/Analysis/BoxIntegral/UnitPartition.lean | theorem tendsto_card_div_pow₂ (hs₁ : IsBounded s)
(hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s) {x y : ℝ} (hx : 0 < x) (hy : x ≤ y) :
Nat.card ↑(s ∩ x⁻¹ • L) ≤ Nat.card ↑(s ∩ y⁻¹ • L) | ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ Nat.card ↑(s ∩ x⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) ≤
Nat.card ↑(s ∩ y⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) | rw [Nat.card_congr (tendsto_card_div_pow₁ s hx.ne'),
Nat.card_congr (tendsto_card_div_pow₁ s (hx.trans_le hy).ne')] | ι : Type u_1
inst✝ : Fintype ι
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₄ : ∀ ⦃x y : ℝ⦄, 0 < x → x ≤ y → x • s ⊆ y • s
x y : ℝ
hx : 0 < x
hy : x ≤ y
⊢ Nat.card ↑(x • s ∩ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) ≤
Nat.card ↑(y • s ∩ ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))) | 5e10a1a1aca1cc38 |
MonoidAlgebra.mapDomain_mul | Mathlib/Algebra/MonoidAlgebra/MapDomain.lean | theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Mul α] [Mul α₂]
{F : Type*} [FunLike F α α₂] [MulHomClass F α α₂] (f : F) (x y : MonoidAlgebra β α) :
mapDomain f (x * y) = mapDomain f x * mapDomain f y | case h_add
α : Type u_3
β : Type u_4
α₂ : Type u_5
inst✝⁴ : Semiring β
inst✝³ : Mul α
inst✝² : Mul α₂
F : Type u_6
inst✝¹ : FunLike F α α₂
inst✝ : MulHomClass F α α₂
f : F
x y : MonoidAlgebra β α
⊢ ∀ (b : α₂) (m₁ m₂ : β),
(sum (mapDomain (⇑f) y) fun a₂ b₂ => single (b * a₂) ((m₁ + m₂) * b₂)) =
(sum (mapDomain... | simp [add_mul] | no goals | 7e14c1d9f46ae3b4 |
Subgroup.exists_finiteIndex_of_leftCoset_cover | Mathlib/GroupTheory/CosetCover.lean | theorem exists_finiteIndex_of_leftCoset_cover : ∃ k ∈ s, (H k).FiniteIndex | G : Type u_1
inst✝ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ∅ = Set.univ
hempty : s = ∅
⊢ False | exact Set.empty_ne_univ hcovers | no goals | a157e4700b8dbc0e |
Subgroup.comap_normalizer_eq_of_surjective | Mathlib/Algebra/Group/Subgroup/Basic.lean | theorem comap_normalizer_eq_of_surjective (H : Subgroup G) {f : N →* G}
(hf : Function.Surjective f) : H.normalizer.comap f = (H.comap f).normalizer :=
le_antisymm (le_normalizer_comap f)
(by
intro x hx
simp only [mem_comap, mem_normalizer_iff] at *
intro n
rcases hf n with ⟨y, rfl⟩
... | G : Type u_1
inst✝¹ : Group G
N : Type u_5
inst✝ : Group N
H : Subgroup G
f : N →* G
hf : Surjective ⇑f
x : N
hx : x ∈ (comap f H).normalizer
⊢ x ∈ comap f H.normalizer | simp only [mem_comap, mem_normalizer_iff] at * | G : Type u_1
inst✝¹ : Group G
N : Type u_5
inst✝ : Group N
H : Subgroup G
f : N →* G
hf : Surjective ⇑f
x : N
hx : ∀ (h : N), f h ∈ H ↔ f (x * h * x⁻¹) ∈ H
⊢ ∀ (h : G), h ∈ H ↔ f x * h * (f x)⁻¹ ∈ H | a9a02fd3dfef1407 |
HasCompactSupport.enorm_le_lintegral_Ici_deriv | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | lemma _root_.HasCompactSupport.enorm_le_lintegral_Ici_deriv
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
‖f x‖ₑ ≤ ∫⁻ y in Iic x, ‖deriv f y‖ₑ | case h.e'_4.h.e'_4.h
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → F
hf : ContDiff ℝ 1 f
h'f : HasCompactSupport f
x : ℝ
I : F →L[ℝ] Completion F := Completion.toComplL
f' : ℝ → Completion F := ⇑I ∘ f
hf' : ContDiff ℝ 1 f'
h'f' : HasCompactSupport f'
this : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic x, ‖d... | simp [I] | no goals | 43d9795b86d3ea3e |
BitVec.and_eq_allOnes_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem and_eq_allOnes_iff {x y : BitVec w} :
x &&& y = allOnes w ↔ x = allOnes w ∧ y = allOnes w | case mpr
w : Nat
x y : BitVec w
h : x = allOnes w ∧ y = allOnes w
⊢ x &&& y = allOnes w | simp [h] | no goals | 0e7a9a27b7e9b2f9 |
MeasureTheory.Filtration.filtrationOfSet_eq_natural | Mathlib/Probability/Process/Filtration.lean | theorem filtrationOfSet_eq_natural [MulZeroOneClass β] [Nontrivial β] {s : ι → Set Ω}
(hsm : ∀ i, MeasurableSet[m] (s i)) :
filtrationOfSet hsm = natural (fun i => (s i).indicator (fun _ => 1 : Ω → β)) fun i =>
stronglyMeasurable_one.indicator (hsm i) | case h.refine_2.intro.intro.intro.intro
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s i)
... | obtain heq | heq | heq | heq := Set.indicator_const_preimage (s n) u (1 : β) | case h.refine_2.intro.intro.intro.intro.inl
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝⁵ : TopologicalSpace β
inst✝⁴ : MetrizableSpace β
mβ : MeasurableSpace β
inst✝³ : BorelSpace β
inst✝² : Preorder ι
inst✝¹ : MulZeroOneClass β
inst✝ : Nontrivial β
s : ι → Set Ω
hsm : ∀ (i : ι), MeasurableSet (s... | 7ffbace60117795e |
MvPFunctor.M.map_dest | Mathlib/Data/PFunctor/Multivariate/M.lean | theorem M.map_dest {α β : TypeVec n} (g : (α ::: P.M α) ⟹ (β ::: P.M β)) (x : P.M α)
(h : ∀ x : P.M α, lastFun g x = (dropFun g <$$> x : P.M β)) :
g <$$> M.dest P x = M.dest P (dropFun g <$$> x) | case e_a
n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ::: P.M α ⟹ β ::: P.M β
x : P.M α
h : ∀ (x : P.M α), lastFun g x = dropFun g <$$> x
⊢ lastFun g = lastFun (dropFun g ::: fun x => dropFun g <$$> x) | simp only [lastFun_appendFun] | case e_a
n : ℕ
P : MvPFunctor.{u} (n + 1)
α β : TypeVec.{u} n
g : α ::: P.M α ⟹ β ::: P.M β
x : P.M α
h : ∀ (x : P.M α), lastFun g x = dropFun g <$$> x
⊢ lastFun g = fun x => dropFun g <$$> x | 3bd96d98f9faee82 |
Ordinal.CNF_foldr | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | theorem CNF_foldr (b o : Ordinal) : (CNF b o).foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 = o | case refine_2
b o✝ o : Ordinal.{u_1}
ho : o ≠ 0
IH : foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o
⊢ foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b o) = o | rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod] | no goals | 596aaa490422ccf7 |
Finsupp.sum_ite_self_eq | Mathlib/Algebra/BigOperators/Finsupp.lean | theorem sum_ite_self_eq [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(f.sum fun x v => ite (a = x) v 0) = f a | α : Type u_1
inst✝¹ : DecidableEq α
N : Type u_16
inst✝ : AddCommMonoid N
f : α →₀ N
a : α
⊢ (f.sum fun x v => if a = x then v else 0) = f a | classical
convert f.sum_ite_eq a fun _ => id
simp [ite_eq_right_iff.2 Eq.symm] | no goals | 5c76290e980ef3f2 |
Ordinal.mem_closure_tfae | Mathlib/SetTheory/Ordinal/Topology.lean | theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
TFAE [a ∈ closure s,
a ∈ closure (s ∩ Iic a),
(s ∩ Iic a).Nonempty ∧ sSup (s ∩ Iic a) = a,
∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
(∀ x hx, f x hx ∈ s) ∧ b... | case intro.intro.intro.intro.refine_2.refine_2
s t : Set Ordinal.{u}
hts : t ⊆ s
hne : t.Nonempty
hbdd : BddAbove t
tfae_1_to_2 : sSup t ∈ closure s → sSup t ∈ closure (s ∩ Iic (sSup t))
tfae_2_to_3 : sSup t ∈ closure (s ∩ Iic (sSup t)) → (s ∩ Iic (sSup t)).Nonempty ∧ sSup (s ∩ Iic (sSup t)) = sSup t
tfae_3_to_4 :
(s... | refine (if_pos hx).symm.trans_le (le_bsup _ _ <| (hlub.1 hx).trans_lt (lt_succ _)) | no goals | f361ec1995f661dc |
Module.finitePresentation_of_ker | Mathlib/Algebra/Module/FinitePresentation.lean | lemma Module.finitePresentation_of_ker [Module.FinitePresentation R N]
(l : M →ₗ[R] N) (hl : Function.Surjective l) [Module.FinitePresentation R (LinearMap.ker l)] :
Module.FinitePresentation R M | case fg_top
R : Type u_1
M : Type u_3
N : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : FinitePresentation R N
l : M →ₗ[R] N
hl : Function.Surjective ⇑l
inst✝ : FinitePresentation R ↥(LinearMap.ker l)
s : Finset M
hs : Submodule.span R ↑s = ⊤
π... | exact Module.FinitePresentation.fg_ker _ (hl.comp H) | no goals | 888cd3f368652bf6 |
LinearMap.bound_of_sphere_bound | Mathlib/Analysis/NormedSpace/RCLike.lean | theorem LinearMap.bound_of_sphere_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ) (f : E →ₗ[𝕜] 𝕜)
(h : ∀ z ∈ sphere (0 : E) r, ‖f z‖ ≤ c) (z : E) : ‖f z‖ ≤ c / r * ‖z‖ | 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
norm_f_z₁ : ‖f z₁‖ ≤ c
r_ne_zero : ↑r ≠ 0
eq : f z = ↑‖z‖ / ↑r... | exact mul_nonneg ((norm_nonneg _).trans norm_f_z₁) (norm_nonneg z) | no goals | 7391d26c013b6c1e |
CategoryTheory.presheafHom_isSheafFor | Mathlib/CategoryTheory/Sites/SheafHom.lean | lemma presheafHom_isSheafFor :
Presieve.IsSheafFor (presheafHom F G) S.arrows | case hex
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
hg : S.arrows g
⊢ app h... | have H := app_cond hG x hx g (𝟙 _) (by simpa using hg) | case hex
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
X : C
S : Sieve X
hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
Y : C
g : Y ⟶ X
hg : S.arrows g
H : app... | 03e610e11f4d0fc5 |
BoundedContinuousFunction.norm_integral_le_mul_norm | Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | lemma norm_integral_le_mul_norm [IsFiniteMeasure μ] (f : X →ᵇ E) :
‖∫ x, f x ∂μ‖ ≤ ENNReal.toReal (μ Set.univ) * ‖f‖ | X : Type u_1
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
μ : Measure X
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : OpensMeasurableSpace X
inst✝⁴ : SecondCountableTopology E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
f : X →ᵇ E
⊢ Integrable (fun... | exact (integrable_norm_iff f.continuous.measurable.aestronglyMeasurable).mpr (f.integrable μ) | no goals | d89709ec32a2079f |
BitVec.toInt_allOnes | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toInt_allOnes : (allOnes w).toInt = if 0 < w then -1 else 0 | case neg
w : Nat
h : ¬w = 0
⊢ (allOnes w).toInt = if 0 < w then -1 else 0 | have : 1 < 2 ^ w := by simp [h] | case neg
w : Nat
h : ¬w = 0
this : 1 < 2 ^ w
⊢ (allOnes w).toInt = if 0 < w then -1 else 0 | f00205704d5f5dc2 |
Plausible.InjectiveFunction.List.applyId_zip_eq | Mathlib/Testing/Plausible/Functions.lean | theorem List.applyId_zip_eq [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs)
(h₁ : xs.length = ys.length) (x y : α) (i : ℕ) (h₂ : xs[i]? = some x) :
List.applyId.{u} (xs.zip ys) x = y ↔ ys[i]? = some y | case cons.zero
α : Type u
inst✝ : DecidableEq α
y x' : α
xs ys : List α
h₀ : (x' :: xs).Nodup
h₁ : (x' :: xs).length = ys.length
xs_ih :
∀ {ys : List α},
xs.Nodup → xs.length = ys.length → ∀ (i : ℕ), xs[i]? = some x' → (applyId (xs.zip ys) x' = y ↔ ys[i]? = some y)
⊢ applyId ((x' :: xs).zip ys) x' = y ↔ ys[0]? = ... | cases ys | case cons.zero.nil
α : Type u
inst✝ : DecidableEq α
y x' : α
xs : List α
h₀ : (x' :: xs).Nodup
xs_ih :
∀ {ys : List α},
xs.Nodup → xs.length = ys.length → ∀ (i : ℕ), xs[i]? = some x' → (applyId (xs.zip ys) x' = y ↔ ys[i]? = some y)
h₁ : (x' :: xs).length = [].length
⊢ applyId ((x' :: xs).zip []) x' = y ↔ [][0]? =... | b43790ed7cfe4c9d |
CategoryTheory.Functor.additive_of_full_essSurj_comp | Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean | lemma additive_of_full_essSurj_comp [Full F] [EssSurj F] (G : D ⥤ E)
[(F ⋙ G).Additive] : G.Additive where
map_add {X Y f g} | case intro.intro
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_5, u_2} D
inst✝⁷ : Category.{u_6, u_3} E
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : Preadditive E
F : C ⥤ D
inst✝³ : F.Additive
inst✝² : F.Full
inst✝¹ : F.EssSurj
G : D ⥤ E
inst✝ : (F ⋙ G).Additive
X ... | simp only [← cancel_mono (G.map (F.objObjPreimageIso Y).inv),
← cancel_epi (G.map (F.objObjPreimageIso X).hom),
Preadditive.add_comp, Preadditive.comp_add, ← Functor.map_comp] | case intro.intro
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_5, u_2} D
inst✝⁷ : Category.{u_6, u_3} E
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : Preadditive E
F : C ⥤ D
inst✝³ : F.Additive
inst✝² : F.Full
inst✝¹ : F.EssSurj
G : D ⥤ E
inst✝ : (F ⋙ G).Additive
X ... | 7c8555e61c3aa8c9 |
SimplexCategory.eq_id_of_mono | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem eq_id_of_mono {x : SimplexCategory} (i : x ⟶ x) [Mono i] : i = 𝟙 _ | case hf
x : SimplexCategory
i : x ⟶ x
inst✝ : Mono i
⊢ Function.Bijective (Hom.toOrderHom i).toFun | dsimp | case hf
x : SimplexCategory
i : x ⟶ x
inst✝ : Mono i
⊢ Function.Bijective ⇑(Hom.toOrderHom i) | 5ceb9935050d299a |
CategoryTheory.Subpresheaf.eq_top_iff_isIso | Mathlib/CategoryTheory/Subpresheaf/Basic.lean | theorem eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι | case mpr.obj.h.h
C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
G : Subpresheaf F
H : IsIso G.ι
U : Cᵒᵖ
x : F.obj U
⊢ (ConcreteCategory.hom (G.ι.app U)) ((ConcreteCategory.hom (inv (G.ι.app U))) x) ∈ G.obj U | exact ((inv (G.ι.app U)) x).2 | no goals | 59961e19256fb918 |
iteratedFDerivWithin_comp_of_eventually_mem | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | theorem iteratedFDerivWithin_comp_of_eventually_mem {t : Set F}
(hg : ContDiffWithinAt 𝕜 n g t (f x)) (hf : ContDiffWithinAt 𝕜 n f s x)
(ht : UniqueDiffOn 𝕜 t) (hs : UniqueDiffOn 𝕜 s) (hxs : x ∈ s) (hst : ∀ᶠ y in 𝓝[s] x, f y ∈ t)
{i : ℕ} (hi : i ≤ n) :
iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
... | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
g : F → G
x : E
n : WithTop ℕ∞
t : Set F
hg : C... | have H₁ : ∀ᶠ u in (𝓝[s] x).smallSets,
HasFTaylorSeriesUpToOn i f (ftaylorSeriesWithin 𝕜 f s) u :=
hf.eventually_hasFTaylorSeriesUpToOn hs hxs hi | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
f : E → F
g : F → G
x : E
n : WithTop ℕ∞
t : Set F
hg : C... | 0ff5459c06f40f96 |
OrderIso.isMax_apply | Mathlib/Order/Hom/Basic.lean | theorem OrderIso.isMax_apply {α β : Type*} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} :
IsMax (f x) ↔ IsMax x | α : Type u_6
β : Type u_7
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
x : α
⊢ IsMax x → IsMax (f x) | conv_lhs => rw [← f.symm_apply_apply x] | α : Type u_6
β : Type u_7
inst✝¹ : Preorder α
inst✝ : Preorder β
f : α ≃o β
x : α
⊢ IsMax (f.symm (f x)) → IsMax (f x) | 4ab86bbfbe5770b3 |
LinearPMap.mem_inverse_graph_snd_eq_zero | Mathlib/LinearAlgebra/LinearPMap.lean | theorem mem_inverse_graph_snd_eq_zero (x : F × E)
(hv : x ∈ (graph f).map (LinearEquiv.prodComm R E F))
(hv' : x.fst = 0) : x.snd = 0 | case intro.intro.intro.intro.refl
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
a : E
b : F
ha : a ∈ f.domain
h1 : ↑f ⟨a, ⋯⟩ = 0
hv' : b = 0
hf : ⟨a, ha⟩ = 0
⊢ a = 0 | simp only [Submodule.mk_eq_zero] at hf | case intro.intro.intro.intro.refl
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
f : E →ₗ.[R] F
a : E
b : F
ha : a ∈ f.domain
h1 : ↑f ⟨a, ⋯⟩ = 0
hv' : b = 0
hf : a = 0
⊢ a = 0 | ebcf3facfad50c42 |
Finset.prodMk_sup'_sup' | Mathlib/Data/Finset/Lattice/Prod.lean | /-- See also `Finset.sup'_prodMap`. -/
lemma prodMk_sup'_sup' (hs : s.Nonempty) (ht : t.Nonempty) (f : ι → α) (g : κ → β) :
(sup' s hs f, sup' t ht g) = sup' (s ×ˢ t) (hs.product ht) (Prod.map f g) :=
eq_of_forall_ge_iff fun i ↦ by
obtain ⟨a, ha⟩ := hs
obtain ⟨b, hb⟩ := ht
simp only [Prod.map, sup'_le... | case intro.intro
ι : Type u_7
κ : Type u_8
α : Type u_9
β : Type u_10
inst✝¹ : SemilatticeSup α
inst✝ : SemilatticeSup β
s : Finset ι
t : Finset κ
f : ι → α
g : κ → β
i : α × β
a : ι
ha : a ∈ s
b : κ
hb : b ∈ t
⊢ ((∀ b ∈ s, f b ≤ i.1) ∧ ∀ b ∈ t, g b ≤ i.2) ↔ ∀ (a : ι) (b : κ), a ∈ s → b ∈ t → f a ≤ i.1 ∧ g b ≤ i.2 | exact ⟨by aesop, fun h ↦ ⟨fun i hi ↦ (h _ _ hi hb).1, fun j hj ↦ (h _ _ ha hj).2⟩⟩ | no goals | 0af939cf04ef6168 |
HomologicalComplex.extend.leftHomologyData.lift_d_comp_eq_zero_iff' | Mathlib/Algebra/Homology/Embedding/ExtendHomology.lean | /-- Auxiliary lemma for `lift_d_comp_eq_zero_iff`. -/
lemma lift_d_comp_eq_zero_iff' ⦃W : C⦄ (f' : K.X i ⟶ cone.pt)
(hf' : f' ≫ cone.ι = K.d i j)
(f'' : (K.extend e).X i' ⟶ cone.pt)
(hf'' : f'' ≫ cone.ι ≫ (extendXIso K e hj').inv = (K.extend e).d i' j')
(φ : cone.pt ⟶ W) :
f' ≫ φ = 0 ↔ f'' ≫ φ = 0 | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
cone : KernelFork (K.d j k)
hcone... | apply Fork.IsLimit.hom_ext hcone | ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{u_4, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
cone : KernelFork (K.d j k)
hcone... | 47e14c7240329723 |
PowerSeries.eq_zero_or_eq_zero_of_mul_eq_zero | Mathlib/RingTheory/PowerSeries/Basic.lean | theorem eq_zero_or_eq_zero_of_mul_eq_zero [NoZeroDivisors R] (φ ψ : R⟦X⟧) (h : φ * ψ = 0) :
φ = 0 ∨ ψ = 0 | R : Type u_1
inst✝¹ : Ring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ * ψ = 0
H : ¬φ = 0
ex : ∃ m, (coeff R m) φ ≠ 0
m : ℕ := Nat.find ex
⊢ ψ = 0 | have hm₁ : coeff R m φ ≠ 0 := Nat.find_spec ex | R : Type u_1
inst✝¹ : Ring R
inst✝ : NoZeroDivisors R
φ ψ : R⟦X⟧
h : φ * ψ = 0
H : ¬φ = 0
ex : ∃ m, (coeff R m) φ ≠ 0
m : ℕ := Nat.find ex
hm₁ : (coeff R m) φ ≠ 0
⊢ ψ = 0 | d0b45343dbc828b7 |
Complex.partialGamma_add_one | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | theorem partialGamma_add_one {s : ℂ} (hs : 0 < s.re) {X : ℝ} (hX : 0 ≤ X) :
partialGamma (s + 1) X = s * partialGamma s X - (-X).exp * X ^ s | s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
F_der_I :
∀ x ∈ Ioo 0 X,
HasDerivAt (fun x => ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x
cont : Continuous fun x => (ofReal ∘ fun y => rexp (-y)) x * ↑x ^ s
der_ible : IntervalIntegrable (fun x => -(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (... | congr 2 | case e_a.e_f
s : ℂ
hs : 0 < s.re
X : ℝ
hX : 0 ≤ X
F_der_I :
∀ x ∈ Ioo 0 X,
HasDerivAt (fun x => ↑(rexp (-x)) * ↑x ^ s) (-(↑(rexp (-x)) * ↑x ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) x
cont : Continuous fun x => (ofReal ∘ fun y => rexp (-y)) x * ↑x ^ s
der_ible : IntervalIntegrable (fun x => -(↑(rexp (-x)) * ↑x ^ ... | 64c3078f05c8af7b |
Fintype.card_embedding_eq | Mathlib/Data/Fintype/CardEmbedding.lean | theorem card_embedding_eq {α β : Type*} [Fintype α] [Fintype β] [emb : Fintype (α ↪ β)] :
‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖ | case refine_3
β : Type u_2
inst✝ : Fintype β
γ : Type u_1
h : Fintype γ
ih : ‖γ ↪ β‖ = ‖β‖.descFactorial ‖γ‖
⊢ ‖Option γ ↪ β‖ = ‖β‖.descFactorial ‖Option γ‖ | rw [card_option, Nat.descFactorial_succ, card_congr (Embedding.optionEmbeddingEquiv γ β),
card_sigma, ← ih] | case refine_3
β : Type u_2
inst✝ : Fintype β
γ : Type u_1
h : Fintype γ
ih : ‖γ ↪ β‖ = ‖β‖.descFactorial ‖γ‖
⊢ ∑ i : γ ↪ β, ‖↑(Set.range ⇑i)ᶜ‖ = (‖β‖ - ‖γ‖) * ‖γ ↪ β‖ | 52dcb38ad480110e |
Ordinal.infinite_pigeonhole | Mathlib/SetTheory/Cardinal/Cofinality.lean | theorem infinite_pigeonhole {β α : Type u} (f : β → α) (h₁ : ℵ₀ ≤ #β) (h₂ : #α < (#β).ord.cof) :
∃ a : α, #(f ⁻¹' {a}) = #β | case intro
β α : Type u
f : β → α
h₁ : ℵ₀ ≤ #β
h₂ : #α < (#β).ord.cof
x : α
h : #β ≤ #↑(f ⁻¹' {x})
⊢ ∃ a, #↑(f ⁻¹' {a}) = #β | refine ⟨x, h.antisymm' ?_⟩ | case intro
β α : Type u
f : β → α
h₁ : ℵ₀ ≤ #β
h₂ : #α < (#β).ord.cof
x : α
h : #β ≤ #↑(f ⁻¹' {x})
⊢ #↑(f ⁻¹' {x}) ≤ #β | 8fa0d4d26a8c4da8 |
MvPolynomial.degree_degLexDegree | Mathlib/RingTheory/MvPolynomial/MonomialOrder/DegLex.lean | theorem degree_degLexDegree : (degLex.degree f).degree = f.totalDegree | case neg.a
σ : Type u_1
inst✝² : LinearOrder σ
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : WellFoundedGT σ
f : MvPolynomial σ R
hf : ¬f = 0
⊢ (f.support.sup fun s => s.sum fun x e => e) ≤ (degLex.degree f).degree | apply Finset.sup_le | case neg.a.a
σ : Type u_1
inst✝² : LinearOrder σ
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : WellFoundedGT σ
f : MvPolynomial σ R
hf : ¬f = 0
⊢ ∀ b ∈ f.support, (b.sum fun x e => e) ≤ (degLex.degree f).degree | 8a62b59720df2b16 |
LieSubmodule.comap_bracket_eq | Mathlib/Algebra/Lie/IdealOperations.lean | theorem comap_bracket_eq [LieModule R L M] (hf₁ : f.ker = ⊥) (hf₂ : N₂ ≤ f.range) :
comap f ⁅I, N₂⁆ = ⁅I, comap f N₂⁆ | R : Type u
L : Type v
M : Type w
M₂ : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M₂
inst✝³ : LieRingModule L M₂
N₂ : LieSubmodule R L M₂
f : M →ₗ⁅R,L⁆ M₂
inst✝² : LieAlgebra R L
inst✝¹ : LieModule R L ... | conv_lhs => rw [← map_comap_eq N₂ f hf₂] | R : Type u
L : Type v
M : Type w
M₂ : Type w₁
inst✝¹⁰ : CommRing R
inst✝⁹ : LieRing L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : AddCommGroup M₂
inst✝⁴ : Module R M₂
inst✝³ : LieRingModule L M₂
N₂ : LieSubmodule R L M₂
f : M →ₗ⁅R,L⁆ M₂
inst✝² : LieAlgebra R L
inst✝¹ : LieModule R L ... | e8e8a110167c4b1f |
List.prev_reverse_eq_next | Mathlib/Data/List/Cycle.lean | theorem prev_reverse_eq_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
prev l.reverse x (mem_reverse.mpr hx) = next l x hx | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
k : ℕ
hk : k < l.length
hx : l[k] ∈ l
lpos : 0 < l.length
key : l.length - 1 - k < l.length
⊢ l.reverse.prev l[k] ⋯ = l.next l[k] hx | rw [← getElem_pmap l.next (fun _ h => h) (by simpa using hk)] | case intro.intro
α : Type u_1
inst✝ : DecidableEq α
l : List α
h : l.Nodup
k : ℕ
hk : k < l.length
hx : l[k] ∈ l
lpos : 0 < l.length
key : l.length - 1 - k < l.length
⊢ l.reverse.prev l[k] ⋯ = (pmap l.next l ⋯)[k] | 223a9f0719f3b161 |
ENNReal.mul_rpow_eq_ite | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | theorem mul_rpow_eq_ite (x y : ℝ≥0∞) (z : ℝ) :
(x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z | case inr
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ∨ 0 < z
⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z | wlog hxy : x ≤ y | case inr.inr
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ∨ 0 < z
this :
∀ (x y : ℝ≥0∞) (z : ℝ),
z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z
hxy : ¬x ≤ y
⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z
x y : ℝ≥0∞
z : ℝ
hz : z < 0 ... | 6f5669cb0b4bcc9c |
Finset.sum_pow_of_commute | Mathlib/Data/Nat/Choose/Multinomial.lean | theorem sum_pow_of_commute (x : α → R) (s : Finset α)
(hc : (s : Set α).Pairwise (Commute on x)) :
∀ n,
s.sum x ^ n =
∑ k : s.sym n,
k.1.1.multinomial *
(k.1.1.map <| x).noncommProd
(Multiset.map_set_pairwise <| hc.mono <| mem_sym_iff.1 k.2) | case insert.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
x : α → R
a : α
s : Finset α
ha : a ∉ s
hc : (↑(insert a s)).Pairwise (Commute on x)
n : ℕ
ih : ∀ (n : ℕ), s.sum x ^ n = ∑ k : { x // x ∈ s.sym n }, ↑(↑↑k).multinomial * (Multiset.map x ↑↑k).noncommProd ⋯
m : { x // x ∈ (insert a s).sym... | rfl | no goals | 602c53a5a4f37e3f |
Real.binEntropy_two_inv_add | Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean | /-- `binEntropy` is symmetric about 1/2. -/
lemma binEntropy_two_inv_add (p : ℝ) : binEntropy (2⁻¹ + p) = binEntropy (2⁻¹ - p) | p : ℝ
⊢ binEntropy (2⁻¹ + p) = binEntropy (2⁻¹ - p) | rw [← binEntropy_one_sub] | p : ℝ
⊢ binEntropy (1 - (2⁻¹ + p)) = binEntropy (2⁻¹ - p) | 2d3b63cd00ded4af |
InfiniteGalois.krullTopology_mem_nhds_one_iff_of_isGalois | Mathlib/FieldTheory/Galois/Profinite.lean | lemma krullTopology_mem_nhds_one_iff_of_isGalois [IsGalois k K] (A : Set (K ≃ₐ[k] K)) :
A ∈ 𝓝 1 ↔ ∃ (L : FiniteGaloisIntermediateField k K), (L.fixingSubgroup : Set _) ⊆ A | k : Type u_3
K : Type u_4
inst✝³ : Field k
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
A : Set (K ≃ₐ[k] K)
⊢ (∃ E, FiniteDimensional k ↥E ∧ Normal k ↥E ∧ ↑E.fixingSubgroup ⊆ A) ↔ ∃ L, ↑L.fixingSubgroup ⊆ A | exact ⟨fun ⟨L, _, hL, hsub⟩ ↦ ⟨{ toIntermediateField := L, isGalois := ⟨⟩ }, hsub⟩,
fun ⟨L, hL⟩ ↦ ⟨L, inferInstance, inferInstance, hL⟩⟩ | no goals | 816baa46f1df36d8 |
StrictMonoOn.Iic_union_Ici | Mathlib/Order/Monotone/Union.lean | theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
(h₂ : StrictMonoOn f (Ici a)) : StrictMono f | α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a : α
f : α → β
h₁ : StrictMonoOn f (Iic a)
h₂ : StrictMonoOn f (Ici a)
⊢ StrictMono f | rw [← strictMonoOn_univ, ← @Iic_union_Ici _ _ a] | α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
a : α
f : α → β
h₁ : StrictMonoOn f (Iic a)
h₂ : StrictMonoOn f (Ici a)
⊢ StrictMonoOn f (Iic a ∪ Ici a) | 2e41e2c2f854a0da |
BoundedContinuousFunction.norm_lt_iff_of_compact | Mathlib/Topology/ContinuousMap/Bounded/Basic.lean | theorem norm_lt_iff_of_compact [CompactSpace α] {f : α →ᵇ β} {M : ℝ} (M0 : 0 < M) :
‖f‖ < M ↔ ∀ x, ‖f x‖ < M | α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : CompactSpace α
f : α →ᵇ β
M : ℝ
M0 : 0 < M
⊢ ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M | simp_rw [norm_def, ← dist_zero_right] | α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : CompactSpace α
f : α →ᵇ β
M : ℝ
M0 : 0 < M
⊢ dist f 0 < M ↔ ∀ (x : α), dist (f x) 0 < M | 4eb6bff83c919205 |
Dynamics.netMaxcard_empty | Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean | @[simp]
lemma netMaxcard_empty {T : X → X} {U : Set (X × X)} {n : ℕ} : netMaxcard T ∅ U n = 0 | X : Type u_1
T : X → X
U : Set (X × X)
n : ℕ
s : Finset X
s_net : s = ∅
⊢ ↑s.card = ⊥ | rw [s_net, Finset.card_empty, CharP.cast_eq_zero, bot_eq_zero'] | no goals | 4c397c9c537e4b8f |
Algebra.adjoin_induction' | Mathlib/RingTheory/Adjoin/Basic.lean | theorem adjoin_induction' {p : adjoin R s → Prop} (mem : ∀ (x) (h : x ∈ s), p ⟨x, subset_adjoin h⟩)
(algebraMap : ∀ r, p (algebraMap R _ r)) (add : ∀ x y, p x → p y → p (x + y))
(mul : ∀ x y, p x → p y → p (x * y)) (x : adjoin R s) : p x :=
Subtype.recOn x fun x hx => by
induction hx using adjoin_inductio... | case add
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
p : ↥(adjoin R s) → Prop
mem : ∀ (x : A) (h : x ∈ s), p ⟨x, ⋯⟩
algebraMap : ∀ (r : R), p ((_root_.algebraMap R ↥(adjoin R s)) r)
add : ∀ (x y : ↥(adjoin R s)), p x → p y → p (x + y)
mul : ∀ (x y : ↥(adjoin R s)), ... | exact add _ _ h₁ h₂ | no goals | adbccfa244789925 |
Polynomial.SplittingFieldAux.splits | Mathlib/FieldTheory/SplittingField/Construction.lean | theorem splits (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f :=
Nat.recOn (motive := fun n => ∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n), Splits (algebraMap K <| SplittingFieldAux n f) f) n
(fun {_} ... | n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : f.natDegree = n.succ
⊢ Splits (algebraMap K (SplittingFieldAux n.succ f)) f | rw [← splits_id_iff_splits, algebraMap_succ, ← map_map, splits_id_iff_splits,
← X_sub_C_mul_removeFactor f fun h => by rw [h] at hf; cases hf] | n✝ : ℕ
K✝ : Type u
inst✝ : Field K✝
n : ℕ
ih :
(fun n =>
∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Splits (algebraMap K (SplittingFieldAux n f)) f)
n
K : Type u
x✝ : Field K
f : K[X]
hf : f.natDegree = n.succ
⊢ Splits (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)... | 7582f06879320b9f |
CategoryTheory.NonPreadditiveAbelian.σ_comp | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | theorem σ_comp {X Y : C} (f : X ⟶ Y) : σ ≫ f = Limits.prod.map f f ≫ σ | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
X Y : C
f : X ⟶ Y
⊢ diag X ≫ prod.map f f ≫ σ = 0 | rw [prod.diag_map_assoc, diag_σ, comp_zero] | no goals | b7d15c6a05369a0f |
Stream'.nats_eq | Mathlib/Data/Stream/Init.lean | theorem nats_eq : nats = cons 0 (map succ nats) | ⊢ nats = 0 :: map succ nats | apply Stream'.ext | case a
⊢ ∀ (n : ℕ), nats.get n = (0 :: map succ nats).get n | e0d9653b559896b4 |
Int.add_mul_ediv_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean | theorem add_mul_ediv_right (a b : Int) {c : Int} (H : c ≠ 0) : (a + b * c) / c = a / c + b :=
suffices ∀ {{a b c : Int}}, 0 < c → (a + b * c).ediv c = a.ediv c + b from
match Int.lt_trichotomy c 0 with
| Or.inl hlt => by
rw [← Int.neg_inj, ← Int.ediv_neg, Int.neg_add, ← Int.ediv_neg, ← Int.neg_mul_neg]
... | case neg
a b c : Int
H✝ : c ≠ 0
k n m : Nat
h✝ : ¬m < n * k.succ
h : n * k.succ ≤ m
H : ∀ {a b : Nat}, a ≤ b → ↑a + -(↑b + 1) = -[b - a+1]
⊢ (↑(n * k.succ) - ↑m.succ).ediv ↑k.succ = ↑n - ↑(m / k.succ + 1) | show ediv (↑(n * succ k) + -((m : Int) + 1)) (succ k) = n + -(↑(m / succ k) + 1 : Int) | case neg
a b c : Int
H✝ : c ≠ 0
k n m : Nat
h✝ : ¬m < n * k.succ
h : n * k.succ ≤ m
H : ∀ {a b : Nat}, a ≤ b → ↑a + -(↑b + 1) = -[b - a+1]
⊢ (↑(n * k.succ) + -(↑m + 1)).ediv ↑k.succ = ↑n + -(↑(m / k.succ) + 1) | b62a45e15f46c8a0 |
SzemerediRegularity.le_sum_distinctPairs_edgeDensity_sq | Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean | lemma le_sum_distinctPairs_edgeDensity_sq (x : {i // i ∈ P.parts.offDiag}) (hε₁ : ε ≤ 1)
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5) :
(G.edgeDensity x.1.1 x.1.2 : ℝ) ^ 2 +
((if G.IsUniform ε x.1.1 x.1.2 then 0 else ε ^ 4 / 3) - ε ^ 5 / 25) ≤
(∑ i ∈ distinctPairs hP ... | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
x : { i // i ∈ P.parts.offDiag }
hε₁ : ε ≤ 1
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ ↑(G.edgeDensity (↑x).... | rw [distinctPairs, ← add_sub_assoc, add_sub_right_comm] | α : Type u_1
inst✝³ : Fintype α
inst✝² : DecidableEq α
P : Finpartition univ
hP : P.IsEquipartition
G : SimpleGraph α
inst✝¹ : DecidableRel G.Adj
ε : ℝ
inst✝ : Nonempty α
x : { i // i ∈ P.parts.offDiag }
hε₁ : ε ≤ 1
hPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α
hPε : 100 ≤ 4 ^ #P.parts * ε ^ 5
⊢ (↑(G.edgeDensity (↑x)... | 287f013480af0bef |
CategoryTheory.PreGaloisCategory.has_non_trivial_subobject_of_not_isConnected_of_not_initial | Mathlib/CategoryTheory/Galois/Basic.lean | /-- An object that is neither initial or connected has a non-trivial subobject. -/
lemma has_non_trivial_subobject_of_not_isConnected_of_not_initial (X : C) (hc : ¬ IsConnected X)
(hi : IsInitial X → False) :
∃ (Y : C) (v : Y ⟶ X), (IsInitial Y → False) ∧ Mono v ∧ (¬ IsIso v) | C : Type u₁
inst✝ : Category.{u₂, u₁} C
X : C
hi : IsInitial X → False
hc : ∀ (Y : C) (v : Y ⟶ X), (IsInitial Y → False) → Mono v → IsIso v
⊢ IsConnected X | exact ⟨hi, fun Y i hm hni ↦ hc Y i hni hm⟩ | no goals | 9fcd084466ecbce4 |
Zsqrtd.nonneg_mul | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem nonneg_mul {a b : ℤ√d} (ha : Nonneg a) (hb : Nonneg b) : Nonneg (a * b) :=
match a, b, nonneg_cases ha, nonneg_cases hb, ha, hb with
| _, _, ⟨_, _, Or.inl rfl⟩, ⟨_, _, Or.inl rfl⟩, _, _ => trivial
| _, _, ⟨x, y, Or.inl rfl⟩, ⟨z, w, Or.inr <| Or.inr rfl⟩, _, hb => nonneg_mul_lem hb
| _, _, ⟨x, y, Or.inl ... | d : ℕ
a b : ℤ√↑d
ha✝ : a.Nonneg
hb✝ : b.Nonneg
x y z w : ℕ
ha : { re := -↑x, im := ↑y }.Nonneg
hb : { re := ↑z, im := -↑w }.Nonneg
⊢ {
re :=
{ re := -↑x, im := ↑y }.re * { re := ↑z, im := -↑w }.re +
↑d * { re := -↑x, im := ↑y }.im * { re := ↑z, im := -↑w }.im,
im :=
{ re := -↑x, im... | simp [add_comm] | no goals | 46d597e584ea62c6 |
Int.natCast_mul | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem natCast_mul (a b : Nat) : ((a * b : Nat) : Int) = (a : Int) * (b : Int) | a b : Nat
⊢ ↑(a * b) = ↑a * ↑b | simp | no goals | 7ed6e7945c7dcf89 |
Complex.Gammaℝ_mul_Gammaℝ_add_one | Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean | /-- Reformulation of the doubling formula in terms of `Gammaℝ`. -/
lemma Gammaℝ_mul_Gammaℝ_add_one (s : ℂ) : Gammaℝ s * Gammaℝ (s + 1) = Gammaℂ s | s : ℂ
⊢ ↑π ^ (-s / 2) * ↑π ^ (-(s + 1) / 2) * (Gamma (s / 2) * Gamma (s / 2 + 1 / 2)) =
2 ^ (1 - s) * (↑π ^ (-1 / 2 - s) * ↑π ^ (1 / 2)) * Gamma s | rw [← cpow_add _ _ (ofReal_ne_zero.mpr pi_ne_zero), Complex.Gamma_mul_Gamma_add_half,
sqrt_eq_rpow, ofReal_cpow pi_pos.le, ofReal_div, ofReal_one, ofReal_ofNat] | s : ℂ
⊢ ↑π ^ (-s / 2 + -(s + 1) / 2) * (Gamma (2 * (s / 2)) * 2 ^ (1 - 2 * (s / 2)) * ↑π ^ (1 / 2)) =
2 ^ (1 - s) * (↑π ^ (-1 / 2 - s) * ↑π ^ (1 / 2)) * Gamma s | fd9946926629d4d4 |
SetTheory.PGame.Domineering.moveRight_smaller | Mathlib/SetTheory/Game/Domineering.lean | theorem moveRight_smaller {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) / 2 < Finset.card b / 2 | b : Board
m : ℤ × ℤ
h : m ∈ right b
⊢ Finset.card (moveRight b m) / 2 < Finset.card b / 2 | simp [← moveRight_card h, lt_add_one] | no goals | c067a6d73c618920 |
CategoryTheory.isSeparating_iff_epi | Mathlib/CategoryTheory/Generator/Basic.lean | theorem isSeparating_iff_epi (𝒢 : Set C)
[∀ A : C, HasCoproduct fun f : ΣG : 𝒢, (G : C) ⟶ A => (f.1 : C)] :
IsSeparating 𝒢 ↔ ∀ A : C, Epi (Sigma.desc (@Sigma.snd 𝒢 fun G => (G : C) ⟶ A)) | case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
𝒢 : Set C
inst✝ : ∀ (A : C), HasCoproduct fun f => ↑f.fst
h : ∀ (A : C), Epi (Sigma.desc Sigma.snd)
X Y : C
f g : X ⟶ Y
hh : ∀ G ∈ 𝒢, ∀ (h : G ⟶ X), h ≫ f = h ≫ g
this : Epi (Sigma.desc Sigma.snd)
j : Discrete ((G : ↑𝒢) × (↑G ⟶ X))
⊢ colimit.ι (Discrete.functor ... | simpa using hh j.as.1.1 j.as.1.2 j.as.2 | no goals | 8ce658f474840320 |
PrimeMultiset.coePNat_nat | Mathlib/Data/PNat/Factors.lean | theorem coePNat_nat (v : PrimeMultiset) : ((v : Multiset ℕ+) : Multiset ℕ) = (v : Multiset ℕ) | v : PrimeMultiset
⊢ Multiset.map Subtype.val (Multiset.map Coe.coe v) = Multiset.map Subtype.val v | rw [Multiset.map_map] | v : PrimeMultiset
⊢ Multiset.map (Subtype.val ∘ Coe.coe) v = Multiset.map Subtype.val v | 8d06f49f08abee13 |
OreLocalization.smul'_char | Mathlib/GroupTheory/OreLocalization/Basic.lean | theorem smul'_char (r₁ : R) (r₂ : X) (s₁ s₂ : S) (u : S) (v : R) (huv : u * r₁ = v * s₂) :
OreLocalization.smul' r₁ s₁ r₂ s₂ = v • r₂ /ₒ (u * s₁) | R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type u_2
inst✝ : MulAction R X
r₁ : R
r₂ : X
s₁ s₂ u : ↥S
v : R
huv : ↑u * r₁ = v * ↑s₂
v₀ : R := oreNum r₁ s₂
u₀ : ↥S := oreDenom r₁ s₂
h₀ : ↑u₀ * r₁ = v₀ * ↑s₂
r₃ : R
s₃ : ↥S
h₃ : ↑s₃ * ↑u₀ = r₃ * ↑u
⊢ r₃ * v * ↑s₂ = r₃ * (↑u * r₁) | rw [mul_assoc, ← huv] | no goals | 158523f845373f5e |
Submodule.iSup_map_single | Mathlib/LinearAlgebra/Pi.lean | theorem iSup_map_single [DecidableEq ι] [Finite ι] :
⨆ i, map (LinearMap.single R φ i : φ i →ₗ[R] (i : ι) → φ i) (p i) = pi Set.univ p | case intro
R : Type u
ι : Type x
inst✝⁴ : Semiring R
φ : ι → Type u_1
inst✝³ : (i : ι) → AddCommMonoid (φ i)
inst✝² : (i : ι) → Module R (φ i)
p : (i : ι) → Submodule R (φ i)
inst✝¹ : DecidableEq ι
inst✝ : Finite ι
val✝ : Fintype ι
⊢ ⨆ i, map (single R φ i) (p i) = pi Set.univ p | refine (iSup_le fun i => ?_).antisymm ?_ | case intro.refine_1
R : Type u
ι : Type x
inst✝⁴ : Semiring R
φ : ι → Type u_1
inst✝³ : (i : ι) → AddCommMonoid (φ i)
inst✝² : (i : ι) → Module R (φ i)
p : (i : ι) → Submodule R (φ i)
inst✝¹ : DecidableEq ι
inst✝ : Finite ι
val✝ : Fintype ι
i : ι
⊢ map (single R φ i) (p i) ≤ pi Set.univ p
case intro.refine_2
R : Type ... | 635e11935f20cbf9 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.derivedLitsInvariant_confirmRupHint | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem derivedLitsInvariant_confirmRupHint {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(rupHints : Array Nat) (i : Fin rupHints.size)
(acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
(ih : ∃ hsize : acc.1.size = n, DerivedLitsInvariant f f_assignments_size a... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
rupHints : Array Nat
i : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
ih : f.DerivedLitsInvariant f_assignments_size acc.fst hsize acc.snd.fst
hsize' : (confirmRupHint f.clauses acc rupHin... | exact (Or.inr ∘ Or.inr ∘ Or.inl ∘ Exists.intro l) rfl | no goals | 0f20fdb04006be12 |
Topology.IsEmbedding.completelyNormalSpace | Mathlib/Topology/Separation/Regular.lean | theorem Topology.IsEmbedding.completelyNormalSpace [TopologicalSpace Y] [CompletelyNormalSpace Y]
{e : X → Y} (he : IsEmbedding e) : CompletelyNormalSpace X | case refine_2
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : CompletelyNormalSpace Y
e : X → Y
he : IsEmbedding e
s t : Set X
hd₁ : Disjoint (closure s) t
hd₂ : Disjoint s (closure t)
⊢ Disjoint (e '' s) (closure (e '' t)) | rwa [← subset_compl_iff_disjoint_right, image_subset_iff, preimage_compl,
← he.closure_eq_preimage_closure_image, subset_compl_iff_disjoint_right] | no goals | f98c51ef2e25fae3 |
LinearMap.IsSymmetric.hasEigenvector_eigenvectorBasis | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
HasEigenvector T (hT.eigenvalues hn i) (hT.eigenvectorBasis hn i) | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hT : T.IsSymmetric
n : ℕ
hn : Module.finrank 𝕜 E = n
i : Fin n
v : E := (hT.eigenvectorBasis hn) i
⊢ HasEigenvector T (↑(hT.eigenvalues hn i)) ((hT.eigenvectorBasis ... | let μ : 𝕜 :=
(hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i
hT.orthogonalFamily_eigenspaces').val | 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hT : T.IsSymmetric
n : ℕ
hn : Module.finrank 𝕜 E = n
i : Fin n
v : E := (hT.eigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisInde... | 3fdf2eae1da3fdea |
InitialSeg.isMin_apply_iff | Mathlib/Order/InitialSeg.lean | theorem isMin_apply_iff [PartialOrder α] (f : α ≤i β) : IsMin (f a) ↔ IsMin a | α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder β
a : α
inst✝ : PartialOrder α
f : (fun x1 x2 => x1 < x2) ≼i fun x1 x2 => x1 < x2
h : IsMin a
b : β
hb : b ≤ f a
⊢ f a ≤ b | obtain ⟨x, rfl⟩ := f.mem_range_of_le hb | case intro
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder β
a : α
inst✝ : PartialOrder α
f : (fun x1 x2 => x1 < x2) ≼i fun x1 x2 => x1 < x2
h : IsMin a
x : α
hb : f x ≤ f a
⊢ f a ≤ f x | 9b5efa0ad03b3af6 |
LinearMap.ofIsCompl_eq | Mathlib/LinearAlgebra/Projection.lean | theorem ofIsCompl_eq (h : IsCompl p q) {φ : p →ₗ[R] F} {ψ : q →ₗ[R] F} {χ : E →ₗ[R] F}
(hφ : ∀ u, φ u = χ u) (hψ : ∀ u, ψ u = χ u) : ofIsCompl h φ ψ = χ | case h.intro.intro.intro
R : Type u_1
inst✝⁴ : Ring R
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module R E
F : Type u_3
inst✝¹ : AddCommGroup F
inst✝ : Module R F
p q : Submodule R E
h : IsCompl p q
φ : ↥p →ₗ[R] F
ψ : ↥q →ₗ[R] F
χ : E →ₗ[R] F
hφ : ∀ (u : ↥p), φ u = χ ↑u
hψ : ∀ (u : ↥q), ψ u = χ ↑u
w✝¹ : ↥p
w✝ : ↥q
... | simp [ofIsCompl, hφ, hψ] | no goals | 635b36cc788f370b |
ContinuousSMul.of_basis_zero | Mathlib/Topology/Algebra/FilterBasis.lean | theorem _root_.ContinuousSMul.of_basis_zero {ι : Type*} [IsTopologicalRing R] [TopologicalSpace M]
[IsTopologicalAddGroup M] {p : ι → Prop} {b : ι → Set M} (h : HasBasis (𝓝 0) p b)
(hsmul : ∀ {i}, p i → ∃ V ∈ 𝓝 (0 : R), ∃ j, p j ∧ V • b j ⊆ b i)
(hsmul_left : ∀ (x₀ : R) {i}, p i → ∃ j, p j ∧ MapsTo (x₀ • ... | case hmul.intro.intro.intro.intro
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : IsTopologicalRing R
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
p : ι → Prop
b : ι → Set M
h : (𝓝 0).HasBasis p b
hsmul : ∀ {i : ... | rintro ⟨v, w⟩ ⟨v_in : v ∈ V, w_in : w ∈ b j⟩ | case hmul.intro.intro.intro.intro.mk.intro
R : Type u_1
M : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : TopologicalSpace R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : IsTopologicalRing R
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
p : ι → Prop
b : ι → Set M
h : (𝓝 0).HasBasis p b
hsmul ... | 4f90af5932a68739 |
EReal.induction₂ | Mathlib/Data/Real/EReal.lean | theorem induction₂ {P : EReal → EReal → Prop} (top_top : P ⊤ ⊤) (top_pos : ∀ x : ℝ, 0 < x → P ⊤ x)
(top_zero : P ⊤ 0) (top_neg : ∀ x : ℝ, x < 0 → P ⊤ x) (top_bot : P ⊤ ⊥)
(pos_top : ∀ x : ℝ, 0 < x → P x ⊤) (pos_bot : ∀ x : ℝ, 0 < x → P x ⊥) (zero_top : P 0 ⊤)
(coe_coe : ∀ x y : ℝ, P x y) (zero_bot : P 0 ⊥) ... | case inl
P : EReal → EReal → Prop
top_top : P ⊤ ⊤
top_pos : ∀ (x : ℝ), 0 < x → P ⊤ ↑x
top_zero : P ⊤ 0
top_neg : ∀ x < 0, P ⊤ ↑x
top_bot : P ⊤ ⊥
pos_top : ∀ (x : ℝ), 0 < x → P ↑x ⊤
pos_bot : ∀ (x : ℝ), 0 < x → P ↑x ⊥
zero_top : P 0 ⊤
coe_coe : ∀ (x y : ℝ), P ↑x ↑y
zero_bot : P 0 ⊥
neg_top : ∀ x < 0, P ↑x ⊤
neg_bot : ∀ ... | exacts [neg_top x hx, zero_top, pos_top x hx] | no goals | 32d20d4f0c0a2e49 |
Polynomial.monomial_add_erase | Mathlib/Algebra/Polynomial/Basic.lean | theorem monomial_add_erase (p : R[X]) (n : ℕ) : monomial n (coeff p n) + p.erase n = p :=
toFinsupp_injective <| by
rcases p with ⟨⟩
rw [toFinsupp_add, toFinsupp_monomial, toFinsupp_erase, coeff]
exact Finsupp.single_add_erase _ _
| R : Type u
inst✝ : Semiring R
p : R[X]
n : ℕ
⊢ ((monomial n) (p.coeff n) + erase n p).toFinsupp = p.toFinsupp | rcases p with ⟨⟩ | case ofFinsupp
R : Type u
inst✝ : Semiring R
n : ℕ
toFinsupp✝ : R[ℕ]
⊢ ((monomial n) ({ toFinsupp := toFinsupp✝ }.coeff n) + erase n { toFinsupp := toFinsupp✝ }).toFinsupp =
{ toFinsupp := toFinsupp✝ }.toFinsupp | 89ce8c69cb49a811 |
Std.Sat.AIG.RefVec.zip_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Zip.lean | theorem zip_decl_eq {aig : AIG α} (target : ZipTarget aig len) :
∀ idx (h1 : idx < aig.decls.size) (h2),
(zip aig target).1.decls[idx]'h2 = aig.decls[idx]'h1 | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
aig : AIG α
target : ZipTarget aig len
⊢ ∀ (idx : Nat) (h1 : idx < aig.decls.size) (h2 : idx < (zip aig target).aig.decls.size),
(zip aig target).aig.decls[idx] = aig.decls[idx] | intros | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
aig : AIG α
target : ZipTarget aig len
idx✝ : Nat
h1✝ : idx✝ < aig.decls.size
h2✝ : idx✝ < (zip aig target).aig.decls.size
⊢ (zip aig target).aig.decls[idx✝] = aig.decls[idx✝] | 8244d107cdd1419b |
Ideal.finprod_heightOneSpectrum_factorization | Mathlib/RingTheory/DedekindDomain/Factorization.lean | theorem finprod_heightOneSpectrum_factorization {I : Ideal R} (hI : I ≠ 0) :
∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I = I | case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ ∀ (p : Associates (Ideal R)),
Irreducible p →
p.count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
p.count (Associates.mk I).factors | intro v hv | case h
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
v : Associates (Ideal R)
hv : Irreducible v
⊢ v.count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors = v.count (Associates.mk I).factors | b3b84ace84306993 |
IntermediateField.adjoin_le_subfield | Mathlib/FieldTheory/IntermediateField/Adjoin/Defs.lean | theorem adjoin_le_subfield {K : Subfield E} (HF : Set.range (algebraMap F E) ⊆ K) (HS : S ⊆ K) :
(adjoin F S).toSubfield ≤ K | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : Subfield E
HF : Set.range ⇑(algebraMap F E) ⊆ ↑K
HS : S ⊆ ↑K
⊢ (adjoin F S).toSubfield ≤ K | apply Subfield.closure_le.mpr | F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
S : Set E
K : Subfield E
HF : Set.range ⇑(algebraMap F E) ⊆ ↑K
HS : S ⊆ ↑K
⊢ Set.range ⇑(algebraMap F E) ∪ S ⊆ ↑K | 22def3c7240358f7 |
CategoryTheory.PreGaloisCategory.nhds_one_has_basis_stabilizers | Mathlib/CategoryTheory/Galois/Topology.lean | /-- The stabilizers of points in the fibers of Galois objects form a neighbourhood basis
of the identity in `Aut F`. -/
lemma nhds_one_has_basis_stabilizers : (nhds (1 : Aut F)).HasBasis (fun _ ↦ True)
(fun X : PointedGaloisObject F ↦ MulAction.stabilizer (Aut F) X.pt) where
mem_iff' S | C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S : Set (Aut F)
⊢ (∃ t ⊆ S, IsOpen t ∧ 1 ∈ t) ↔ ∃ i, True ∧ ↑(MulAction.stabilizer (Aut F) i.pt) ⊆ S | refine ⟨?_, ?_⟩ | case refine_1
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
S : Set (Aut F)
⊢ (∃ t ⊆ S, IsOpen t ∧ 1 ∈ t) → ∃ i, True ∧ ↑(MulAction.stabilizer (Aut F) i.pt) ⊆ S
case refine_2
C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCatego... | 420621663a2a4989 |
AnalyticAt.preimage_of_nhdsNE | Mathlib/Analysis/Analytic/IsolatedZeros.lean | theorem AnalyticAt.preimage_of_nhdsNE {x : 𝕜} {f : 𝕜 → E} {s : Set E} (hfx : AnalyticAt 𝕜 f x)
(h₂f : ¬EventuallyConst f (𝓝 x)) (hs : s ∈ 𝓝[≠] f x) :
f ⁻¹' s ∈ 𝓝[≠] x | case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
s : Set E
hfx : AnalyticAt 𝕜 f x
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
h : ¬f ⁻¹' s ∈ 𝓝[≠] x
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z ∈ insert (f x... | rw [Set.mem_insert_iff] at h₂z | case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
x : 𝕜
f : 𝕜 → E
s : Set E
hfx : AnalyticAt 𝕜 f x
hs : s ∈ 𝓝[≠] f x
this : ∀ᶠ (z : 𝕜) in 𝓝 x, f z ∈ insert (f x) s
h : ¬f ⁻¹' s ∈ 𝓝[≠] x
z : 𝕜
h₁z : f z ∈ s → False
h₂z : f z = f x ∨ f z ∈... | d911f70f139eb823 |
MeasureTheory.isProbabilityMeasure_map | Mathlib/MeasureTheory/Measure/Typeclasses.lean | theorem isProbabilityMeasure_map {f : α → β} (hf : AEMeasurable f μ) :
IsProbabilityMeasure (map f μ) :=
⟨by simp [map_apply_of_aemeasurable, hf]⟩
| α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : α → β
hf : AEMeasurable f μ
⊢ (Measure.map f μ) univ = 1 | simp [map_apply_of_aemeasurable, hf] | no goals | 95b2589a4bebd7a6 |
MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory | X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
Ssep : ∀ (n : ℕ), Metric.AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), Metric.AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n ... | rw [ENNReal.inv_lt_inv, Nat.cast_lt] | X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed s}
s : Set X
S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}
Ssep : ∀ (n : ℕ), Metric.AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), Metric.AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n ... | 603b48ab3190787f |
CategoryTheory.GrothendieckTopology.Plus.res_mk_eq_mk_pullback | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | theorem res_mk_eq_mk_pullback {Y X : C} {P : Cᵒᵖ ⥤ D} {S : J.Cover X} (x : Meq P S) (f : Y ⟶ X) :
(J.plusObj P).map f.op (mk x) = mk (x.pullback f) | case h
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShape ... | apply (Meq.equiv P _).injective | case h.a
C : Type u
inst✝⁵ : Category.{v, u} C
J : GrothendieckTopology C
D : Type w
inst✝⁴ : Category.{max v u, w} D
FD : D → D → Type u_1
CD : D → Type (max v u)
inst✝³ : (X Y : D) → FunLike (FD X Y) (CD X) (CD Y)
instCC : ConcreteCategory D FD
inst✝² : PreservesLimits (forget D)
inst✝¹ : ∀ (X : C), HasColimitsOfShap... | 4a32216862819df6 |
MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem haveLebesgueDecomposition_of_finiteMeasure [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
HaveLebesgueDecomposition μ ν where
lebesgue_decomposition | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | refine ⟨hξm.add (measurable_const.indicator hE₁), fun A hA ↦ ?_⟩ | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : ℕ → ℝ≥0∞
h✝ : Monotone g
hg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))
f : ℕ → α → ℝ≥0∞
hf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ
hf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g ... | 0e340e69a0abebdf |
AbsoluteValue.not_isNontrivial_iff | Mathlib/Algebra/Order/AbsoluteValue/Basic.lean | lemma not_isNontrivial_iff (v : AbsoluteValue R S) :
¬ v.IsNontrivial ↔ ∀ x ≠ 0, v x = 1 | R : Type u_5
inst✝¹ : Semiring R
S : Type u_6
inst✝ : OrderedSemiring S
v : AbsoluteValue R S
⊢ (∀ (x : R), x ≠ 0 → v x = 1) ↔ ∀ (x : R), x ≠ 0 → v x = 1 | rfl | no goals | 86edd53a032b0895 |
Std.Sat.AIG.RefVec.fold.denote_go_and | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Fold.lean | theorem denote_go_and {aig : AIG α} (acc : AIG.Ref aig) (curr : Nat) (hcurr : curr ≤ len)
(input : RefVec aig len) :
⟦
(go aig acc curr len input mkAndCached).aig,
(go aig acc curr len input mkAndCached).ref,
assign
⟧
=
(
⟦aig, acc, assign⟧
∧
(∀ (idx : Nat) (h... | case isTrue.mpr
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
(aig.mkAndCached { lhs := acc, ... | constructor | case isTrue.mpr.left
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
acc : aig.Ref
curr : Nat
hcurr : curr ≤ len
input : aig.RefVec len
res : Entrypoint α
h✝ : curr < len
hgo :
go (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig
(aig.mkAndCached { lhs := ... | 2185fe5626c5bad5 |
Std.DHashMap.Internal.Raw₀.getKeyD_erase | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKeyD_erase [EquivBEq α] [LawfulHashable α] (h : m.1.WF) {k a fallback : α} :
(m.erase k).getKeyD a fallback = if k == a then fallback else m.getKeyD a fallback | α : Type u
β : α → Type v
m : Raw₀ α β
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.val.WF
k a fallback : α
⊢ (m.erase k).getKeyD a fallback = if (k == a) = true then fallback else m.getKeyD a fallback | simp_to_model [erase] using List.getKeyD_eraseKey | no goals | 543384042f4eaee8 |
FermatLastTheoremForThreeGen.Solution.exists_minimal | Mathlib/NumberTheory/FLT/Three.lean | /-- If there is a solution then there is a minimal one. -/
lemma Solution.exists_minimal : ∃ (S₁ : Solution hζ), S₁.isMinimal | K : Type u_1
inst✝ : Field K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S : Solution hζ
⊢ ∃ S₁, S₁.isMinimal | classical
let T := {n | ∃ (S' : Solution hζ), S'.multiplicity = n}
rcases Nat.find_spec (⟨S.multiplicity, ⟨S, rfl⟩⟩ : T.Nonempty) with ⟨S₁, hS₁⟩
exact ⟨S₁, fun S'' ↦ hS₁ ▸ Nat.find_min' _ ⟨S'', rfl⟩⟩ | no goals | bd5d5a3a7e2e42fd |
Equiv.Perm.ofSubtype_eq_iff | Mathlib/GroupTheory/Perm/Support.lean | /-- A permutation c is the extension of a restriction of g to s
iff its support is contained in s and its restriction is that of g -/
lemma ofSubtype_eq_iff {g c : Equiv.Perm α} {s : Finset α}
(hg : ∀ x, x ∈ s ↔ g x ∈ s) :
ofSubtype (g.subtypePerm hg) = c ↔
c.support ≤ s ∧
∀ (hc' : ∀ x, x ∈ s ↔ c ... | case mp
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
s : Finset α
hg : ∀ (x : α), x ∈ s ↔ g x ∈ s
h : ∀ (x : α), (ofSubtype (g.subtypePerm hg)) x = c x
⊢ c.support ≤ s ∧ ((∀ (x : α), x ∈ s ↔ c x ∈ s) → ∀ a ∈ s, c a = g a) | constructor | case mp.left
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
s : Finset α
hg : ∀ (x : α), x ∈ s ↔ g x ∈ s
h : ∀ (x : α), (ofSubtype (g.subtypePerm hg)) x = c x
⊢ c.support ≤ s
case mp.right
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
g c : Perm α
s : Finset α
hg : ∀ (x : α), x ∈ s ↔ g x ∈ ... | ab1e7a47010d0039 |
UnitAddCircle.mem_approxAddOrderOf_iff | Mathlib/NumberTheory/WellApproximable.lean | 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 | case mpr.intro.intro.intro
δ : ℝ
x : UnitAddCircle
n : ℕ
hn : 0 < n
m : ℕ
hm₁ : m < n
hm₂ : gcd m n = 1
hx : ‖x - ↑(↑m / ↑n)‖ < δ
⊢ ∃ b, (∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n) = b) ∧ ‖x - b‖ < δ | exact ⟨↑((m : ℝ) / n), ⟨m, hm₁, hm₂, rfl⟩, hx⟩ | no goals | 6c2242c810a53b2d |
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