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
getValueCast! k' l
|
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 ⋯) (f (getValueCast? k l))).get! else (getValueCast? k' l).get!
|
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 : 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
⊢ (getValueCast? k' l).get! = (getValueCast? k' l).get!
|
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 (erase a s ∪ singleton a b) = lookup x s
|
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 then a else 0) g i +
∑ x ∈ s, (addRightEmbedding fun t => if t = i then a else 0) g x =
n ∧
∀ (i_1 : ι), ¬(addRightEmbedding fun t => if t = i then a else 0) g i_1 = 0 → i_1 = i ∨ i_1 ∈ s
|
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
⊢ (addRightEmbedding fun t => if t = i then a else 0) g i +
∑ x ∈ s, (addRightEmbedding fun t => if t = i then a else 0) g x =
n ∧
∀ (i_1 : ι), ¬(addRightEmbedding fun t => if t = i then a else 0) g i_1 = 0 → i_1 = i ∨ i_1 ∈ s
|
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 ((aeval f) q), y + z = v
|
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, h₂, h => by
have am : a ∈ b :: l₂ := h.subset (mem_cons_self _ _)
have bm : b ∈ a :: l₁ := h.symm.subset (mem_cons_self _ _)
have ab : a = b
|
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 idx hidx).cast ⋯
|
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_cauchyPowerSeries _ _ _
r_pos := ENNReal.coe_pos.2 hR
hasSum := fun {w} hw => by
have hw' : c + w ∈ ball c R
|
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 => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))
|
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 => (cauchyPowerSeries f c (↑R) n) fun x => w)
((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)
|
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.add_assoc, Nat.add_left_comm, Nat.add_le_add_iff_left]
using Nat.add_le_add_right (@AM_GM a b) 4
|
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, ∀ᶠ (x : ι) in l, (φ x).rOut < ε
ε : ℝ
hε : 0 < ε
i : ι
hi : (φ i).rOut < ε
⊢ (support fun x => (φ i).normed μ x) ∈ 𝒫 ball c ε
|
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, ∀ᶠ (x : ι) in l, (φ x).rOut < ε
ε : ℝ
hε : 0 < ε
i : ι
hi : (φ i).rOut < ε
⊢ ball c (φ i).rOut ∈ 𝒫 ball c ε
|
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))
(algebraMap_left : ∀ (r) (x) (hx : x ∈ s), p (algebraMap R A r) x (algebraMap_mem _ r)
(subset_adjoin hx))
(algebraMap_right : ∀ (r) (x) (hx : x ∈ s), p x (algebraMap R A r) (subset_adjoin hx)
(algebraMap_mem _ r))
(add_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x + y) z (add_mem hx hy) hz)
(add_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y + z) hx (add_mem hy hz))
(mul_left : ∀ x y z hx hy hz, p x z hx hz → p y z hy hz → p (x * y) z (mul_mem hx hy) hz)
(mul_right : ∀ x y z hx hy hz, p x y hx hy → p x z hx hz → p x (y * z) hx (mul_mem hy hz))
{x y : A} (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) :
p x y hx hy
|
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) ⋯ ⋯
algebraMap_left : ∀ (r : R) (x : A) (hx : x ∈ s), p ((algebraMap R A) r) x ⋯ ⋯
algebraMap_right : ∀ (r : R) (x : A) (hx : x ∈ s), p x ((algebraMap R A) r) ⋯ ⋯
add_left :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
∀ (x y z : A) (hx : x ∈ adjoin R s) (hy : y ∈ adjoin R s) (hz : z ∈ adjoin R s),
p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y z : A
hz : z ∈ s
r✝ : R
⊢ p ((algebraMap R A) r✝) z ⋯ ⋯
|
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 : IsCoboundedUnder (fun x1 x2 => x1 ≥ x2) f v
U : R
hU : ∀ᶠ (x : α) in f, u x ≤ U
V : R
hV : ∃ᶠ (x : α) in f, v x ≤ V
⊢ IsCoboundedUnder (fun x1 x2 => x1 ≥ x2) f (u + v)
|
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 : IsCoboundedUnder (fun x1 x2 => x1 ≥ x2) f v
U : R
hU : ∀ᶠ (x : α) in f, u x ≤ U
V : R
hV : ∃ᶠ (x : α) in f, v x ≤ V
⊢ ∃ᶠ (x : α) in f, (u + v) x ≤ U + V
|
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 (⇑f) y) fun a₂ b₂ => single (b * a₂) (m₁ * b₂)) +
sum (mapDomain (⇑f) y) fun a₂ b₂ => single (b * a₂) (m₂ * b₂)
|
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⟩
simp [hx y])
|
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, ‖deriv f' y‖ₑ
y : ℝ
⊢ ‖deriv f y‖ₑ = ‖I (deriv f y)‖ₑ
|
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)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x => 1) ⁻¹' u = t
⊢ MeasurableSet t
|
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 i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x => 1) ⁻¹' u = t
heq : ((s n).indicator fun x => 1) ⁻¹' u = Set.univ
⊢ MeasurableSet t
case h.refine_2.intro.intro.intro.intro.inr.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 i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x => 1) ⁻¹' u = t
heq : ((s n).indicator fun x => 1) ⁻¹' u = s n
⊢ MeasurableSet t
case h.refine_2.intro.intro.intro.intro.inr.inr.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 i)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x => 1) ⁻¹' u = t
heq : ((s n).indicator fun x => 1) ⁻¹' u = (s n)ᶜ
⊢ MeasurableSet t
case h.refine_2.intro.intro.intro.intro.inr.inr.inr
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)
i : ι
t✝ : Set Ω
n : ι
ht : MeasurableSet t✝
t : Set Ω
hn : n ≤ i
u : Set β
left✝ : MeasurableSet u
hu' : ((s n).indicator fun x => 1) ⁻¹' u = t
heq : ((s n).indicator fun x => 1) ⁻¹' u ∈ {∅}
⊢ MeasurableSet t
|
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) ∧ bsup.{u, u} o f = a,
∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ ⨆ i, f i = a]
|
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 ∩ Iic (sSup t)).Nonempty ∧ sSup (s ∩ Iic (sSup t)) = sSup t →
∃ t_1 ⊆ s, t_1.Nonempty ∧ BddAbove t_1 ∧ sSup t_1 = sSup t
hlub : IsLUB t (sSup t)
y : Ordinal.{u}
hyt : y ∈ t
x : Ordinal.{u}
hx : x ∈ t
⊢ x ≤ (succ (sSup t)).bsup fun x x_1 => if x ∈ t then x else y
|
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 : ({ x // x ∈ s } →₀ R) →ₗ[R] M := linearCombination R Subtype.val
H : Function.Surjective ⇑π
⊢ (LinearMap.ker (l ∘ₗ π)).FG
|
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 * f z₁
⊢ 0 ≤ c * ‖z‖
|
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 hG x hx g = (x g hg).app (op (Over.mk (𝟙 Y)))
|
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 hG x hx g ≫ G.map (𝟙 Y).op = F.map (𝟙 Y).op ≫ (x (𝟙 Y ≫ g) ⋯).app (op (Over.mk (𝟙 Y)))
⊢ app hG x hx g = (x g hg).app (op (Over.mk (𝟙 Y)))
|
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 x => ‖f x‖) μ
|
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]? = some y
|
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]? = some y
case cons.zero.cons
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)
head✝ : α
tail✝ : List α
h₁ : (x' :: xs).length = (head✝ :: tail✝).length
⊢ applyId ((x' :: xs).zip (head✝ :: tail✝)) x' = y ↔ (head✝ :: tail✝)[0]? = some y
|
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 Y : D
f g : X ⟶ Y
f' : F.objPreimage X ⟶ F.objPreimage Y
hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv
g' : F.objPreimage X ⟶ F.objPreimage Y
hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv
⊢ G.map (f + g) = G.map f + G.map g
|
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 Y : D
f g : X ⟶ Y
f' : F.objPreimage X ⟶ F.objPreimage Y
hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv
g' : F.objPreimage X ⟶ F.objPreimage Y
hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv
⊢ G.map
((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv +
(F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv) =
G.map ((F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv) +
G.map ((F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv)
|
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 =
(ftaylorSeriesWithin 𝕜 g t (f x)).taylorComp (ftaylorSeriesWithin 𝕜 f s x) i
|
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 : 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 : E) in 𝓝[s] x, f y ∈ t
i : ℕ
hi : ↑i ≤ n
hxt : f x ∈ t
hf_tendsto : Tendsto f (𝓝[s] x) (𝓝[t] f x)
⊢ ∃ u,
x ∈ u ∧
IsOpen u ∧
HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) (s ∩ u) ∧
HasFTaylorSeriesUpToOn (↑i) g (ftaylorSeriesWithin 𝕜 g t) (f '' (s ∩ u))
|
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 : 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 : E) in 𝓝[s] x, f y ∈ t
i : ℕ
hi : ↑i ≤ n
hxt : f x ∈ t
hf_tendsto : Tendsto f (𝓝[s] x) (𝓝[t] f x)
H₁ : ∀ᶠ (u : Set E) in (𝓝[s] x).smallSets, HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) u
⊢ ∃ u,
x ∈ u ∧
IsOpen u ∧
HasFTaylorSeriesUpToOn (↑i) f (ftaylorSeriesWithin 𝕜 f s) (s ∩ u) ∧
HasFTaylorSeriesUpToOn (↑i) g (ftaylorSeriesWithin 𝕜 g t) (f '' (s ∩ u))
|
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_iff, mem_product, and_imp, Prod.forall, Prod.le_def]
exact ⟨by aesop, fun h ↦ ⟨fun i hi ↦ (h _ _ hi hb).1, fun j hj ↦ (h _ _ ha hj).2⟩⟩
|
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 : IsLimit cone
W : C
f' : K.X i ⟶ cone.pt
hf' : f' ≫ Fork.ι cone = K.d i j
f'' : (K.extend e).X i' ⟶ cone.pt
hf'' : f'' ≫ Fork.ι cone ≫ (K.extendXIso e hj').inv = (K.extend e).d i' j'
φ : cone.pt ⟶ W
hij : c.Rel i j
hi'' : e.f i = i'
⊢ (K.extendXIso e hi'').hom ≫ f' = f''
|
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 : IsLimit cone
W : C
f' : K.X i ⟶ cone.pt
hf' : f' ≫ Fork.ι cone = K.d i j
f'' : (K.extend e).X i' ⟶ cone.pt
hf'' : f'' ≫ Fork.ι cone ≫ (K.extendXIso e hj').inv = (K.extend e).d i' j'
φ : cone.pt ⟶ W
hij : c.Rel i j
hi'' : e.f i = i'
⊢ ((K.extendXIso e hi'').hom ≫ f') ≫ Fork.ι cone = f'' ≫ Fork.ι cone
|
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 (-x)) * (s * ↑x ^ (s - 1))) volume 0 X
int_eval :
(∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * ↑x ^ s) + -∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) =
-((ofReal ∘ fun y => rexp (-y)) X * ↑X ^ s - (ofReal ∘ fun y => rexp (-y)) 0 * ↑0 ^ s)
this : (fun x => ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x => s * ↑(rexp (-x)) * ↑x ^ (s - 1)
t : ∫ (x : ℝ) in 0 ..X, s * (↑(rexp (-x)) * ↑x ^ (s - 1)) = s * ∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * ↑x ^ (s - 1)
⊢ (∫ (x : ℝ) in 0 ..X, s * ↑(rexp (-x)) * ↑x ^ (s - 1)) - (ofReal ∘ fun y => rexp (-y)) X * ↑X ^ s =
(∫ (x : ℝ) in 0 ..X, s * (↑(rexp (-x)) * ↑x ^ (s - 1))) - ↑(rexp (-X)) * ↑X ^ s
|
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 ^ s) + ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) volume 0 X
int_eval :
(∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * ↑x ^ s) + -∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * (s * ↑x ^ (s - 1)) =
-((ofReal ∘ fun y => rexp (-y)) X * ↑X ^ s - (ofReal ∘ fun y => rexp (-y)) 0 * ↑0 ^ s)
this : (fun x => ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x => s * ↑(rexp (-x)) * ↑x ^ (s - 1)
t : ∫ (x : ℝ) in 0 ..X, s * (↑(rexp (-x)) * ↑x ^ (s - 1)) = s * ∫ (x : ℝ) in 0 ..X, ↑(rexp (-x)) * ↑x ^ (s - 1)
⊢ (fun x => s * ↑(rexp (-x)) * ↑x ^ (s - 1)) = fun x => s * (↑(rexp (-x)) * ↑x ^ (s - 1))
|
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 M₂
I : LieIdeal R L
inst✝ : LieModule R L M
hf₁ : f.ker = ⊥
hf₂ : N₂ ≤ f.range
⊢ comap f ⁅I, N₂⁆ = ⁅I, comap f N₂⁆
|
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 M₂
I : LieIdeal R L
inst✝ : LieModule R L M
hf₁ : f.ker = ⊥
hf₂ : N₂ ≤ f.range
⊢ comap f ⁅I, map f (comap f N₂)⁆ = ⁅I, comap f N₂⁆
|
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 ∨ 0 < z
hxy : x ≤ y
⊢ (x * y) ^ z = if (x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0) ∧ z < 0 then ⊤ else x ^ z * y ^ z
|
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 n }
⊢ x a ^ Multiset.count a ↑↑m * ↑(Multiset.filter (fun x => a ≠ x) ↑↑m).multinomial *
(Multiset.map x (Multiset.filter (fun a_1 => ¬a = a_1) ↑↑m)).noncommProd ⋯ =
x a ^ ↑((symInsertEquiv ha) m).fst * ↑(↑↑((symInsertEquiv ha) m).snd).multinomial *
(Multiset.map x ↑↑((symInsertEquiv ha) m).snd).noncommProd ⋯
|
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_induction with
| mem _ h => exact mem _ h
| algebraMap _ => exact algebraMap _
| mul _ _ _ _ h₁ h₂ => exact mul _ _ h₁ h₂
| add _ _ _ _ h₁ h₂ => exact add _ _ h₁ h₂
|
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)), p x → p y → p (x * y)
x✝¹ : ↥(adjoin R s)
x x✝ y✝ : A
hx✝ : x✝ ∈ adjoin R s
hy✝ : y✝ ∈ adjoin R s
h₁ : p ⟨x✝, hx✝⟩
h₂ : p ⟨y✝, hy✝⟩
⊢ p ⟨x✝ + y✝, ⋯⟩
|
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 {_} _ _ hf =>
splits_of_degree_le_one _
(le_trans degree_le_natDegree <| hf.symm ▸ WithBot.coe_le_coe.2 zero_le_one))
fun n ih {K} _ f hf => by
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]
exact splits_mul _ (splits_X_sub_C _) (ih _ (natDegree_removeFactor' 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 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))
((X - C (AdjoinRoot.root f.factor)) * 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]
exact this (Int.neg_pos_of_neg hlt)
| Or.inr (Or.inl HEq) => absurd HEq H
| Or.inr (Or.inr hgt) => this hgt
suffices ∀ {k n : Nat} {a : Int}, (a + n * k.succ).ediv k.succ = a.ediv k.succ + n from
fun a b c H => match c, eq_succ_of_zero_lt H, b with
| _, ⟨_, rfl⟩, ofNat _ => this
| _, ⟨k, rfl⟩, -[n+1] => show (a - n.succ * k.succ).ediv k.succ = a.ediv k.succ - n.succ by
rw [← Int.add_sub_cancel (ediv ..), ← this, Int.sub_add_cancel]
fun {k n} => @fun
| ofNat _ => congrArg ofNat <| Nat.add_mul_div_right _ _ k.succ_pos
| -[m+1] => by
show ((n * k.succ : Nat) - m.succ : Int).ediv k.succ = n - (m / k.succ + 1 : Nat)
by_cases h : m < n * k.succ
· rw [← Int.ofNat_sub h, ← Int.ofNat_sub ((Nat.div_lt_iff_lt_mul k.succ_pos).2 h)]
apply congrArg ofNat
rw [Nat.mul_comm, Nat.mul_sub_div]; rwa [Nat.mul_comm]
· have h := Nat.not_lt.1 h
have H {a b : Nat} (h : a ≤ b) : (a : Int) + -((b : Int) + 1) = -[b - a +1]
|
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 G ε x, G.edgeDensity i.1 i.2 ^ 2 : ℝ) / 16 ^ #P.parts
|
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).1 (↑x).2) ^ 2 + ((if G.IsUniform ε (↑x).1 (↑x).2 then 0 else ε ^ 4 / 3) - ε ^ 5 / 25) ≤
(∑ i ∈ SzemerediRegularity.distinctPairs hP G ε x, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ #P.parts
|
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).1 (↑x).2) ^ 2 - ε ^ 5 / 25 + if G.IsUniform ε (↑x).1 (↑x).2 then 0 else ε ^ 4 / 3) ≤
(∑ i ∈ (chunk hP G ε ⋯).parts ×ˢ (chunk hP G ε ⋯).parts, ↑(G.edgeDensity i.1 i.2) ^ 2) / 16 ^ #P.parts
|
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 rfl⟩, ⟨z, w, Or.inr <| Or.inl rfl⟩, _, hb => nonneg_mul_lem hb
| _, _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ⟨z, w, Or.inl rfl⟩, ha, _ => by
rw [mul_comm]; exact nonneg_mul_lem ha
| _, _, ⟨x, y, Or.inr <| Or.inl rfl⟩, ⟨z, w, Or.inl rfl⟩, ha, _ => by
rw [mul_comm]; exact nonneg_mul_lem ha
| _, _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ⟨z, w, Or.inr <| Or.inr rfl⟩, ha, hb => by
rw [calc
(⟨-x, y⟩ * ⟨-z, w⟩ : ℤ√d) = ⟨_, _⟩ := rfl
_ = ⟨x * z + d * y * w, -(x * w + y * z)⟩
|
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 := ↑y }.re * { re := ↑z, im := -↑w }.im +
{ re := -↑x, im := ↑y }.im * { re := ↑z, im := -↑w }.re } =
{ re := -(↑x * ↑z + ↑d * ↑y * ↑w), im := ↑x * ↑w + ↑y * ↑z }
|
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 fun b => ↑b.fst) j ≫ Sigma.desc Sigma.snd ≫ f =
colimit.ι (Discrete.functor fun b => ↑b.fst) j ≫ Sigma.desc Sigma.snd ≫ g
|
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 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
⊢ pi Set.univ p ≤ ⨆ i, map (single R φ i) (p i)
|
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 acc.1 hsize acc.2.1) :
let rupHint_res := (confirmRupHint f.clauses) acc rupHints[i]
∃ hsize : rupHint_res.1.size = n, DerivedLitsInvariant f f_assignments_size rupHint_res.1 hsize rupHint_res.2.1
|
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 rupHints[i]).fst.size = n
h✝ : ¬(acc.2.2.snd || acc.2.2.fst) = true
c : DefaultClause n
rupHint_clause_eq_c : f.clauses[rupHints[↑i]]? = some (some c)
l : Literal (PosFin n)
⊢ ReduceResult.reducedToUnit l = ReduceResult.encounteredBoth ∨
ReduceResult.reducedToUnit l = ReduceResult.reducedToEmpty ∨
(∃ l_1, ReduceResult.reducedToUnit l = ReduceResult.reducedToUnit l_1) ∨
ReduceResult.reducedToUnit l = ReduceResult.reducedToNonunit
|
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 hn) i)
|
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.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
⊢ HasEigenvector T (↑(hT.eigenvalues hn i)) ((hT.eigenvectorBasis hn) i)
|
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
right✝ : ∀ (r : ↥p) (s : ↥q), ↑r + ↑s = ↑w✝¹ + ↑w✝ → r = w✝¹ ∧ s = w✝
⊢ (ofIsCompl h φ ψ) (↑w✝¹ + ↑w✝) = χ (↑w✝¹ + ↑w✝)
|
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 • ·) (b j) (b i))
(hsmul_right : ∀ (m₀ : M) {i}, p i → ∀ᶠ x in 𝓝 (0 : R), x • m₀ ∈ b i) : ContinuousSMul R M
|
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 : ι}, p i → ∃ V ∈ 𝓝 0, ∃ j, p j ∧ V • b j ⊆ b i
hsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j, p j ∧ MapsTo (fun x => x₀ • x) (b j) (b i)
hsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i
i : ι
hi : p i
V : Set R
V_in : V ∈ 𝓝 0
j : ι
hj : p j
hVj : V • b j ⊆ b i
⊢ V ×ˢ b j ⊆ {x | (fun x => x.1 • x.2 ∈ b i) x}
|
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 : ∀ {i : ι}, p i → ∃ V ∈ 𝓝 0, ∃ j, p j ∧ V • b j ⊆ b i
hsmul_left : ∀ (x₀ : R) {i : ι}, p i → ∃ j, p j ∧ MapsTo (fun x => x₀ • x) (b j) (b i)
hsmul_right : ∀ (m₀ : M) {i : ι}, p i → ∀ᶠ (x : R) in 𝓝 0, x • m₀ ∈ b i
i : ι
hi : p i
V : Set R
V_in : V ∈ 𝓝 0
j : ι
hj : p j
hVj : V • b j ⊆ b i
v : R
w : M
v_in : v ∈ V
w_in : w ∈ b j
⊢ (v, w) ∈ {x | (fun x => x.1 • x.2 ∈ b i) x}
|
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 ⊥) (neg_top : ∀ x : ℝ, x < 0 → P x ⊤)
(neg_bot : ∀ x : ℝ, x < 0 → P x ⊥) (bot_top : P ⊥ ⊤) (bot_pos : ∀ x : ℝ, 0 < x → P ⊥ x)
(bot_zero : P ⊥ 0) (bot_neg : ∀ x : ℝ, x < 0 → P ⊥ x) (bot_bot : P ⊥ ⊥) : ∀ x y, P x y
| ⊥, ⊥ => bot_bot
| ⊥, (y : ℝ) => by
rcases lt_trichotomy y 0 with (hy | rfl | hy)
exacts [bot_neg y hy, bot_zero, bot_pos y hy]
| ⊥, ⊤ => bot_top
| (x : ℝ), ⊥ => by
rcases lt_trichotomy x 0 with (hx | rfl | hx)
exacts [neg_bot x hx, zero_bot, pos_bot x hx]
| (x : ℝ), (y : ℝ) => coe_coe _ _
| (x : ℝ), ⊤ => by
rcases lt_trichotomy x 0 with (hx | rfl | hx)
exacts [neg_top x hx, zero_top, pos_top x hx]
| ⊤, ⊥ => top_bot
| ⊤, (y : ℝ) => by
rcases lt_trichotomy y 0 with (hy | rfl | hy)
exacts [top_neg y hy, top_zero, top_pos y hy]
| ⊤, ⊤ => top_top
|
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 : ∀ x < 0, P ↑x ⊥
bot_top : P ⊥ ⊤
bot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x
bot_zero : P ⊥ 0
bot_neg : ∀ x < 0, P ⊥ ↑x
bot_bot : P ⊥ ⊥
x : ℝ
hx : x < 0
⊢ P ↑x ⊤
case inr.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 : ∀ x < 0, P ↑x ⊥
bot_top : P ⊥ ⊤
bot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x
bot_zero : P ⊥ 0
bot_neg : ∀ x < 0, P ⊥ ↑x
bot_bot : P ⊥ ⊥
⊢ P ↑0 ⊤
case inr.inr
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 : ∀ x < 0, P ↑x ⊥
bot_top : P ⊥ ⊤
bot_pos : ∀ (x : ℝ), 0 < x → P ⊥ ↑x
bot_zero : P ⊥ 0
bot_neg : ∀ x < 0, P ⊥ ↑x
bot_bot : P ⊥ ⊥
x : ℝ
hx : 0 < x
⊢ P ↑x ⊤
|
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 : GaloisCategory C
inst✝ : FiberFunctor F
S : Set (Aut F)
⊢ (∃ i, True ∧ ↑(MulAction.stabilizer (Aut F) i.pt) ⊆ S) → ∃ t ⊆ S, IsOpen t ∧ 1 ∈ t
|
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) s
⊢ f z = 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 ∈ s
⊢ f z = f x
|
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 : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ⊤
r n i j : ℕ
hj : i < j
⊢ (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
|
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 : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ⊤
r n i j : ℕ
hj : i < j
⊢ 2 * i + 1 + r < 2 * j + r
|
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 (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
Y X : C
P : Cᵒᵖ ⥤ D
S : J.Cover X
x : Meq P S
f : Y ⟶ X
⊢ (ConcreteCategory.hom ((J.diagramPullback P f).app (op S))) ((Meq.equiv P S).symm x) =
(Meq.equiv P (S.pullback f)).symm (x.pullback f)
|
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), HasColimitsOfShape (J.Cover X)ᵒᵖ D
inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)
Y X : C
P : Cᵒᵖ ⥤ D
S : J.Cover X
x : Meq P S
f : Y ⟶ X
⊢ (Meq.equiv P (unop ((J.pullback f).op.obj (op S))))
((ConcreteCategory.hom ((J.diagramPullback P f).app (op S))) ((Meq.equiv P S).symm x)) =
(Meq.equiv P (unop ((J.pullback f).op.obj (op S)))) ((Meq.equiv P (S.pullback f)).symm (x.pullback f))
|
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 n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
⊢ (ξ + E.indicator fun x => ↑ε) ∈ measurableLE ν μ
|
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 n
ξ : α → ℝ≥0∞ := ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ : ξ = ⨆ n, ⨆ k, ⨆ (_ : k ≤ n), f k
hξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν
hξm : Measurable ξ
hξle : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A, ξ a ∂ν ≤ μ A
hle : ν.withDensity ξ ≤ μ
this : IsFiniteMeasure (ν.withDensity ξ)
μ₁ : Measure α := μ - ν.withDensity ξ
hμ₁ : μ₁ = μ - ν.withDensity ξ
h : ¬(μ₁, ξ).1 ⟂ₘ ν
ε : ℝ≥0
hε₁ : 0 < ε
E : Set α
hE₁ : MeasurableSet E
hE₂ : 0 < ν E
hE₃ : ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ (μ - ν.withDensity ξ) (A ∩ E)
hε₂ : ∀ (A : Set α), MeasurableSet A → ∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν ≤ μ (A ∩ E)
A : Set α
hA : MeasurableSet A
⊢ ∫⁻ (x : α) in A, (ξ + E.indicator fun x => ↑ε) x ∂ν ≤ μ A
|
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) (hidx1 : idx < len), curr ≤ idx → ⟦aig, input.get idx hidx1, assign⟧)
)
|
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, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
res
hacc : ⟦assign, { aig := aig, ref := acc }⟧ = true
hrest : ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true
⊢ (⟦assign, { aig := aig, ref := acc }⟧ = true ∧ ⟦assign, { aig := aig, ref := input.get curr h✝ }⟧ = true) ∧
∀ (idx : Nat) (hidx1 : idx < len),
curr + 1 ≤ idx →
⟦assign,
{ aig := (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig,
ref := (input.get idx hidx1).cast ⋯ }⟧ =
true
|
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 := acc, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
res
hacc : ⟦assign, { aig := aig, ref := acc }⟧ = true
hrest : ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true
⊢ ⟦assign, { aig := aig, ref := acc }⟧ = true ∧ ⟦assign, { aig := aig, ref := input.get curr h✝ }⟧ = true
case isTrue.mpr.right
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, rhs := input.get curr h✝ }).ref (curr + 1) len (input.cast ⋯) mkAndCached =
res
hacc : ⟦assign, { aig := aig, ref := acc }⟧ = true
hrest : ∀ (idx : Nat) (hidx1 : idx < len), curr ≤ idx → ⟦assign, { aig := aig, ref := input.get idx hidx1 }⟧ = true
⊢ ∀ (idx : Nat) (hidx1 : idx < len),
curr + 1 ≤ idx →
⟦assign,
{ aig := (aig.mkAndCached { lhs := acc, rhs := input.get curr h✝ }).aig,
ref := (input.get idx hidx1).cast ⋯ }⟧ =
true
|
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 x ∈ s), c.subtypePerm hc' = g.subtypePerm hg
|
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 ∈ s
h : ∀ (x : α), (ofSubtype (g.subtypePerm hg)) x = c x
⊢ (∀ (x : α), x ∈ s ↔ c x ∈ s) → ∀ a ∈ s, c a = g a
|
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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