mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
⊢ ∑ i, (e₂.val i).degree.succ < ∑ i, (e.val i).degree.succ

Tactic:
refine Fintype.sum_strictMono <| Pi.lt_def.2 ⟨fun j ↦ ?_, i, ?_⟩

Goal state after:
case refine_1
R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
j : Fin n
⊢ (fun i ↦ (e₂.val i).degree.succ) j ≤ (fun i ↦ (e.val i).degree.succ) j

case refine_2
R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
⊢ (e₂.val i).degree.succ < (e.val i).degree.succ
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
⊢ e.val i ≠ 0

Tactic:
simpa using

Goal state after:
R₀ : Type u_1
inst✝² : CommRing R₀
n : ℕ
R : Type u_6
inst✝¹ : CommRing R
inst✝ : Algebra R₀ R
c : R
i : Fin n
e : InductionObj R n
hi : c = (e.val i).leadingCoeff
hc : c ≠ 0
q₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)
q₂ : R →ₐ[R₀] R ⧸ Ideal.span {c} := Ideal.Quotient.mkₐ R₀ (Ideal.span {c})
q₂_surjective : Surjective ⇑q₂
e₁ : InductionObj (Localization.Away c) n :=
  { val := Polynomial.C (IsLocalization.Away.invSelf c) • ⇑(mapRingHom ↑q₁) ∘ e.val }
e₂ : InductionObj (R ⧸ Ideal.span {c}) n := { val := ⇑(mapRingHom ↑q₂) ∘ e.val }
degBound_e₁_le : e₁.degBound ≤ e.degBound
⊢ ¬e.val i = 0
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case pos
X : Type u_1
inst✝ : TopologicalSpace X
U : Set X
ι : Type u_3
F : ι → locallyFinsuppWithin U ℤ
this : Function.support F = Function.support fun i ↦ ⇑(F i)
h : (Function.support F).Finite
⊢ ∑ n ∈ h.toFinset, ⇑(F n) = ∑ᶠ (i : ι), ⇑(F i)

Tactic:
have h₂ : (fun i ↦ (F i : X → ℤ)).support.Finite := by simp_all

Goal state after:
case pos
X : Type u_1
inst✝ : TopologicalSpace X
U : Set X
ι : Type u_3
F : ι → locallyFinsuppWithin U ℤ
this : Function.support F = Function.support fun i ↦ ⇑(F i)
h : (Function.support F).Finite
h₂ : (Function.support fun i ↦ ⇑(F i)).Finite
⊢ ∑ n ∈ h.toFinset, ⇑(F n) = ∑ᶠ (i : ι), ⇑(F i)
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U : Set X
ι : Type u_3
F : ι → locallyFinsuppWithin U ℤ
⊢ ⇑(∑ᶠ (i : ι), F i) = ∑ᶠ (i : ι), ⇑(F i)

Tactic:
have : F.support = (fun i ↦ (F i : X → ℤ)).support := by
    simp [Set.ext_iff, DFunLike.ext_iff, funext_iff]
  by_cases h : F.support.Finite
  · rw [finsum_eq_sum F h, Function.locallyFinsuppWithin.coe_sum]
    have h₂ : (fun i ↦ (F i : X → ℤ)).support.Finite := by simp_all
    simp_all [finsum_eq_sum _ h₂]
  · simp_all [finsum_of_infinite_support]

Goal state after:
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case pos
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : f x < 0
⊢ max (n • f x) 0 = n • max (f x) 0

Tactic:
simpa using

Goal state after:
case pos
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : f x < 0
⊢ n • f x ≤ 0
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case neg
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : ¬f x < 0
⊢ max (n • f x) 0 = n • max (f x) 0

Tactic:
simpa using

Goal state after:
case neg
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : ¬f x < 0
⊢ 0 ≤ n • f x
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case pos
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : -f x < 0
⊢ max (-(n • f x)) 0 = n • max (-f x) 0

Tactic:
simpa using

Goal state after:
case pos
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : -f x < 0
⊢ 0 ≤ n • f x
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: case neg
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : ¬-f x < 0
⊢ max (-(n • f x)) 0 = n • max (-f x) 0

Tactic:
simpa using

Goal state after:
case neg
X : Type u_1
inst✝³ : TopologicalSpace X
U : Set X
Y : Type u_2
inst✝² : AddCommGroup Y
inst✝¹ : LinearOrder Y
inst✝ : IsOrderedAddMonoid Y
n : ℕ
f : locallyFinsuppWithin U Y
x : X
h : ¬-f x < 0
⊢ n • f x ≤ 0
Module: Mathlib.Topology.LocallyFinsupp

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : DecidableEq X
D : locallyFinsupp X ℤ
h : (support D).Finite
w : X
s : Finset X := h.toFinset
hs : s = h.toFinset
hw : w ∉ s
⊢ w ∉ support D

Tactic:
simpa using

Goal state after:
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : DecidableEq X
D : locallyFinsupp X ℤ
h : (support D).Finite
w : X
s : Finset X := h.toFinset
hs : s = h.toFinset
hw : w ∉ s
⊢ w ∉ support D
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: X Y S : Scheme
f✝ : X ⟶ Y
sX : X ⟶ S
sY : Y ⟶ S
R✝ A : CommRingCat
inst✝ : IsLocallyNoetherian X
R : CommRingCat
hR : IsNoetherianRing ↑R
I : Ideal ↑R
hI : I.IsPrime
J : Ideal ↑R := RingHom.ker (algebraMap (↑R) (Localization.AtPrime I))
hJ : ∀ (x : ↑R), x ∈ J ↔ ∃ y, ↑y * x = 0
f : (x : ↑R) → x ∈ J → ↥I.primeCompl
hf : ∀ (x : ↑R) (a : x ∈ J), ↑(f x a) * x = 0
s : Finset ↑R
hs : Ideal.span ↑s = J
hs' : ↑s ⊆ ↑J
x y : ↑R
e : y * x = 0
hy : y ∉ I
⊢ ↑(∏ x ∈ s.attach, f ↑x ⋯) ^ 1 * x = 0

Tactic:
rw [pow_one, mul_comm, ← smul_eq_mul, ← Submodule.mem_annihilator_span_singleton]

Goal state after:
X Y S : Scheme
f✝ : X ⟶ Y
sX : X ⟶ S
sY : Y ⟶ S
R✝ A : CommRingCat
inst✝ : IsLocallyNoetherian X
R : CommRingCat
hR : IsNoetherianRing ↑R
I : Ideal ↑R
hI : I.IsPrime
J : Ideal ↑R := RingHom.ker (algebraMap (↑R) (Localization.AtPrime I))
hJ : ∀ (x : ↑R), x ∈ J ↔ ∃ y, ↑y * x = 0
f : (x : ↑R) → x ∈ J → ↥I.primeCompl
hf : ∀ (x : ↑R) (a : x ∈ J), ↑(f x a) * x = 0
s : Finset ↑R
hs : Ideal.span ↑s = J
hs' : ↑s ⊆ ↑J
x y : ↑R
e : y * x = 0
hy : y ∉ I
⊢ x ∈ (↑R ∙ ↑(∏ x ∈ s.attach, f ↑x ⋯)).annihilator
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: case e
X Y : Scheme
x : ↥X
inst✝ : X.IsGermInjectiveAt x
f g : X ⟶ Y
e : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g
⊢ f x = g x

Tactic:
simpa using

Goal state after:
case e
X Y : Scheme
x : ↥X
inst✝ : X.IsGermInjectiveAt x
f g : X ⟶ Y
e : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g
⊢ f x = g x
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: X : Scheme
R A : CommRingCat
U : X.Opens
x : ↥X
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
this : Algebra ↑R ↑A := (CommRingCat.Hom.hom φRA).toAlgebra
s : Finset ↑A
hs : Algebra.adjoin ↑R ↑s = ⊤
W : ↑A → TopologicalSpace.Opens ↥X
hxW : ∀ (t : ↑A), x ∈ W t
f : (t : ↑A) → ToType (X.presheaf.obj (Opposite.op (W t)))
hf : ∀ (t : ↑A), (ConcreteCategory.hom (X.presheaf.germ (W t) x ⋯)) (f t) = (ConcreteCategory.hom φ) t
⊢ x ∈ s.inf (SetLike.coe ∘ W)

Tactic:
simpa using

Goal state after:
X : Scheme
R A : CommRingCat
U : X.Opens
x : ↥X
hxU : x ∈ U
φ : A ⟶ X.presheaf.stalk x
φRA : R ⟶ A
φRX : R ⟶ Γ(X, U)
e : φRA ≫ φ = φRX ≫ X.presheaf.germ U x hxU
this : Algebra ↑R ↑A := (CommRingCat.Hom.hom φRA).toAlgebra
s : Finset ↑A
hs : Algebra.adjoin ↑R ↑s = ⊤
W : ↑A → TopologicalSpace.Opens ↥X
hxW : ∀ (t : ↑A), x ∈ W t
f : (t : ↑A) → ToType (X.presheaf.obj (Opposite.op (W t)))
hf : ∀ (t : ↑A), (ConcreteCategory.hom (X.presheaf.germ (W t) x ⋯)) (f t) = (ConcreteCategory.hom φ) t
⊢ ∀ i ∈ s, x ∈ W i
Module: Mathlib.AlgebraicGeometry.SpreadingOut

Goal state before:
pp: case refine_3
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
φ : Spec (X.presheaf.stalk x) ⟶ Y
h : φ ≫ sY = X.fromSpecStalk x ≫ sX
this :
  ∃ U,
    ∃ (hxU : x ∈ U),
      ∃ f,
        Spec.map (Scheme.stalkClosedPointTo φ) ≫ Y.fromSpecStalk (φ (IsLocalRing.closedPoint ↑(X.presheaf.stalk x))) =
            U.fromSpecStalkOfMem x hxU ≫ f ∧
          f ≫ sY = U.ι ≫ sX
⊢ ∃ U, ∃ (hxU : x ∈ U), ∃ f, φ = U.fromSpecStalkOfMem x hxU ≫ f ∧ f ≫ sY = U.ι ≫ sX

Tactic:
simpa using

Goal state after:
case refine_3
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
inst✝¹ : LocallyOfFiniteType sY
x : ↥X
inst✝ : X.IsGermInjectiveAt x
φ : Spec (X.presheaf.stalk x) ⟶ Y
h : φ ≫ sY = X.fromSpecStalk x ≫ sX
this :
  ∃ U,
    ∃ (hxU : x ∈ U),
      ∃ f,
        Spec.map (Scheme.stalkClosedPointTo φ) ≫ Y.fromSpecStalk (φ (IsLocalRing.closedPoint ↑(X.presheaf.stalk x))) =
            U.fromSpecStalkOfMem x hxU ≫ f ∧
          f ≫ sY = U.ι ≫ sX
⊢ ∃ U, ∃ (hxU : x ∈ U), ∃ f, φ = U.fromSpecStalkOfMem x hxU ≫ f ∧ f ≫ sY = U.ι ≫ sX
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: X : Scheme
inst✝ : IrreducibleSpace ↥X
U : X.Opens
h : Nonempty ↥↑U
⊢ (Set.univ ∩ ↑U).Nonempty

Tactic:
simpa using

Goal state after:
X : Scheme
inst✝ : IrreducibleSpace ↥X
U : X.Opens
h : Nonempty ↥↑U
⊢ (↑U).Nonempty
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: X✝ : Scheme
X : Scheme
inst✝ : IsIntegral X
U : X.Opens
hU : IsAffineOpen U
h : Nonempty ↥↑U
⊢ (Set.univ ∩ ↑U).Nonempty

Tactic:
simpa using

Goal state after:
X✝ : Scheme
X : Scheme
inst✝ : IsIntegral X
U : X.Opens
hU : IsAffineOpen U
h : Nonempty ↥↑U
⊢ (↑U).Nonempty
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: X : Scheme
inst✝¹ : IsIntegral X
U : X.Opens
hU : IsAffineOpen U
inst✝ : Nonempty ↥↑U
⊢ (Set.univ ∩ ↑U).Nonempty

Tactic:
simpa using

Goal state after:
X : Scheme
inst✝¹ : IsIntegral X
U : X.Opens
hU : IsAffineOpen U
inst✝ : Nonempty ↥↑U
⊢ (↑U).Nonempty
Module: Mathlib.AlgebraicGeometry.FunctionField

Goal state before:
pp: X : Scheme
inst✝ : IsIntegral X
f : ↑X.functionField
hf : f ≠ 0
U : Opens ↥X
hU : genericPoint ↥X ∈ U
g : ToType (X.presheaf.obj (op U))
hg : (ConcreteCategory.hom (X.presheaf.germ U (genericPoint ↥X) hU)) g = f
A : Opens ↥X
hA : A ∈ X.affineOpens
hxA : genericPoint ↥X ∈ ↑A
hAU : ↑A ⊆ ↑U
this✝ : Nonempty ↥↑A
gA : ↑Γ(X, A) := (ConcreteCategory.hom (X.presheaf.map (homOfLE hAU).op)) g
h_germ_gA : (ConcreteCategory.hom (X.presheaf.germ A (genericPoint ↥X) hxA)) gA = f
hxV : genericPoint ↥X ∈ X.basicOpen gA
this : Nonempty ↥↑(X.basicOpen gA)
⊢ (ConcreteCategory.hom (X.germToFunctionField (X.basicOpen gA)))
      ((ConcreteCategory.hom (X.presheaf.map ⋯.hom.op)) gA) =
    f

Tactic:
simpa using

Goal state after:
X : Scheme
inst✝ : IsIntegral X
f : ↑X.functionField
hf : f ≠ 0
U : Opens ↥X
hU : genericPoint ↥X ∈ U
g : ToType (X.presheaf.obj (op U))
hg : (ConcreteCategory.hom (X.presheaf.germ U (genericPoint ↥X) hU)) g = f
A : Opens ↥X
hA : A ∈ X.affineOpens
hxA : genericPoint ↥X ∈ ↑A
hAU : ↑A ⊆ ↑U
this✝ : Nonempty ↥↑A
gA : ↑Γ(X, A) := (ConcreteCategory.hom (X.presheaf.map (homOfLE hAU).op)) g
h_germ_gA : (ConcreteCategory.hom (X.presheaf.germ A (genericPoint ↥X) hxA)) gA = f
hxV : genericPoint ↥X ∈ X.basicOpen gA
this : Nonempty ↥↑(X.basicOpen gA)
⊢ (ConcreteCategory.hom (X.presheaf.germ A (genericPoint ↥X) ⋯)) gA = f
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: case mpr
X Y : Scheme
U : X.Opens
hU : Dense ↑U
f : ↑U ⟶ Y
hV : Dense ↑U
g : ↑U ⟶ Y
e :
  { domain := U, dense_domain := hU, hom := f }.hom =
    (X.isoOfEq ⋯).hom ≫ { domain := U, dense_domain := hV, hom := g }.hom
⊢ f = g

Tactic:
simpa using

Goal state after:
case mpr
X Y : Scheme
U : X.Opens
hU : Dense ↑U
f : ↑U ⟶ Y
hV : Dense ↑U
g : ↑U ⟶ Y
e :
  { domain := U, dense_domain := hU, hom := f }.hom =
    (X.isoOfEq ⋯).hom ≫ { domain := U, dense_domain := hV, hom := g }.hom
⊢ f = g
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: case refine_1
X Y : Scheme
inst✝¹ : IrreducibleSpace ↥X
x : ↥X
inst✝ : X.IsGermInjectiveAt x
f g : X.PartialMap Y
hxf : x ∈ f.domain
hxg : x ∈ g.domain
hdense : Dense (↑f.domain ⊓ ↑g.domain)
H :
  (f.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯ =
    (g.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯
this : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ⟨x, ⋯⟩
⊢ (↑(f.domain ⊓ g.domain)).fromSpecStalk ⟨x, ⋯⟩ ≫ X.homOfLE ⋯ ≫ f.hom =
    (↑(f.domain ⊓ g.domain)).fromSpecStalk ⟨x, ⋯⟩ ≫ X.homOfLE ⋯ ≫ g.hom

Tactic:
simpa using

Goal state after:
case refine_1
X Y : Scheme
inst✝¹ : IrreducibleSpace ↥X
x : ↥X
inst✝ : X.IsGermInjectiveAt x
f g : X.PartialMap Y
hxf : x ∈ f.domain
hxg : x ∈ g.domain
hdense : Dense (↑f.domain ⊓ ↑g.domain)
H :
  (f.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯ =
    (g.restrict (f.domain ⊓ g.domain) hdense ⋯).fromSpecStalkOfMem ⋯
this : (↑(f.domain ⊓ g.domain)).IsGermInjectiveAt ⟨x, ⋯⟩
⊢ (↑(f.domain ⊓ g.domain)).fromSpecStalk ⟨x, ⋯⟩ ≫ X.homOfLE ⋯ ≫ f.hom =
    (↑(f.domain ⊓ g.domain)).fromSpecStalk ⟨x, ⋯⟩ ≫ X.homOfLE ⋯ ≫ g.hom
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y S : Scheme
inst✝⁵ : X.Over S
inst✝⁴ : Y.Over S
inst✝³ : IsReduced X
inst✝² : IsSeparated (Y ↘ S)
f g : X.PartialMap Y
inst✝¹ : IsOver S f
inst✝ : IsOver S g
W : X.Opens
hW : Dense ↑W
hWl : W ≤ f.domain
hWr : W ≤ g.domain
x✝ : f.equiv g
V : X.Opens
hV : Dense ↑V
hVl : V ≤ f.domain
hVr : V ≤ g.domain
e : (f.restrict V hV hVl).hom = (g.restrict V hV hVr).hom
this : IsDominant (X.homOfLE ⋯)
⊢ X.homOfLE ⋯ ≫ (f.restrict W hW hWl).hom = X.homOfLE ⋯ ≫ (g.restrict W hW hWr).hom

Tactic:
simpa using

Goal state after:
X Y S : Scheme
inst✝⁵ : X.Over S
inst✝⁴ : Y.Over S
inst✝³ : IsReduced X
inst✝² : IsSeparated (Y ↘ S)
f g : X.PartialMap Y
inst✝¹ : IsOver S f
inst✝ : IsOver S g
W : X.Opens
hW : Dense ↑W
hWl : W ≤ f.domain
hWr : W ≤ g.domain
x✝ : f.equiv g
V : X.Opens
hV : Dense ↑V
hVl : V ≤ f.domain
hVr : V ≤ g.domain
e : (f.restrict V hV hVl).hom = (g.restrict V hV hVr).hom
this : IsDominant (X.homOfLE ⋯)
⊢ X.homOfLE ⋯ ≫ f.hom = X.homOfLE ⋯ ≫ g.hom
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y S : Scheme
inst✝³ : X.Over S
inst✝² : Y.Over S
inst✝¹ : IsReduced X
inst✝ : S.IsSeparated
f : X.PartialMap Y
x✝ : RationalMap.IsOver S f.toRationalMap
U : X.Opens
hU : Dense ↑U
hU' : U ≤ f.domain
H : IsOver S (f.restrict U hU hU')
this : IsDominant (X.homOfLE hU')
⊢ X.homOfLE hU' ≫ (f.compHom (Y ↘ S)).hom = X.homOfLE hU' ≫ f.domain.ι ≫ X ↘ S

Tactic:
simpa using

Goal state after:
X Y S : Scheme
inst✝³ : X.Over S
inst✝² : Y.Over S
inst✝¹ : IsReduced X
inst✝ : S.IsSeparated
f : X.PartialMap Y
x✝ : RationalMap.IsOver S f.toRationalMap
U : X.Opens
hU : Dense ↑U
hU' : U ≤ f.domain
H : IsOver S (f.restrict U hU hU')
this : IsDominant (X.homOfLE hU')
⊢ X.homOfLE hU' ≫ f.hom ≫ Y ↘ S = ↑U ↘ S
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofilteredOrEmpty I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
Z : (i : I) → Set ↥(D.obj i)
hZc : ∀ (i : I), IsClosed (Z i)
hZne : ∀ (i : I), (Z i).Nonempty
hZcpt : ∀ (i : I), IsCompact (Z i)
hmapsTo : ∀ {i i' : I} (f : i ⟶ i'), Set.MapsTo (⇑(D.map f)) (Z i) (Z i')
X Y : I
f : X ⟶ Y
⊢ TopologicalSpace.Closeds.closure (⇑(D.map f) '' ↑{ carrier := Z X, isClosed' := ⋯ }) ≤
    (vanishingIdeal { carrier := Z Y, isClosed' := ⋯ }).support

Tactic:
simpa using

Goal state after:
I : Type u
inst✝² : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝¹ : IsCofilteredOrEmpty I
inst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
Z : (i : I) → Set ↥(D.obj i)
hZc : ∀ (i : I), IsClosed (Z i)
hZne : ∀ (i : I), (Z i).Nonempty
hZcpt : ∀ (i : I), IsCompact (Z i)
hmapsTo : ∀ {i i' : I} (f : i ⟶ i'), Set.MapsTo (⇑(D.map f)) (Z i) (Z i')
X Y : I
f : X ⟶ Y
⊢ Z X ⊆ (fun a ↦ (D.map f) a) ⁻¹' Z Y
Module: Mathlib.AlgebraicGeometry.Birational.RationalMap

Goal state before:
pp: X Y : Scheme
inst✝¹ : IsReduced X
inst✝ : Y.IsSeparated
f : X.PartialMap Y
⊢ (f.toRationalMap.toPartialMap.restrict f.domain ⋯ ⋯).hom = (X.isoOfEq ⋯).hom ≫ f.hom

Tactic:
simpa using

Goal state after:
X Y : Scheme
inst✝¹ : IsReduced X
inst✝ : Y.IsSeparated
f : X.PartialMap Y
⊢ X.homOfLE ⋯ ≫ f.toRationalMap.toPartialMap.hom = f.hom
Module: Mathlib.AlgebraicGeometry.Birational.Birational

Goal state before:
pp: case mpr
X Y : Scheme
U₁ : X.Opens
hU₁ : Dense ↑U₁
V₂ : Y.Opens
hU₂✝ : Dense ↑V₂
hV₁ : Dense ↑U₁
g : ↑U₁ ≅ ↑V₂
hU₂ : Dense ↑V₂
f : ↑U₁ ≅ ↑V₂
e : f = X.isoOfEq ⋯ ≪≫ g ≪≫ Y.isoOfEq ⋯
⊢ { source := U₁, dense_source := hU₁, target := V₂, dense_target := hU₂, iso := f } =
    { source := U₁, dense_source := hV₁, target := V₂, dense_target := hU₂✝, iso := g }

Tactic:
simpa using

Goal state after:
case mpr
X Y : Scheme
U₁ : X.Opens
hU₁ : Dense ↑U₁
V₂ : Y.Opens
hU₂✝ : Dense ↑V₂
hV₁ : Dense ↑U₁
g : ↑U₁ ≅ ↑V₂
hU₂ : Dense ↑V₂
f : ↑U₁ ≅ ↑V₂
e : f = X.isoOfEq ⋯ ≪≫ g ≪≫ Y.isoOfEq ⋯
⊢ f = g
Module: Mathlib.AlgebraicGeometry.Birational.Birational

Goal state before:
pp: X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
f : X.PartialIso Y
hf : IsOver sX sY f
⊢ IsOver sY sX f.symm

Tactic:
simpa using

Goal state after:
X Y S : Scheme
sX : X ⟶ S
sY : Y ⟶ S
f : X.PartialIso Y
hf : IsOver sX sY f
⊢ f.source.ι ≫ sX = f.iso.hom ≫ f.target.ι ≫ sY
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g : Y.PartialMap Z
⊢ (f.domain.ι ''ᵁ f.hom ⁻¹ᵁ g.domain).carrier.Nonempty

Tactic:
simpa using

Goal state after:
X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g : Y.PartialMap Z
⊢ (⇑f.hom ⁻¹' ↑g.domain).Nonempty
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g : Y.PartialMap Z
V : Y.Opens
hV : Dense ↑V
hV' : V ≤ g.domain
⊢ (f.domain.ι ''ᵁ f.hom ⁻¹ᵁ V).carrier.Nonempty

Tactic:
simpa using

Goal state after:
X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g : Y.PartialMap Z
V : Y.Opens
hV : Dense ↑V
hV' : V ≤ g.domain
⊢ (⇑f.hom ⁻¹' ↑V).Nonempty
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝³ : PreirreducibleSpace ↥X
inst✝² : Nonempty ↥Y
f₁ f₂ : X.PartialMap Y
inst✝¹ : IsDominant f₁.hom
inst✝ : IsDominant f₂.hom
g : Y.PartialMap Z
W : X.Opens
hW : Dense ↑W
hW₁ : W ≤ f₁.domain
hW₂ : W ≤ f₂.domain
e : (f₁.restrict W hW hW₁).hom = (f₂.restrict W hW hW₂).hom
⊢ (f₁.restrict W hW hW₁).hom = (X.isoOfEq ⋯).hom ≫ (f₂.restrict W hW hW₂).hom

Tactic:
simpa using

Goal state after:
X Y Z : Scheme
inst✝³ : PreirreducibleSpace ↥X
inst✝² : Nonempty ↥Y
f₁ f₂ : X.PartialMap Y
inst✝¹ : IsDominant f₁.hom
inst✝ : IsDominant f₂.hom
g : Y.PartialMap Z
W : X.Opens
hW : Dense ↑W
hW₁ : W ≤ f₁.domain
hW₂ : W ≤ f₂.domain
e : (f₁.restrict W hW hW₁).hom = (f₂.restrict W hW hW₂).hom
⊢ X.homOfLE hW₁ ≫ f₁.hom = X.homOfLE hW₂ ≫ f₂.hom
Module: Mathlib.AlgebraicGeometry.Birational.Composition

Goal state before:
pp: X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g₁ g₂ : Y.PartialMap Z
W : Y.Opens
hW : Dense ↑W
hW₁ : W ≤ g₁.domain
hW₂ : W ≤ g₂.domain
e : (g₁.restrict W hW hW₁).hom = (g₂.restrict W hW hW₂).hom
⊢ (g₁.restrict W hW hW₁).hom = (Y.isoOfEq ⋯).hom ≫ (g₂.restrict W hW hW₂).hom

Tactic:
simpa using

Goal state after:
X Y Z : Scheme
inst✝² : PreirreducibleSpace ↥X
inst✝¹ : Nonempty ↥Y
f : X.PartialMap Y
inst✝ : IsDominant f.hom
g₁ g₂ : Y.PartialMap Z
W : Y.Opens
hW : Dense ↑W
hW₁ : W ≤ g₁.domain
hW₂ : W ≤ g₂.domain
e : (g₁.restrict W hW hW₁).hom = (g₂.restrict W hW hW₂).hom
⊢ Y.homOfLE hW₁ ≫ g₁.hom = Y.homOfLE hW₂ ≫ g₂.hom
Module: Mathlib.AlgebraicGeometry.AffineTransitionLimit

Goal state before:
pp: I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
i : I
U : (D.obj i).Opens
hU : c.π.app i ⁻¹ᵁ U = ⊤
H : ∀ (j : I) (fji : j ⟶ i), D.map fji ⁻¹ᵁ U ≠ ⊤
s : ↥c.pt
hs : ∀ (i_1 : I) (hij : i_1 ⟶ i), (c.π.app i_1) s ∈ (↑(D.map hij ⁻¹ᵁ U))ᶜ
⊢ s ∉ c.π.app i ⁻¹ᵁ U

Tactic:
simpa using

Goal state after:
I : Type u
inst✝³ : Category.{u, u} I
D : I ⥤ Scheme
c : Cone D
hc : IsLimit c
inst✝² : IsCofiltered I
inst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)
inst✝ : ∀ (i : I), CompactSpace ↥(D.obj i)
i : I
U : (D.obj i).Opens
hU : c.π.app i ⁻¹ᵁ U = ⊤
H : ∀ (j : I) (fji : j ⟶ i), D.map fji ⁻¹ᵁ U ≠ ⊤
s : ↥c.pt
hs : ∀ (i_1 : I) (hij : i_1 ⟶ i), (c.π.app i_1) s ∈ (↑(D.map hij ⁻¹ᵁ U))ᶜ
⊢ (TopCat.Hom.hom (c.π.app i).base) s ∉ U