awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
left : (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
⊢ (fun (x : R) ↦ x ^ n) '' (↑I : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R) ∧
    (fun (x : R) ↦ x ^ n) '' (↑J : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
have right : (fun (x : R) ↦ x ^ n) '' (↑J : Set R) ⊆ (↑(I ^ n ⊔ J ^ n) : Set R)
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₂ : I ⊔ J = ⊤
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
rw₂ : I ⊔ J = ⊤
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I ⊔ J) : Set R)) ∅
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
rw₂ : I ⊔ J = ⊤
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I ⊔ J) : Set R)) ∅
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₄ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∩ zeroLocus (↑J : Set R)) ∅
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
rw₂ : I ⊔ J = ⊤
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I ⊔ J) : Set R)) ∅
rw₄ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∩ zeroLocus (↑J : Set R)) ∅
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₅ : sᶜ ∩ zeroLocus (↑J : Set R) = ∅
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
rw₂ : I ⊔ J = ⊤
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I ⊔ J) : Set R)) ∅
rw₄ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∩ zeroLocus (↑J : Set R)) ∅
rw₅ : sᶜ ∩ zeroLocus (↑J : Set R) = ∅
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₆ : sᶜ ∩ s = ∅
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
rw : Eq (α := Ideal R) (Ideal.span (↑I : Set R) ⊔ Ideal.span (↑J : Set R)) ⊤
rw₁ : I ⊔ Ideal.span (↑J : Set R) = ⊤
rw₂ : I ⊔ J = ⊤
rw₃ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑(I ⊔ J) : Set R)) ∅
rw₄ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R) ∩ zeroLocus (↑J : Set R)) ∅
rw₅ : sᶜ ∩ zeroLocus (↑J : Set R) = ∅
rw₆ : sᶜ ∩ s = ∅
⊢ Eq (α := Ideal R) (Ideal.span ((↑I : Set R) ∪ (↑J : Set R))) ⊤
have rw₇ : Eq.{u + 1} (α := Set (PrimeSpectrum R)) ∅ ∅
Tactic: rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have this₁ : I ^ n ⊔ J ^ n = ⊤
Tactic: have : I ^ n ⊔ J ^ n = ⊤ := by
    rw [eq_top_iff, ← Ideal.span_pow_eq_top (I ∪ J : Set R) _ n, Ideal.span_le, Set.image_union,
      Set.union_subset_iff]
    constructor
    · rintro _ ⟨x, hx, rfl⟩; exact Ideal.mem_sup_left (Ideal.pow_mem_pow hx n)
    · rintro _ ⟨x, hx, rfl⟩; exact Ideal.mem_sup_right (Ideal.pow_mem_pow hx n)
    · rw [Ideal.span_union, Ideal.span_eq, Ideal.span_eq, ← zeroLocus_empty_iff_eq_top,
        zeroLocus_sup, hI', hJ', Set.compl_inter_self]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
this : I ^ n ⊔ J ^ n = ⊤
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have this₁ : ∃ y ∈ I ^ n, ∃ z ∈ J ^ n, y + z = 1
Tactic: rw [Ideal.eq_top_iff_one, Submodule.mem_sup] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
this : ∃ y ∈ I ^ n, ∃ z ∈ J ^ n, y + z = 1
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : ∀ (x : R) (hx : x ∈ I ^ n) (y : R) (hy : y ∈ J ^ n) (add : x + y = 1),
  ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: obtain ⟨x, hx, y, hy, add⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have mul : x * y = 0
Tactic: have mul : x * y = 0 := hn (Ideal.mul_mem_mul hx hy)
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ s = (↑(basicOpen x) : Set (PrimeSpectrum R))
have refine_1 : s ⊆ (↑(basicOpen x) : Set (PrimeSpectrum R))
Tactic: refine subset_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
refine_1 : s ⊆ (↑(basicOpen x) : Set (PrimeSpectrum R))
⊢ s = (↑(basicOpen x) : Set (PrimeSpectrum R))
have refine_2 : (↑(basicOpen x) : Set (PrimeSpectrum R)) ⊆ s
Tactic: refine subset_antisymm ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ s ⊆ (↑(basicOpen x) : Set (PrimeSpectrum R))
have refine_1 : Subset (α := Set (PrimeSpectrum R)) (zeroLocus (↑J : Set R)) (zeroLocus {y})
Tactic: rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ (↑(basicOpen x) : Set (PrimeSpectrum R)) ⊆ s
have refine_2 : Subset (α := Set (PrimeSpectrum R)) (zeroLocus (↑I : Set R)) (zeroLocus {x})
Tactic: rw [basicOpen_eq_zeroLocus_compl, Set.compl_subset_comm, ← hI']
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (hypothesis)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have this₁ : s = (↑(basicOpen x) : Set (PrimeSpectrum R))
Tactic: have : s = basicOpen x := by
    refine subset_antisymm ?_ ?_
    · rw [← hJ', basicOpen_eq_zeroLocus_of_mul_add _ _ mul add]
      exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hy)
    · rw [basicOpen_eq_zeroLocus_compl, Set.compl_subset_comm, ← hI']
      exact zeroLocus_anti_mono (Set.singleton_subset_iff.mpr <| Ideal.pow_le_self hnz hx)
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
this : s = (↑(basicOpen x) : Set (PrimeSpectrum R))
⊢ ∃ (e : R) (f : R),
    e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))
have inr : sᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
Tactic: refine ⟨x, y, mul, add, this, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
this : s = (↑(basicOpen x) : Set (PrimeSpectrum R))
⊢ sᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
have inr : (↑(basicOpen x) : Set (PrimeSpectrum R))ᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
Tactic: rw [this, basicOpen_eq_zeroLocus_of_mul_add _ _ mul add, basicOpen_eq_zeroLocus_compl]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
this : s = (↑(basicOpen x) : Set (PrimeSpectrum R))
inr : (↑(basicOpen x) : Set (PrimeSpectrum R))ᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
⊢ sᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
have inr₁ : (zeroLocus {y})ᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
Tactic: rw [this, basicOpen_eq_zeroLocus_of_mul_add _ _ mul add, basicOpen_eq_zeroLocus_compl]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
h : Nontrivial R
I : Ideal R
hI : I.FG
J : Ideal R
hJ : J.FG
hI' : zeroLocus (↑I : Set R) = sᶜ
hJ' : zeroLocus (↑J : Set R) = s
this_1 : I * J ≤ nilradical R
n : ℕ
hn : I ^ n * J ^ n ≤ ⊥
hnz : n ≠ 0
x : R
hx : x ∈ I ^ n
y : R
hy : y ∈ J ^ n
add : x + y = 1
mul : x * y = 0
this : s = (↑(basicOpen x) : Set (PrimeSpectrum R))
inr : (↑(basicOpen x) : Set (PrimeSpectrum R))ᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
inr₁ : (zeroLocus {y})ᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
⊢ sᶜ = (↑(basicOpen y) : Set (PrimeSpectrum R))
have inr₂ : Eq (α := Set (PrimeSpectrum R)) (zeroLocus {y})ᶜ (zeroLocus {y})ᶜ
Tactic: rw [this, basicOpen_eq_zeroLocus_of_mul_add _ _ mul add, basicOpen_eq_zeroLocus_compl]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
hs : IsClopen s
⊢ ∃! e : R, IsIdempotentElem e ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
have refine : ∀ (y₁ y₂ : R),
  IsIdempotentElem y₁ ∧ s = (↑(basicOpen y₁) : Set (PrimeSpectrum R)) →
    IsIdempotentElem y₂ ∧ s = (↑(basicOpen y₂) : Set (PrimeSpectrum R)) → y₁ = y₂
Tactic: refine existsUnique_of_exists_of_unique (exists_idempotent_basicOpen_eq_of_isClopen hs) ?_
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ IsClopen s ↔ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
have refine_1 : ∀ (h : IsClopen s), ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
Tactic: refine ⟨fun h ↦ ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
refine_1 : ∀ (h : IsClopen s), ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
⊢ IsClopen s ↔ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
have refine_2 : (∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))) → IsClopen s
Tactic: refine ⟨fun h ↦ ?_, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
h : IsClopen s
⊢ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
have refine_1 : ∀ (e f : R)
  (h :
    e * f = 0 ∧
      e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R)) ∧ sᶜ = (↑(basicOpen f) : Set (PrimeSpectrum R))),
  ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = (↑(basicOpen e) : Set (PrimeSpectrum R))
Tactic: have ⟨e, f, h⟩ := exists_mul_eq_zero_add_eq_one_basicOpen_eq_of_isClopen h
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ IsClopen s ↔ ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}
have rw : (∃ (b : R) (a : R), a * b = 0 ∧ a + b = 1 ∧ s = (↑(basicOpen a) : Set (PrimeSpectrum R))) ↔
  ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}
Tactic: rw [isClopen_iff_mul_add, exists_comm]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
⊢ (∃ (b : R) (a : R), a * b = 0 ∧ a + b = 1 ∧ s = (↑(basicOpen a) : Set (PrimeSpectrum R))) ↔
    ∃ (e : R) (f : R), e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}
have refine : ∀ (e f : R),
  f * e = 0 ∧ f + e = 1 ∧ s = (↑(basicOpen f) : Set (PrimeSpectrum R)) ↔ e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}
Tactic: refine exists₂_congr fun e f ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
e f : R
⊢ f * e = 0 ∧ f + e = 1 ∧ s = (↑(basicOpen f) : Set (PrimeSpectrum R)) ↔ e * f = 0 ∧ e + f = 1 ∧ s = zeroLocus {e}
have rw : e * f = 0 ∧ e + f = 1 → (s = (↑(basicOpen f) : Set (PrimeSpectrum R)) ↔ s = zeroLocus {e})
Tactic: rw [mul_comm, add_comm, ← and_assoc, ← and_assoc, and_congr_right]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
inst : CommSemiring R
s : Set (PrimeSpectrum R)
e f : R
mul : e * f = 0
add : e + f = 1
⊢ s = (↑(basicOpen f) : Set (PrimeSpectrum R)) ↔ s = zeroLocus {e}
have rw : s = (↑(basicOpen f) : Set (PrimeSpectrum R)) ↔ s = (↑(basicOpen f) : Set (PrimeSpectrum R))
Tactic: rw [zeroLocus_eq_basicOpen_of_mul_add e f mul add]
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
S : Type v
inst : CommSemiring R
inst_1 : CommSemiring S
⊢ Function.Bijective (β := Clopens (PrimeSpectrum R)) fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
    { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := ⋯ }
have refine_1 : ∀ (x x : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) (x : R × R) (hx : x.1 * x.2 = 0 ∧ x.1 + x.2 = 1) (y : R × R)
  (hy : y.1 * y.2 = 0 ∧ y.1 + y.2 = 1)
  (eq :
    Eq.{u + 1} (α := Clopens (PrimeSpectrum R))
      ((fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
          { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := sorry :
            Clopens (PrimeSpectrum R) })
        (Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) x hx))
      ((fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
          { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := sorry :
            Clopens (PrimeSpectrum R) })
        (Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) y hy))),
  (↑(Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) x hx) : R × R).1 =
    (↑(Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) y hy) : R × R).1
Tactic: refine ⟨fun ⟨x, hx⟩ ⟨y, hy⟩ eq ↦ mul_eq_zero_add_eq_one_ext_left ?_, fun s ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.Spectrum.Prime.Topology (goal)
Goal:
R : Type u
S : Type v
inst : CommSemiring R
inst_1 : CommSemiring S
refine_1 :
  ∀ (x x : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) (x : R × R) (hx : x.1 * x.2 = 0 ∧ x.1 + x.2 = 1) (y : R × R)
    (hy : y.1 * y.2 = 0 ∧ y.1 + y.2 = 1)
    (eq :
      Eq.{u + 1} (α := Clopens (PrimeSpectrum R))
        ((fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
            { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := ⋯ :
              Clopens (PrimeSpectrum R) })
          (Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) x hx))
        ((fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
            { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := ⋯ :
              Clopens (PrimeSpectrum R) })
          (Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) y hy))),
    (↑(Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) x hx) : R × R).1 =
      (↑(Subtype.mk (p := fun (e : R × R) ↦ e.1 * e.2 = 0 ∧ e.1 + e.2 = 1) y hy) : R × R).1
⊢ Function.Bijective (β := Clopens (PrimeSpectrum R)) fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
    { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := ⋯ }
have refine_2 : ∀ (s : Clopens (PrimeSpectrum R)),
  ∃ (a : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }),
    (fun (e : { e : R × R // e.1 * e.2 = 0 ∧ e.1 + e.2 = 1 }) ↦
          { carrier := (↑(basicOpen (↑e : R × R).1) : Set (PrimeSpectrum R)), isClopen' := sorry :
            Clopens (PrimeSpectrum R) })
        a =
      s
Tactic: refine ⟨fun ⟨x, hx⟩ ⟨y, hy⟩ eq ↦ mul_eq_zero_add_eq_one_ext_left ?_, fun s ↦ ?_⟩