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 |
|---|---|---|---|---|---|---|
Array.map_eq_empty_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem map_eq_empty_iff {f : α → β} {l : Array α} : map f l = #[] ↔ l = #[] | α : Type u_1
β : Type u_2
f : α → β
l : Array α
⊢ map f l = #[] ↔ l = #[] | cases l | case mk
α : Type u_1
β : Type u_2
f : α → β
toList✝ : List α
⊢ map f { toList := toList✝ } = #[] ↔ { toList := toList✝ } = #[] | 848eb1af69850b56 |
PowerBasis.adjoin_gen_eq_top | Mathlib/RingTheory/PowerBasis.lean | theorem adjoin_gen_eq_top (B : PowerBasis R S) : adjoin R ({B.gen} : Set S) = ⊤ | R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
B : PowerBasis R S
⊢ Set.range ⇑B.basis ⊆ ↑(Subalgebra.toSubmodule (adjoin R {B.gen})) | rintro x ⟨i, rfl⟩ | case intro
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
B : PowerBasis R S
i : Fin B.dim
⊢ B.basis i ∈ ↑(Subalgebra.toSubmodule (adjoin R {B.gen})) | db45fc28b296c790 |
Matrix.col_smul | Mathlib/Data/Matrix/RowCol.lean | theorem col_smul [SMul R α] (x : R) (v : m → α) : col ι (x • v) = x • col ι v | case a
m : Type u_2
R : Type u_5
α : Type v
ι : Type u_6
inst✝ : SMul R α
x : R
v : m → α
i✝ : m
j✝ : ι
⊢ col ι (x • v) i✝ j✝ = (x • col ι v) i✝ j✝ | rfl | no goals | 4c942164e640a5ce |
Ordinal.iSup_iterate_eq_nfp | Mathlib/SetTheory/Ordinal/FixedPoint.lean | theorem iSup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) (a : Ordinal.{u}) :
⨆ n : ℕ, f^[n] a = nfp f a | case a
f : Ordinal.{u} → Ordinal.{u}
a : Ordinal.{u}
⊢ ⨆ n, f^[n] a ≤ nfp f a | rw [Ordinal.iSup_le_iff] | case a
f : Ordinal.{u} → Ordinal.{u}
a : Ordinal.{u}
⊢ ∀ (i : ℕ), f^[i] a ≤ nfp f a | c976a13d93c931d3 |
AmpleSet.preimage | Mathlib/Analysis/Convex/AmpleSet.lean | theorem preimage {s : Set F} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) : AmpleSet (L ⁻¹' s) | F : Type u_1
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module ℝ F
inst✝³ : TopologicalSpace F
E : Type u_2
inst✝² : AddCommGroup E
inst✝¹ : Module ℝ E
inst✝ : TopologicalSpace E
s : Set F
h : AmpleSet s
L : E ≃ᵃL[ℝ] F
⊢ AmpleSet (⇑L.symm '' s) | exact h.image L.symm | no goals | 453d407fec960729 |
WithSeminorms.isVonNBounded_iff_seminorm_bounded | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | theorem WithSeminorms.isVonNBounded_iff_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
Bornology.IsVonNBounded 𝕜 s ↔ ∀ i : ι, ∃ r > 0, ∀ x ∈ s, p i x < r | 𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
⊢ Bornology.IsVonNBounded 𝕜 s ↔ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x < r | rw [hp.isVonNBounded_iff_finset_seminorm_bounded] | 𝕜 : Type u_1
E : Type u_5
ι : Type u_8
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
s : Set E
hp : WithSeminorms p
⊢ (∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r) ↔ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p i) x ... | 24b13a155e892387 |
RCLike.exists_norm_eq_mul_self | Mathlib/Analysis/RCLike/Basic.lean | lemma exists_norm_eq_mul_self (x : K) : ∃ c, ‖c‖ = 1 ∧ ↑‖x‖ = c * x | case inr
K : Type u_1
inst✝ : RCLike K
x : K
hx : x ≠ 0
⊢ ∃ c, ‖c‖ = 1 ∧ ↑‖x‖ = c * x | exact ⟨‖x‖ / x, by simp [norm_ne_zero_iff.2, hx]⟩ | no goals | cb316a5885a4b0a9 |
Real.sInf_nonpos | Mathlib/Data/Real/Archimedean.lean | /-- As `sInf s = 0` when `s` is a set of reals that's either empty or unbounded below,
it suffices to show that all elements of `s` are nonpositive to show that `sInf s ≤ 0`. -/
lemma sInf_nonpos (hs : ∀ x ∈ s, x ≤ 0) : sInf s ≤ 0 | case inl
hs : ∀ x ∈ ∅, x ≤ 0
⊢ sInf ∅ ≤ 0 | exact sInf_empty.le | no goals | 7972a2fb30da1c10 |
Finset.inductive_claim_mul | Mathlib/Combinatorics/Additive/SmallTripling.lean | @[to_additive]
private lemma inductive_claim_mul (hm : 3 ≤ m)
(h : ∀ ε : Fin 3 → ℤ, (∀ i, |ε i| = 1) → #((finRange 3).map fun i ↦ A ^ ε i).prod ≤ k * #A)
(ε : Fin m → ℤ) (hε : ∀ i, |ε i| = 1) :
#((finRange m).map fun i ↦ A ^ ε i).prod ≤ k ^ (m - 2) * #A | G : Type u_1
inst✝¹ : DecidableEq G
inst✝ : Group G
A : Finset G
k : ℝ
m✝ : ℕ
h : ∀ (ε : Fin 3 → ℤ), (∀ (i : Fin 3), |ε i| = 1) → ↑(#(List.map (fun i => A ^ ε i) (finRange 3)).prod) ≤ k * ↑(#A)
m : ℕ
hm : 3 ≤ m + 1
ih :
∀ (ε : Fin (m + 1) → ℤ),
(∀ (i : Fin (m + 1)), |ε i| = 1) →
↑(#(List.map (fun i => A ^ ε... | positivity | no goals | 75599e5c7cd8b516 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_blastDivSubtractShift_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastDivSubtractShift_r (aig : AIG α) (assign : α → Bool) (lhs rhs : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d : AIG.RefVec aig w) (wn wr : Nat)
(q r : AIG.RefVec aig w) (qbv rbv : BitVec w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(h... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀... | rw [AIG.LawfulVecOperator.denote_mem_prefix (f := AIG.RefVec.ite)] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀... | 249a96ab3a68ac7e |
LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure | Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean | theorem LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure [NormedSpace 𝕜 E]
(f : E →ₗ[𝕜] E) (hf : LipschitzWith 1 f) (g : E →L[𝕜] LinearMap.eqLocus f 1)
(hg_proj : ∀ x : LinearMap.eqLocus f 1, g x = x)
(hg_ker : (LinearMap.ker g : Set E) ⊆ closure (LinearMap.range (f - 1))) (x : E) :
Tendsto (b... | 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : LipschitzWith 1 ⇑f
g : E →L[𝕜] ↥(eqLocus f 1)
hg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x
hg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))
y : E
hy : g y = 0
z : E
hz : IsFixedPt (⇑f) z
this : Tendsto (f... | simpa [hy, hgz, birkhoffAverage, birkhoffSum, Finset.sum_add_distrib, smul_add]
using this.add (hz.tendsto_birkhoffAverage 𝕜 _root_.id) | no goals | ffcebdb2b0436857 |
Subgroup.coe_iSup_of_directed | Mathlib/Algebra/Group/Subgroup/Lattice.lean | theorem coe_iSup_of_directed {ι} [Nonempty ι] {S : ι → Subgroup G} (hS : Directed (· ≤ ·) S) :
((⨆ i, S i : Subgroup G) : Set G) = ⋃ i, S i :=
Set.ext fun x ↦ by simp [mem_iSup_of_directed hS]
| G : Type u_1
inst✝¹ : Group G
ι : Sort u_2
inst✝ : Nonempty ι
S : ι → Subgroup G
hS : Directed (fun x1 x2 => x1 ≤ x2) S
x : G
⊢ x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i) | simp [mem_iSup_of_directed hS] | no goals | 5acd702f11b8ea0c |
IsDiscreteValuationRing.of_ufd_of_unique_irreducible | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | theorem of_ufd_of_unique_irreducible {R : Type u} [CommRing R] [IsDomain R]
[UniqueFactorizationMonoid R] (h₁ : ∃ p : R, Irreducible p)
(h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q) :
IsDiscreteValuationRing R | R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime | haveI PID : IsPrincipalIdealRing R := aux_pid_of_ufd_of_unique_irreducible R h₁ h₂ | R : Type u
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₁ : ∃ p, Irreducible p
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
PID : IsPrincipalIdealRing R
⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime | 9253614c2b77dfb3 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.assignmentsInvariant_insertRatUnits | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RatAddSound.lean | theorem assignmentsInvariant_insertRatUnits {n : Nat} (f : DefaultFormula n)
(hf : f.ratUnits = #[] ∧ AssignmentsInvariant f) (units : CNF.Clause (PosFin n)) :
AssignmentsInvariant (insertRatUnits f units).1 | case inr.inr.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitI... | let j1_unit := unit (insertRatUnits f units).1.ratUnits[j1] | case inr.inr.intro.intro.intro.intro.intro.intro.intro
n : Nat
f : DefaultFormula n
hf : f.ratUnits = #[] ∧ f.AssignmentsInvariant
units : CNF.Clause (PosFin n)
h :
let assignments := (f.insertRatUnits units).fst.assignments;
let_fun hsize := ⋯;
let ratUnits := (f.insertRatUnits units).fst.ratUnits;
InsertUnitI... | b6d274376b14ee0b |
Module.finrank_eq_card_chooseBasisIndex | Mathlib/LinearAlgebra/Dimension/Free.lean | theorem _root_.Module.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
finrank R M = Fintype.card (ChooseBasisIndex R M) | R : Type u
M : Type v
inst✝⁵ : Semiring R
inst✝⁴ : StrongRankCondition R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : Free R M
inst✝ : Module.Finite R M
⊢ finrank R M = Fintype.card (ChooseBasisIndex R M) | simp [finrank, rank_eq_card_chooseBasisIndex] | no goals | 3372258a41c862e1 |
MeasureTheory.projectiveFamilyFun_union | Mathlib/MeasureTheory/Constructions/ProjectiveFamilyContent.lean | lemma projectiveFamilyFun_union (hP : IsProjectiveMeasureFamily P)
(hs : s ∈ measurableCylinders α) (ht : t ∈ measurableCylinders α) (hst : Disjoint s t) :
projectiveFamilyFun P (s ∪ t) = projectiveFamilyFun P s + projectiveFamilyFun P t | ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : { x // x ∈ I }) → α ↑i)
hS ... | rw [hs_eq, ht_eq] | ι : Type u_1
α : ι → Type u_2
mα : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
s t : Set ((i : ι) → α i)
hP : IsProjectiveMeasureFamily P
hs : s ∈ measurableCylinders α
ht : t ∈ measurableCylinders α
hst : Disjoint s t
I : Finset ι
S : Set ((i : { x // x ∈ I }) → α ↑i)
hS ... | db52bbf98927a594 |
Submonoid.LocalizationMap.map_mul_left | Mathlib/GroupTheory/MonoidLocalization/Basic.lean | theorem map_mul_left (z) : k.toMap (g (f.sec z).2) * f.map hy k z = k.toMap (g (f.sec z).1) | M : Type u_1
inst✝³ : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝² : CommMonoid N
P : Type u_3
inst✝¹ : CommMonoid P
f : S.LocalizationMap N
g : M →* P
T : Submonoid P
hy : ∀ (y : ↥S), g ↑y ∈ T
Q : Type u_4
inst✝ : CommMonoid Q
k : T.LocalizationMap Q
z : N
⊢ k.toMap (g ↑(f.sec z).2) * (f.map hy k) z = k.toMap (g (... | rw [mul_comm, f.map_mul_right] | no goals | 55c096010f70a6e4 |
Action.rightDual_ρ | Mathlib/CategoryTheory/Action/Monoidal.lean | theorem rightDual_ρ [RightRigidCategory V] (h : H) : Xᘁ.ρ h = (X.ρ (h⁻¹ : H))ᘁ | V : Type (u + 1)
inst✝³ : LargeCategory V
inst✝² : MonoidalCategory V
H : Type u
inst✝¹ : Group H
X : Action V H
inst✝ : RightRigidCategory V
h : H
⊢ Xᘁ.ρ h = X.ρ (inv h)ᘁ
case x
V : Type (u + 1)
inst✝³ : LargeCategory V
inst✝² : MonoidalCategory V
H : Type u
inst✝¹ : Group H
X : Action V H
inst✝ : RightRigidCategory ... | rfl | no goals | 467c60d5375160b7 |
AlgebraicIndependent.map | Mathlib/RingTheory/AlgebraicIndependent/Basic.lean | theorem map {f : A →ₐ[R] A'} (hf_inj : Set.InjOn f (adjoin R (range x))) :
AlgebraicIndependent R (f ∘ x) | ι : Type u_1
R : Type u_3
A : Type u_5
A' : Type u_6
x : ι → A
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : CommRing A'
inst✝¹ : Algebra R A
inst✝ : Algebra R A'
hx : AlgebraicIndependent R x
f : A →ₐ[R] A'
hf_inj : InjOn ⇑f ↑(adjoin R (range x))
this : aeval (⇑f ∘ x) = f.comp (aeval x)
p : MvPolynomial ι R
⊢ (aeva... | rw [AlgHom.mem_range] | ι : Type u_1
R : Type u_3
A : Type u_5
A' : Type u_6
x : ι → A
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : CommRing A'
inst✝¹ : Algebra R A
inst✝ : Algebra R A'
hx : AlgebraicIndependent R x
f : A →ₐ[R] A'
hf_inj : InjOn ⇑f ↑(adjoin R (range x))
this : aeval (⇑f ∘ x) = f.comp (aeval x)
p : MvPolynomial ι R
⊢ ∃ x_1... | 7f136c5ea980fb7c |
CategoryTheory.MorphismProperty.Over.map_comp | Mathlib/CategoryTheory/MorphismProperty/OverAdjunction.lean | lemma Over.map_comp {X Y Z : T} {f : X ⟶ Y} (hf : P f) {g : Y ⟶ Z} (hg : P g) :
map Q (P.comp_mem f g hf hg) = map Q hf ⋙ map Q hg | T : Type u_1
inst✝² : Category.{u_2, u_1} T
P Q : MorphismProperty T
inst✝¹ : Q.IsMultiplicative
inst✝ : P.IsStableUnderComposition
X Y Z : T
f : X ⟶ Y
hf : P f
g : Y ⟶ Z
hg : P g
⊢ map Q ⋯ = map Q hf ⋙ map Q hg | fapply Functor.ext | case h_obj
T : Type u_1
inst✝² : Category.{u_2, u_1} T
P Q : MorphismProperty T
inst✝¹ : Q.IsMultiplicative
inst✝ : P.IsStableUnderComposition
X Y Z : T
f : X ⟶ Y
hf : P f
g : Y ⟶ Z
hg : P g
⊢ ∀ (X_1 : P.Over Q X), (map Q ⋯).obj X_1 = (map Q hf ⋙ map Q hg).obj X_1
case h_map
T : Type u_1
inst✝² : Category.{u_2, u_1} T... | c1f7af44f70dc7be |
Affine.Simplex.mongePlane_comm | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem mongePlane_comm {n : ℕ} (s : Simplex ℝ P (n + 2)) (i₁ i₂ : Fin (n + 3)) :
s.mongePlane i₁ i₂ = s.mongePlane i₂ i₁ | case e_a.e_direction.e_K.h.mpr.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
r : ℝ
⊢ ∃ a, a • (s.points i₁ -ᵥ s.points i₂) = r • (s.points i₂ -ᵥ s.points i₁) | use -r | case h
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ
s : Simplex ℝ P (n + 2)
i₁ i₂ : Fin (n + 3)
r : ℝ
⊢ -r • (s.points i₁ -ᵥ s.points i₂) = r • (s.points i₂ -ᵥ s.points i₁) | d24be0dc0bad5776 |
Nat.count_le_card | Mathlib/Data/Nat/Count.lean | theorem count_le_card (hp : (setOf p).Finite) (n : ℕ) : count p n ≤ #hp.toFinset | p : ℕ → Prop
inst✝ : DecidablePred p
hp : (setOf p).Finite
n : ℕ
⊢ count p n ≤ #hp.toFinset | rw [count_eq_card_filter_range] | p : ℕ → Prop
inst✝ : DecidablePred p
hp : (setOf p).Finite
n : ℕ
⊢ #(filter (fun x => p x) (range n)) ≤ #hp.toFinset | 5754070b25886974 |
IsSMulRegular.pow | Mathlib/Algebra/Regular/SMul.lean | theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) | R : Type u_1
M : Type u_3
a : R
inst✝¹ : Monoid R
inst✝ : MulAction R M
n : ℕ
ra : IsSMulRegular M a
⊢ IsSMulRegular M (a ^ n) | induction n with
| zero => rw [pow_zero]; simp only [one]
| succ n hn =>
rw [pow_succ']
exact (ra.smul_iff (a ^ n)).mpr hn | no goals | c5cc644d33ce662f |
Concept.ext | Mathlib/Order/Concept.lean | theorem ext (h : c.fst = d.fst) : c = d | α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
h : c.toProd.1 = d.toProd.1
⊢ c = d | obtain ⟨⟨s₁, t₁⟩, h₁, _⟩ := c | case mk.mk
α : Type u_2
β : Type u_3
r : α → β → Prop
d : Concept α β r
s₁ : Set α
t₁ : Set β
h₁ : intentClosure r (s₁, t₁).1 = (s₁, t₁).2
closure_snd✝ : extentClosure r (s₁, t₁).2 = (s₁, t₁).1
h : { toProd := (s₁, t₁), closure_fst := h₁, closure_snd := closure_snd✝ }.toProd.1 = d.toProd.1
⊢ { toProd := (s₁, t₁), closu... | dd6b7dc0117169a9 |
CategoryTheory.types_ext | Mathlib/CategoryTheory/Types.lean | theorem types_ext {α β : Type u} (f g : α ⟶ β) (h : ∀ a : α, f a = g a) : f = g | α β : Type u
f g : α ⟶ β
h : ∀ (a : α), f a = g a
⊢ f = g | funext x | case h
α β : Type u
f g : α ⟶ β
h : ∀ (a : α), f a = g a
x : α
⊢ f x = g x | 52ba5f8e3d9f54b5 |
PMF.filter_apply | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | theorem filter_apply (a : α) :
(p.filter s h) a = s.indicator p a * (∑' a', (s.indicator p) a')⁻¹ | α : Type u_1
p : PMF α
s : Set α
h : ∃ a ∈ s, a ∈ p.support
a : α
⊢ (p.filter s h) a = s.indicator (⇑p) a * (∑' (a' : α), s.indicator (⇑p) a')⁻¹ | rw [filter, normalize_apply] | no goals | e4a535050cb08343 |
IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_two | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1)))
[hpri : Fact (p : ℕ).Prime] [IsCyclotomicExtension {p ^ (k + 1)} K L]
(hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (hs : s ≤ k)
(htwo : p ^ (k - s + 1) ≠ 2) : norm K (ζ ^ (p : ℕ) ^ s - 1) = (p : K) ^ (... | case refine_2
p : ℕ+
K : Type u
L : Type v
inst✝³ : Field L
ζ : L
inst✝² : Field K
inst✝¹ : Algebra K L
k s : ℕ
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hpri : Fact (Nat.Prime ↑p)
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hs : s ≤ k
htwo : p ^ (k - s + 1) ≠ 2
hirr₁ ... | exact add_mem (IntermediateField.mem_adjoin_simple_self K η) (one_mem _) | no goals | 460ee0dff43c909a |
Finsupp.support_mul_subset_left | Mathlib/Data/Finsupp/Pointwise.lean | lemma support_mul_subset_left {g₁ g₂ : α →₀ β} :
(g₁ * g₂).support ⊆ g₁.support := fun x hx => by
aesop
| α : Type u₁
β : Type u₂
inst✝ : MulZeroClass β
g₁ g₂ : α →₀ β
x : α
hx : x ∈ (g₁ * g₂).support
⊢ x ∈ g₁.support | aesop | no goals | 561127566dbeba86 |
GaloisConnection.isLUB_l_image | Mathlib/Order/GaloisConnection/Basic.lean | theorem isLUB_l_image {s : Set α} {a : α} (h : IsLUB s a) : IsLUB (l '' s) (l a) :=
⟨gc.monotone_l.mem_upperBounds_image h.left, fun b hb =>
gc.l_le <| h.right <| by rwa [gc.upperBounds_l_image] at hb⟩
| α : Type u
β : Type v
inst✝¹ : Preorder α
inst✝ : Preorder β
l : α → β
u : β → α
gc : GaloisConnection l u
s : Set α
a : α
h : IsLUB s a
b : β
hb : b ∈ upperBounds (l '' s)
⊢ u b ∈ upperBounds s | rwa [gc.upperBounds_l_image] at hb | no goals | ac8229e41b2880ed |
Sigma.Lex.mono | Mathlib/Data/Sigma/Lex.lean | theorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ i a b, s₁ i a b → s₂ i a b) {a b : Σ i, α i}
(h : Lex r₁ s₁ a b) : Lex r₂ s₂ a b | case right
ι : Type u_1
α : ι → Type u_2
r₁ r₂ : ι → ι → Prop
s₁ s₂ : (i : ι) → α i → α i → Prop
hr : ∀ (a b : ι), r₁ a b → r₂ a b
hs : ∀ (i : ι) (a b : α i), s₁ i a b → s₂ i a b
i✝ : ι
a b : α i✝
hab : s₁ i✝ a b
⊢ Lex r₂ s₂ ⟨i✝, a⟩ ⟨i✝, b⟩ | exact Lex.right _ _ (hs _ _ _ hab) | no goals | 8256ad01d1a76a61 |
derivWithin.scomp | Mathlib/Analysis/Calculus/Deriv/Comp.lean | theorem derivWithin.scomp (hg : DifferentiableWithinAt 𝕜' g₁ t' (h x))
(hh : DifferentiableWithinAt 𝕜 h s x) (hs : MapsTo h s t') :
derivWithin (g₁ ∘ h) s x = derivWithin h s x • derivWithin g₁ t' (h x) | 𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F
hg : Dif... | rcases uniqueDiffWithinAt_or_nhdsWithin_eq_bot s x with hxs | hxs | case inl
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
x : 𝕜
s : Set 𝕜
𝕜' : Type u_1
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedAlgebra 𝕜 𝕜'
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
t' : Set 𝕜'
h : 𝕜 → 𝕜'
g₁ : 𝕜' → F... | fe039bb728aee7e6 |
List.mem_iff_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem mem_iff_append {a : α} {l : List α} : a ∈ l ↔ ∃ s t : List α, l = s ++ a :: t :=
⟨append_of_mem, fun ⟨s, t, e⟩ => e ▸ by simp⟩
| α : Type u_1
a : α
l : List α
x✝ : ∃ s t, l = s ++ a :: t
s t : List α
e : l = s ++ a :: t
⊢ a ∈ s ++ a :: t | simp | no goals | 92e482803a9d9461 |
WeierstrassCurve.Jacobian.addZ_of_Z_eq_zero_left | Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean | lemma addZ_of_Z_eq_zero_left {P Q : Fin 3 → R} (hPz : P z = 0) : addZ P Q = P x * Q z * Q z | R : Type r
inst✝ : CommRing R
P Q : Fin 3 → R
hPz : P z = 0
⊢ addZ P Q = P x * Q z * Q z | rw [addZ, hPz] | R : Type r
inst✝ : CommRing R
P Q : Fin 3 → R
hPz : P z = 0
⊢ P x * Q z ^ 2 - Q x * 0 ^ 2 = P x * Q z * Q z | 28ca0f25643f1375 |
Pell.xy_modEq_yn | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem xy_modEq_yn (n) :
∀ k, xn a1 (n * k) ≡ xn a1 n ^ k [MOD yn a1 n ^ 2] ∧ yn a1 (n * k) ≡
k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]
| 0 => by constructor <;> simpa using Nat.ModEq.refl _
| k + 1 => by
let ⟨hx, hy⟩ := xy_modEq_yn n k
have L : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n... | a : ℕ
a1 : 1 < a
n : ℕ
⊢ xn a1 (n * 0) ≡ xn a1 n ^ 0 [MOD yn a1 n ^ 2] ∧ yn a1 (n * 0) ≡ 0 * xn a1 n ^ (0 - 1) * yn a1 n [MOD yn a1 n ^ 3] | constructor <;> simpa using Nat.ModEq.refl _ | no goals | 032161580a61a378 |
ProbabilityTheory.Kernel.rnDeriv_withDensity | Mathlib/Probability/Kernel/RadonNikodym.lean | lemma rnDeriv_withDensity [IsFiniteKernel κ] {f : α → γ → ℝ≥0∞} [IsFiniteKernel (withDensity κ f)]
(hf : Measurable (Function.uncurry f)) (a : α) :
(κ.withDensity f).rnDeriv κ a =ᵐ[κ a] f a | α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel (κ.withDensity f)
hf : Measurable (Function.uncurry f)
a : α
h_ae : (κ.withDensity f).rnDeriv κ a =ᶠ[ae (κ a)] ... | filter_upwards [h_ae, (κ a).rnDeriv_withDensity (hf' a)] with x hx1 hx2 | case h
α : Type u_1
γ : Type u_2
mα : MeasurableSpace α
mγ : MeasurableSpace γ
κ : Kernel α γ
hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ
inst✝¹ : IsFiniteKernel κ
f : α → γ → ℝ≥0∞
inst✝ : IsFiniteKernel (κ.withDensity f)
hf : Measurable (Function.uncurry f)
a : α
h_ae : (κ.withDensity f).rnDeriv κ a =ᶠ[ae ... | c01bf246993d3f91 |
Relation.church_rosser | Mathlib/Logic/Relation.lean | theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
(hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c | case tail.intro.intro.single.intro.intro
α : Type u_1
r : α → α → Prop
a✝² b✝¹ c✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac✝ : ReflTransGen r a✝² c✝
d e : α
a✝¹ : ReflTransGen r a✝² d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
a : α
hea : ReflTransGen r e a
hfa :... | exact ⟨c, hea.trans hac, hbc⟩ | no goals | b76c57a616feebb5 |
Set.encard_singleton | Mathlib/Data/Set/Card.lean | theorem encard_singleton (e : α) : ({e} : Set α).encard = 1 | α : Type u_1
e : α
⊢ {e}.encard = 1 | rw [encard, ENat.card_eq_coe_fintype_card, Fintype.card_ofSubsingleton, Nat.cast_one] | no goals | f97a161f23285fb3 |
EquicontinuousAt.tendsto_of_mem_closure | Mathlib/Topology/UniformSpace/Equicontinuity.lean | theorem EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
{s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
(hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
Tendsto (F · x) l (𝓝 z) | case intro.intro.intro.intro.intro.intro
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x => F x y) l (𝓝 ... | filter_upwards [hs y hys (ball_mem_nhds _ hV)] with i hi | case h
ι : Type u_1
X : Type u_3
α : Type u_6
tX : TopologicalSpace X
uα : UniformSpace α
l : Filter ι
F : ι → X → α
f : X → α
s : Set X
x : X
z : α
hF : EquicontinuousAt F x
hf : ∀ i ∈ 𝓤 α, ∀ᶠ (x : X) in 𝓝[s] x, f x ∈ {y | (y, z) ∈ id i}
hs : ∀ y ∈ s, Tendsto (fun x => F x y) l (𝓝 (f y))
hx : (𝓝[s] x).NeBot
U : Se... | b75bfbc553aec3e3 |
MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg' | Mathlib/MeasureTheory/Integral/SetToL1.lean | theorem setToSimpleFunc_nonneg' (T : Set α → G' →L[ℝ] G'')
(hT_nonneg : ∀ s, MeasurableSet s → μ s < ∞ → ∀ x, 0 ≤ x → 0 ≤ T s x) (f : α →ₛ G') (hf : 0 ≤ f)
(hfi : Integrable f μ) : 0 ≤ setToSimpleFunc T f | case neg.intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x ... | rw [← hy] | case neg.intro
α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝³ : NormedLatticeAddCommGroup G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : NormedLatticeAddCommGroup G'
inst✝ : NormedSpace ℝ G'
T : Set α → G' →L[ℝ] G''
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x ... | fd6a8699634a7fec |
MulAction.fst_mem_orbit_of_mem_orbit | Mathlib/GroupTheory/GroupAction/Basic.lean | @[to_additive]
lemma fst_mem_orbit_of_mem_orbit {x y : α × β} (h : x ∈ MulAction.orbit M y) :
x.1 ∈ MulAction.orbit M y.1 | M : Type u
inst✝² : Monoid M
α : Type v
inst✝¹ : MulAction M α
β : Type u_1
inst✝ : MulAction M β
x y : α × β
h : x ∈ orbit M y
⊢ x.1 ∈ orbit M y.1 | rcases h with ⟨g, rfl⟩ | case intro
M : Type u
inst✝² : Monoid M
α : Type v
inst✝¹ : MulAction M α
β : Type u_1
inst✝ : MulAction M β
y : α × β
g : M
⊢ ((fun m => m • y) g).1 ∈ orbit M y.1 | 53250cb36fb5a2ab |
Equiv.Perm.mem_sumCongrHom_range_of_perm_mapsTo_inl | Mathlib/GroupTheory/Perm/Finite.lean | theorem mem_sumCongrHom_range_of_perm_mapsTo_inl {m n : Type*} [Finite m] [Finite n]
{σ : Perm (m ⊕ n)} (h : Set.MapsTo σ (Set.range Sum.inl) (Set.range Sum.inl)) :
σ ∈ (sumCongrHom m n).range | m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)
h1 : ∀ (x : m ⊕ n), (∃ a, Sum.inl a = x) → ∃ a, Sum.inl a = σ x
h3 : ∀ (x : m ⊕ n), (∃ b, Sum.inr b = x) → ∃ b, Sum.inr b = σ x
⊢ σ ∈ (sumCongrHom m n).range | let σ₁' := subtypePermOfFintype σ h1 | m : Type u_1
n : Type u_2
inst✝¹ : Finite m
inst✝ : Finite n
σ : Perm (m ⊕ n)
h : Set.MapsTo (⇑σ) (Set.range Sum.inl) (Set.range Sum.inl)
h1 : ∀ (x : m ⊕ n), (∃ a, Sum.inl a = x) → ∃ a, Sum.inl a = σ x
h3 : ∀ (x : m ⊕ n), (∃ b, Sum.inr b = x) → ∃ b, Sum.inr b = σ x
σ₁' : Perm { x // ∃ a, Sum.inl a = x } := σ.subtypePer... | 6361a7f565bfff0a |
Submodule.goursat | Mathlib/LinearAlgebra/Goursat.lean | /-- **Goursat's lemma** for an arbitrary submodule of a product.
If `L` is a submodule of `M × N`, then there exist submodules `M'' ≤ M' ≤ M` and `N'' ≤ N' ≤ N` such
that `L ≤ M' × N'`, and `L` is (the image in `M × N` of) the preimage of the graph of an `R`-linear
isomorphism `M' ⧸ M'' ≃ N' ⧸ N''`. -/
lemma goursat :... | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
⊢ ∃ M' N' M'' N'' e, L = map (M'.subtype.prodMap N'.su... | let P : L →ₗ[R] M' := (LinearMap.fst ..).submoduleMap L | R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
L : Submodule R (M × N)
M' : Submodule R M := map (LinearMap.fst R M N) L
N' : Submodule R N := map (LinearMap.snd R M N) L
P : ↥L →ₗ[R] ↥M' := (LinearMap.fst R M N).submoduleMap... | 8bafbf13c72f3711 |
HasFDerivWithinAt.mul_const | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | theorem HasFDerivWithinAt.mul_const (hc : HasFDerivWithinAt c c' s x) (d : 𝔸') :
HasFDerivWithinAt (fun y => c y * d) (d • c') s x | case h.e'_12.h.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
s : Set E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c : E → 𝔸'
c' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
d : 𝔸'
e_8✝ : SeminormedAddCommGr... | ext z | case h.e'_12.h.h.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
s : Set E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c : E → 𝔸'
c' : E →L[𝕜] 𝔸'
hc : HasFDerivWithinAt c c' s x
d : 𝔸'
e_8✝ : SeminormedAddComm... | 3c70500a9fa5ebfd |
IsCyclotomicExtension.discr_prime_pow_ne_two | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | theorem discr_prime_pow_ne_two [IsCyclotomicExtension {p ^ (k + 1)} K L] [hp : Fact (p : ℕ).Prime]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K))
(hk : p ^ (k + 1) ≠ 2) : discr K (hζ.powerBasis K).basis =
(-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ... | case e_a.inr
p : ℕ+
k : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hp : Fact (Nat.Prime ↑p)
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hk : p ^ (k + 1) ≠ 2
hne : NeZero ↑↑(p ^ (k + 1))... | replace hp2 : (p : ℕ) ≠ 2 := by rwa [Ne, ← coe_two, PNat.coe_inj] | case e_a.inr
p : ℕ+
k : ℕ
K : Type u
L : Type v
ζ : L
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L
hp : Fact (Nat.Prime ↑p)
hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))
hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)
hk : p ^ (k + 1) ≠ 2
hne : NeZero ↑↑(p ^ (k + 1))... | 0ff08dcef08cb104 |
Matrix.nonsing_inv_nonsing_inv | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | theorem nonsing_inv_nonsing_inv (h : IsUnit A.det) : A⁻¹⁻¹ = A :=
calc
A⁻¹⁻¹ = 1 * A⁻¹⁻¹ | n : Type u'
α : Type v
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : CommRing α
A : Matrix n n α
h : IsUnit A.det
⊢ A * A⁻¹ * A⁻¹⁻¹ = A | rw [Matrix.mul_assoc, A⁻¹.mul_nonsing_inv (A.isUnit_nonsing_inv_det h), Matrix.mul_one] | no goals | 2a08da7f87c7ce96 |
Std.DHashMap.Internal.Raw₀.getKey?_insertMany_list_of_contains_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/RawLemmas.lean | theorem getKey?_insertMany_list_of_contains_eq_false [EquivBEq α] [LawfulHashable α] (h : m.1.WF)
{l : List ((a : α) × β a)} {k : α}
(h' : (l.map Sigma.fst).contains k = false) :
(m.insertMany l).1.getKey? k = m.getKey? k | 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
l : List ((a : α) × β a)
k : α
h' : (List.map Sigma.fst l).contains k = false
⊢ (m.insertMany l).val.getKey? k = m.getKey? k | simp_to_model [insertMany] using List.getKey?_insertList_of_contains_eq_false | no goals | 264cca27489e9b1c |
EMetric.cauchySeq_iff_le_tendsto_0 | Mathlib/Topology/Instances/ENNReal/Lemmas.lean | theorem EMetric.cauchySeq_iff_le_tendsto_0 [Nonempty β] [SemilatticeSup β] {s : β → α} :
CauchySeq s ↔ ∃ b : β → ℝ≥0∞, (∀ n m N : β, N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧
Tendsto b atTop (𝓝 0) := EMetric.cauchySeq_iff.trans <| by
constructor
· intro hs
/- `s` is Cauchy sequence. Let `b n` be the... | case mp
α : Type u_1
β : Type u_2
inst✝² : PseudoEMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
s : β → α
⊢ (∀ ε > 0, ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε) →
∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0) | intro hs | case mp
α : Type u_1
β : Type u_2
inst✝² : PseudoEMetricSpace α
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
s : β → α
hs : ∀ ε > 0, ∃ N, ∀ (m : β), N ≤ m → ∀ (n : β), N ≤ n → edist (s m) (s n) < ε
⊢ ∃ b, (∀ (n m N : β), N ≤ n → N ≤ m → edist (s n) (s m) ≤ b N) ∧ Tendsto b atTop (𝓝 0) | 99e59b2095ac4f59 |
Std.DHashMap.Internal.List.hashesTo_cons | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/HashesTo.lean | theorem hashesTo_cons [BEq α] [Hashable α] {i : Nat} {size : Nat} {l : List ((a : α) × β a)} {k : α}
{v : β k} (h : (h' : 0 < size) → (mkIdx size h' (hash k)).1.toNat = i) :
HashesTo l i size → HashesTo (⟨k, v⟩ :: l) i size | case inl
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : Hashable α
i size : Nat
l : List ((a : α) × β a)
k : α
v : β k
h : ∀ (h' : 0 < size), (mkIdx size h' (hash k)).val.toNat = i
x✝ : HashesTo l i size
ih : ∀ (h : 0 < size) (p : (a : α) × β a), p ∈ l → (mkIdx size h (hash p.fst)).val.toNat = i
h' : 0 < size
⊢ (mkId... | exact h h' | no goals | 41e73158cdb01ddf |
LinearMap.rTensor_injective_of_fg | Mathlib/RingTheory/Flat/Basic.lean | theorem _root_.LinearMap.rTensor_injective_of_fg {f : N →ₗ[R] P}
(h : ∀ (N' : Submodule R N) (P' : Submodule R P),
N'.FG → P'.FG → ∀ h : N' ≤ P'.comap f, Function.Injective ((f.restrict h).rTensor M)) :
Function.Injective (f.rTensor M) := fun x y eq ↦ by
have ⟨N', Nfg, sub⟩ := Submodule.exists_fg_le_su... | R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
f : N →ₗ[R] P
h :
∀ (N' : Submodule R N) (P' : Submodule R P),
N'.FG → P'.FG → ∀ (h : N' ≤ comap f P'), Func... | have ⟨N', Nfg, sub⟩ := Submodule.exists_fg_le_subset_range_rTensor_subtype {x, y} (by simp) | R : Type u
M : Type v
N : Type u_1
P : Type u_2
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
f : N →ₗ[R] P
h :
∀ (N' : Submodule R N) (P' : Submodule R P),
N'.FG → P'.FG → ∀ (h : N' ≤ comap f P'), Func... | f4fa23b28dfe7ac1 |
Nat.totient_gcd_mul_totient_mul | Mathlib/Data/Nat/Totient.lean | theorem totient_gcd_mul_totient_mul (a b : ℕ) : φ (a.gcd b) * φ (a * b) = φ a * φ b * a.gcd b | a b a1 a2 b1 b2 c1 c2 : ℕ
h1 : b1 ∣ a1
h2 : b2 ∣ a2
⊢ a1 / b1 * c1 * (a2 / b2 * c2) = a1 * a2 / (b1 * b2) * (c1 * c2) | calc
a1 / b1 * c1 * (a2 / b2 * c2) = a1 / b1 * (a2 / b2) * (c1 * c2) := by apply mul_mul_mul_comm
_ = a1 * a2 / (b1 * b2) * (c1 * c2) := by
congr 1
exact div_mul_div_comm h1 h2 | no goals | 35f1994c784b46d9 |
CategoryTheory.mono_to_simple_zero_of_not_iso | Mathlib/CategoryTheory/Simple.lean | theorem mono_to_simple_zero_of_not_iso {X Y : C} [Simple Y] {f : X ⟶ Y} [Mono f]
(w : IsIso f → False) : f = 0 | C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
X Y : C
inst✝¹ : Simple Y
f : X ⟶ Y
inst✝ : Mono f
w : IsIso f → False
h : ¬f = 0
⊢ False | exact w (isIso_of_mono_of_nonzero h) | no goals | aa45b01f535803f9 |
IsDenseInducing.extend_Z_bilin | Mathlib/Topology/Algebra/UniformGroup/Basic.lean | theorem extend_Z_bilin : Continuous (extend (de.prodMap df) (fun p : β × δ => φ p.1 p.2)) | case mk.right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : Topol... | have key := extend_Z_bilin_key de df hφ W'_nhd x₀ y₀ | case mk.right
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁴ : TopologicalSpace α
inst✝¹³ : AddCommGroup α
inst✝¹² : IsTopologicalAddGroup α
inst✝¹¹ : TopologicalSpace β
inst✝¹⁰ : AddCommGroup β
inst✝⁹ : TopologicalSpace γ
inst✝⁸ : AddCommGroup γ
inst✝⁷ : IsTopologicalAddGroup γ
inst✝⁶ : Topol... | 973370763b8f9daf |
PadicInt.fwdDiff_mahlerSeries | Mathlib/NumberTheory/Padics/MahlerBasis.lean | /--
The coefficients of a Mahler series can be recovered from the sum by taking forward differences at
`0`.
-/
lemma fwdDiff_mahlerSeries (ha : Tendsto a atTop (𝓝 0)) (n) :
Δ_[1]^[n] (mahlerSeries a) (0 : ℤ_[p]) = a n :=
calc Δ_[1]^[n] (mahlerSeries a) 0
-- throw away terms after the n'th
_ = Δ_[1]^[n] (fun ... | p : ℕ
hp : Fact (Nat.Prime p)
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℚ_[p] E
inst✝¹ : IsUltrametricDist E
inst✝ : CompleteSpace E
a : ℕ → E
ha : Tendsto a atTop (𝓝 0)
n : ℕ
⊢ ∑ j ∈ range (n + 1), Δ_[1]^[n] (fun k => ↑(k.choose j)) 0 • a j = a n | simp only [fwdDiff_iter_choose_zero, ite_smul, one_smul, zero_smul, sum_ite_eq,
Finset.mem_range, lt_add_iff_pos_right, zero_lt_one, ↓reduceIte] | no goals | e38eeb570cb71904 |
Monotone.cauchySeq_alternating_series_of_tendsto_zero | Mathlib/Analysis/SpecificLimits/Normed.lean | theorem Monotone.cauchySeq_alternating_series_of_tendsto_zero (hfa : Monotone f)
(hf0 : Tendsto f atTop (𝓝 0)) : CauchySeq fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i | f : ℕ → ℝ
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i | simp_rw [mul_comm] | f : ℕ → ℝ
hfa : Monotone f
hf0 : Tendsto f atTop (𝓝 0)
⊢ CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x | da025471c6845571 |
Besicovitch.exists_closedBall_covering_tsum_measure_le | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ]
[Measure.OuterRegular μ] {ε : ℝ≥0∞} (hε : ε ≠ 0) (f : α → Set ℝ) (s : Set α)
(hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty) :
∃ (t : Set α) (r : α → ℝ), t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧
(s ⊆ ⋃ x ∈ t, closedBa... | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ : μ.OuterR... | refine ⟨t0 ∪ ⋃ i : Fin N, ((↑) : s' → α) '' S i, r, ?_, ?_, ?_, ?_, ?_⟩ | case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
α : Type u_1
inst✝⁶ : MetricSpace α
inst✝⁵ : SecondCountableTopology α
inst✝⁴ : MeasurableSpace α
inst✝³ : OpensMeasurableSpace α
inst✝² : HasBesicovitchCovering α
μ : Measure α
inst✝¹ : SFinite μ
inst✝ :... | 181682c2ed74f98b |
OrderIso.strictConvexOn_symm | Mathlib/Analysis/Convex/Function.lean | theorem OrderIso.strictConvexOn_symm (f : α ≃o β) (hf : StrictConcaveOn 𝕜 univ f) :
StrictConvexOn 𝕜 univ f.symm | 𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConcaveOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1
⊢ f.symm... | obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩ | case intro
𝕜 : Type u_1
α : Type u_4
β : Type u_5
inst✝⁴ : OrderedSemiring 𝕜
inst✝³ : OrderedAddCommMonoid α
inst✝² : SMul 𝕜 α
inst✝¹ : OrderedAddCommMonoid β
inst✝ : SMul 𝕜 β
f : α ≃o β
hf : StrictConcaveOn 𝕜 univ ⇑f
x : β
x✝¹ : x ∈ univ
y : β
x✝ : y ∈ univ
hxy : x ≠ y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b =... | 3c23ef5cd4d8b1a4 |
WithTop.iSup_coe_eq_top | Mathlib/Order/ConditionallyCompleteLattice/Indexed.lean | lemma iSup_coe_eq_top : ⨆ x, (f x : WithTop α) = ⊤ ↔ ¬BddAbove (range f) | case refine_1
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ b < ⊤, ∃ i, b < ↑(f i)
r : α
⊢ ∃ y ∈ range f, r < y | rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩ | case refine_1.intro
α : Type u_1
ι : Sort u_4
inst✝ : ConditionallyCompleteLinearOrderBot α
f : ι → α
hf : ∀ b < ⊤, ∃ i, b < ↑(f i)
r : α
i : ι
hi : ↑r < ↑(f i)
⊢ ∃ y ∈ range f, r < y | a8bcd3637912f636 |
TopCat.partialSections.closed | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | theorem partialSections.closed [∀ j : J, T2Space (F.obj j)] {G : Finset J}
(H : Finset (FiniteDiagramArrow G)) : IsClosed (partialSections F H) | J : Type u
inst✝¹ : SmallCategory J
F : J ⥤ TopCat
inst✝ : ∀ (j : J), T2Space ↑(F.obj j)
G : Finset J
H : Finset (TopCat.FiniteDiagramArrow G)
⊢ IsClosed (partialSections F H) | have :
partialSections F H =
⋂ (f : FiniteDiagramArrow G) (_ : f ∈ H), {u | F.map f.2.2.2.2 (u f.1) = u f.2.1} := by
ext1
simp only [Set.mem_iInter, Set.mem_setOf_eq]
rfl | J : Type u
inst✝¹ : SmallCategory J
F : J ⥤ TopCat
inst✝ : ∀ (j : J), T2Space ↑(F.obj j)
G : Finset J
H : Finset (TopCat.FiniteDiagramArrow G)
this : partialSections F H = ⋂ f ∈ H, {u | (ConcreteCategory.hom (F.map f.snd.snd.snd.snd)) (u f.fst) = u f.snd.fst}
⊢ IsClosed (partialSections F H) | 8fc61b38c5d26173 |
Std.DHashMap.Raw.Const.get_erase | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/RawLemmas.lean | theorem get_erase [EquivBEq α] [LawfulHashable α] (h : m.WF) {k a : α} {h'} :
get (m.erase k) a h' = get m a (mem_of_mem_erase h h') | α : Type u
inst✝³ : BEq α
inst✝² : Hashable α
β : Type v
m : Raw α fun x => β
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
h : m.WF
k a : α
h' : a ∈ m.erase k
⊢ get (m.erase k) a h' = get m a ⋯ | simp_to_raw using Raw₀.Const.get_erase ⟨m, _⟩ | no goals | 0a746a06c43767e6 |
List.takeWhile_go_toArray | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/ToArray.lean | theorem takeWhile_go_toArray (p : α → Bool) (l : List α) (i : Nat) :
Array.takeWhile.go p l.toArray i r = r ++ (takeWhile p (l.drop i)).toArray | case cons.zero
α : Type u_1
p : α → Bool
a : α
l : List α
ih : ∀ {r : Array α} (i : Nat), takeWhile.go p l.toArray i r = r ++ (takeWhile p (drop i l)).toArray
r : Array α
⊢ (if p a = true then r ++ (a :: takeWhile p l).toArray else r) =
r ++ (if p a = true then a :: takeWhile p l else []).toArray | split <;> simp | no goals | 20ee17ae559c4cd9 |
Complex.horizontalSegment_eq | Mathlib/Data/Complex/Basic.lean | /-- A real segment `[a₁, a₂]` translated by `b * I` is the complex line segment. -/
lemma horizontalSegment_eq (a₁ a₂ b : ℝ) :
(fun (x : ℝ) ↦ x + b * I) '' [[a₁, a₂]] = [[a₁, a₂]] ×ℂ {b} | case h.mp.intro.intro
a₁ a₂ b : ℝ
x : ℂ
x₁ : ℝ
hx₁ : x₁ ∈ [[a₁, a₂]]
hx₁' : (fun x => ↑x + ↑b * I) x₁ = x
⊢ x ∈ ⇑equivRealProd ⁻¹' [[a₁, a₂]] ×ˢ {b} | simp [← hx₁', mem_preimage, mem_prod, hx₁] | no goals | 69651e1e9083f8c8 |
Array.Perm.trans | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Perm.lean | theorem Perm.trans {l₁ l₂ l₃ : Array α} (h₁ : l₁ ~ l₂) (h₂ : l₂ ~ l₃) : l₁ ~ l₃ | case mk.mk.mk
α : Type u_1
toList✝² toList✝¹ toList✝ : List α
h₁ : toList✝² ~ toList✝¹
h₂ : toList✝¹ ~ toList✝
⊢ { toList := toList✝² } ~ { toList := toList✝ } | simpa using h₁.trans h₂ | no goals | 01ea126d68abf836 |
Finset.Nonempty.smul_zero | Mathlib/Algebra/GroupWithZero/Pointwise/Finset.lean | lemma Nonempty.smul_zero (hs : s.Nonempty) : s • (0 : Finset β) = 0 :=
s.smul_zero_subset.antisymm <| by simpa [mem_smul] using hs
| α : Type u_1
β : Type u_2
inst✝² : DecidableEq β
inst✝¹ : Zero β
inst✝ : SMulZeroClass α β
s : Finset α
hs : s.Nonempty
⊢ 0 ⊆ s • 0 | simpa [mem_smul] using hs | no goals | 4d1fdd8ac002d540 |
CategoryTheory.isIso_of_coyoneda_map_bijective | Mathlib/CategoryTheory/Yoneda.lean | lemma isIso_of_coyoneda_map_bijective {X Y : C} (f : X ⟶ Y)
(hf : ∀ (T : C), Function.Bijective (fun (x : Y ⟶ T) => f ≫ x)) :
IsIso f | C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
f : X ⟶ Y
hf : ∀ (T : C), Function.Bijective fun x => f ≫ x
⊢ IsIso f | obtain ⟨g, hg : f ≫ g = 𝟙 X⟩ := (hf X).2 (𝟙 X) | case intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y : C
f : X ⟶ Y
hf : ∀ (T : C), Function.Bijective fun x => f ≫ x
g : Y ⟶ X
hg : f ≫ g = 𝟙 X
⊢ IsIso f | b1106f1d61595e54 |
IsUnit.not_isNilpotent | Mathlib/RingTheory/Nilpotent/Basic.lean | lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
¬ IsNilpotent x | R : Type u_1
inst✝¹ : Ring R
inst✝ : Nontrivial R
x : R
hx : IsUnit x
H : IsNilpotent x
⊢ False | simpa using H.isUnit_add_right_of_commute hx.neg (by simp) | no goals | adaf8be103e18642 |
Finset.sum_smul_vsub_const_eq_weightedVSub | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | theorem sum_smul_vsub_const_eq_weightedVSub (w : ι → k) (p₁ : ι → P) (p₂ : P)
(h : ∑ i ∈ s, w i = 0) : (∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.weightedVSub p₁ w | k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
w : ι → k
p₁ : ι → P
p₂ : P
h : ∑ i ∈ s, w i = 0
⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSub p₁) w | rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, sub_zero] | no goals | 78f3aa55ebc5b128 |
Affine.Triangle.dist_orthocenter_reflection_circumcenter | Mathlib/Geometry/Euclidean/MongePoint.lean | theorem dist_orthocenter_reflection_circumcenter (t : Triangle ℝ P) {i₁ i₂ : Fin 3} (h : i₁ ≠ i₂) :
dist t.orthocenter (reflection (affineSpan ℝ (t.points '' {i₁, i₂})) t.circumcenter) =
t.circumradius | V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : {i₁, i₂} ⊆ univ
⊢ -(∑ x : Fin (0 + 2 + 1),
(∑ x_1 : Fin (0 + 2 + 1),
((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ th... | obtain ⟨i₃, hi₃, hi₃₁, hi₃₂⟩ :
∃ i₃, univ \ ({i₁, i₂} : Finset (Fin 3)) = {i₃} ∧ i₃ ≠ i₁ ∧ i₃ ≠ i₂ := by
decide +revert | case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
t : Triangle ℝ P
i₁ i₂ : Fin 3
h : i₁ ≠ i₂
hu : {i₁, i₂} ⊆ univ
i₃ : Fin 3
hi₃ : univ \ {i₁, i₂} = {i₃}
hi₃₁ : i₃ ≠ i₁
hi₃₂ : i₃ ≠ i₂
⊢ -(∑ x : Fin (0 + 2 + 1)... | cf2667ffc8b63c24 |
MvPolynomial.aeval_sumElim_pderiv_inl | Mathlib/Algebra/MvPolynomial/PDeriv.lean | lemma aeval_sumElim_pderiv_inl {S τ : Type*} [CommRing S] [Algebra R S]
(p : MvPolynomial (σ ⊕ τ) R) (f : τ → S) (j : σ) :
aeval (Sum.elim X (C ∘ f)) ((pderiv (Sum.inl j)) p) =
(pderiv j) ((aeval (Sum.elim X (C ∘ f))) p) | R : Type u
σ : Type v
inst✝² : CommSemiring R
S : Type u_1
τ : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
p : MvPolynomial (σ ⊕ τ) R
f : τ → S
j : σ
⊢ (aeval (Sum.elim X (⇑C ∘ f))) ((pderiv (Sum.inl j)) p) = (pderiv j) ((aeval (Sum.elim X (⇑C ∘ f))) p) | induction' p using MvPolynomial.induction_on with a p q hp hq p q h | case h_C
R : Type u
σ : Type v
inst✝² : CommSemiring R
S : Type u_1
τ : Type u_2
inst✝¹ : CommRing S
inst✝ : Algebra R S
f : τ → S
j : σ
a : R
⊢ (aeval (Sum.elim X (⇑C ∘ f))) ((pderiv (Sum.inl j)) (C a)) = (pderiv j) ((aeval (Sum.elim X (⇑C ∘ f))) (C a))
case h_add
R : Type u
σ : Type v
inst✝² : CommSemiring R
S : Typ... | cea34ff12d9793df |
SemiNormedGrp.explicitCokernelDesc_unique | Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean | theorem explicitCokernelDesc_unique {X Y Z : SemiNormedGrp.{u}} {f : X ⟶ Y} {g : Y ⟶ Z}
(w : f ≫ g = 0) (e : explicitCokernel f ⟶ Z) (he : explicitCokernelπ f ≫ e = g) :
e = explicitCokernelDesc w | case x
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ ∀ (j : WalkingParallelPair), (cokernelCocone f).ι.app j ≫ e = (Cofork.ofπ g ⋯).ι.app j | rintro (_ | _) | case x.zero
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokernel f ⟶ Z
he : explicitCokernelπ f ≫ e = g
⊢ (cokernelCocone f).ι.app WalkingParallelPair.zero ≫ e = (Cofork.ofπ g ⋯).ι.app WalkingParallelPair.zero
case x.one
X Y Z : SemiNormedGrp
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
e : explicitCokern... | d83ba6fbc8e92dee |
Stream'.Seq.coinduction2 | Mathlib/Data/Seq/Seq.lean | theorem coinduction2 (s) (f g : Seq α → Seq β)
(H :
∀ s,
BisimO (fun s1 s2 : Seq β => ∃ s : Seq α, s1 = f s ∧ s2 = g s) (destruct (f s))
(destruct (g s))) :
f s = g s | α : Type u
β : Type v
s : Seq α
f g : Seq α → Seq β
H : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct
⊢ f s = g s | refine eq_of_bisim (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) ?_ ⟨s, rfl, rfl⟩ | α : Type u
β : Type v
s : Seq α
f g : Seq α → Seq β
H : ∀ (s : Seq α), BisimO (fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s) (f s).destruct (g s).destruct
⊢ IsBisimulation fun s1 s2 => ∃ s, s1 = f s ∧ s2 = g s | b52202dbb190e447 |
Associates.FactorSet.unique | Mathlib/RingTheory/UniqueFactorizationDomain/FactorSet.lean | theorem FactorSet.unique [Nontrivial α] {p q : FactorSet α} (h : p.prod = q.prod) : p = q | case coe.top
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : UniqueFactorizationMonoid α
inst✝ : Nontrivial α
a✝ : Multiset { a // Irreducible a }
h : prod ↑a✝ = prod ⊤
⊢ ↑a✝ = ⊤ | rw [← FactorSet.prod_eq_zero_iff, h, Associates.prod_top] | no goals | 6deae4149576f9a7 |
boundedSub_of_lipschitzWith_sub | Mathlib/Topology/Bornology/BoundedOperation.lean | lemma boundedSub_of_lipschitzWith_sub [PseudoMetricSpace R] [Sub R] {K : NNReal}
(lip : LipschitzWith K (fun (p : R × R) ↦ p.1 - p.2)) :
BoundedSub R where
isBounded_sub {s t} s_bdd t_bdd | case h.e'_3.h
R : Type u_1
inst✝¹ : PseudoMetricSpace R
inst✝ : Sub R
K : ℝ≥0
lip : LipschitzWith K fun p => p.1 - p.2
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
bdd : Bornology.IsBounded (s ×ˢ t)
p : R
⊢ p ∈ s - t ↔ p ∈ (fun p => p.1 - p.2) '' s ×ˢ t | simp only [Set.mem_image, Set.mem_prod, Prod.exists] | case h.e'_3.h
R : Type u_1
inst✝¹ : PseudoMetricSpace R
inst✝ : Sub R
K : ℝ≥0
lip : LipschitzWith K fun p => p.1 - p.2
s t : Set R
s_bdd : Bornology.IsBounded s
t_bdd : Bornology.IsBounded t
bdd : Bornology.IsBounded (s ×ˢ t)
p : R
⊢ p ∈ s - t ↔ ∃ a b, (a ∈ s ∧ b ∈ t) ∧ a - b = p | 52ebfc0f9d3d4b6e |
Nat.Prime.dvd_iff_eq | Mathlib/Data/Nat/Prime/Basic.lean | theorem Prime.dvd_iff_eq {p a : ℕ} (hp : p.Prime) (a1 : a ≠ 1) : a ∣ p ↔ p = a | case intro
a : ℕ
a1 : a ≠ 1
j : ℕ
hp : Prime (a * j)
⊢ a * j = a | rcases prime_mul_iff.mp hp with (⟨_, rfl⟩ | ⟨_, rfl⟩) | case intro.inl.intro
a : ℕ
a1 : a ≠ 1
left✝ : Prime a
hp : Prime (a * 1)
⊢ a * 1 = a
case intro.inr.intro
j : ℕ
left✝ : Prime j
a1 : 1 ≠ 1
hp : Prime (1 * j)
⊢ 1 * j = 1 | 82e61d1e37cd62f6 |
mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors | Mathlib/RingTheory/ChainOfDivisors.lean | theorem mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors {m p : M} {n : N} (hm : m ≠ 0)
(hn : n ≠ 0) (hp : p ∈ normalizedFactors m) {d : { l : M // l ∣ m } ≃ { l : N // l ∣ n }}
(hd : ∀ l l', (d l : N) ∣ d l' ↔ (l : M) ∣ (l' : M)) :
↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩) ∈ normalizedFactors ... | M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n ... | rw [← Associates.prime_mk, this] | M : Type u_1
inst✝⁵ : CancelCommMonoidWithZero M
N : Type u_2
inst✝⁴ : CancelCommMonoidWithZero N
inst✝³ : Subsingleton Mˣ
inst✝² : Subsingleton Nˣ
inst✝¹ : UniqueFactorizationMonoid M
inst✝ : UniqueFactorizationMonoid N
m p : M
n : N
hm : m ≠ 0
hn : n ≠ 0
hp : p ∈ normalizedFactors m
d : { l // l ∣ m } ≃ { l // l ∣ n ... | f1fd55130dfcc672 |
Affine.Simplex.circumsphere_reindex | Mathlib/Geometry/Euclidean/Circumcenter.lean | theorem circumsphere_reindex {m n : ℕ} (s : Simplex ℝ P m) (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).circumsphere = s.circumsphere | case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
m n : ℕ
s : Simplex ℝ P m
e : Fin (m + 1) ≃ Fin (n + 1)
⊢ Set.range (s.reindex e).points ⊆ Metric.sphere (s.reindex e).circumsphere.center (s.reindex e).circumsphere.ra... | exact (s.reindex e).circumsphere_unique_dist_eq.1.2 | no goals | 4450e8b52c9c2999 |
Bimod.id_whiskerRight_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem id_whiskerRight_bimod {X Y Z : Mon_ C} {M : Bimod X Y} {N : Bimod Y Z} :
whiskerRight (𝟙 M) N = 𝟙 (M.tensorBimod N) | case h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N : Bimod Y Z
⊢ (whisker... | apply Limits.coequalizer.hom_ext | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N : Bimod Y Z
⊢ coequa... | 4fbec8c485b362b6 |
LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div
(hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l))
(hfS : ∀ s : ℝ, 1 < s → LSeriesSummable f s) :
Tendsto (fun s : ℝ ↦ (s - 1) * LSeries f s) (𝓝[>] 1) (𝓝 l) | f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
h₁ : ∀ {C ε : ℝ}, Tendsto (fun s => (s - 1) * s * C + s * ε) (𝓝[>] 1) (𝓝 ε)
⊢ IsBoundedUnder (fun x1 x2 => x1 ≤ x2) (𝓝[>] 1) fun s => ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖ | obtain ⟨C, _, hC₂⟩ :=
LSeries_tendsto_sub_mul_nhds_one_of_tendsto_sum_div_aux₃ hlim hfS zero_lt_one | case intro.intro
f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
hfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s
h₁ : ∀ {C ε : ℝ}, Tendsto (fun s => (s - 1) * s * C + s * ε) (𝓝[>] 1) (𝓝 ε)
C : ℝ
left✝ : C ≥ 0
hC₂ : (fun s => ‖(↑s - 1) * LSeries f ↑s - ↑s * l‖) ≤ᶠ[𝓝[>] 1] fun s => (s - ... | e99bfc7281febeca |
ValuationSubring.ofPrime_idealOfLE | Mathlib/RingTheory/Valuation/ValuationSubring.lean | theorem ofPrime_idealOfLE (R S : ValuationSubring K) (h : R ≤ S) :
ofPrime R (idealOfLE R S h) = S | case pos
K : Type u
inst✝ : Field K
R S : ValuationSubring K
h : R ≤ S
x : K
hx : x ∈ S
hr : x ∈ R
⊢ x ∈ R.ofPrime (R.idealOfLE S h) | exact R.le_ofPrime _ hr | no goals | cdd30d279244a00d |
Rat.add_den_dvd_lcm | Mathlib/Data/Rat/Lemmas.lean | theorem add_den_dvd_lcm (q₁ q₂ : ℚ) : (q₁ + q₂).den ∣ q₁.den.lcm q₂.den | q₁ q₂ : ℚ
⊢ (q₁ + q₂).den ∣ q₁.den.lcm q₂.den | rw [add_def, normalize_eq, Nat.div_dvd_iff_dvd_mul (Nat.gcd_dvd_right _ _)
(Nat.gcd_ne_zero_right (by simp)), ← Nat.gcd_mul_lcm,
mul_dvd_mul_iff_right (Nat.lcm_ne_zero (by simp) (by simp)), Nat.dvd_gcd_iff] | q₁ q₂ : ℚ
⊢ q₁.den.gcd q₂.den ∣ (q₁.num * ↑q₂.den + q₂.num * ↑q₁.den).natAbs ∧
q₁.den.gcd q₂.den ∣ q₁.den.gcd q₂.den * q₁.den.lcm q₂.den | aca82c817232fd95 |
WType.infinite_of_nonempty_of_isEmpty | Mathlib/Data/W/Basic.lean | theorem infinite_of_nonempty_of_isEmpty (a b : α) [ha : Nonempty (β a)] [he : IsEmpty (β b)] :
Infinite (WType β) :=
⟨by
intro hf
have hba : b ≠ a := fun h => ha.elim (IsEmpty.elim' (show IsEmpty (β a) from h ▸ he))
refine
not_injective_infinite_finite
(fun n : ℕ =>
show WType ... | α : Type u_1
β : α → Type u_2
a b : α
ha : Nonempty (β a)
he : IsEmpty (β b)
hf : Finite (WType β)
hba : b ≠ a
⊢ False | refine
not_injective_infinite_finite
(fun n : ℕ =>
show WType β from Nat.recOn n ⟨b, IsEmpty.elim' he⟩ fun _ ih => ⟨a, fun _ => ih⟩)
?_ | α : Type u_1
β : α → Type u_2
a b : α
ha : Nonempty (β a)
he : IsEmpty (β b)
hf : Finite (WType β)
hba : b ≠ a
⊢ Function.Injective fun n =>
let_fun this := Nat.recOn n (mk b he.elim') fun x ih => mk a fun x => ih;
this | 38f01095073e301a |
Polynomial.exists_eq_polynomial | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | theorem exists_eq_polynomial [Semiring Fq] {d : ℕ} {m : ℕ} (hm : Fintype.card Fq ^ d ≤ m)
(b : Fq[X]) (hb : natDegree b ≤ d) (A : Fin m.succ → Fq[X])
(hA : ∀ i, degree (A i) < degree b) : ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀ | Fq : Type u_1
inst✝¹ : Fintype Fq
inst✝ : Semiring Fq
d m : ℕ
hm : Fintype.card Fq ^ d ≤ m
b : Fq[X]
hb : b.natDegree ≤ d
A : Fin m.succ → Fq[X]
hA : ∀ (i : Fin m.succ), (A i).degree < b.degree
f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff ↑j
⊢ Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) | simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m) | no goals | 0d4d3843c54f2d08 |
Representation.smul_ofModule_asModule | Mathlib/RepresentationTheory/Basic.lean | theorem smul_ofModule_asModule (r : MonoidAlgebra k G) (m : (ofModule M).asModule) :
(RestrictScalars.addEquiv k _ _) ((ofModule M).asModuleEquiv (r • m)) =
r • (RestrictScalars.addEquiv k _ _) ((ofModule M).asModuleEquiv (G := G) m) | k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : Monoid G
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : Module (MonoidAlgebra k G) M
r : MonoidAlgebra k G
m : (ofModule M).asModule
⊢ (RestrictScalars.addEquiv k (MonoidAlgebra k G) M) ((ofModule M).asModuleEquiv (r • m)) =
r • (RestrictScalars.addEquiv k (... | dsimp | k : Type u_1
G : Type u_2
inst✝³ : CommSemiring k
inst✝² : Monoid G
M : Type u_4
inst✝¹ : AddCommMonoid M
inst✝ : Module (MonoidAlgebra k G) M
r : MonoidAlgebra k G
m : (ofModule M).asModule
⊢ (RestrictScalars.addEquiv k (MonoidAlgebra k G) M) (((ofModule M).asAlgebraHom r) ((ofModule M).asModuleEquiv m)) =
r • (Re... | fdc15e6ebd6d6319 |
Std.DHashMap.Internal.AssocList.get!_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/AssocList/Lemmas.lean | theorem get!_eq {β : Type v} [BEq α] [Inhabited β] {l : AssocList α (fun _ => β)} {a : α} :
l.get! a = getValue! a l.toList | α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : Inhabited β
l : AssocList α fun x => β
a : α
⊢ get! a l = getValue! a l.toList | induction l | case nil
α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : Inhabited β
a : α
⊢ get! a nil = getValue! a nil.toList
case cons
α : Type u
β : Type v
inst✝¹ : BEq α
inst✝ : Inhabited β
a key✝ : α
value✝ : β
tail✝ : AssocList α fun x => β
tail_ih✝ : get! a tail✝ = getValue! a tail✝.toList
⊢ get! a (cons key✝ value✝ tail✝) = ge... | 9d230ccc18a33628 |
RootPairing.RootPositiveForm.zero_lt_apply_root_root_iff | Mathlib/LinearAlgebra/RootSystem/RootPositive.lean | @[simp]
lemma zero_lt_apply_root_root_iff
(hi : P.root i ∈ span S (range P.root) := subset_span (mem_range_self i))
(hj : P.root j ∈ span S (range P.root) := subset_span (mem_range_self j)) :
0 < B.posForm ⟨P.root i, hi⟩ ⟨P.root j, hj⟩ ↔ 0 < P.pairingIn S i j | ι : Type u_1
R : Type u_2
S : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : LinearOrderedCommRing S
inst✝⁹ : CommRing R
inst✝⁸ : Algebra S R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
inst✝³ : P.IsValuedIn S
B : RootPositiveForm S P
i j : ι
inst✝² : Fa... | rw [this] | no goals | ea06a8689494296c |
ordinaryHypergeometric_radius_top_of_neg_nat₁ | Mathlib/Analysis/SpecialFunctions/OrdinaryHypergeometric.lean | theorem ordinaryHypergeometric_radius_top_of_neg_nat₁ {k : ℕ} :
(ordinaryHypergeometricSeries 𝔸 (-(k : 𝕂)) b c).radius = ⊤ | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
b c : 𝕂
k : ℕ
⊢ (ordinaryHypergeometricSeries 𝔸 (-↑k) b c).radius = ⊤ | refine FormalMultilinearSeries.radius_eq_top_of_forall_image_add_eq_zero _ (1 + k) fun n ↦ ?_ | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
b c : 𝕂
k n : ℕ
⊢ ordinaryHypergeometricSeries 𝔸 (-↑k) b c (n + (1 + k)) = 0 | 24537d6a825608ad |
contDiffAt_succ_iff_hasFDerivAt | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem contDiffAt_succ_iff_hasFDerivAt {n : ℕ} :
ContDiffAt 𝕜 (n + 1) f x ↔ ∃ f' : E → E →L[𝕜] F,
(∃ u ∈ 𝓝 x, ∀ x ∈ u, HasFDerivAt f (f' x) x) ∧ ContDiffAt 𝕜 n f' x | 𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
⊢ (∃ u ∈ 𝓝 x,
(↑n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffWithi... | constructor | case mp
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
n : ℕ
⊢ (∃ u ∈ 𝓝 x,
(↑n = ω → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContD... | 3893ab4aba3eed00 |
Int.mul_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem mul_add : ∀ a b c : Int, a * (b + c) = a * b + a * c
| (m:Nat), (n:Nat), (k:Nat) => by simp [Nat.left_distrib]
| (m:Nat), (n:Nat), -[k+1] => by
simp [negOfNat_eq_subNatNat_zero]; rw [← subNatNat_add]; rfl
| (m:Nat), -[n+1], (k:Nat) => by
simp [negOfNat_eq_subNatNat_zero]; rw [Int.add_comm, ← sub... | m n k : Nat
⊢ -[m+1] * (↑n + ↑k) = -[m+1] * ↑n + -[m+1] * ↑k | simp [Nat.mul_comm] | m n k : Nat
⊢ negOfNat ((n + k) * (m + 1)) = negOfNat (n * (m + 1) + k * (m + 1)) | 27d26162776a9945 |
MvPolynomial.eval₂_rename_prod_mk | Mathlib/Algebra/MvPolynomial/Rename.lean | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | case h_X
σ : Type u_1
τ : Type u_2
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
g : σ × τ → S
i : σ
p p✝ : MvPolynomial τ R
n✝ : τ
a✝ : eval₂ f g ((rename (Prod.mk i)) p✝) = eval₂ f (fun j => g (i, j)) p✝
⊢ eval₂ f g ((rename (Prod.mk i)) (p✝ * X n✝)) = eval₂ f (fun j => g (i, j)... | simp [*] | no goals | 0e3a0fdfd9f8acb2 |
continuousAt_const_cpow | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem continuousAt_const_cpow {a b : ℂ} (ha : a ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b | case h
a b✝ : ℂ
ha : a ≠ 0
b : ℂ
⊢ a ^ b = cexp (Complex.log a * b) | rw [cpow_def_of_ne_zero ha] | no goals | 149a41b9e818c468 |
ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunction_le_one | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma IsMeasurableRatCDF.stieltjesFunction_le_one (a : α) (x : ℝ) :
hf.stieltjesFunction a x ≤ 1 | case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ⨅ r, ↑(hf.stieltjesFunction a) ↑↑r ≤ 1 | simp_rw [IsMeasurableRatCDF.stieltjesFunction_eq] | case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ⨅ r, f a ↑r ≤ 1 | 969ac33d4dcd9898 |
EReal.preimage_coe_Ici | Mathlib/Data/Real/EReal.lean | @[simp]
lemma preimage_coe_Ici (x : ℝ) : Real.toEReal ⁻¹' Ici x = Ici x | x : ℝ
⊢ WithTop.some ⁻¹' (WithBot.some ⁻¹' Ici ↑↑x) = Ici x | simp only [WithBot.preimage_coe_Ici, WithTop.preimage_coe_Ici] | no goals | ee4b8f2d6e459258 |
Liouville.exists_one_le_pow_mul_dist | Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean | theorem exists_one_le_pow_mul_dist {Z N R : Type*} [PseudoMetricSpace R] {d : N → ℝ}
{j : Z → N → R} {f : R → R} {α : R} {ε M : ℝ}
-- denominators are positive
(d0 : ∀ a : N, 1 ≤ d a)
(e0 : 0 < ε)
-- function is Lipschitz at α
(B : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M)... | case neg
Z : Type u_1
N : Type u_2
R : Type u_3
inst✝ : PseudoMetricSpace R
d : N → ℝ
j : Z → N → R
f : R → R
α : R
ε M : ℝ
d0 : ∀ (a : N), 1 ≤ d a
e0 : 0 < ε
B : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M
L : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))
me0 : 0 < 1 ... | exact mul_le_mul_of_nonneg_left (le_max_right _ M) dist_nonneg | no goals | be11f7cf63f4a593 |
Finset.sup_mem_of_nonempty | Mathlib/Data/Finset/Lattice/Fold.lean | theorem sup_mem_of_nonempty (hs : s.Nonempty) : s.sup f ∈ f '' s | case insert.inr.inl
α : Type u_2
ι : Type u_5
inst✝¹ : LinearOrder α
inst✝ : OrderBot α
s✝ : Finset ι
f : ι → α
a : ι
s : Finset ι
a✝ : a ∉ s
h : s.Nonempty → s.sup f ∈ f '' ↑s
hs✝ : (insert a s).Nonempty
hs : s.Nonempty
ha : f a ≤ s.sup f
⊢ f a ⊔ s.sup f ∈ f '' insert a ↑s | rw [sup_eq_right.mpr ha] | case insert.inr.inl
α : Type u_2
ι : Type u_5
inst✝¹ : LinearOrder α
inst✝ : OrderBot α
s✝ : Finset ι
f : ι → α
a : ι
s : Finset ι
a✝ : a ∉ s
h : s.Nonempty → s.sup f ∈ f '' ↑s
hs✝ : (insert a s).Nonempty
hs : s.Nonempty
ha : f a ≤ s.sup f
⊢ s.sup f ∈ f '' insert a ↑s | b97132ba29f9d49c |
not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi | Mathlib/Analysis/Calculus/MeanValue.lean | theorem not_differentiableWithinAt_of_deriv_tendsto_atTop_Ioi (f : ℝ → ℝ) {a : ℝ}
(hf : Tendsto (deriv f) (𝓝[>] a) atTop) : ¬ DifferentiableWithinAt ℝ f (Ioi a) a | f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a) a + 1) ⊔ 0 < derivWithin f (Ioi a) x
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] a, slope f a ... | rw [(nhds_basis_Ioo _).tendsto_right_iff] at hdiff | f : ℝ → ℝ
a : ℝ
hf : Tendsto (derivWithin f (Ioi a)) (𝓝[>] a) atTop
hcont_at_a : ContinuousWithinAt f (Ici a) a
hdiff :
∀ (i : ℝ × ℝ),
i.1 < derivWithin f (Ioi a) a ∧ derivWithin f (Ioi a) a < i.2 → ∀ᶠ (x : ℝ) in 𝓝[>] a, slope f a x ∈ Ioo i.1 i.2
h₀ : ∀ᶠ (b : ℝ) in 𝓝[>] a, ∀ x ∈ Ioc a b, (derivWithin f (Ioi a)... | d40ee29628b6175f |
MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const | Mathlib/MeasureTheory/Measure/Restrict.lean | lemma NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
{β : Type*} [MeasurableSpace β] {b : β} {f : α → β} {s : Set α}
(f_mble : NullMeasurable f (μ.restrict s)) (hs : f =ᵐ[Measure.restrict μ sᶜ] (fun _ ↦ b))
{t : Set β} (t_mble : MeasurableSet t) (ht : b ∉ t) :
μ (f ⁻¹'... | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) = μ (f ⁻¹' t ∩ s) | apply le_antisymm _ (measure_mono inter_subset_left) | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : μ ({a | ¬f a = b} ∩ sᶜ) = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ μ (f ⁻¹' t) ≤ μ (f ⁻¹' t ∩ s) | 38ba84a38a03d7e4 |
IsPRadical.injective_comp_of_pNilradical_eq_bot | Mathlib/FieldTheory/IsPerfectClosure.lean | theorem injective_comp_of_pNilradical_eq_bot [IsPRadical i p] (h : pNilradical M p = ⊥) :
Function.Injective fun f : L →+* M ↦ f.comp i := fun f g heq ↦ by
ext x
obtain ⟨n, y, hx⟩ := IsPRadical.pow_mem i p x
apply_fun _ using pow_expChar_pow_inj_of_pNilradical_eq_bot M p h n
simpa only [← map_pow, ← hx] usi... | K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁴ : CommRing K
inst✝³ : CommRing L
inst✝² : CommRing M
i : K →+* L
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : IsPRadical i p
h : pNilradical M p = ⊥
f g : L →+* M
heq : (fun f => f.comp i) f = (fun f => f.comp i) g
⊢ f = g | ext x | case a
K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁴ : CommRing K
inst✝³ : CommRing L
inst✝² : CommRing M
i : K →+* L
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : IsPRadical i p
h : pNilradical M p = ⊥
f g : L →+* M
heq : (fun f => f.comp i) f = (fun f => f.comp i) g
x : L
⊢ f x = g x | 35d04cc8b6fcd746 |
Int.mem_box | Mathlib/Order/Interval/Finset/Box.lean | @[simp] lemma mem_box : ∀ {n}, x ∈ box n ↔ max x.1.natAbs x.2.natAbs = n
| 0 => by simp [Prod.ext_iff]
| n + 1 => by
simp [box_succ_eq_sdiff, Prod.le_def]
omega
| x : ℤ × ℤ
n : ℕ
⊢ x ∈ box (n + 1) ↔ x.1.natAbs ⊔ x.2.natAbs = n + 1 | simp [box_succ_eq_sdiff, Prod.le_def] | x : ℤ × ℤ
n : ℕ
⊢ ((-1 ≤ x.1 + ↑n ∧ -1 ≤ x.2 + ↑n) ∧ x.1 ≤ ↑n + 1 ∧ x.2 ≤ ↑n + 1) ∧ (-↑n ≤ x.1 → -↑n ≤ x.2 → x.1 ≤ ↑n → ↑n < x.2) ↔
x.1.natAbs ⊔ x.2.natAbs = n + 1 | 0dc2795aae101904 |
preNormEDS_even | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma preNormEDS_even (m : ℤ) : preNormEDS b c d (2 * m) =
preNormEDS b c d (m - 1) ^ 2 * preNormEDS b c d m * preNormEDS b c d (m + 2) -
preNormEDS b c d (m - 2) * preNormEDS b c d m * preNormEDS b c d (m + 1) ^ 2 | case neg
R : Type u
inst✝ : CommRing R
b c d : R
h :
∀ (n : ℕ),
preNormEDS b c d (2 * ↑n) =
preNormEDS b c d (↑n - 1) ^ 2 * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 2) -
preNormEDS b c d (↑n - 2) * preNormEDS b c d ↑n * preNormEDS b c d (↑n + 1) ^ 2
m : ℕ
⊢ -(preNormEDS b c d (↑m - 1) ^ 2 * preN... | ring1 | no goals | 04b24e60b18bcf70 |
Finset.prod_prod_bipartiteAbove_eq_prod_prod_bipartiteBelow | Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | theorem prod_prod_bipartiteAbove_eq_prod_prod_bipartiteBelow
[CommMonoid R] (f : α → β → R) [∀ a b, Decidable (r a b)] :
∏ a ∈ s, ∏ b ∈ t.bipartiteAbove r a, f a b = ∏ b ∈ t, ∏ a ∈ s.bipartiteBelow r b, f a b | R : Type u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
inst✝¹ : CommMonoid R
f : α → β → R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
⊢ ∏ a ∈ s, ∏ b ∈ bipartiteAbove r t a, f a b = ∏ b ∈ t, ∏ a ∈ bipartiteBelow r s b, f a b | simp_rw [bipartiteAbove, bipartiteBelow, prod_filter] | R : Type u_1
α : Type u_2
β : Type u_3
r : α → β → Prop
s : Finset α
t : Finset β
inst✝¹ : CommMonoid R
f : α → β → R
inst✝ : (a : α) → (b : β) → Decidable (r a b)
⊢ (∏ x ∈ s, ∏ a ∈ t, if r x a then f x a else 1) = ∏ x ∈ t, ∏ a ∈ s, if r a x then f a x else 1 | a7996e2b74256135 |
contDiffWithinAt_localInvariantProp_of_le | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | theorem contDiffWithinAt_localInvariantProp_of_le (n m : WithTop ℕ∞) (hmn : m ≤ n) :
(contDiffGroupoid n I).LocalInvariantProp (contDiffGroupoid n I')
(ContDiffWithinAtProp I I' m) where
is_local {s x u f} u_open xu | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | have : I x = (I ∘ e.symm ∘ I.symm) (I (e x)) := by simp only [hx, mfld_simps] | 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
H : Type u_3
inst✝³ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝ : TopologicalSpace H'
I' : ModelWithCorn... | 841928255da4a913 |
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