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 |
|---|---|---|---|---|---|---|
CategoryTheory.IsGrothendieckAbelian.generatingMonomorphisms.pushouts_ofLE_le_largerSubobject | Mathlib/CategoryTheory/Abelian/GrothendieckCategory/EnoughInjectives.lean | lemma pushouts_ofLE_le_largerSubobject (A : Subobject X) :
(generatingMonomorphisms G).pushouts
(Subobject.ofLE _ _ (le_largerSubobject hG A)) | C : Type u
inst✝¹ : Category.{v, u} C
G : C
inst✝ : Abelian C
hG : IsSeparator G
X : C
⊢ largerSubobject hG ⊤ = ⊤ | simp | no goals | 46561d78655c5ec9 |
Std.Sat.AIG.RefVec.map.go_get_aux | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Map.lean | theorem go_get_aux {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len) (s : RefVec aig curr)
(input : RefVec aig len) (f : (aig : AIG α) → Ref aig → Entrypoint α)
[LawfulOperator α Ref f] [LawfulMapOperator α f] :
-- The hfoo here is a trick to make the dependent type gods happy.
∀ (idx : Nat) (hidx : idx <... | α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx : idx < curr
res : RefVecEntry α len
hgo :
(if hidx... | split at hgo | case isTrue
α : Type
inst✝³ : Hashable α
inst✝² : DecidableEq α
len : Nat
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
input : aig.RefVec len
f : (aig : AIG α) → aig.Ref → Entrypoint α
inst✝¹ : LawfulOperator α Ref f
inst✝ : LawfulMapOperator α f
idx : Nat
hidx : idx < curr
res : RefVecEntry α len
h✝ :... | bc7701018f2eb538 |
Array.all_bne | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem all_bne [BEq α] {xs : Array α} : (xs.all fun x => a != x) = !xs.contains a | case mk
α : Type u_1
a : α
inst✝ : BEq α
toList✝ : List α
⊢ ({ toList := toList✝ }.all fun x => a != x) = !{ toList := toList✝ }.contains a | simp [List.all_bne] | no goals | c92fa9e7bb968d58 |
CategoryTheory.Abelian.coimage_image_factorisation | Mathlib/CategoryTheory/Abelian/Images.lean | theorem coimage_image_factorisation : coimage.π f ≫ coimageImageComparison f ≫ image.ι f = f | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasKernels C
inst✝ : HasCokernels C
P Q : C
f : P ⟶ Q
⊢ coimage.π f ≫ coimageImageComparison f ≫ image.ι f = f | simp [coimageImageComparison] | no goals | 313b2e29ada163d8 |
Std.DHashMap.Raw.Const.getKey!_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem getKey!_insertManyIfNewUnit_list_of_not_mem_of_contains_eq_false
[EquivBEq α] [LawfulHashable α] [Inhabited α] (h : m.WF) {l : List α} {k : α} :
¬ k ∈ m → l.contains k = false → getKey! (insertManyIfNewUnit m l) k = default | α : Type u
inst✝⁴ : BEq α
inst✝³ : Hashable α
m : Raw α fun x => Unit
inst✝² : EquivBEq α
inst✝¹ : LawfulHashable α
inst✝ : Inhabited α
h : m.WF
l : List α
k : α
⊢ m.contains k = false → l.contains k = false → (insertManyIfNewUnit m l).getKey! k = default | simp_to_raw using
Raw₀.Const.getKey!_insertManyIfNewUnit_list_of_contains_eq_false_of_contains_eq_false | no goals | 74040a703261549f |
Int.Matrix.N_le_P_add_one | Mathlib/NumberTheory/SiegelsLemma.lean | private lemma N_le_P_add_one (i : α) : N i ≤ P i + 1 | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
i : α
⊢ ∀ i_1 ∈ univ, 0 ≤ ↑B * (A i i_1)⁺ | intro j _ | α : Type u_1
β : Type u_2
inst✝¹ : Fintype α
inst✝ : Fintype β
A : Matrix α β ℤ
i : α
j : β
a✝ : j ∈ univ
⊢ 0 ≤ ↑B * (A i j)⁺ | 0bb8b2854f9a32ee |
SimpleGraph.Walk.reverse_append | Mathlib/Combinatorics/SimpleGraph/Walk.lean | theorem reverse_append {u v w : V} (p : G.Walk u v) (q : G.Walk v w) :
(p.append q).reverse = q.reverse.append p.reverse | V : Type u
G : SimpleGraph V
u v w : V
p : G.Walk u v
q : G.Walk v w
⊢ (p.append q).reverse = q.reverse.append p.reverse | simp [reverse] | no goals | b0725c05b600108b |
Finsupp.finite_of_nat_weight_le | Mathlib/Data/Finsupp/Weight.lean | theorem finite_of_nat_weight_le [Finite σ] (w : σ → ℕ) (hw : ∀ x, w x ≠ 0) (n : ℕ) :
{d : σ →₀ ℕ | weight w d ≤ n}.Finite | case h.h.h
σ : Type u_1
inst✝ : Finite σ
w : σ → ℕ
hw : ∀ (x : σ), w x ≠ 0
n : ℕ
fg : Finset ((σ →₀ ℕ) × (σ →₀ ℕ)) := Finset.antidiagonal (equivFunOnFinite.symm (Function.const σ n))
hfg : fg = Finset.antidiagonal (equivFunOnFinite.symm (Function.const σ n))
d : σ →₀ ℕ
hd : d ∈ {d | (weight w) d ≤ n}
x : σ
⊢ (weight w)... | simpa only [Set.mem_setOf_eq] using hd | no goals | f50712ae8705d3f4 |
HasFPowerSeriesWithinOnBall.changeOrigin | Mathlib/Analysis/Analytic/ChangeOrigin.lean | theorem HasFPowerSeriesWithinOnBall.changeOrigin (hf : HasFPowerSeriesWithinOnBall f p s x r)
(h : ‖y‖ₑ < r) (hy : x + y ∈ insert x s) :
HasFPowerSeriesWithinOnBall f (p.changeOrigin y) s (x + y) (r - ‖y‖ₑ) where
r_le | case h'y
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x y : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinO... | have : insert (x + y) s ⊆ insert (x + y) (insert x s) := by
apply insert_subset_insert (subset_insert _ _) | case h'y
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x y : E
r : ℝ≥0∞
hf : HasFPowerSeriesWithinO... | 8ade1d76cafb166c |
MultilinearMap.domCoprod_alternization | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | theorem MultilinearMap.domCoprod_alternization [DecidableEq ιa] [DecidableEq ιb]
(a : MultilinearMap R' (fun _ : ιa => Mᵢ) N₁) (b : MultilinearMap R' (fun _ : ιb => Mᵢ) N₂) :
MultilinearMap.alternatization (MultilinearMap.domCoprod a b) =
a.alternatization.domCoprod (MultilinearMap.alternatization b) | case a
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ... | rw [domCoprod_coe, MultilinearMap.alternatization_coe,
Finset.sum_partition (QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range)] | case a
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ... | 300c825287cae793 |
Stream'.WSeq.dropn_ofSeq | Mathlib/Data/Seq/WSeq.lean | theorem dropn_ofSeq (s : Seq α) : ∀ n, drop (ofSeq s) n = ofSeq (s.drop n)
| 0 => rfl
| n + 1 => by
simp only [drop, Nat.add_eq, Nat.add_zero, Seq.drop]
rw [dropn_ofSeq s n, tail_ofSeq]
| α : Type u
s : Seq α
n : ℕ
⊢ ((↑s).drop n).tail = ↑(s.drop n).tail | rw [dropn_ofSeq s n, tail_ofSeq] | no goals | a75cf49b7235eeec |
Module.Finite.of_localizationSpan_finite' | Mathlib/RingTheory/Localization/Finiteness.lean | theorem of_localizationSpan_finite' (t : Finset R) (ht : Ideal.span (t : Set R) = ⊤)
{Mₚ : ∀ (_ : t), Type*} [∀ (g : t), AddCommMonoid (Mₚ g)] [∀ (g : t), Module R (Mₚ g)]
{Rₚ : ∀ (_ : t), Type u} [∀ (g : t), CommRing (Rₚ g)] [∀ (g : t), Algebra R (Rₚ g)]
[∀ (g : t), IsLocalization.Away g.val (Rₚ g)]
[∀... | case h.intro.mk.intro.intro.mk.intro
R : Type u
inst✝¹⁰ : CommRing R
M : Type w
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
t : Finset R
ht : Ideal.span ↑t = ⊤
Mₚ : { x // x ∈ t } → Type u_1
inst✝⁷ : (g : { x // x ∈ t }) → AddCommMonoid (Mₚ g)
inst✝⁶ : (g : { x // x ∈ t }) → Module R (Mₚ g)
Rₚ : { x // x ∈ t } → Type ... | rw [Submonoid.smul_def] at hn₂ | case h.intro.mk.intro.intro.mk.intro
R : Type u
inst✝¹⁰ : CommRing R
M : Type w
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
t : Finset R
ht : Ideal.span ↑t = ⊤
Mₚ : { x // x ∈ t } → Type u_1
inst✝⁷ : (g : { x // x ∈ t }) → AddCommMonoid (Mₚ g)
inst✝⁶ : (g : { x // x ∈ t }) → Module R (Mₚ g)
Rₚ : { x // x ∈ t } → Type ... | c6847175b7a0f55d |
Nat.add_mul_mod_self_left | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Div/Basic.lean | theorem add_mul_mod_self_left (x y z : Nat) : (x + y * z) % y = x % y | x y z : Nat
⊢ (x + y * 0) % y = x % y | rw [Nat.mul_zero, Nat.add_zero] | no goals | 79d60d00b9740ceb |
Array.rel_of_isEqvAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/DecidableEq.lean | theorem rel_of_isEqvAux
{r : α → α → Bool} {a b : Array α} (hsz : a.size = b.size) {i : Nat} (hi : i ≤ a.size)
(heqv : Array.isEqvAux a b hsz r i hi)
{j : Nat} (hj : j < i) : r (a[j]'(Nat.lt_of_lt_of_le hj hi)) (b[j]'(Nat.lt_of_lt_of_le hj (hsz ▸ hi))) | α : Type u_1
r : α → α → Bool
a b : Array α
hsz : a.size = b.size
j i : Nat
ih : ∀ (hi : i ≤ a.size), a.isEqvAux b hsz r i hi = true → ∀ (hj : j < i), r a[j] b[j] = true
hi : i + 1 ≤ a.size
hj : j < i + 1
heqv : r a[i] b[i] = true ∧ a.isEqvAux b hsz r i ⋯ = true
hj' : ¬j < i
⊢ r a[j] b[j] = true | replace hj' : j = i := Nat.eq_of_le_of_lt_succ (Nat.not_lt.mp hj') hj | α : Type u_1
r : α → α → Bool
a b : Array α
hsz : a.size = b.size
j i : Nat
ih : ∀ (hi : i ≤ a.size), a.isEqvAux b hsz r i hi = true → ∀ (hj : j < i), r a[j] b[j] = true
hi : i + 1 ≤ a.size
hj : j < i + 1
heqv : r a[i] b[i] = true ∧ a.isEqvAux b hsz r i ⋯ = true
hj' : j = i
⊢ r a[j] b[j] = true | 5b3c341e7edc7d99 |
specializes_TFAE | Mathlib/Topology/Inseparable.lean | theorem specializes_TFAE (x y : X) :
TFAE [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : Set X , IsOpen s → y ∈ s → x ∈ s,
∀ s : Set X , IsClosed s → x ∈ s → y ∈ s,
y ∈ closure ({ x } : Set X),
closure ({ y } : Set X) ⊆ closure { x },
ClusterPt y (pure x)] | X : Type u_1
inst✝ : TopologicalSpace X
x y : X
tfae_1_to_2 : x ⤳ y → pure x ≤ 𝓝 y
tfae_2_to_3 : pure x ≤ 𝓝 y → ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s
tfae_3_to_4 : (∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s) → ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s
tfae_4_to_5 : (∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s) → y ∈ cl... | tfae_have 5 ↔ 7 := by
rw [mem_closure_iff_clusterPt, principal_singleton] | X : Type u_1
inst✝ : TopologicalSpace X
x y : X
tfae_1_to_2 : x ⤳ y → pure x ≤ 𝓝 y
tfae_2_to_3 : pure x ≤ 𝓝 y → ∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s
tfae_3_to_4 : (∀ (s : Set X), IsOpen s → y ∈ s → x ∈ s) → ∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s
tfae_4_to_5 : (∀ (s : Set X), IsClosed s → x ∈ s → y ∈ s) → y ∈ cl... | 5fadf2ff9737a10e |
tangentCone_eq_univ | Mathlib/Analysis/Calculus/TangentCone.lean | theorem tangentCone_eq_univ {s : Set 𝕜} {x : 𝕜} (hx : (𝓝[s \ {x}] x).NeBot) :
tangentConeAt 𝕜 s x = univ | case inr
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
x : 𝕜
hx : (𝓝[s \ {x}] x).NeBot
y : 𝕜
hy : y ≠ 0
⊢ y ∈ tangentConeAt 𝕜 s x | obtain ⟨u, -, u_pos, u_lim⟩ :
∃ u, StrictAnti u ∧ (∀ (n : ℕ), 0 < u n) ∧ Tendsto u atTop (𝓝 (0 : ℝ)) :=
exists_seq_strictAnti_tendsto (0 : ℝ) | case inr.intro.intro.intro
𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
s : Set 𝕜
x : 𝕜
hx : (𝓝[s \ {x}] x).NeBot
y : 𝕜
hy : y ≠ 0
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ y ∈ tangentConeAt 𝕜 s x | aeb42d5f7b21be66 |
Filter.isUnit_iff | Mathlib/Order/Filter/Pointwise.lean | theorem isUnit_iff : IsUnit f ↔ ∃ a, f = pure a ∧ IsUnit a | case mp
α : Type u_2
inst✝ : DivisionMonoid α
f : Filter α
⊢ IsUnit f → ∃ a, f = pure a ∧ IsUnit a | rintro ⟨u, rfl⟩ | case mp.intro
α : Type u_2
inst✝ : DivisionMonoid α
u : (Filter α)ˣ
⊢ ∃ a, ↑u = pure a ∧ IsUnit a | 3fbe884b3f6b48f5 |
multiple_mem_adjoin_of_mem_localization_adjoin | Mathlib/RingTheory/RingHom/Finite.lean | theorem multiple_mem_adjoin_of_mem_localization_adjoin [Algebra R' S] [Algebra R S]
[IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S)
(hx : x ∈ Algebra.adjoin R' s) : ∃ t : M, t • x ∈ Algebra.adjoin R s | R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid R
R' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : Algebra R R'
inst✝³ : Algebra R' S
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R R' S
inst✝ : IsLocalization M R'
s : Set S
x : S
hx : x ∈ Subalgebra.toSubmodule (Algebra.adjoin R' s)
⊢ ∃ t, t • x ∈ Subalgebra... | simp_rw [Algebra.adjoin_eq_span] at hx ⊢ | R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid R
R' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : Algebra R R'
inst✝³ : Algebra R' S
inst✝² : Algebra R S
inst✝¹ : IsScalarTower R R' S
inst✝ : IsLocalization M R'
s : Set S
x : S
hx : x ∈ Submodule.span R' ↑(Submonoid.closure s)
⊢ ∃ t, t • x ∈ Submodule.span... | d8cf0e986fe384fb |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.confirmRupHint_preserves_motive | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem confirmRupHint_preserves_motive {n : Nat} (f : DefaultFormula n) (rupHints : Array Nat)
(idx : Fin rupHints.size) (acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool)
(ih : ConfirmRupHintFoldEntailsMotive f idx.1 acc) :
ConfirmRupHintFoldEntailsMotive f (idx.1 + 1) ((confirmRupHint f.claus... | n : Nat
f : DefaultFormula n
rupHints : Array Nat
idx : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
h1 : Limplies (PosFin n) f acc.fst
h2 : acc.snd.snd.fst = true → Incompatible (PosFin n) acc.fst f
acc2_eq_false : acc.2.2.snd = false ∧ acc.2.2.fst = false
x✝¹... | rcases incompatible_of_reducedToEmpty c acc.1 heq p with pc | pacc | case inl
n : Nat
f : DefaultFormula n
rupHints : Array Nat
idx : Fin rupHints.size
acc : Array Assignment × CNF.Clause (PosFin n) × Bool × Bool
hsize : acc.fst.size = n
h1 : Limplies (PosFin n) f acc.fst
h2 : acc.snd.snd.fst = true → Incompatible (PosFin n) acc.fst f
acc2_eq_false : acc.2.2.snd = false ∧ acc.2.2.fst = ... | 610b60376ef97394 |
Std.Sat.AIG.RefVec.zip.denote_go | Mathlib/.lake/packages/lean4/src/lean/Std/Sat/AIG/RefVecOperator/Zip.lean | theorem denote_go {aig : AIG α} (curr : Nat) (hcurr : curr ≤ len) (s : RefVec aig curr)
(lhs rhs : RefVec aig len) (f : (aig : AIG α) → BinaryInput aig → Entrypoint α)
[LawfulOperator α BinaryInput f] [chainable : LawfulZipOperator α f] :
∀ (idx : Nat) (hidx1 : idx < len),
curr ≤ idx
→
⟦... | case isTrue.inr
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
lhs rhs : aig.RefVec len
f : (aig : AIG α) → aig.BinaryInput → Entrypoint α
inst✝ : LawfulOperator α BinaryInput f
chainable : LawfulZipOperator α f
idx : Nat
hid... | rw [denote_go] | case isTrue.inr
α : Type
inst✝² : Hashable α
inst✝¹ : DecidableEq α
len : Nat
assign : α → Bool
aig : AIG α
curr : Nat
hcurr : curr ≤ len
s : aig.RefVec curr
lhs rhs : aig.RefVec len
f : (aig : AIG α) → aig.BinaryInput → Entrypoint α
inst✝ : LawfulOperator α BinaryInput f
chainable : LawfulZipOperator α f
idx : Nat
hid... | c73fc5664cc58f9a |
MeasureTheory.IsStoppingTime.measurableSet_gt | Mathlib/Probability/Process/Stopping.lean | theorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | i < τ ω} | Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → ι
hτ : IsStoppingTime f τ
i : ι
⊢ {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ | ext1 ω | case h
Ω : Type u_1
ι : Type u_3
m : MeasurableSpace Ω
inst✝ : LinearOrder ι
f : Filtration ι m
τ : Ω → ι
hτ : IsStoppingTime f τ
i : ι
ω : Ω
⊢ ω ∈ {ω | i < τ ω} ↔ ω ∈ {ω | τ ω ≤ i}ᶜ | 94005878f9f6b04b |
IsLocalization.exist_integer_multiples | Mathlib/RingTheory/Localization/Integer.lean | theorem exist_integer_multiples {ι : Type*} (s : Finset ι) (f : ι → S) :
∃ b : M, ∀ i ∈ s, IsLocalization.IsInteger R ((b : R) • f i) | case refine_2.e_a
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
ι : Type u_4
s : Finset ι
f : ι → S
this : (a : Prop) → Decidable a
i : ι
hi : i ∈ s
⊢ ↑(∏ j ∈ s.erase i, (sec M (f j)).2) * ↑(sec M (f i)).2 = ∏ x ∈ s, M.subtype (... | rw [mul_comm,Submonoid.coe_finset_prod,
← Finset.prod_insert (f := fun i => ((sec M (f i)).snd : R)) (s.not_mem_erase i),
Finset.insert_erase hi] | case refine_2.e_a
R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
ι : Type u_4
s : Finset ι
f : ι → S
this : (a : Prop) → Decidable a
i : ι
hi : i ∈ s
⊢ ∏ x ∈ s, ↑(sec M (f x)).2 = ∏ x ∈ s, M.subtype (sec M (f x)).2 | 4b72d52ad57436e3 |
MeasureTheory.SimpleFunc.nearestPtInd_le | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N | case succ
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
N : ℕ
ihN : (nearestPtInd e N) x ≤ N
⊢ (nearestPtInd e (N + 1)) x ≤ N + 1 | simp only [nearestPtInd_succ] | case succ
α : Type u_1
inst✝² : MeasurableSpace α
inst✝¹ : PseudoEMetricSpace α
inst✝ : OpensMeasurableSpace α
e : ℕ → α
x : α
N : ℕ
ihN : (nearestPtInd e N) x ≤ N
⊢ (if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else (nearestPtInd e N) x) ≤ N + 1 | fa6e32de6ae6a38c |
CategoryTheory.GradedObject.Monoidal.ιTensorObj₃'_tensorHom | Mathlib/CategoryTheory/GradedObject/Monoidal.lean | @[reassoc (attr := simp)]
lemma ιTensorObj₃'_tensorHom (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃)
(i₁ i₂ i₃ j : I) (h : i₁ + i₂ + i₃ = j) :
ιTensorObj₃' X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ tensorHom (tensorHom f₁ f₂) f₃ j =
((f₁ i₁ ⊗ f₂ i₂) ⊗ f₃ i₃) ≫ ιTensorObj₃' Y₁ Y₂ Y₃ i₁ i₂ i₃ j h | I : Type u
inst✝⁶ : AddMonoid I
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ Y₁ Y₂ Y₃ : GradedObject I C
inst✝³ : X₁.HasTensor X₂
inst✝² : (tensorObj X₁ X₂).HasTensor X₃
inst✝¹ : Y₁.HasTensor Y₂
inst✝ : (tensorObj Y₁ Y₂).HasTensor Y₃
f₁ : X₁ ⟶ Y₁
f₂ : X₂ ⟶ Y₂
f₃ : X₃ ⟶ Y₃
i₁ i₂ i₃ j ... | rw [ιTensorObj₃'_eq _ _ _ i₁ i₂ i₃ j h _ rfl,
ιTensorObj₃'_eq _ _ _ i₁ i₂ i₃ j h _ rfl, assoc, ι_tensorHom,
← tensorHom_id, ← tensorHom_id, ← MonoidalCategory.tensor_comp_assoc, id_comp,
ι_tensorHom, ← MonoidalCategory.tensor_comp_assoc, comp_id] | no goals | adfe20f4057b38e1 |
InnerProductSpace.Core.cauchy_schwarz_aux | Mathlib/Analysis/InnerProductSpace/Defs.lean | theorem cauchy_schwarz_aux (x y : F) : normSqF (⟪x, y⟫ • x - ⟪x, x⟫ • y)
= normSqF x * (normSqF x * normSqF y - ‖⟪x, y⟫‖ ^ 2) | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ normSq (⟪x, y⟫_𝕜 • x - ⟪x, x⟫_𝕜 • y) = normSq x * (normSq x * normSq y - ‖⟪x, y⟫_𝕜‖ ^ 2) | rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self] | 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪⟪x, y⟫_𝕜 • x - ⟪x, x⟫_𝕜 • y, ⟪x, y⟫_𝕜 • x - ⟪x, x⟫_𝕜 • y⟫_𝕜 = ↑(normSq x * (normSq x * normSq y - ‖⟪x, y⟫_𝕜‖ ^ 2)) | 6177653f1f9af9b1 |
Sym.exists_cons_of_mem | Mathlib/Data/Sym/Basic.lean | theorem exists_cons_of_mem {s : Sym α (n + 1)} {a : α} (h : a ∈ s) : ∃ t, s = a ::ₛ t | α : Type u_1
n : ℕ
s : Sym α (n + 1)
a : α
h✝ : a ∈ s
m : Multiset α
h : (↑s).card = (a ::ₘ m).card
⊢ m.card = n | rw [s.2, Multiset.card_cons, add_left_inj] at h | α : Type u_1
n : ℕ
s : Sym α (n + 1)
a : α
h✝ : a ∈ s
m : Multiset α
h : n = m.card
⊢ m.card = n | 2e8d9d989ef3719a |
ENNReal.lintegral_prod_norm_pow_le | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_prod_norm_pow_le {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
(s : Finset ι) {f : ι → α → ℝ≥0∞} (hf : ∀ i ∈ s, AEMeasurable (f i) μ)
{p : ι → ℝ} (hp : ∑ i ∈ s, p i = 1) (h2p : ∀ i ∈ s, 0 ≤ p i) :
∫⁻ a, ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ a, f i a ∂μ) ^ p i | α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
(∀ i ∈ s, AEMeasurable (f i) μ) →
∀ {p : ι → ℝ},
∑ i ∈ s, p i = 1 →
(∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ ins... | simp [h2i₀] | no goals | c6067aa6ebb0db88 |
Turing.PartrecToTM2.codeSupp'_supports | Mathlib/Computability/TMToPartrec.lean | theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S | case comp.refine_1
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H'✝ : trStmts₁ (trNormal (f.comp g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'... | exact ⟨h.2.2, H.2⟩ | no goals | 5fb2278c84599fe0 |
PowerSeries.intValuation_eq_of_coe | Mathlib/RingTheory/LaurentSeries.lean | theorem intValuation_eq_of_coe (P : K[X]) :
(Polynomial.idealX K).intValuation P = (idealX K).intValuation (P : K⟦X⟧) | case neg
K : Type u_2
inst✝ : Field K
P : K[X]
hP : ¬P = 0
span_ne_zero : Ideal.span {P} ≠ 0 ∧ Ideal.span {Polynomial.X} ≠ 0
span_ne_zero' : Ideal.span {↑P} ≠ 0 ∧ (idealX K).asIdeal ≠ 0
⊢ (Associates.mk (Ideal.span {Polynomial.X})).count (Associates.mk (Ideal.span {P})).factors =
(Associates.mk (idealX K).asIdeal).... | classical
rw [count_associates_factors_eq (span_ne_zero).1
(Ideal.span_singleton_prime Polynomial.X_ne_zero|>.mpr prime_X) (span_ne_zero).2,
count_associates_factors_eq]
on_goal 1 => convert (normalized_count_X_eq_of_coe hP).symm
exacts [count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,... | no goals | ca855a45c5eb2633 |
NNReal.iSup_eq_zero | Mathlib/Data/NNReal/Defs.lean | @[simp] lemma iSup_eq_zero (hf : BddAbove (range f)) : ⨆ i, f i = 0 ↔ ∀ i, f i = 0 | case inl
ι : Sort u_1
f : ι → ℝ≥0
hf : BddAbove (range f)
h✝ : IsEmpty ι
⊢ ⨆ i, f i = 0 ↔ ∀ (i : ι), f i = 0 | simp | no goals | 3052fd25f3a1e24e |
equicontinuousWithinAt_iInf_dom | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem equicontinuousWithinAt_iInf_dom {t : κ → TopologicalSpace X'} {F : ι → X' → α}
{S : Set X'} {x₀ : X'} {k : κ} (hk : EquicontinuousWithinAt (tX := t k) F S x₀) :
EquicontinuousWithinAt (tX := ⨅ k, t k) F S x₀ | ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
S : Set X'
x₀ : X'
k : κ
hk : ContinuousWithinAt (⇑ofFun ∘ swap F) S x₀
⊢ ContinuousWithinAt (⇑ofFun ∘ swap F) S x₀ | unfold ContinuousWithinAt nhdsWithin at hk ⊢ | ι : Type u_1
κ : Type u_2
X' : Type u_4
α : Type u_6
uα : UniformSpace α
t : κ → TopologicalSpace X'
F : ι → X' → α
S : Set X'
x₀ : X'
k : κ
hk : Tendsto (⇑ofFun ∘ swap F) (𝓝 x₀ ⊓ 𝓟 S) (𝓝 ((⇑ofFun ∘ swap F) x₀))
⊢ Tendsto (⇑ofFun ∘ swap F) (𝓝 x₀ ⊓ 𝓟 S) (𝓝 ((⇑ofFun ∘ swap F) x₀)) | 1d280b8ff53f76a1 |
Module.exists_isPrincipal_quotient_of_finite | Mathlib/RingTheory/Finiteness/TensorProduct.lean | lemma Module.exists_isPrincipal_quotient_of_finite :
∃ N : Submodule R M, N ≠ ⊤ ∧ Submodule.IsPrincipal (⊤ : Submodule R (M ⧸ N)) | case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : Nontrivial M
n : ℕ
f : Fin n → M
hf : Submodule.span R (Set.range f) = ⊤
s : Set ℕ := {m | Submodule.span R (f '' (Fin.val ⁻¹' Set.Iio m)) ≠ ⊤}
hns : ∀ x ∈ s, x < n
hs₁ : s.Nonem... | have := not_not.mp (not_mem_of_csSup_lt (Order.lt_succ _) hs₂) | case intro.intro
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module.Finite R M
inst✝ : Nontrivial M
n : ℕ
f : Fin n → M
hf : Submodule.span R (Set.range f) = ⊤
s : Set ℕ := {m | Submodule.span R (f '' (Fin.val ⁻¹' Set.Iio m)) ≠ ⊤}
hns : ∀ x ∈ s, x < n
hs₁ : s.Nonem... | 2361a252546487f4 |
Subgroup.card_quotient_dvd_card | Mathlib/GroupTheory/Coset/Card.lean | theorem card_quotient_dvd_card (s : Subgroup α) : Nat.card (α ⧸ s) ∣ Nat.card α | α : Type u_1
inst✝ : Group α
s : Subgroup α
⊢ Nat.card (α ⧸ s) ∣ Nat.card α | simp [card_eq_card_quotient_mul_card_subgroup s, @dvd_mul_right ℕ] | no goals | 96bf62bc4ce8e3ee |
StrictMonoOn.union | Mathlib/Order/Monotone/Union.lean | theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoOn f s)
(h₂ : StrictMonoOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : StrictMonoOn f (s ∪ t) | case inl
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrder α
inst✝ : Preorder β
f : α → β
s t : Set α
c : α
h₁ : StrictMonoOn f s
h₂ : StrictMonoOn f t
hs : IsGreatest s c
ht : IsLeast t c
x : α
hxc : x ≤ c
h✝ : x ∈ s
⊢ x ∈ s | assumption | no goals | 23ab84e137b57813 |
Equiv.Perm.exists_fixed_point_of_prime' | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem exists_fixed_point_of_prime' {p n : ℕ} [hp : Fact p.Prime] (hα : p ∣ Fintype.card α)
{σ : Perm α} (hσ : σ ^ p ^ n = 1) {a : α} (ha : σ a = a) : ∃ b : α, σ b = b ∧ b ≠ a | α : Type u_1
inst✝ : Fintype α
p n : ℕ
hp : Fact (Nat.Prime p)
hα : p ∣ Fintype.card α
σ : Perm α
hσ : σ ^ p ^ n = 1
a : α
ha : σ a = a
⊢ ∃ b, σ b = b ∧ b ≠ a | classical
have h : ∀ b : α, b ∈ σ.supportᶜ ↔ σ b = b := fun b => by
rw [Finset.mem_compl, mem_support, Classical.not_not]
obtain ⟨b, hb1, hb2⟩ := Finset.exists_ne_of_one_lt_card (hp.out.one_lt.trans_le
(Nat.le_of_dvd (Finset.card_pos.mpr ⟨a, (h a).mpr ha⟩) (Nat.modEq_zero_iff_dvd.mp
((card_compl_suppo... | no goals | 43f9b6d28acf06ab |
InverseSystem.unique_pEquivOn | Mathlib/Order/DirectedInverseSystem.lean | theorem unique_pEquivOn (hs : IsLowerSet s) {e₁ e₂ : PEquivOn f equivSucc s} : e₁ = e₂ | case mk.mk.equiv.h.refine_1.H.h.mk.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e... | have hi : i ∈ s := hs (le_succ i) hi | case mk.mk.equiv.h.refine_1.H.h.mk.inr
ι : Type u_6
F : ι → Type u_7
X : ι → Type u_8
inst✝² : LinearOrder ι
f : ⦃i j : ι⦄ → i ≤ j → F j → F i
inst✝¹ : SuccOrder ι
equivSucc : ⦃i : ι⦄ → ¬IsMax i → F i⁺ ≃ F i × X i
s : Set ι
inst✝ : WellFoundedLT ι
hs : IsLowerSet s
e₁ : (i : ↑s) → F ↑i ≃ piLT X ↑i
nat₁ : IsNatEquiv f e... | dcc32ff31e12d17b |
TrivSqZeroExt.snd_invOf | Mathlib/Algebra/TrivSqZeroExt.lean | theorem snd_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] :
(⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) | R : Type u
M : Type v
inst✝⁶ : AddCommGroup M
inst✝⁵ : Semiring R
inst✝⁴ : Module Rᵐᵒᵖ M
inst✝³ : Module R M
inst✝² : SMulCommClass R Rᵐᵒᵖ M
x : tsze R M
inst✝¹ : Invertible x
inst✝ : Invertible x.fst
⊢ (⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) | letI := invertibleOfInvertibleFst x | R : Type u
M : Type v
inst✝⁶ : AddCommGroup M
inst✝⁵ : Semiring R
inst✝⁴ : Module Rᵐᵒᵖ M
inst✝³ : Module R M
inst✝² : SMulCommClass R Rᵐᵒᵖ M
x : tsze R M
inst✝¹ : Invertible x
inst✝ : Invertible x.fst
this : Invertible x := x.invertibleOfInvertibleFst
⊢ (⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) | 6eb77e53e928a5a0 |
surjective_algebraMap_of_linearMap | Mathlib/Algebra/Algebra/Basic.lean | theorem surjective_algebraMap_of_linearMap (hb : Surjective b) :
Surjective (algebraMap F E) := fun x ↦ by
obtain ⟨x, rfl⟩ := hb x
obtain ⟨y, hy⟩ := hb (b 1 * b 1)
refine ⟨x * y, ?_⟩
obtain ⟨z, hz⟩ := hb 1
apply_fun (x • z • ·) at hy
rwa [← map_smul, smul_eq_mul, mul_comm, ← smul_mul_assoc, ← map_smul _... | case intro.intro
F : Type u_1
E : Type u_2
inst✝² : CommSemiring F
inst✝¹ : Semiring E
inst✝ : Algebra F E
b : F →ₗ[F] E
hb : Surjective ⇑b
x y : F
hy : b y = b 1 * b 1
⊢ (algebraMap F E) (x * y) = b x | obtain ⟨z, hz⟩ := hb 1 | case intro.intro.intro
F : Type u_1
E : Type u_2
inst✝² : CommSemiring F
inst✝¹ : Semiring E
inst✝ : Algebra F E
b : F →ₗ[F] E
hb : Surjective ⇑b
x y : F
hy : b y = b 1 * b 1
z : F
hz : b z = 1
⊢ (algebraMap F E) (x * y) = b x | d08b12c22edf4723 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastAdd.denote_mkFullAdderCarry | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Add.lean | theorem denote_mkFullAdderCarry (assign : α → Bool) (aig : AIG α) (input : FullAdderInput aig) :
⟦mkFullAdderCarry aig input, assign⟧
=
((⟦aig, input.lhs, assign⟧ ^^ ⟦aig, input.rhs, assign⟧) && ⟦aig, input.cin, assign⟧ ||
⟦aig, input.lhs, assign⟧ && ⟦aig, input.rhs, assign⟧)
| α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : FullAdderInput aig
⊢ input.rhs.gate < (aig.mkXorCached { lhs := input.lhs, rhs := input.rhs }).aig.decls.size | apply LawfulOperator.lt_size_of_lt_aig_size (f := mkXorCached) | case h
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
assign : α → Bool
aig : AIG α
input : FullAdderInput aig
⊢ input.rhs.gate < aig.decls.size | 0b90c2014a503ffb |
Real.contDiffAt_log | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | theorem contDiffAt_log {n : WithTop ℕ∞} {x : ℝ} : ContDiffAt ℝ n log x ↔ x ≠ 0 | n : WithTop ℕ∞
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x < 0
⊢ ContDiffAt ℝ n (log ∘ fun y => -y) x | apply ContDiffAt.comp | case hg
n : WithTop ℕ∞
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x < 0
⊢ ContDiffAt ℝ n log (-x)
case hf
n : WithTop ℕ∞
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x < 0
⊢ ContDiffAt ℝ n (fun y => -y) x | 7fd2de582049795f |
ContinuousSMul.of_basis_zero | Mathlib/Topology/Algebra/FilterBasis.lean | theorem _root_.ContinuousSMul.of_basis_zero {ι : Type*} [IsTopologicalRing R] [TopologicalSpace M]
[IsTopologicalAddGroup M] {p : ι → Prop} {b : ι → Set M} (h : HasBasis (𝓝 0) p b)
(hsmul : ∀ {i}, p i → ∃ V ∈ 𝓝 (0 : R), ∃ j, p j ∧ V • b j ⊆ b i)
(hsmul_left : ∀ (x₀ : R) {i}, p i → ∃ j, p j ∧ MapsTo (x₀ • ... | case hmulright
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 ... | rcases hsmul_left x₀ hi with ⟨j, hj, hji⟩ | case hmulright.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... | 4f90af5932a68739 |
recursion' | Mathlib/Data/Real/Pi/Irrational.lean | /--
Auxiliary for the proof that `π` is irrational.
While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction... | θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
⊢ (∫ (x : ℝ) in -1 ..1, (1 - x ^ 2) ^ (n + 1) * cos (x * θ)) * θ ^ 2 =
-(2 * 2 * ((↑n + 1) * (0 ^ n * cos θ))) + 2 * (↑n + 1) * (2 * ↑n + 1) * I n θ - 4 * (↑n + 1) * ↑n * I (n - 1) ... | let v₁ (x : ℝ) : ℝ := sin (x * θ) | θ : ℝ
n : ℕ
f : ℝ → ℝ := fun x => 1 - x ^ 2
u₁ : ℝ → ℝ := fun x => f x ^ (n + 1)
u₁' : ℝ → ℝ := fun x => -(2 * (↑n + 1) * x * f x ^ n)
v₁ : ℝ → ℝ := fun x => sin (x * θ)
⊢ (∫ (x : ℝ) in -1 ..1, (1 - x ^ 2) ^ (n + 1) * cos (x * θ)) * θ ^ 2 =
-(2 * 2 * ((↑n + 1) * (0 ^ n * cos θ))) + 2 * (↑n + 1) * (2 * ↑n + 1) * I n... | 834ac76d1ec60c71 |
AlgebraicTopology.DoldKan.QInfty_idem | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | theorem QInfty_idem : (QInfty : K[X] ⟶ _) ≫ QInfty = QInfty | C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
⊢ QInfty ≫ QInfty = QInfty | ext n | case h
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ
⊢ (QInfty ≫ QInfty).f n = QInfty.f n | 479fb647b1062b18 |
Sum.Lex.Ioc_inr_inr | Mathlib/Data/Sum/Interval.lean | lemma Ioc_inr_inr :
Ioc (inrₗ b₁ : α ⊕ₗ β) (inrₗ b₂) = (Ioc b₁ b₂).map (Embedding.inr.trans toLex.toEmbedding) | α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
b₁ b₂ : β
⊢ Ioc (inrₗ b₁) (inrₗ b₂) = map toLex.toEmbedding (map Embedding.inr (Ioc b₁ b₂)) | rfl | no goals | a2d01e3cede2febb |
FirstOrder.Language.DirectLimit.exists_quotient_mk'_sigma_mk'_eq | Mathlib/ModelTheory/DirectLimit.lean | theorem exists_quotient_mk'_sigma_mk'_eq {α : Type*} [Finite α] (x : α → DirectLimit G f) :
∃ (i : ι) (y : α → G i), x = fun a => ⟦.mk f i (y a)⟧ | case intro.h
L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝² : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝¹ : Nonempty ι
α : Type u_1
inst✝ : Finite α
x : α → DirectLimit G f
i : ι
h... | generalize_proofs r | case intro.h
L : Language
ι : Type v
inst✝⁵ : Preorder ι
G : ι → Type w
inst✝⁴ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
inst✝³ : IsDirected ι fun x1 x2 => x1 ≤ x2
inst✝² : DirectedSystem G fun i j h => ⇑(f i j h)
inst✝¹ : Nonempty ι
α : Type u_1
inst✝ : Finite α
x : α → DirectLimit G f
i : ι
h... | b5e6e1b0da835dab |
AnalyticAt.eventually_constant_or_nhds_le_map_nhds | Mathlib/Analysis/Complex/OpenMapping.lean | theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : AnalyticAt ℂ g z₀) :
(∀ᶠ z in 𝓝 z₀, g z = g z₀) ∨ 𝓝 (g z₀) ≤ map g (𝓝 z₀) | case neg.h.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t => z₀ + t • z
gray : E → ℂ → ℂ := fun z => g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
z : E
hz : z ∈ sphere 0 1
hrz : ¬∀ᶠ (t : ℂ) in 𝓝 0,... | refine h7.trans (map_mono ?_) | case neg.h.intro.intro
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
g : E → ℂ
z₀ : E
hg : AnalyticAt ℂ g z₀
ray : E → ℂ → E := fun z t => z₀ + t • z
gray : E → ℂ → ℂ := fun z => g ∘ ray z
r : ℝ
hr : r > 0
hgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}
z : E
hz : z ∈ sphere 0 1
hrz : ¬∀ᶠ (t : ℂ) in 𝓝 0,... | bbddb65fce2e7708 |
Subalgebra.LinearDisjoint.of_linearDisjoint_finite_left | Mathlib/RingTheory/LinearDisjoint.lean | theorem of_linearDisjoint_finite_left [Algebra.IsIntegral R A]
(H : ∀ A' : Subalgebra R S, A' ≤ A → [Module.Finite R A'] → A'.LinearDisjoint B) :
A.LinearDisjoint B | case intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (t... | rw [← hx', ← hy'] | case intro.intro.intro.intro.intro.intro
R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [inst : Module.Finite R ↥A'], A'.LinearDisjoint B
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (t... | 1d56e99b263dd6b5 |
CategoryTheory.Limits.kernelSubobjectMap_arrow | Mathlib/CategoryTheory/Subobject/Limits.lean | theorem kernelSubobjectMap_arrow (sq : Arrow.mk f ⟶ Arrow.mk f') :
kernelSubobjectMap sq ≫ (kernelSubobject f').arrow = (kernelSubobject f).arrow ≫ sq.left | C : Type u
inst✝³ : Category.{v, u} C
X Y : C
inst✝² : HasZeroMorphisms C
f : X ⟶ Y
inst✝¹ : HasKernel f
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasKernel f'
sq : Arrow.mk f ⟶ Arrow.mk f'
⊢ kernelSubobjectMap sq ≫ (kernelSubobject f').arrow = (kernelSubobject f).arrow ≫ sq.left | simp [kernelSubobjectMap] | no goals | e835958b7230a57b |
CategoryTheory.PresheafHom.isAmalgamation_iff | Mathlib/CategoryTheory/Sites/SheafHom.lean | lemma PresheafHom.isAmalgamation_iff {X : C} (S : Sieve X)
(x : Presieve.FamilyOfElements (presheafHom F G) S.arrows)
(hx : x.Compatible) (y : (presheafHom F G).obj (op X)) :
x.IsAmalgamation y ↔ ∀ (Y : C) (g : Y ⟶ X) (hg : S g),
y.app (op (Over.mk g)) = (x g hg).app (op (Over.mk (𝟙 Y))) | case mpr.w.h.op
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
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
y : (presheafHom F G).obj (op X)
h : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y.app (op (Over.mk g)) = (x g hg).app (op... | simp only [Functor.map_id, FunctorToTypes.map_id_apply] at H | case mpr.w.h.op
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
x : Presieve.FamilyOfElements (presheafHom F G) S.arrows
hx : x.Compatible
y : (presheafHom F G).obj (op X)
h : ∀ (Y : C) (g : Y ⟶ X) (hg : S.arrows g), y.app (op (Over.mk g)) = (x g hg).app (op... | e8bc65961d66b442 |
IsLUB.frequently_mem | Mathlib/Topology/Order/IsLUB.lean | theorem IsLUB.frequently_mem {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty) :
∃ᶠ x in 𝓝[≤] a, x ∈ s | case intro.inl
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
s : Set α
a' : α
ha' : a' ∈ s
ha : IsLUB s a'
h : ∀ᶠ (x : α) in 𝓝[≤] a', ¬(fun x => x ∈ s) x
⊢ False | exact h.self_of_nhdsWithin le_rfl ha' | no goals | 77540bd2791cdbf8 |
BoxIntegral.HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO | Mathlib/Analysis/BoxIntegral/Basic.lean | theorem HasIntegral.of_bRiemann_eq_false_of_forall_isLittleO (hl : l.bRiemann = false)
(B : ι →ᵇᵃ[I] ℝ) (hB0 : ∀ J, 0 ≤ B J) (g : ι →ᵇᵃ[I] F) (s : Set ℝⁿ) (hs : s.Countable)
(hlH : s.Nonempty → l.bHenstock = true)
(H₁ : ∀ (c : ℝ≥0), ∀ x ∈ Box.Icc I ∩ s, ∀ ε > (0 : ℝ),
∃ δ > 0, ∀ J ≤ I, Box.Icc J ⊆ Met... | ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s... | have ε0' := half_pos ε0 | ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
hl : l.bRiemann = false
B : ι →ᵇᵃ[↑I] ℝ
hB0 : ∀ (J : Box ι), 0 ≤ B J
g : ι →ᵇᵃ[↑I] F
s... | f7fbe41cacd6cc55 |
MvPolynomial.vars_prod | Mathlib/Algebra/MvPolynomial/Variables.lean | theorem vars_prod {ι : Type*} [DecidableEq σ] {s : Finset ι} (f : ι → MvPolynomial σ R) :
(∏ i ∈ s, f i).vars ⊆ s.biUnion fun i => (f i).vars | R : Type u
σ : Type u_1
inst✝¹ : CommSemiring R
ι : Type u_3
inst✝ : DecidableEq σ
s : Finset ι
f : ι → MvPolynomial σ R
⊢ (∏ i ∈ s, f i).vars ⊆ s.biUnion fun i => (f i).vars | classical
induction s using Finset.induction_on with
| empty => simp
| insert hs hsub =>
simp only [hs, Finset.biUnion_insert, Finset.prod_insert, not_false_iff]
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset_union (Finset.Subset.refl _) hsub | no goals | 744e5df48190c626 |
List.map_inj | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem map_inj : map f = map g ↔ f = g | case mpr
α✝¹ : Type u_1
α✝ : Type u_2
f g : α✝¹ → α✝
⊢ f = g → map f = map g | intro h | case mpr
α✝¹ : Type u_1
α✝ : Type u_2
f g : α✝¹ → α✝
h : f = g
⊢ map f = map g | 707d036b1fa4a9fd |
CompleteLattice.IsSupClosedCompact.wellFoundedGT | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :
WellFoundedGT α where
wf | case intro
α : Type u_2
inst✝ : CompleteLattice α
h : IsSupClosedCompact α
a : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 > x2
this : sSup (range ⇑a) ∈ range ⇑a
n : ℕ
hn : a n = sSup (range ⇑a)
h' : sSup (range ⇑a) < a (n + 1)
⊢ False | apply lt_irrefl (a (n + 1)) | case intro
α : Type u_2
inst✝ : CompleteLattice α
h : IsSupClosedCompact α
a : (fun x1 x2 => x1 > x2) ↪r fun x1 x2 => x1 > x2
this : sSup (range ⇑a) ∈ range ⇑a
n : ℕ
hn : a n = sSup (range ⇑a)
h' : sSup (range ⇑a) < a (n + 1)
⊢ a (n + 1) < a (n + 1) | 723b42b10b75548c |
Rel.comp_left_id | Mathlib/Data/Rel.lean | theorem comp_left_id (r : Rel α β) : @Eq α • r = r | case a.h.a
α : Type u_1
β : Type u_2
r : Rel α β
x : α
x✝ : β
⊢ (∃ y, x = y ∧ r y x✝) ↔ r x x✝ | simp | no goals | bfec415a2b680ffd |
Polynomial.card_support_le_one_of_eraseLead_eq_zero | Mathlib/Algebra/Polynomial/EraseLead.lean | theorem card_support_le_one_of_eraseLead_eq_zero (h : f.eraseLead = 0) : #f.support ≤ 1 | case pos
R : Type u_1
inst✝ : Semiring R
f : R[X]
h : f.eraseLead = 0
hpz : f = 0
⊢ #f.support ≤ 1
case neg
R : Type u_1
inst✝ : Semiring R
f : R[X]
h : f.eraseLead = 0
hpz : ¬f = 0
⊢ #f.support ≤ 1 | case pos => simp [hpz] | case neg
R : Type u_1
inst✝ : Semiring R
f : R[X]
h : f.eraseLead = 0
hpz : ¬f = 0
⊢ #f.support ≤ 1 | 768dacd9e4a6cbf0 |
Polynomial.coeff_le_of_roots_le | Mathlib/Topology/Algebra/Polynomial.lean | theorem coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ) (h1 : p.Monic)
(h2 : Splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * p.natDegree.choose i | case inr.inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : Splits f p
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ ‖(map f p).roots.esymm ((map f p).natDegree - i)‖ ≤ B ^ ((map f p).natDegree - i) * ↑((map f p).nat... | apply ((norm_multiset_sum_le _).trans <| sum_le_card_nsmul _ _ fun r hr => _).trans | case inr.inr
F : Type u_3
K : Type u_4
inst✝¹ : CommRing F
inst✝ : NormedField K
p : F[X]
f : F →+* K
B : ℝ
i : ℕ
h1 : p.Monic
h2 : Splits f p
h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B
hB : 0 ≤ B
hi : i ≤ (map f p).natDegree
⊢ (Multiset.map (fun x => ‖x‖) (Multiset.map prod (powersetCard ((map f p).natDegree - i) (map f p).r... | 15177c00698e7d89 |
contMDiffOn_model_symm | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | theorem contMDiffOn_model_symm : ContMDiffOn 𝓘(𝕜, E) I n I.symm (range I) | 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : WithTop ℕ∞
⊢ ContMDiffOn 𝓘(𝕜, E) I n (↑I.symm) (range ↑I) | rw [contMDiffOn_iff] | 𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
H : Type u_3
inst✝ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
n : WithTop ℕ∞
⊢ ContinuousOn (↑I.symm) (range ↑I) ∧
∀ (x : E) (y : H),
ContDiffOn 𝕜 n (↑(extChartAt I y) ∘ ↑I.symm ∘ ↑(ex... | 6d82224688faa495 |
QPF.recF_eq_of_Wequiv | Mathlib/Data/QPF/Univariate/Basic.lean | theorem recF_eq_of_Wequiv {α : Type u} (u : F α → α) (x y : q.P.W) :
Wequiv x y → recF u x = recF u y | F : Type u → Type u
q : QPF F
α : Type u
u : F α → α
x y : (P F).W
h : Wequiv x y
⊢ recF u x = recF u y | induction h with
| ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp_def, ih]
| abs a f a' f' h => simp only [recF_eq', abs_map, h]
| trans x y z _ _ ih₁ ih₂ => exact Eq.trans ih₁ ih₂ | no goals | 1f7bc3a7baf99498 |
List.eraseP_eq_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Erase.lean | theorem eraseP_eq_iff {p} {l : List α} :
l.eraseP p = l' ↔
((∀ a ∈ l, ¬ p a) ∧ l = l') ∨
∃ a l₁ l₂, (∀ b ∈ l₁, ¬ p b) ∧ p a ∧ l = l₁ ++ a :: l₂ ∧ l' = l₁ ++ l₂ | case inl.mpr.inr.intro.intro.intro.intro.intro.intro
α : Type u_1
p : α → Bool
a : α
l₁ l₂ : List α
h₁ : ∀ (b : α), b ∈ l₁ → ¬p b = true
h₂ : p a = true
h : eraseP p (l₁ ++ a :: l₂) = l₁ ++ a :: l₂
⊢ eraseP p (l₁ ++ a :: l₂) = l₁ ++ l₂ | rw [eraseP_append_right _ h₁, eraseP_cons_of_pos h₂] | no goals | e94e910f43eabd04 |
Nat.factors_multiset_prod_of_irreducible | Mathlib/RingTheory/UniqueFactorizationDomain/Nat.lean | lemma factors_multiset_prod_of_irreducible {s : Multiset ℕ} (h : ∀ x : ℕ, x ∈ s → Irreducible x) :
normalizedFactors s.prod = s | s : Multiset ℕ
h : ∀ x ∈ s, Irreducible x
⊢ 0 ∉ s | exact fun con ↦ not_irreducible_zero (h 0 con) | no goals | e02ee213c35c91ee |
SimplexCategory.epi_iff_surjective | Mathlib/AlgebraicTopology/SimplexCategory/Basic.lean | theorem epi_iff_surjective {n m : SimplexCategory} {f : n ⟶ m} :
Epi f ↔ Function.Surjective f.toOrderHom | n m : SimplexCategory
f : n ⟶ m
⊢ Epi (skeletalFunctor.map f) ↔ Function.Surjective ⇑(Hom.toOrderHom f) | simp only [skeletalFunctor_obj, skeletalFunctor_map,
NonemptyFinLinOrd.epi_iff_surjective, NonemptyFinLinOrd.coe_of, ConcreteCategory.hom_ofHom] | no goals | 870fbe7dc1ef15f2 |
Complex.cpow_add | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | theorem cpow_add {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y + z) = x ^ y * x ^ z | x y z : ℂ
hx : x ≠ 0
⊢ (if x = 0 then if y + z = 0 then 1 else 0 else cexp (log x * (y + z))) =
if x = 0 then
if z = 0 then if x = 0 then if y = 0 then 1 * 1 else 0 * 1 else cexp (log x * y) * 1
else if x = 0 then if y = 0 then 1 * 0 else 0 * 0 else cexp (log x * y) * 0
else
if x = 0 then if y... | simp_all [exp_add, mul_add] | no goals | 164bdee86be9dc80 |
SeparationQuotient.map_unique | Mathlib/Topology/UniformSpace/Separation.lean | theorem map_unique {f : α → β} (hf : UniformContinuous f)
{g : SeparationQuotient α → SeparationQuotient β} (comm : mk ∘ f = g ∘ mk) : map f = g | case h.mk
α : Type u
β : Type v
inst✝¹ : UniformSpace α
inst✝ : UniformSpace β
f : α → β
hf : UniformContinuous f
g : SeparationQuotient α → SeparationQuotient β
comm : mk ∘ f = g ∘ mk
x✝ : SeparationQuotient α
a : α
⊢ map f (Quot.mk (⇑(inseparableSetoid α)) a) = g (Quot.mk (⇑(inseparableSetoid α)) a) | calc
map f ⟦a⟧ = ⟦f a⟧ := map_mk hf a
_ = g ⟦a⟧ := congr_fun comm a | no goals | a8f8ee368b4c204b |
ContinuousLinearMap.derivWithin_of_bilinear | Mathlib/Analysis/Calculus/Deriv/Mul.lean | theorem derivWithin_of_bilinear
(hu : DifferentiableWithinAt 𝕜 u s x) (hv : DifferentiableWithinAt 𝕜 v s x) :
derivWithin (fun y => B (u y) (v y)) s x =
B (u x) (derivWithin v s x) + B (derivWithin u s x) (v x) | case inr
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
E : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
G : Type u_1
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
x : 𝕜
s : Set 𝕜
B : E →L[𝕜] F →L[𝕜] G
u : 𝕜 → E
v : 𝕜 →... | simp [derivWithin_zero_of_isolated hxs] | no goals | 3f13d7399d8fc5eb |
Ideal.subset_union_prime' | Mathlib/RingTheory/Ideal/Operations.lean | theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} {a b : ι}
(hp : ∀ i ∈ s, IsPrime (f i)) {I : Ideal R} :
((I : Set R) ⊆ f a ∪ f b ∪ ⋃ i ∈ (↑s : Set ι), f i) ↔ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i | case pos.h.inr.inr.intro.intro
ι : Type u_1
R : Type u
inst✝ : CommRing R
f : ι → Ideal R
I : Ideal R
n : ℕ
a b i : ι
t : Finset ι
hit : i ∉ t
hn : t.card = n
h : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i_1 ∈ ↑(insert i t), ↑(f i_1)
hp : (f i).IsPrime ∧ ∀ x ∈ t, (f x).IsPrime
Ht : ¬∃ j ∈ t, f j ≤ f i
Ha : f a ≤ f i
h' : ↑I ⊆ ↑(f i) ∪... | exact Or.inr ⟨k, Finset.mem_insert_of_mem hkt, ih⟩ | no goals | fa41770a7d2d9fd9 |
QuadraticMap.polar_smul_left | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | theorem polar_smul_left (a : R) (x y : M) : polar Q (a • x) y = a • polar Q x y | R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
a : R
x y : M
⊢ polar (⇑Q) (a • x) y = a • polar (⇑Q) x y | obtain ⟨B, h⟩ := Q.exists_companion | case intro
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
a : R
x y : M
B : BilinMap R M N
h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y
⊢ polar (⇑Q) (a • x) y = a • polar (⇑Q) x y | a8d4b5545dba06f2 |
SzemerediRegularity.card_aux₂ | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | theorem card_aux₂ (hP : P.IsEquipartition) (hu : u ∈ P.parts) (hucard : #u ≠ m * 4 ^ #P.parts + a) :
(4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ m * 4 ^ #P.parts + a
⊢ Fintype.card α / #P.parts / 4 ^ #P.parts * 4 ^ #P.parts ≤ Fintype.card α / #P.parts | exact Nat.div_mul_le_self _ _ | no goals | c434f1e0b19cc540 |
LSeries.abscissaOfAbsConv_le_one_of_isBigO_one | Mathlib/NumberTheory/LSeries/Convergence.lean | /-- If `f` is `O(1)`, then the abscissa of absolute convergence of `f` is bounded above by `1`. -/
lemma LSeries.abscissaOfAbsConv_le_one_of_isBigO_one {f : ℕ → ℂ} (h : f =O[atTop] fun _ ↦ (1 : ℝ)) :
abscissaOfAbsConv f ≤ 1 | f : ℕ → ℂ
h : f =O[atTop] fun x => 1
⊢ f =O[atTop] fun n => ↑n ^ 0 | simpa using h | no goals | c2ae3049c3586a9d |
PiTensorProduct.reindex_trans | Mathlib/LinearAlgebra/PiTensorProduct.lean | theorem reindex_trans (e : ι ≃ ι₂) (e' : ι₂ ≃ ι₃) :
(reindex R s e).trans (reindex R _ e') = reindex R s (e.trans e') | case a
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝² : CommSemiring R
s : ι → Type u_7
inst✝¹ : (i : ι) → AddCommMonoid (s i)
inst✝ : (i : ι) → Module R (s i)
e : ι ≃ ι₂
e' : ι₂ ≃ ι₃
⊢ ↑(reindex R s e ≪≫ₗ reindex R (fun i => s (e.symm i)) e') = ↑(reindex R s (e.trans e')) | ext f | case a.H.H
ι : Type u_1
ι₂ : Type u_2
ι₃ : Type u_3
R : Type u_4
inst✝² : CommSemiring R
s : ι → Type u_7
inst✝¹ : (i : ι) → AddCommMonoid (s i)
inst✝ : (i : ι) → Module R (s i)
e : ι ≃ ι₂
e' : ι₂ ≃ ι₃
f : (i : ι) → s i
⊢ ((↑(reindex R s e ≪≫ₗ reindex R (fun i => s (e.symm i)) e')).compMultilinearMap (tprod R)) f =
... | cd44f97b1bbd1a25 |
PresheafOfModules.Sheafify.smul_add | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | protected lemma smul_add : smul α φ r (m + m') = smul α φ r m + smul α φ r m' | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
i... | let S := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m' | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.val
inst✝³ : Presheaf.IsLocallyInjective J α
inst✝² : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrp
φ : M₀.presheaf ⟶ A.val
inst✝¹ : Presheaf.IsLocallyInjective J φ
i... | e45a9bac6c91f436 |
GenLoop.homotopicTo | Mathlib/Topology/Homotopy/HomotopyGroup.lean | theorem homotopicTo (i : N) {p q : Ω^ N X x} :
Homotopic p q → (toLoop i p).Homotopic (toLoop i q) | N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i : N
p q : ↑(Ω^ N X x)
⊢ Homotopic p q → Path.Homotopic (toLoop i p) (toLoop i q) | refine Nonempty.map fun H => ⟨⟨⟨fun t => ⟨homotopyTo i H t, ?_⟩, ?_⟩, ?_, ?_⟩, ?_⟩ | case refine_1
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i : N
p q : ↑(Ω^ N X x)
H : (↑p).HomotopyRel (↑q) (Cube.boundary N)
t : ↑I × ↑I
⊢ (homotopyTo i H) t ∈ Ω^ { j // j ≠ i } X x
case refine_2
N : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace X
x : X
inst✝ : DecidableEq N
i ... | 5948809eedd27d88 |
IsBaseChange.ofEquiv | Mathlib/RingTheory/IsTensorProduct.lean | theorem IsBaseChange.ofEquiv (e : M ≃ₗ[R] N) : IsBaseChange R e.toLinearMap | case smul.h.h
R : Type u_1
M : Type v₁
N : Type v₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : CommSemiring R
inst✝¹ : Module R M
inst✝ : Module R N
e : M ≃ₗ[R] N
Q : Type (max v₁ v₂ u_1)
I₁ : AddCommMonoid Q
I₂ I₃ : Module R Q
I₄ : IsScalarTower R R Q
g : M →ₗ[R] Q
r : R
q : Q
⊢ SMul.smul r q = SMul.smu... | show (by let _ := I₂; exact r • q) = (by let _ := I₃; exact r • q) | case smul.h.h
R : Type u_1
M : Type v₁
N : Type v₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : CommSemiring R
inst✝¹ : Module R M
inst✝ : Module R N
e : M ≃ₗ[R] N
Q : Type (max v₁ v₂ u_1)
I₁ : AddCommMonoid Q
I₂ I₃ : Module R Q
I₄ : IsScalarTower R R Q
g : M →ₗ[R] Q
r : R
q : Q
⊢ (let x := I₂;
r • q)... | 4bf0cf3eb1d464b0 |
QPF.has_good_supp_iff | Mathlib/Data/QPF/Univariate/Basic.lean | theorem has_good_supp_iff {α : Type u} (x : F α) :
(∀ p, Liftp p x ↔ ∀ u ∈ supp x, p u) ↔
∃ a f, abs ⟨a, f⟩ = x ∧ ∀ a' f', abs ⟨a', f'⟩ = x → f '' univ ⊆ f' '' univ | case mpr.intro.intro.intro.mpr.a
F : Type u → Type u
q : QPF F
α : Type u
x : F α
a : (P F).A
f : (P F).B a → α
xeq : abs ⟨a, f⟩ = x
h : ∀ (a' : (P F).A) (f' : (P F).B a' → α), abs ⟨a', f'⟩ = x → f '' univ ⊆ f' '' univ
p : α → Prop
h' : ∀ u ∈ supp x, p u
i : (P F).B a
⊢ f i ∈ supp x | rw [mem_supp] | case mpr.intro.intro.intro.mpr.a
F : Type u → Type u
q : QPF F
α : Type u
x : F α
a : (P F).A
f : (P F).B a → α
xeq : abs ⟨a, f⟩ = x
h : ∀ (a' : (P F).A) (f' : (P F).B a' → α), abs ⟨a', f'⟩ = x → f '' univ ⊆ f' '' univ
p : α → Prop
h' : ∀ u ∈ supp x, p u
i : (P F).B a
⊢ ∀ (a : (P F).A) (f_1 : (P F).B a → α), abs ⟨a, f_... | 8f4503c995a7a699 |
EisensteinSeries.vecMul_SL2_mem_gammaSet | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean | /-- Right-multiplying by `γ ∈ SL(2, ℤ)` sends `gammaSet N a` to `gammaSet N (a ᵥ* γ)`. -/
lemma vecMul_SL2_mem_gammaSet {v : Fin 2 → ℤ} (hv : v ∈ gammaSet N a) (γ : SL(2, ℤ)) :
v ᵥ* γ ∈ gammaSet N (a ᵥ* γ) | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | have := RingHom.map_vecMul (m := Fin 2) (n := Fin 2) (Int.castRingHom (ZMod N)) γ v | N : ℕ
a : Fin 2 → ZMod N
v : Fin 2 → ℤ
hv : v ∈ gammaSet N a
γ : SL(2, ℤ)
this :
∀ (i : Fin 2),
(Int.castRingHom (ZMod N)) ((v ᵥ* ↑γ) i) =
(⇑(Int.castRingHom (ZMod N)) ∘ v ᵥ* (↑γ).map ⇑(Int.castRingHom (ZMod N))) i
⊢ Int.cast ∘ (v ᵥ* ↑γ) = a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ) | 8f69b5e51edc526b |
eventually_mem_of_tendsto_nhdsWithin | Mathlib/Topology/ContinuousOn.lean | theorem eventually_mem_of_tendsto_nhdsWithin {f : β → α} {a : α} {s : Set α} {l : Filter β}
(h : Tendsto f l (𝓝[s] a)) : ∀ᶠ i in l, f i ∈ s | α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
f : β → α
a : α
s : Set α
l : Filter β
h : Tendsto f l (𝓝[s] a)
⊢ ∀ᶠ (i : β) in l, f i ∈ s | simp_rw [nhdsWithin_eq, tendsto_iInf, mem_setOf_eq, tendsto_principal, mem_inter_iff,
eventually_and] at h | α : Type u_1
β : Type u_2
inst✝ : TopologicalSpace α
f : β → α
a : α
s : Set α
l : Filter β
h : ∀ (i : Set α), a ∈ i ∧ IsOpen i → (∀ᶠ (x : β) in l, f x ∈ i) ∧ ∀ᶠ (x : β) in l, f x ∈ s
⊢ ∀ᶠ (i : β) in l, f i ∈ s | 3b6fbca2eb91aa31 |
Traversable.length_toList | Mathlib/Control/Fold.lean | theorem length_toList {xs : t α} : length xs = List.length (toList xs) | α : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
ys : List α
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := 0 } ys).down = ys.length | rw [← Nat.add_zero ys.length] | α : Type u
t : Type u → Type u
inst✝¹ : Traversable t
inst✝ : LawfulTraversable t
xs : t α
ys : List α
⊢ (List.foldl (fun l x => { down := l.down + 1 }) { down := 0 } ys).down = ys.length + 0 | 9d29c0089df9fd87 |
MeasureTheory.Measure.addHaar_preimage_linearEquiv | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem addHaar_preimage_linearEquiv (f : E ≃ₗ[ℝ] E) (s : Set E) :
μ (f ⁻¹' s) = ENNReal.ofReal |LinearMap.det (f.symm : E →ₗ[ℝ] E)| * μ s | case h.e'_3.h.e'_5.h.e'_1.h.e'_4
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E ≃ₗ[ℝ] E
s : Set E
A : LinearMap.det ↑f ≠ 0
⊢ LinearMap.det ↑f.symm = (LinearMap.det ↑f)⁻¹ | simp only [LinearEquiv.det_coe_symm] | no goals | 545fd0dda826b1af |
AffineSubspace.wSameSide_iff_exists_left | Mathlib/Analysis/Convex/Side.lean | theorem wSameSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p₁ ∈ s) :
s.WSameSide x y ↔ x ∈ s ∨ ∃ p₂ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂) | case mp.intro.intro.intro.intro.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
h : p₁ ∈ s
p₁' : P
hp₁' : p₁' ∈ s
p₂' : P
hp₂' : p₂' ∈ s
h0 : x = p₁'
⊢ p₁' ∈ s ∨ ∃ p₂ ∈ s, SameRay R (p₁' -ᵥ p₁) (... | exact Or.inl hp₁' | no goals | b828f1ac322a3263 |
List.min?_mem | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean | theorem min?_mem [Min α] (min_eq_or : ∀ a b : α, min a b = a ∨ min a b = b) :
{xs : List α} → xs.min? = some a → a ∈ xs | α : Type u_1
a : α
inst✝ : Min α
min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b
⊢ ∀ {xs : List α}, xs.min? = some a → a ∈ xs | intro xs | α : Type u_1
a : α
inst✝ : Min α
min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b
xs : List α
⊢ xs.min? = some a → a ∈ xs | f362dcf8097d474b |
Algebra.FinitePresentation.ker_fg_of_mvPolynomial | Mathlib/RingTheory/FinitePresentation.lean | theorem ker_fg_of_mvPolynomial {n : ℕ} (f : MvPolynomial (Fin n) R →ₐ[R] A)
(hf : Function.Surjective f) [FinitePresentation R A] : f.toRingHom.ker.FG | case refine_1.intro.hy.a
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ↑s =... | apply Set.mem_union_left | case refine_1.intro.hy.a.a
R : Type w₁
A : Type w₂
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
n : ℕ
f : MvPolynomial (Fin n) R →ₐ[R] A
hf : Surjective ⇑f
inst✝ : FinitePresentation R A
m : ℕ
f' : MvPolynomial (Fin m) R →ₐ[R] A
hf' : Surjective ⇑f'
s : Finset (MvPolynomial (Fin m) R)
hs : Ideal.span ↑s... | 37c27acde7e67205 |
HahnSeries.embDomain_mul | Mathlib/RingTheory/HahnSeries/Multiplication.lean | theorem embDomain_mul [NonUnitalNonAssocSemiring R] (f : Γ ↪o Γ')
(hf : ∀ x y, f (x + y) = f x + f y) (x y : HahnSeries Γ R) :
embDomain f (x * y) = embDomain f x * embDomain f y | case h.mk
Γ : Type u_1
R : Type u_3
inst✝² : OrderedCancelAddCommMonoid Γ
Γ' : Type u_6
inst✝¹ : OrderedCancelAddCommMonoid Γ'
inst✝ : NonUnitalNonAssocSemiring R
f : Γ ↪o Γ'
hf : ∀ (x y : Γ), f (x + y) = f x + f y
x y : HahnSeries Γ R
g : Γ
i j : Γ'
hij : (i, j) ∈ Finset.map (f.prodMap f.toEmbedding) (addAntidiagonal ... | simp only [exists_prop, mem_map, Prod.mk.inj_iff, mem_addAntidiagonal,
Function.Embedding.coe_prodMap, mem_support, Prod.exists] at hij | case h.mk
Γ : Type u_1
R : Type u_3
inst✝² : OrderedCancelAddCommMonoid Γ
Γ' : Type u_6
inst✝¹ : OrderedCancelAddCommMonoid Γ'
inst✝ : NonUnitalNonAssocSemiring R
f : Γ ↪o Γ'
hf : ∀ (x y : Γ), f (x + y) = f x + f y
x y : HahnSeries Γ R
g : Γ
i j : Γ'
hij : ∃ a b, (x.coeff a ≠ 0 ∧ y.coeff b ≠ 0 ∧ a + b = g) ∧ Prod.map ⇑... | 074c4ef2eea46220 |
Equiv.Perm.toList_formPerm_nontrivial | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | theorem toList_formPerm_nontrivial (l : List α) (hl : 2 ≤ l.length) (hn : Nodup l) :
toList (formPerm l) (l.get ⟨0, (zero_lt_two.trans_le hl)⟩) = l | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
l : List α
hl : 2 ≤ l.length
hn : l.Nodup
⊢ l.formPerm.toList (l.get ⟨0, ⋯⟩) = l | have hc : l.formPerm.IsCycle := List.isCycle_formPerm hn hl | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
l : List α
hl : 2 ≤ l.length
hn : l.Nodup
hc : l.formPerm.IsCycle
⊢ l.formPerm.toList (l.get ⟨0, ⋯⟩) = l | abc2f1fc358fb73a |
IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul | Mathlib/MeasureTheory/Measure/Doubling.lean | theorem eventually_measure_le_scaling_constant_mul (K : ℝ) :
∀ᶠ r in 𝓝[>] 0, ∀ x, μ (closedBall x (K * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) | case h
α : Type u_1
inst✝² : PseudoMetricSpace α
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : IsUnifLocDoublingMeasure μ
K r : ℝ
hr : ∀ (x : α), ∀ t ≤ K, μ (closedBall x (t * r)) ≤ ↑(Classical.choose ⋯) * μ (closedBall x r)
x : α
⊢ μ (closedBall x (K * r)) ≤ ↑(scalingConstantOf μ K) * μ (closedBall x r) | exact (hr x K le_rfl).trans (mul_le_mul_right' (ENNReal.coe_le_coe.2 (le_max_left _ _)) _) | no goals | fed0ccf33c901042 |
Cardinal.sum_lt_prod | Mathlib/SetTheory/Cardinal/Basic.lean | theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
lt_of_not_ge fun ⟨F⟩ => by
have : Inhabited (∀ i : ι, (g i).out) | ι : Type u_1
f g : ι → Cardinal.{u_2}
H : ∀ (i : ι), f i < g i
x✝ : sum f ≥ prod g
F : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)
this : Inhabited ((i : ι) → Quotient.out (g i))
G : (i : ι) × Quotient.out (f i) → (i : ι) → Quotient.out (g i) := invFun ⇑F
sG : Surjective G
C : (i : ι) → Quotient.out (... | let ⟨⟨i, a⟩, h⟩ := sG C | ι : Type u_1
f g : ι → Cardinal.{u_2}
H : ∀ (i : ι), f i < g i
x✝ : sum f ≥ prod g
F : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)
this : Inhabited ((i : ι) → Quotient.out (g i))
G : (i : ι) × Quotient.out (f i) → (i : ι) → Quotient.out (g i) := invFun ⇑F
sG : Surjective G
C : (i : ι) → Quotient.out (... | b3952dc06d646f41 |
Array.findIdx?_eq_fst_find?_zipIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Find.lean | theorem findIdx?_eq_fst_find?_zipIdx {xs : Array α} {p : α → Bool} :
xs.findIdx? p = (xs.zipIdx.find? fun ⟨x, _⟩ => p x).map (·.2) | case mk
α : Type u_1
p : α → Bool
xs : List α
⊢ findIdx? p { toList := xs } =
Option.map (fun x => x.snd)
(find?
(fun x =>
match x with
| (x, snd) => p x)
{ toList := xs }.zipIdx) | simp [List.findIdx?_eq_fst_find?_zipIdx] | no goals | bc53022505b06e06 |
TopologicalSpace.IsTopologicalBasis.sigma | Mathlib/Topology/Bases.lean | theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
(hs : ∀ i, IsTopologicalBasis (s i)) :
IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) | case h.e'_4.h.a
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → TopologicalSpace (E i)
s : (i : ι) → Set (Set (E i))
hs : ∀ (i : ι), IsTopologicalBasis (s i)
a : (i : ι) × E i
x✝ : Set ((i : ι) × E i)
⊢ x✝ ∈ ⋃ i, (fun u => Sigma.mk i '' u) '' s i ∧ a ∈ x✝ ↔
x✝ ∈ (fun i => Sigma.mk a.fst '' i) '' {i | i ∈ s a.fst ∧ ... | aesop | no goals | 23df735b20448821 |
Polynomial.lift_of_splits | Mathlib/RingTheory/Adjoin/Field.lean | theorem Polynomial.lift_of_splits {F K L : Type*} [Field F] [Field K] [Field L] [Algebra F K]
[Algebra F L] (s : Finset K) : (∀ x ∈ s, IsIntegral F x ∧
Splits (algebraMap F L) (minpoly F x)) → Nonempty (Algebra.adjoin F (s : Set K) →ₐ[F] L) | case refine_2.intro.intro.intro
F : Type u_2
K : Type u_3
L : Type u_4
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra F K
inst✝ : Algebra F L
s✝ : Finset K
a : K
s : Finset K
x✝ : a ∉ s
H1 : IsIntegral F a
H2 : Splits (algebraMap F L) (minpoly F a)
H3 : ∀ x ∈ s, IsIntegral F x
a✝ : ∀ x ∈ s, Splits ... | letI := fieldOfFiniteDimensional F Ks | case refine_2.intro.intro.intro
F : Type u_2
K : Type u_3
L : Type u_4
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra F K
inst✝ : Algebra F L
s✝ : Finset K
a : K
s : Finset K
x✝ : a ∉ s
H1 : IsIntegral F a
H2 : Splits (algebraMap F L) (minpoly F a)
H3 : ∀ x ∈ s, IsIntegral F x
a✝ : ∀ x ∈ s, Splits ... | 491397bb377c234f |
KummerDedekind.emultiplicity_factors_map_eq_emultiplicity | Mathlib/NumberTheory/KummerDedekind.lean | theorem emultiplicity_factors_map_eq_emultiplicity
(hI : IsMaximal I) (hI' : I ≠ ⊥)
(hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) {J : Ideal S}
(hJ : J ∈ normalizedFactors (I.map (algebraMap R S))) :
emultiplicity J (I.map (algebraMap R S)) =
emultiplicity (↑(normali... | R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
x : S
I : Ideal R
inst✝³ : IsDomain R
inst✝² : IsIntegrallyClosed R
inst✝¹ : IsDedekindDomain S
inst✝ : NoZeroSMulDivisors R S
hI : I.IsMaximal
hI' : I ≠ ⊥
hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤
hx' : _root_.IsIntegral R... | rw [normalizedFactorsMapEquivNormalizedFactorsMinPolyMk, Equiv.coe_trans, Function.comp_apply,
emultiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_emultiplicity,
normalizedFactorsEquivOfQuotEquiv_emultiplicity_eq_emultiplicity] | no goals | 461f6a21a0c51bc8 |
CharP.cast_eq_mod | Mathlib/Algebra/CharP/Defs.lean | lemma cast_eq_mod (p : ℕ) [CharP R p] (k : ℕ) : (k : R) = (k % p : ℕ) :=
calc
(k : R) = ↑(k % p + p * (k / p)) | R : Type u_1
inst✝¹ : NonAssocRing R
p : ℕ
inst✝ : CharP R p
k : ℕ
⊢ ↑(k % p + p * (k / p)) = ↑(k % p) | simp [cast_eq_zero] | no goals | c6e910e6b26e3e3f |
Complex.GammaSeq_tendsto_Gamma | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) | case zero
s : ℂ
hs : 0 < s.re
⊢ Tendsto s.GammaSeq atTop (𝓝 (Gamma s)) | refine Tendsto.congr' ?_ (approx_Gamma_integral_tendsto_Gamma_integral hs) | case zero
s : ℂ
hs : 0 < s.re
⊢ (fun n => ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) =ᶠ[atTop] s.GammaSeq | dcb07666a18cd782 |
WithBot.unbotD_zero_mul | Mathlib/Algebra/Order/Ring/WithTop.lean | @[simp]
lemma unbotD_zero_mul (a b : WithBot α) : (a * b).unbotD 0 = a.unbotD 0 * b.unbotD 0 | case neg.bot
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : MulZeroClass α
b : WithBot α
hb : ¬b = 0
ha : ¬⊥ = 0
⊢ unbotD 0 (⊥ * b) = unbotD 0 ⊥ * unbotD 0 b | rw [bot_mul hb, unbotD_bot, zero_mul] | no goals | ade949097f543e93 |
norm_commutator_sub_one_le | Mathlib/Analysis/Normed/Field/Basic.lean | lemma norm_commutator_sub_one_le (ha : a ≠ 0) (hb : b ≠ 0) :
‖a * b * a⁻¹ * b⁻¹ - 1‖ ≤ 2 * ‖a‖⁻¹ * ‖b‖⁻¹ * ‖a - 1‖ * ‖b - 1‖ | α : Type u_2
inst✝ : NormedDivisionRing α
a b : α
ha : a ≠ 0
hb : b ≠ 0
⊢ ‖a * b * a⁻¹ * b⁻¹ - 1‖ ≤ 2 * ‖a‖⁻¹ * ‖b‖⁻¹ * ‖a - 1‖ * ‖b - 1‖ | simpa using norm_commutator_units_sub_one_le (.mk0 a ha) (.mk0 b hb) | no goals | 35ce02bd0103ddc3 |
Finset.fold_cons | Mathlib/Data/Finset/Fold.lean | theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f | α : Type u_1
β : Type u_2
op : β → β → β
hc : Std.Commutative op
ha : Std.Associative op
f : α → β
b : β
s : Finset α
a : α
h : a ∉ s
⊢ fold op b f (cons a s h) = op (f a) (fold op b f s) | dsimp only [fold] | α : Type u_1
β : Type u_2
op : β → β → β
hc : Std.Commutative op
ha : Std.Associative op
f : α → β
b : β
s : Finset α
a : α
h : a ∉ s
⊢ Multiset.fold op b (Multiset.map f (cons a s h).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val)) | 2f440cb1e4952ade |
Ideal.natAbs_det_equiv | Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean | theorem natAbs_det_equiv (I : Ideal S) {E : Type*} [EquivLike E S I] [AddEquivClass E S I] (e : E) :
Int.natAbs
(LinearMap.det
((Submodule.subtype I).restrictScalars ℤ ∘ₗ AddMonoidHom.toIntLinearMap (e : S →+ I))) =
Ideal.absNorm I | S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisInde... | intro i | S : Type u_1
inst✝⁶ : CommRing S
inst✝⁵ : Nontrivial S
inst✝⁴ : IsDedekindDomain S
inst✝³ : Module.Free ℤ S
inst✝² : Module.Finite ℤ S
I : Ideal S
E : Type u_2
inst✝¹ : EquivLike E S ↥I
inst✝ : AddEquivClass E S ↥I
e : E
hI : ¬I = ⊥
ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S
b : Basis (Module.Free.ChooseBasisInde... | 87e25ff10e007725 |
CFC.negPart_smul_of_nonpos | Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/PosPart.lean | lemma negPart_smul_of_nonpos {r : ℝ} (hr : r ≤ 0) {a : A} : (r • a)⁻ = -r • a⁺ | A : Type u_1
inst✝⁸ : NonUnitalRing A
inst✝⁷ : Module ℝ A
inst✝⁶ : SMulCommClass ℝ A A
inst✝⁵ : IsScalarTower ℝ A A
inst✝⁴ : StarRing A
inst✝³ : TopologicalSpace A
inst✝² : NonUnitalContinuousFunctionalCalculus ℝ IsSelfAdjoint
inst✝¹ : T2Space A
inst✝ : StarModule ℝ A
r : ℝ
hr : r ≤ 0
a : A
⊢ (r • a)⁻ = -r • a⁺ | conv_lhs => rw [← neg_neg r, neg_smul, negPart_neg, posPart_smul_of_nonneg (by simpa)] | no goals | a0e53fc11b5c57a9 |
MvPolynomial.IsHomogeneous.eval₂ | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | lemma eval₂ (hφ : φ.IsHomogeneous m) (f : R →+* MvPolynomial τ S) (g : σ → MvPolynomial τ S)
(hf : ∀ r, (f r).IsHomogeneous 0) (hg : ∀ i, (g i).IsHomogeneous n) :
(eval₂ f g φ).IsHomogeneous (n * m) | case convert_6
σ : Type u_1
τ : Type u_2
R : Type u_3
S : Type u_4
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
φ : MvPolynomial σ R
m n : ℕ
hφ : φ.IsHomogeneous m
f : R →+* MvPolynomial τ S
g : σ → MvPolynomial τ S
hf : ∀ (r : R), (f r).IsHomogeneous 0
hg : ∀ (i : σ), (g i).IsHomogeneous n
i : σ →₀ ℕ
hi : i ∈ φ.supp... | apply (hg k).pow | no goals | db8e0d9dfe5d44a4 |
intervalIntegral.integral_add | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | theorem integral_add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) :
∫ x in a..b, f x + g x ∂μ = (∫ x in a..b, f x ∂μ) + ∫ x in a..b, g x ∂μ | E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f g : ℝ → E
μ : Measure ℝ
hf : IntervalIntegrable f μ a b
hg : IntervalIntegrable g μ a b
⊢ ∫ (x : ℝ) in a..b, f x + g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in a..b, g x ∂μ | simp only [intervalIntegral_eq_integral_uIoc, integral_add hf.def' hg.def', smul_add] | no goals | a1d513cb5e8c13b1 |
WittVector.bind₁_zero_wittPolynomial | Mathlib/RingTheory/WittVector/IsPoly.lean | theorem bind₁_zero_wittPolynomial [Fact p.Prime] (n : ℕ) :
bind₁ (0 : ℕ → MvPolynomial ℕ R) (wittPolynomial p R n) = 0 | p : ℕ
R : Type u
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
n : ℕ
⊢ (bind₁ 0) (wittPolynomial p R n) = 0 | rw [← aeval_eq_bind₁, aeval_zero, constantCoeff_wittPolynomial, RingHom.map_zero] | no goals | 1d11913afaf1c232 |
MulChar.sum_one_eq_card_units | Mathlib/NumberTheory/MulChar/Basic.lean | theorem sum_one_eq_card_units [DecidableEq R] :
(∑ a, (1 : MulChar R R') a) = Fintype.card Rˣ | R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
⊢ ∑ a : R, 1 a = ↑(Fintype.card Rˣ) | calc
(∑ a, (1 : MulChar R R') a) = ∑ a : R, if IsUnit a then 1 else 0 :=
Finset.sum_congr rfl fun a _ => ?_
_ = ((Finset.univ : Finset R).filter IsUnit).card := Finset.sum_boole _ _
_ = (Finset.univ.map ⟨((↑) : Rˣ → R), Units.ext⟩).card := ?_
_ = Fintype.card Rˣ := congr_arg _ (Finset.card_map _) | case calc_1
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
a : R
x✝ : a ∈ Finset.univ
⊢ 1 a = if IsUnit a then 1 else 0
case calc_2
R : Type u_1
inst✝³ : CommMonoid R
inst✝² : Fintype R
R' : Type u_2
inst✝¹ : CommRing R'
inst✝ : DecidableEq R
⊢ ↑(Finset.f... | 4d1d31a95db53ae1 |
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