mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
G : C
inst✝⁵ : Abelian C
hG : IsSeparator G
X : C
inst✝⁴ : IsGrothendieckAbelian.{w, v, u} C
A₀ : Subobject X
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hJ : HasCardinalLT (Subobject X) (Cardinal.mk J)
⊢ largerSubobject hG ⊤ = ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
⊢ g ≫ πObj I κ f = I.homFamily ⟨Arrow.mk i, hi⟩ ≫ b

Tactic:
sq.w

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y A B : C
i : A ⟶ B
hi : I i
f : X ⟶ Y
this✝¹ : ∀ (X Y : C) (p : X ⟶ Y), HasColimitsOfShape (Discrete (FunctorObjIndex I.homFamily p)) C
this✝ : HasPushouts C
this : PreservesColimit (relativeCellComplexιObj I κ f).F (coyoneda.obj (Opposite.op A))
g : A ⟶ obj I κ f
b : B ⟶ Y
sq : CommSq g i (πObj I κ f) b
j : κ.ord.ToType
t : A ⟶ (relativeCellComplexιObj I κ f).F.obj j
ht : t ≫ (relativeCellComplexιObj I κ f).incl.app j = g
⊢ i ≫ b = I.homFamily ⟨Arrow.mk i, hi⟩ ≫ b
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case inl.zero
p : ℕ
hp : Fact (Nat.Prime p)
⊢ padicValRat p (0 ^ 0) = ↑0 * padicValRat p 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case inl.succ
p : ℕ
hp : Fact (Nat.Prime p)
n✝ : ℕ
⊢ padicValRat p (0 ^ (n✝ + 1)) = ↑(n✝ + 1) * padicValRat p 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case nat
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
n✝ : ℕ
⊢ padicValRat p (q ^ ↑n✝) = ↑n✝ * padicValRat p q

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case neg
p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
a✝ : ∀ (n : ℕ), padicValRat p (q ^ ↑n) = ↑n * padicValRat p q
n✝ : ℕ
⊢ padicValRat p (q ^ (-↑n✝)) = -↑n✝ * padicValRat p q

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
⊢ I.rlp.llp f → ((coproducts.{w, v, u} I).pushouts.transfiniteCompositionsOfShape κ.ord.ToType).retracts f

Tactic:
hf

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
hf : I.rlp.llp f
⊢ ((coproducts.{w, v, u} I).pushouts.transfiniteCompositionsOfShape κ.ord.ToType).retracts f
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
hf : I.rlp.llp f
⊢ ιObj I κ f ≫ πObj I κ f = f ≫ 𝟙 Y

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
G : C
inst✝¹ : Abelian C
inst✝ : IsGrothendieckAbelian.{w, v, u} C
hG : IsSeparator G
⊢ (transfiniteCompositions.{w, v, u} (coproducts.{w, v, u} (monomorphisms C)).pushouts).retracts ≤ monomorphisms C

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q : ℚ
h : padicNorm p q = 0
⊢ q = 0

Tactic:
by
  apply by_contradiction; intro hq
  unfold padicNorm at h; rw [if_neg hq] at h
  apply absurd h
  apply zpow_ne_zero
  exact mod_cast hp.1.ne_zero

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: case inl
p : ℕ
hp : Fact (Nat.Prime p)
⊢ padicNorm p ↑0 ≤ 1

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNorm

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z : ℤ
hz : z ≠ 0
⊢ -padicValRat p ↑z ≤ 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q ≠ padicValRat p r
h1 : min (padicValRat p q) (padicValRat p r) ≤ padicValRat p (q + r)
h2 : min (padicValRat p (q + r)) (padicValRat p (-r)) ≤ padicValRat p (q + r + -r)
h3 : min (padicValRat p (r + q)) (padicValRat p (-q)) ≤ padicValRat p (r + q + -q)
⊢ padicValRat p (q + r) = min (padicValRat p q) (padicValRat p r)

Tactic:
rw [add_neg_cancel_right, padicValRat.neg] at h2 h3

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
q r : ℚ
hqr : q + r ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hval : padicValRat p q ≠ padicValRat p r
h1 : min (padicValRat p q) (padicValRat p r) ≤ padicValRat p (q + r)
h2 : min (padicValRat p (q + r)) (padicValRat p r) ≤ padicValRat p q
h3 : min (padicValRat p (r + q)) (padicValRat p q) ≤ padicValRat p r
⊢ padicValRat p (q + r) = min (padicValRat p q) (padicValRat p r)
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p j : ℕ
hp : Fact (Nat.Prime p)
F : ℕ → ℚ
S : Finset ℕ
hn1 : ∀ (i : ℕ), 0 < F i
k : ℕ
hF : ∀ i ∈ {k}, padicValRat p (F j) < padicValRat p (F i)
⊢ k ∈ {k}

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ 1 ∈ {x | ‖x‖ ≤ 1}

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x y : ℤ_[p]
⊢ 0 ∈ {x | ‖x‖ ≤ 1}

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : ℤ_[p]
⊢ ⟨↑k, ⋯⟩ = k

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℤ
⊢ ↑z1 = ↑z2 ↔ z1 = z2

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case inl
p b : ℕ
hp : Fact (Nat.Prime p)
h : b ∣ 0
⊢ padicValNat p (0 / b) = padicValNat p 0 - padicValNat p b

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ ‖↑p ^ n‖ = ↑p ^ (-↑n)

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case zero
p : ℕ
⊢ padicValNat p 0 ≤ log p 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case succ.zero
n : ℕ
⊢ padicValNat 0 (n + 1) ≤ log 0 (n + 1)

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case succ.succ.zero
n : ℕ
⊢ padicValNat (0 + 1) (n + 1) ≤ log (0 + 1) (n + 1)

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
ε : ℚ
hε : 0 < ε
k : ℕ
hk : ↑p ^ (-↑k) < ↑ε
⊢ ↑p = ↑↑p

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: case inl
p : ℕ
hp : Fact (Nat.Prime p)
⊢ 0 ≤ (↑0).valuation

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: case inl
p : ℕ
hp : Fact (Nat.Prime p)
⊢ Finset.image (fun t ↦ p ^ (t + 1)) (Finset.range (padicValNat p 0)) ⊆ (divisors 0).erase 1

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicVal.Basic

Goal state before:
pp: p n : ℕ
hp : Fact (Nat.Prime p)
⊢ emultiplicity p (p * n)! = emultiplicity p n ! + ↑n

Tactic:
exact Prime.emultiplicity_factorial_mul hp.out

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
z1 z2 : ℤ_[p]
hz2 : ‖z2‖ < 1
⊢ ‖z1 * z2‖ = ‖z1‖ * ‖z2‖

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
u : ℤ_[p]ˣ
⊢ IsUnit ↑u

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
⊢ ↑x = ↑x * ↑p ^ (-↑x.valuation + ↑x.valuation)

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ (if 0 = 0 then 0 else exp (-padicValRat p 0)) = 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
⊢ (if 1 = 0 then 0 else exp (-padicValRat p 1)) = 1

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
x : ℚ
⊢ (padicValuation p) x = 0 ↔ x = 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case pos
p : ℕ
inst✝ : Fact (Nat.Prime p)
x : ℤ
h✝ : x = 0
⊢ 0 ≤ 1

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case zero
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ∃ a, (padicValuation p) a = 0

Tactic:
simp

Goal state after:
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: case mp
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
h : padicNorm p (↑f (stationaryPoint hf)) = 0
ε : ℚ
hε : ε > 0
⊢ ∀ j ≥ stationaryPoint hf, padicNorm p (↑(f - 0) j) < ε

Tactic:
intro j

Goal state after:
case mp
p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
h : padicNorm p (↑f (stationaryPoint hf)) = 0
ε : ℚ
hε : ε > 0
j : ℕ
⊢ j ≥ stationaryPoint hf → padicNorm p (↑(f - 0) j) < ε
Module: Mathlib.NumberTheory.Padics.PadicIntegers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x✝ y : ℤ_[p]
x : ℚ_[p]
hx✝ : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : ↑n = -x.valuation
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
hx : x ≠ 0
⊢ ‖x‖ * ↑p ^ (-↑n) = 1

Tactic:
Padic.norm_eq_zpow_neg_valuation hx,

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
x✝ y : ℤ_[p]
x : ℚ_[p]
hx✝ : ¬‖x‖ ≤ 1
n : ℕ := (-x.valuation).toNat
hn : n = (-x.valuation).toNat
hn_coe : ↑n = -x.valuation
a : ℚ_[p] := x * ↑p ^ n
ha : a = x * ↑p ^ n
hx : x ≠ 0
⊢ ↑p ^ (-x.valuation) * ↑p ^ (-↑n) = 1
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
inst✝ : Fact (Nat.Prime p)
f : PadicSeq p
hf : ¬f ≈ 0
v1 v2 : ℕ
⊢ padicNorm p (↑f (stationaryPoint hf)) = padicNorm p (↑f (max v1 (max v2 (stationaryPoint hf))))

Tactic:
by
  apply stationaryPoint_spec hf
  · apply le_trans
    · apply le_max_right v2
    · apply le_max_right
  · exact le_rfl

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
hf : c.fst = c.snd
Z : C
φ : Z ⟶ c.pt
h : (φ ≫ c.fst) ≫ f = (φ ≫ c.snd) ≫ f
⊢ φ ≫ c.fst = φ ≫ c.snd

Tactic:
hf

Goal state after:
case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
hf : c.fst = c.snd
Z : C
φ : Z ⟶ c.pt
h : (φ ≫ c.fst) ≫ f = (φ ≫ c.snd) ≫ f
⊢ φ ≫ c.snd = φ ≫ c.snd
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
h : c.fst = c.snd
⊢ 𝟙 X ≫ f = 𝟙 X ≫ f

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mp.refine_1
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
h : c.fst = c.snd
φ : X ⟶ c.pt
hφ₁ : φ ≫ c.fst = 𝟙 X
hφ₂ : φ ≫ c.snd = 𝟙 X
⊢ (c.fst ≫ φ) ≫ c.fst = 𝟙 c.pt ≫ c.fst

Tactic:
simp only [assoc, hφ₁, id_comp, comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PullbackCone f f
hc : IsLimit c
a✝ : IsIso c.fst
⊢ 𝟙 X ≫ f = 𝟙 X ≫ f

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
Z : C
φ : c.pt ⟶ Z
h : f ≫ c.inl ≫ φ = f ≫ c.inr ≫ φ
⊢ c.inl ≫ φ = c.inr ≫ φ

Tactic:
hf

Goal state after:
case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
hf : c.inl = c.inr
Z : C
φ : c.pt ⟶ Z
h : f ≫ c.inl ≫ φ = f ≫ c.inr ≫ φ
⊢ c.inr ≫ φ = c.inr ≫ φ
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
h : c.inl = c.inr
⊢ f ≫ 𝟙 Y = f ≫ 𝟙 Y

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.EpiMono

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X Y : C
f : X ⟶ Y
c : PushoutCocone f f
hc : IsColimit c
a✝ : IsIso c.inl
⊢ f ≫ 𝟙 Y = f ≫ 𝟙 Y

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
P Q : Cᵒᵖ ⥤ A
M N : A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
g : P ⟶ Φ.skyscraperPresheaf M
⊢ ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (x : Φ.fiber.obj X),
    P.map f.op ≫ g.app (op X) ≫ Pi.π (fun t ↦ M) x =
      g.app (op Y) ≫ Pi.π (fun t ↦ M) ((ConcreteCategory.hom (Φ.fiber.map f)) x)

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
ι : Type u_1
inst✝ : Small.{w, u_1} ι
U : ι → C
f : (i : ι) → U i ⟶ X
hf : Sieve.ofArrows U f ∈ J X
Φ : P.FullSubcategory
Z Y : C
i : ι
p : Z ⟶ U i
z : Φ.obj.fiber.obj Z
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) ((ConcreteCategory.hom (Φ.obj.fiber.map p)) z) =
    (ConcreteCategory.hom (Φ.obj.fiber.map (p ≫ f i))) z

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
P : Cᵒᵖ ⥤ A
M : A
f : Φ.presheafFiber.obj P ⟶ M
⊢ (Φ.skyscraperPresheafAdjunction.homEquiv P M) f = Φ.skyscraperPresheafHomEquiv f

Tactic:
by
  simp [skyscraperPresheafAdjunction]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
ι : Type u_1
inst✝ : Small.{w, u_1} ι
U : ι → C
f : (i : ι) → U i ⟶ X
hf : Sieve.ofArrows U f ∈ J X
Φ : P.FullSubcategory
x : Φ.obj.fiber.obj X
⊢ ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x

pp: case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
ι : Type u_1
inst✝ : Small.{w, u_1} ι
U : ι → C
f : (i : ι) → U i ⟶ X
hf : ∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X), ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x
⊢ Sieve.ofArrows U f ∈ J X

Tactic:
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
    exact ⟨i, Φ.obj.fiber.map p z, by simp⟩

Goal state after:
case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
X : C
ι : Type u_1
inst✝ : Small.{w, u_1} ι
U : ι → C
f : (i : ι) → U i ⟶ X
hf : ∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X), ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x
⊢ Sieve.ofArrows U f ∈ J X
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
P : Cᵒᵖ ⥤ A
M : A
f : P ⟶ Φ.skyscraperPresheaf M
⊢ (Φ.skyscraperPresheafAdjunction.homEquiv P M).symm f = Φ.skyscraperPresheafHomEquiv.symm f

Tactic:
by
  simp [skyscraperPresheafAdjunction]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Skyscraper

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ : Cᵒᵖ ⥤ A
f : P₁ ⟶ P₂
⊢ J.W f → IsIso (Φ.presheafFiber.map f)

Tactic:
hf

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : HasProducts A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
P₁ P₂ : Cᵒᵖ ⥤ A
f : P₁ ⟶ P₂
hf : J.W f
⊢ IsIso (Φ.presheafFiber.map f)
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
inst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf : Sieve.ofArrows U f ∈ J X
Φ : P.FullSubcategory
Z Y : C
i : ι
p : Z ⟶ U i
z : Φ.obj.fiber.obj Z
⊢ (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) ((ConcreteCategory.hom (Φ.obj.fiber.map p)) z) =
    (ConcreteCategory.hom (Φ.obj.fiber.map (p ≫ f i))) z

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Conservative

Goal state before:
pp: case refine_1
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
inst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf : Sieve.ofArrows U f ∈ J X
Φ : P.FullSubcategory
x : Φ.obj.fiber.obj X
⊢ ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x

pp: case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
inst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf : ∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X), ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x
⊢ Sieve.ofArrows U f ∈ J X

Tactic:
· obtain ⟨Z, _, ⟨_, p, _, ⟨i⟩, rfl⟩, z, rfl⟩ := Φ.obj.jointly_surjective _ hf x
    exact ⟨i, Φ.obj.fiber.map p z, by simp⟩

Goal state after:
case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
P : ObjectProperty J.Point
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : HasSheafify J (Type w)
inst✝¹ : J.WEqualsLocallyBijective (Type w)
hP : P.IsConservativeFamilyOfPoints
inst✝ : ObjectProperty.Small.{w, max u w, max (max u v) (w + 1)} P
X : C
ι : Type u_1
U : ι → C
f : (i : ι) → U i ⟶ X
hf : ∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X), ∃ i y, (ConcreteCategory.hom (Φ.obj.fiber.map (f i))) y = x
⊢ Sieve.ofArrows U f ∈ J X
Module: Mathlib.NumberTheory.Padics.PadicNumbers

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
a : PadicSeq p
⊢ ∀ (k : ℕ), padicNorm p (↑(-a) k) = padicNorm p (↑a k)

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: case refine_2
X S : Scheme
f : X ⟶ S
inst✝² : Etale f
Ω : Type u
inst✝¹ : Field Ω
inst✝ : IsSepClosed Ω
x : ↥X
a : S.residueField (f x) ⟶ CommRingCat.of Ω
hx : f x = ((SpecToEquivOfField Ω S).symm ⟨f x, a⟩) default
m : ↑(S.residueField (f x)) →+* ↑(X.residueField x) := CommRingCat.Hom.hom (Hom.residueFieldMap f x)
algInst✝¹ : Algebra ↑(S.residueField (f x)) ↑(X.residueField x) := m.toAlgebra
algInst✝ : Algebra ↑(S.residueField (f x)) ↑(CommRingCat.of Ω) := (CommRingCat.Hom.hom a).toAlgebra
b : ↑(X.residueField x) →ₐ[↑(S.residueField (f x))] Ω := IsSepClosed.lift
this : Hom.residueFieldMap f x ≫ CommRingCat.ofHom b.toRingHom = a
⊢ (Spec.map (CommRingCat.ofHom b.toRingHom) ≫ X.fromSpecResidueField x) default = x

Tactic:
· dsimp
    apply fromSpecResidueField_apply

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: S : Scheme
Ω : Type u
inst✝¹ : Field Ω
inst✝ : IsSepClosed Ω
s : Spec (CommRingCat.of Ω) ⟶ S
Y : Scheme
f : Y ⟶ S
x✝ : Etale f
y : Spec (CommRingCat.of Ω) ⟶ Y
hy : y ≫ f = s
⊢ y default ∈ ⊤

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.EtalePoint

Goal state before:
pp: S : Scheme
Ω : Type u
inst✝¹ : Field Ω
inst✝ : IsSepClosed Ω
s : Spec (CommRingCat.of Ω) ⟶ S
Y : Scheme
f : Y ⟶ S
x✝ : Etale f
R : CommRingCat
j : Spec (CommRingCat.of ↑R) ⟶ Y
w✝ : IsOpenImmersion j
y' : Spec (CommRingCat.of Ω) ⟶ Spec (CommRingCat.of ↑R)
hy : (y' ≫ j) ≫ f = s
⊢ ⊤ j

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k✝ i j k : ι
⊢ yoneda.map (pullback₃.p₂ ⋯ (f j) (f k)) ≫ f j = yoneda.map (pullback₃.p₃ ⋯ (f j) (f k)) ≫ f k

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k✝ i j k : ι
⊢ yoneda.map (pullback₃.p₂ ⋯ (f j) (f k)) ≫ f j = yoneda.map (pullback₃.p₁ ⋯ (f j) (f k)) ≫ f i

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k✝ i j k : ι
⊢ (⋯.lift₃ (f k) (f i) (pullback₃.p₂ ⋯ (f j) (f k)) (pullback₃.p₃ ⋯ (f j) (f k)) (pullback₃.p₁ ⋯ (f j) (f k)) ⋯ ⋯ ≫
        pullback.snd (⋯.fst' (f k)) (⋯.fst' (f i))) ≫
      ⋯.fst' (f i) =
    (pullback.fst (⋯.fst' (f j)) (⋯.fst' (f k)) ≫ ⋯.symmetry ⋯) ≫ ⋯.fst' (f i)

Tactic:
simp

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.Representability

Goal state before:
pp: F : Sheaf zariskiTopology (Type u)
ι : Type u
X : ι → Scheme
f : (i : ι) → yoneda.obj (X i) ⟶ F.obj
hf : ∀ (i : ι), IsOpenImmersion.presheaf (f i)
i✝ j✝ k✝ i j k : ι
⊢ (⋯.lift₃ (f k) (f i) (pullback₃.p₂ ⋯ (f j) (f k)) (pullback₃.p₃ ⋯ (f j) (f k)) (pullback₃.p₁ ⋯ (f j) (f k)) ⋯ ⋯ ≫
        pullback.snd (⋯.fst' (f k)) (⋯.fst' (f i))) ≫
      ⋯.snd (f i) =
    (pullback.fst (⋯.fst' (f j)) (⋯.fst' (f k)) ≫ ⋯.symmetry ⋯) ≫ ⋯.snd (f i)

Tactic:
simp

Goal state after: