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

Goal state before:
pp: case neg
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
k : Fin n
a✝ : k ∈ Finset.univ
hkj : ¬k = j
⊢ (c.val k).degree ≤ (c.val k).degree

Tactic:
·

Goal state after:
case neg
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
k : Fin n
a✝ : k ∈ Finset.univ
hkj : ¬k = j
⊢ (c.val k).degree ≤ (c.val k).degree
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, ↑t = U
H : ∀ t ∈ s, IsCompactOpenCovered f ↑t
t : Finset ↑s
ht : U ⊆ ⋃ i ∈ t, ↑↑i
⊢ (SetLike.coe '' ↑(Finset.image Subtype.val t)).Finite

pp: case refine_3
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, ↑t = U
H : ∀ t ∈ s, IsCompactOpenCovered f ↑t
t : Finset ↑s
ht : U ⊆ ⋃ i ∈ t, ↑↑i
⊢ ∀ t_1 ∈ SetLike.coe '' ↑(Finset.image Subtype.val t), IsCompactOpenCovered f t_1

Tactic:
·

Goal state after:
case refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, ↑t = U
H : ∀ t ∈ s, IsCompactOpenCovered f ↑t
t : Finset ↑s
ht : U ⊆ ⋃ i ∈ t, ↑↑i
⊢ (SetLike.coe '' ↑(Finset.image Subtype.val t)).Finite
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_3
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, ↑t = U
H : ∀ t ∈ s, IsCompactOpenCovered f ↑t
t : Finset ↑s
ht : U ⊆ ⋃ i ∈ t, ↑↑i
⊢ ∀ t_1 ∈ SetLike.coe '' ↑(Finset.image Subtype.val t), IsCompactOpenCovered f t_1

Tactic:
·

Goal state after:
case refine_3
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
s : Set (Opens S)
hs : ⋃ t ∈ s, ↑t = U
H : ∀ t ∈ s, IsCompactOpenCovered f ↑t
t : Finset ↑s
ht : U ⊆ ⋃ i ∈ t, ↑↑i
⊢ ∀ t_1 ∈ SetLike.coe '' ↑(Finset.image Subtype.val t), IsCompactOpenCovered f t_1
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_1
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
⊢ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t = U

pp: case refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
⊢ ∀ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, IsCompactOpenCovered f ↑t

Tactic:
·

Goal state after:
case refine_1
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
⊢ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t = U
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_1.refine_1
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
x : S
⊢ x ∈ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t → x ∈ U

pp: case refine_1.refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
x : S
hx : x ∈ U
⊢ x ∈ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t

Tactic:
·

Goal state after:
case refine_1.refine_1
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
x : S
⊢ x ∈ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t → x ∈ U
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_1.refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
x : S
hx : x ∈ U
⊢ x ∈ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t

Tactic:
·

Goal state after:
case refine_1.refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
x : S
hx : x ∈ U
⊢ x ∈ ⋃ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, ↑t
Module: Mathlib.Topology.Sets.CompactOpenCovered

Goal state before:
pp: case refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
⊢ ∀ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, IsCompactOpenCovered f ↑t

Tactic:
·

Goal state after:
case refine_2
S : Type u_1
ι : Type u_2
X : ι → Type u_3
f : (i : ι) → X i → S
inst✝¹ : (i : ι) → TopologicalSpace (X i)
inst✝ : TopologicalSpace S
U : Set S
hU : IsCompact U
Us : (x : S) → x ∈ U → Set S
hU' : ∀ (x : S) (a : x ∈ U), Us x a ⊆ U
hUx : ∀ (x : S) (a : x ∈ U), x ∈ Us x a
hUo : ∀ (x : S) (a : x ∈ U), IsOpen (Us x a)
hU'' : ∀ (x : S) (a : x ∈ U), IsCompactOpenCovered f (Us x a)
⊢ ∀ t ∈ {x | ∃ x_1, ∃ (h : x_1 ∈ U), Us x_1 h = ↑x}, IsCompactOpenCovered f ↑t
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case hab.refine_2
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
⊢ ⋃ i, coeff(c'.val i) ⊆ ↑(coeffSubmodule R₀ c)

Tactic:
·

Goal state after:
case hab.refine_2
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
⊢ ⋃ i, coeff(c'.val i) ⊆ ↑(coeffSubmodule R₀ c)
Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case neg
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝¹ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k✝ : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
k : Fin n
h✝ : ¬k = j
⊢ coeff(c.val k) ⊆ ↑(coeffSubmodule R₀ c)

Tactic:
·

Goal state after:
case neg
R✝ : Type u_2
inst✝³ : CommRing R✝
n : ℕ
R : Type u_2
inst✝² : CommRing R
c : InductionObj R n
i j : Fin n
hi : (c.val i).Monic
hle : (c.val i).degree ≤ (c.val j).degree
hne : i ≠ j
H :
  ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],
    Statement R₀ R n { val := Function.update c.val j (c.val j %ₘ c.val i) }
R₀ : Type u_1
inst✝¹ : CommRing R₀
inst✝ : Algebra R₀ R
f : R[X]
h✝¹ : Nontrivial R
S : ConstructibleSetData R
hS :
  comap Polynomial.C ''
      (zeroLocus (Set.range { val := Function.update c.val j (c.val j %ₘ c.val i) }.val) \ zeroLocus {f}) =
    S.toSet
hS' :
  ∀ C ∈ S,
    C.n ≤ { val := Function.update c.val j (c.val j %ₘ c.val i) }.degBound ∧
      ∀ (i_1 : Fin C.n),
        C.g i_1 ∈
          coeffSubmodule R₀ { val := Function.update c.val j (c.val j %ₘ c.val i) } ^
            { val := Function.update c.val j (c.val j %ₘ c.val i) }.powBound
C : BasicConstructibleSetData R
hC : C ∈ S
c' : InductionObj R n := { val := Function.update c.val j (c.val j %ₘ c.val i) }
deg_bound₁ : c'.degBound ≤ c.degBound
k✝ : Fin C.n
hij : coeffSubmodule R₀ c' ≤ coeffSubmodule R₀ c ^ (c.val j).degree.succ
hi' : c.val i = 1
k : Fin n
h✝ : ¬k = j
⊢ coeff(c.val k) ⊆ ↑(coeffSubmodule R₀ c)
Module: Mathlib.AlgebraicGeometry.Cover.QuasiCompact

Goal state before:
pp: case inr
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
this :
  ∀ {U' : T.Opens},
    IsAffineOpen U' →
      (∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U) →
        IsCompactOpenCovered (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) ↑U'
h : ¬∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U
⊢ IsCompactOpenCovered (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) ↑U'

pp: S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
h : ∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U
⊢ IsCompactOpenCovered (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) ↑U'

Tactic:
·

Goal state after:
case inr
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
this :
  ∀ {U' : T.Opens},
    IsAffineOpen U' →
      (∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U) →
        IsCompactOpenCovered (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) ↑U'
h : ¬∃ U, IsAffineOpen U ∧ ⇑f '' ↑U' ⊆ ↑U
⊢ IsCompactOpenCovered (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) ↑U'
Module: Mathlib.AlgebraicGeometry.Cover.QuasiCompact

Goal state before:
pp: case refine_1
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
U : S.Opens
hU : IsAffineOpen U
hsub : ⇑f '' ↑U' ⊆ ↑U
s : Set 𝒰.I₀
hf : s.Finite
V : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥(𝒰.X x)) i)
hc : ∀ (i : 𝒰.I₀) (h : i ∈ s), IsCompact (V i h).carrier
heq : ⋃ i, ⋃ (h : i ∈ s), (fun x ↦ ⇑(𝒰.f x)) i '' ↑(V i h) = ↑U
i : (PreZeroHypercover.pullback₁ f 𝒰).I₀
hi : i ∈ s
⊢ IsCompact ((fun i hi ↦ pullback.fst f (𝒰.f i) ⁻¹ᵁ U' ⊓ pullback.snd f (𝒰.f i) ⁻¹ᵁ V i hi) i hi).carrier

pp: case refine_2
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
U : S.Opens
hU : IsAffineOpen U
hsub : ⇑f '' ↑U' ⊆ ↑U
s : Set 𝒰.I₀
hf : s.Finite
V : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥(𝒰.X x)) i)
hc : ∀ (i : 𝒰.I₀) (h : i ∈ s), IsCompact (V i h).carrier
heq : ⋃ i, ⋃ (h : i ∈ s), (fun x ↦ ⇑(𝒰.f x)) i '' ↑(V i h) = ↑U
⊢ ⋃ i,
      ⋃ (h : i ∈ s),
        (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) i ''
          ↑((fun i hi ↦ pullback.fst f (𝒰.f i) ⁻¹ᵁ U' ⊓ pullback.snd f (𝒰.f i) ⁻¹ᵁ V i hi) i h) =
    ↑U'

Tactic:
·

Goal state after:
case refine_1
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
U : S.Opens
hU : IsAffineOpen U
hsub : ⇑f '' ↑U' ⊆ ↑U
s : Set 𝒰.I₀
hf : s.Finite
V : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥(𝒰.X x)) i)
hc : ∀ (i : 𝒰.I₀) (h : i ∈ s), IsCompact (V i h).carrier
heq : ⋃ i, ⋃ (h : i ∈ s), (fun x ↦ ⇑(𝒰.f x)) i '' ↑(V i h) = ↑U
i : (PreZeroHypercover.pullback₁ f 𝒰).I₀
hi : i ∈ s
⊢ IsCompact ((fun i hi ↦ pullback.fst f (𝒰.f i) ⁻¹ᵁ U' ⊓ pullback.snd f (𝒰.f i) ⁻¹ᵁ V i hi) i hi).carrier
Module: Mathlib.AlgebraicGeometry.Cover.QuasiCompact

Goal state before:
pp: case refine_2
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
U : S.Opens
hU : IsAffineOpen U
hsub : ⇑f '' ↑U' ⊆ ↑U
s : Set 𝒰.I₀
hf : s.Finite
V : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥(𝒰.X x)) i)
hc : ∀ (i : 𝒰.I₀) (h : i ∈ s), IsCompact (V i h).carrier
heq : ⋃ i, ⋃ (h : i ∈ s), (fun x ↦ ⇑(𝒰.f x)) i '' ↑(V i h) = ↑U
⊢ ⋃ i,
      ⋃ (h : i ∈ s),
        (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) i ''
          ↑((fun i hi ↦ pullback.fst f (𝒰.f i) ⁻¹ᵁ U' ⊓ pullback.snd f (𝒰.f i) ⁻¹ᵁ V i hi) i h) =
    ↑U'

Tactic:
·

Goal state after:
case refine_2
S : Scheme
𝒰✝ 𝒰 : PreZeroHypercover S
K : Precoverage Scheme
inst✝ : QuasiCompactCover 𝒰
T : Scheme
f : T ⟶ S
U' : T.Opens
hU' : IsAffineOpen U'
U : S.Opens
hU : IsAffineOpen U
hsub : ⇑f '' ↑U' ⊆ ↑U
s : Set 𝒰.I₀
hf : s.Finite
V : (i : 𝒰.I₀) → i ∈ s → TopologicalSpace.Opens ((fun x ↦ ↥(𝒰.X x)) i)
hc : ∀ (i : 𝒰.I₀) (h : i ∈ s), IsCompact (V i h).carrier
heq : ⋃ i, ⋃ (h : i ∈ s), (fun x ↦ ⇑(𝒰.f x)) i '' ↑(V i h) = ↑U
⊢ ⋃ i,
      ⋃ (h : i ∈ s),
        (fun x ↦ ⇑((PreZeroHypercover.pullback₁ f 𝒰).f x)) i ''
          ↑((fun i hi ↦ pullback.fst f (𝒰.f i) ⁻¹ᵁ U' ⊓ pullback.snd f (𝒰.f i) ⁻¹ᵁ V i hi) i h) =
    ↑U'
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_2
P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
heq :
  (MorphismProperty.Over.map ⊤ ⋯).map iso1.hom ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app (d.cocone j).pt =
    d.transitionMap hij
iso2 : (d.cocone i).pt.left ≅ pullback (d.cocone j).pt.hom (Cover.trans 𝒰 hij) :=
  (MorphismProperty.Over.forget P ⊤ (𝒰.X i) ⋙ CategoryTheory.Over.forget (𝒰.X i)).mapIso iso1
⊢ pullback.snd (d.cocone j).pt.hom (Cover.trans 𝒰 hij) ≫ (Iso.refl (𝒰.X i)).hom = iso2.symm.hom ≫ (d.cocone i).pt.hom

Tactic:
·

Goal state after:
case refine_2
P : MorphismProperty Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝² : Category.{v_2, u_2} 𝒰.I₀
inst✝¹ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
i j : 𝒰.I₀
hij : i ⟶ j
iso1 : (d.cocone i).pt ≅ ((MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)).mapCocone (d.cocone j)).pt :=
  (d.isColimit i).coconePointsIsoOfNatIso
    (isColimitOfPreserves (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij)) (d.isColimit j))
    (D.isoWhiskerLeft (MorphismProperty.Over.pullbackComp (Cover.trans 𝒰 hij) (𝒰.f j) (𝒰.f i) ⋯) ≪≫
      (D.associator (MorphismProperty.Over.pullback P ⊤ (𝒰.f j))
          (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))).symm)
heq :
  (MorphismProperty.Over.map ⊤ ⋯).map iso1.hom ≫
      (MorphismProperty.Over.mapPullbackAdj P ⊤ (Cover.trans 𝒰 hij) ⋯ trivial).counit.app (d.cocone j).pt =
    d.transitionMap hij
iso2 : (d.cocone i).pt.left ≅ pullback (d.cocone j).pt.hom (Cover.trans 𝒰 hij) :=
  (MorphismProperty.Over.forget P ⊤ (𝒰.X i) ⋙ CategoryTheory.Over.forget (𝒰.X i)).mapIso iso1
⊢ pullback.snd (d.cocone j).pt.hom (Cover.trans 𝒰 hij) ≫ (Iso.refl (𝒰.X i)).hom = iso2.symm.hom ≫ (d.cocone i).pt.hom
Module: Mathlib.AlgebraicGeometry.ColimitsOver

Goal state before:
pp: case refine_1
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.95} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.95} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : (a : J) → (D.obj a).left.OpenCover := fun a ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).hom 𝒰
a : J
⊢ D.obj a ⟶ ((Functor.const J).obj d.glued).obj a

pp: case refine_2
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.95} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.95} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : (a : J) → (D.obj a).left.OpenCover := fun a ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).hom 𝒰
a b : J
f : a ⟶ b
⊢ D.map f ≫ ?refine_1 = ?refine_1 ≫ ((Functor.const J).obj d.glued).map f

Tactic:
·

Goal state after:
case refine_1
P : MorphismProperty Scheme
inst✝⁸ : P.IsStableUnderBaseChange
inst✝⁷ : P.IsMultiplicative
S : Scheme
J : Type u_1
inst✝⁶ : Category.{v_1, u_1} J
D : J ⥤ P.Over ⊤ S
𝒰 : S.OpenCover
inst✝⁵ : Category.{v_2, ?u.95} 𝒰.I₀
inst✝⁴ : LocallyDirected 𝒰
d : ColimitGluingData D 𝒰
inst✝³ :
  ∀ {i j : 𝒰.I₀} (hij : i ⟶ j), PreservesColimitsOfShape J (MorphismProperty.Over.pullback P ⊤ (Cover.trans 𝒰 hij))
inst✝² : Quiver.IsThin 𝒰.I₀
inst✝¹ : Small.{u, ?u.95} 𝒰.I₀
inst✝ : IsZariskiLocalAtTarget P
𝒱 : (a : J) → (D.obj a).left.OpenCover := fun a ↦ Precoverage.ZeroHypercover.pullback₁ (D.obj a).hom 𝒰
a : J
⊢ D.obj a ⟶ ((Functor.const J).obj d.glued).obj a