[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Etale.QuasiFinite).

[User]
The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Etale.QuasiFinite.Algebra`:
State 0: ⊢ lemma Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver
    {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
    (p : Ideal R) [p.IsPrime] (q : Ideal S) [q.IsPrime] [q.LiesOver p]
    [Algebra.FiniteType R S]
    [Algebra.QuasiFiniteAt R q] :
    ∃ s ∉ q, ∃ hs : IsIntegral R s, (∀ q' : Ideal S, q'.IsPrime → q' ≠ q → q'.LiesOver p → s ∈ q') ∧
      ∀ (q' : Ideal (integralClosure R S)), q'.IsPrime →
        q' ≠ q.under _ → q'.LiesOver p → ⟨s, hs⟩ ∈ q'
Tactic: obtain ⟨s₁, hs₁q, hs₁⟩ := Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver (R := R) p q
State 1: ⊢ <goal after obtain ⟨s₁, hs₁q, hs₁⟩ := Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver (R := R) p q>
Tactic: obtain ⟨s₂, hs₂q, hs₂⟩ := Algebra.ZariskisMainProperty.of_finiteType (R := R) q
State 2: ⊢ <goal after obtain ⟨s₂, hs₂q, hs₂⟩ := Algebra.ZariskisMainProperty.of_finiteType (R := R) q>
Tactic: obtain ⟨s₃, hs₃⟩ := hs₂.2 (algebraMap _ _ s₁)
State 3: ⊢ <goal after obtain ⟨s₃, hs₃⟩ := hs₂.2 (algebraMap _ _ s₁)>
Tactic: obtain ⟨s₃, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.exists_mk'_eq (.powers s₂) s₃
State 4: ⊢ <goal after obtain ⟨s₃, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.exists_mk'_eq (.powers s₂) s₃>
Tactic: obtain ⟨m, hm⟩ : ∃ m, ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) := by
State 5: ⊢ <goal after obtain ⟨m, hm⟩ : ∃ m, ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) := by>
Tactic: simpa [IsLocalization.Away.map, IsLocalization.map_mk', IsLocalization.mk'_eq_iff_eq_mul,  ← map_mul, ← map_pow, IsLocalization.eq_iff_exists (.powers s₂.1),  Submonoid.mem_powers_iff] using hs₃  wlog hm0 : 0 < m generalizing m  · refine this (m + 1) (by grind) (by simp)
State 6: ⊢ <goal after simpa [IsLocalization.Away.map, IsLocalization.map_mk', IsLocalization.mk'_eq_iff_eq_mul,>
Tactic: have hs₃q : s₃.1 ∉ q := fun h ↦ (show ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) ∉ q from q.primeCompl.mul_mem  (pow_mem hs₂q _) (mul_mem hs₁q (pow_mem hs₂q _))) (hm ▸ Ideal.mul_mem_left _ _ h)
State 7: ⊢ <goal after have hs₃q : s₃.1 ∉ q := fun h ↦ (show ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) ∉ q from q.primeCompl.mul_mem>
Tactic: refine ⟨↑s₂ ^ m * ↑s₃, q.primeCompl.mul_mem (pow_mem hs₂q _) hs₃q, (s₂ ^ m * s₃).2,  fun q' _ hq'q _ ↦ hm ▸ Ideal.mul_mem_left _ _ (Ideal.mul_mem_right _ _ (hs₁ q' ‹_› hq'q ‹_›)),  fun q' _ hq'q _ ↦ ?_⟩  let : Algebra (integralClosure R S) (Localization.Away s₂.1) := OreLocalization.instAlgebra  let e : Localization.Away s₂ ≃ₐ[integralClosure R S] Localization.Away s₂.1 :=  .ofBijective (Localization.awayMapₐ (Algebra.ofId _ _) s₂) hs₂  let q's : Ideal (Localization.Away s₂) := q'.map (algebraMap _ _)  by_contra H
State 8: ⊢ <goal after refine ⟨↑s₂ ^ m * ↑s₃, q.primeCompl.mul_mem (pow_mem hs₂q _) hs₃q, (s₂ ^ m * s₃).2,>
Tactic: have hq's : Disjoint (Submonoid.powers s₂ : Set (integralClosure R S)) ↑q' := by
State 9: ⊢ <goal after have hq's : Disjoint (Submonoid.powers s₂ : Set (integralClosure R S)) ↑q' := by>
Tactic: rw [Ideal.disjoint_powers_iff_notMem _ (Ideal.IsPrime.isRadical ‹_›)]
State 10: ⊢ <goal after rw [Ideal.disjoint_powers_iff_notMem _ (Ideal.IsPrime.isRadical ‹_›)]>
Tactic: contrapose H; exact Ideal.mul_mem_right s₃ _ (Ideal.pow_mem_of_mem _ H m hm0)
State 11: ⊢ <goal after contrapose H; exact Ideal.mul_mem_right s₃ _ (Ideal.pow_mem_of_mem _ H m hm0)>
Tactic: have : q's.IsPrime := IsLocalization.isPrime_of_isPrime_disjoint (.powers s₂) _ _ ‹_› hq's
State 12: ⊢ <goal after have : q's.IsPrime := IsLocalization.isPrime_of_isPrime_disjoint (.powers s₂) _ _ ‹_› hq's>
Tactic: have : q's.LiesOver q' := ⟨(IsLocalization.comap_map_of_isPrime_disjoint _ _ ‹_› hq's).symm⟩
State 13: ⊢ <goal after have : q's.LiesOver q' := ⟨(IsLocalization.comap_map_of_isPrime_disjoint _ _ ‹_› hq's).symm⟩>
Tactic: have : q's.LiesOver p := .trans _ q' _
State 14: ⊢ <goal after have : q's.LiesOver p := .trans _ q' _>
Tactic: have := hs₁ (q's.comap (e.symm.toAlgHom.comp (IsScalarTower.toAlgHom _ _ _)).toRingHom)  inferInstance (by
State 15: ⊢ <goal after have := hs₁ (q's.comap (e.symm.toAlgHom.comp (IsScalarTower.toAlgHom _ _ _)).toRingHom)>
Tactic: contrapose hq'q
State 16: ⊢ <goal after contrapose hq'q>
Tactic: rw [← hq'q, Ideal.under, Ideal.comap_comap, AlgHom.toRingHom_eq_coe,  AlgHom.comp_algebraMap, q's.over_def q'])  (inferInstanceAs ((q's.comap ((e.symm.toAlgHom.comp  (IsScalarTower.toAlgHom _ _ _)).restrictScalars R)).LiesOver _))
State 17: ⊢ <goal after rw [← hq'q, Ideal.under, Ideal.comap_comap, AlgHom.toRingHom_eq_coe,>
Tactic: have : e.symm (algebraMap S (Localization.Away  ((integralClosure R S).val.toRingHom s₂)) s₁) ∈ q's := by
State 18: ⊢ <goal after have : e.symm (algebraMap S (Localization.Away>
Tactic: simpa using this
State 19: ⊢ <goal after simpa using this>
Tactic: rw [← hs₃, ← Ideal.IsPrime.mul_mem_left_iff (x := algebraMap _ _ (s₂ ^ n))] at this  · dsimp [Localization.awayMap, IsLocalization.Away.map] at this
State 20: ⊢ <goal after rw [← hs₃, ← Ideal.IsPrime.mul_mem_left_iff (x := algebraMap _ _ (s₂ ^ n))] at this>
Tactic: rw [IsLocalization.map_mk', ← e.symm.commutes, ← map_mul,  IsScalarTower.algebraMap_eq _ S _] at this  replace this : e.symm ((algebraMap _ (Localization.Away s₂.1)) s₃) ∈ q's := by
State 21: ⊢ <goal after rw [IsLocalization.map_mk', ← e.symm.commutes, ← map_mul,>
Tactic: simpa [-map_mul, -map_pow, -AlgEquiv.commutes] using this  replace this : s₃ ∈ q' := by simpa [← Ideal.mem_comap, ← q's.over_def q'] using this
State 22: ⊢ <goal after simpa [-map_mul, -map_pow, -AlgEquiv.commutes] using this>
Tactic: exact H (Ideal.mul_mem_left _ (s₂ ^ m) this)  · rw [map_pow]; exact Ideal.notMem_of_isUnit _ (.pow _ (IsLocalization.Away.algebraMap_isUnit _))  #adaptation_note /-- The maxHeartbeats bump is required after leanprover/lean4#12564. -/  set_option maxHeartbeats 400000 in -- see adaptation note
```

Question: what is the next tactic after the last recorded state, and why?

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.RingTheory.Etale.QuasiFinite).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following proof-state transcript was extracted from a real mathlib proof. Use it to answer the question below.

```text
Proof-state transcript for `Mathlib.RingTheory.Etale.QuasiFinite.Algebra`:
State 0: ⊢ lemma Algebra.exists_notMem_and_isIntegral_forall_mem_of_ne_of_liesOver
    {R S : Type*} [CommRing R] [CommRing S] [Algebra R S]
    (p : Ideal R) [p.IsPrime] (q : Ideal S) [q.IsPrime] [q.LiesOver p]
    [Algebra.FiniteType R S]
    [Algebra.QuasiFiniteAt R q] :
    ∃ s ∉ q, ∃ hs : IsIntegral R s, (∀ q' : Ideal S, q'.IsPrime → q' ≠ q → q'.LiesOver p → s ∈ q') ∧
      ∀ (q' : Ideal (integralClosure R S)), q'.IsPrime →
        q' ≠ q.under _ → q'.LiesOver p → ⟨s, hs⟩ ∈ q'
Tactic: obtain ⟨s₁, hs₁q, hs₁⟩ := Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver (R := R) p q
State 1: ⊢ <goal after obtain ⟨s₁, hs₁q, hs₁⟩ := Ideal.exists_not_mem_forall_mem_of_ne_of_liesOver (R := R) p q>
Tactic: obtain ⟨s₂, hs₂q, hs₂⟩ := Algebra.ZariskisMainProperty.of_finiteType (R := R) q
State 2: ⊢ <goal after obtain ⟨s₂, hs₂q, hs₂⟩ := Algebra.ZariskisMainProperty.of_finiteType (R := R) q>
Tactic: obtain ⟨s₃, hs₃⟩ := hs₂.2 (algebraMap _ _ s₁)
State 3: ⊢ <goal after obtain ⟨s₃, hs₃⟩ := hs₂.2 (algebraMap _ _ s₁)>
Tactic: obtain ⟨s₃, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.exists_mk'_eq (.powers s₂) s₃
State 4: ⊢ <goal after obtain ⟨s₃, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.exists_mk'_eq (.powers s₂) s₃>
Tactic: obtain ⟨m, hm⟩ : ∃ m, ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) := by
State 5: ⊢ <goal after obtain ⟨m, hm⟩ : ∃ m, ↑s₂ ^ m * ↑s₃ = ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) := by>
Tactic: simpa [IsLocalization.Away.map, IsLocalization.map_mk', IsLocalization.mk'_eq_iff_eq_mul,  ← map_mul, ← map_pow, IsLocalization.eq_iff_exists (.powers s₂.1),  Submonoid.mem_powers_iff] using hs₃  wlog hm0 : 0 < m generalizing m  · refine this (m + 1) (by grind) (by simp)
State 6: ⊢ <goal after simpa [IsLocalization.Away.map, IsLocalization.map_mk', IsLocalization.mk'_eq_iff_eq_mul,>
Tactic: have hs₃q : s₃.1 ∉ q := fun h ↦ (show ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) ∉ q from q.primeCompl.mul_mem  (pow_mem hs₂q _) (mul_mem hs₁q (pow_mem hs₂q _))) (hm ▸ Ideal.mul_mem_left _ _ h)
State 7: ⊢ <goal after have hs₃q : s₃.1 ∉ q := fun h ↦ (show ↑s₂ ^ m * (s₁ * ↑s₂ ^ n) ∉ q from q.primeCompl.mul_mem>
Tactic: refine ⟨↑s₂ ^ m * ↑s₃, q.primeCompl.mul_mem (pow_mem hs₂q _) hs₃q, (s₂ ^ m * s₃).2,  fun q' _ hq'q _ ↦ hm ▸ Ideal.mul_mem_left _ _ (Ideal.mul_mem_right _ _ (hs₁ q' ‹_› hq'q ‹_›)),  fun q' _ hq'q _ ↦ ?_⟩  let : Algebra (integralClosure R S) (Localization.Away s₂.1) := OreLocalization.instAlgebra  let e : Localization.Away s₂ ≃ₐ[integralClosure R S] Localization.Away s₂.1 :=  .ofBijective (Localization.awayMapₐ (Algebra.ofId _ _) s₂) hs₂  let q's : Ideal (Localization.Away s₂) := q'.map (algebraMap _ _)  by_contra H
State 8: ⊢ <goal after refine ⟨↑s₂ ^ m * ↑s₃, q.primeCompl.mul_mem (pow_mem hs₂q _) hs₃q, (s₂ ^ m * s₃).2,>
Tactic: have hq's : Disjoint (Submonoid.powers s₂ : Set (integralClosure R S)) ↑q' := by
State 9: ⊢ <goal after have hq's : Disjoint (Submonoid.powers s₂ : Set (integralClosure R S)) ↑q' := by>
Tactic: rw [Ideal.disjoint_powers_iff_notMem _ (Ideal.IsPrime.isRadical ‹_›)]
State 10: ⊢ <goal after rw [Ideal.disjoint_powers_iff_notMem _ (Ideal.IsPrime.isRadical ‹_›)]>
Tactic: contrapose H; exact Ideal.mul_mem_right s₃ _ (Ideal.pow_mem_of_mem _ H m hm0)
State 11: ⊢ <goal after contrapose H; exact Ideal.mul_mem_right s₃ _ (Ideal.pow_mem_of_mem _ H m hm0)>
Tactic: have : q's.IsPrime := IsLocalization.isPrime_of_isPrime_disjoint (.powers s₂) _ _ ‹_› hq's
State 12: ⊢ <goal after have : q's.IsPrime := IsLocalization.isPrime_of_isPrime_disjoint (.powers s₂) _ _ ‹_› hq's>
Tactic: have : q's.LiesOver q' := ⟨(IsLocalization.comap_map_of_isPrime_disjoint _ _ ‹_› hq's).symm⟩
State 13: ⊢ <goal after have : q's.LiesOver q' := ⟨(IsLocalization.comap_map_of_isPrime_disjoint _ _ ‹_› hq's).symm⟩>
Tactic: have : q's.LiesOver p := .trans _ q' _
State 14: ⊢ <goal after have : q's.LiesOver p := .trans _ q' _>
Tactic: have := hs₁ (q's.comap (e.symm.toAlgHom.comp (IsScalarTower.toAlgHom _ _ _)).toRingHom)  inferInstance (by
State 15: ⊢ <goal after have := hs₁ (q's.comap (e.symm.toAlgHom.comp (IsScalarTower.toAlgHom _ _ _)).toRingHom)>
Tactic: contrapose hq'q
State 16: ⊢ <goal after contrapose hq'q>
Tactic: rw [← hq'q, Ideal.under, Ideal.comap_comap, AlgHom.toRingHom_eq_coe,  AlgHom.comp_algebraMap, q's.over_def q'])  (inferInstanceAs ((q's.comap ((e.symm.toAlgHom.comp  (IsScalarTower.toAlgHom _ _ _)).restrictScalars R)).LiesOver _))
State 17: ⊢ <goal after rw [← hq'q, Ideal.under, Ideal.comap_comap, AlgHom.toRingHom_eq_coe,>
Tactic: have : e.symm (algebraMap S (Localization.Away  ((integralClosure R S).val.toRingHom s₂)) s₁) ∈ q's := by
State 18: ⊢ <goal after have : e.symm (algebraMap S (Localization.Away>
Tactic: simpa using this
State 19: ⊢ <goal after simpa using this>
Tactic: rw [← hs₃, ← Ideal.IsPrime.mul_mem_left_iff (x := algebraMap _ _ (s₂ ^ n))] at this  · dsimp [Localization.awayMap, IsLocalization.Away.map] at this
State 20: ⊢ <goal after rw [← hs₃, ← Ideal.IsPrime.mul_mem_left_iff (x := algebraMap _ _ (s₂ ^ n))] at this>
Tactic: rw [IsLocalization.map_mk', ← e.symm.commutes, ← map_mul,  IsScalarTower.algebraMap_eq _ S _] at this  replace this : e.symm ((algebraMap _ (Localization.Away s₂.1)) s₃) ∈ q's := by
State 21: ⊢ <goal after rw [IsLocalization.map_mk', ← e.symm.commutes, ← map_mul,>
Tactic: simpa [-map_mul, -map_pow, -AlgEquiv.commutes] using this  replace this : s₃ ∈ q' := by simpa [← Ideal.mem_comap, ← q's.over_def q'] using this
State 22: ⊢ <goal after simpa [-map_mul, -map_pow, -AlgEquiv.commutes] using this>
Tactic: exact H (Ideal.mul_mem_left _ (s₂ ^ m) this)  · rw [map_pow]; exact Ideal.notMem_of_isUnit _ (.pow _ (IsLocalization.Away.algebraMap_isUnit _))  #adaptation_note /-- The maxHeartbeats bump is required after leanprover/lean4#12564. -/  set_option maxHeartbeats 400000 in -- see adaptation note
```

Question: what is the next tactic after the last recorded state, and why?[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.BigOperators.Finprod).

[User]
Complete the following proof.

## Imports
```lean
import Mathlib.Algebra.FiniteSupport.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
```

Open namespaces: Function, Set, Classical, in, Batteries.ExtendedBinder

## Surrounding context in Mathlib/Algebra/BigOperators/Finprod.lean
```lean
-- Mathlib.Algebra.BigOperators.Finprod.finprod_eq_prod_plift_of_mulSupport_toFinset_subset (L178-L184)
theorem finprod_eq_prod_plift_of_mulSupport_toFinset_subset {f : α → M}
    (hf : HasFiniteMulSupport (f ∘ PLift.down)) {s : Finset (PLift α)} (hs : hf.toFinset ⊆ s) :
    ∏ᶠ i, f i = ∏ i ∈ s, f i.down

-- Mathlib.Algebra.BigOperators.Finprod.finprod_eq_prod_plift_of_mulSupport_subset (L186-L191)
theorem finprod_eq_prod_plift_of_mulSupport_subset {f : α → M} {s : Finset (PLift α)}
    (hs : mulSupport (f ∘ PLift.down) ⊆ s) : ∏ᶠ i, f i = ∏ i ∈ s, f i.down

-- Mathlib.Algebra.BigOperators.Finprod.finprod_one (L193-L197)
theorem finprod_one : (∏ᶠ _ : α, (1 : M)) = 1
```

Relevant lemmas available in scope: `one_lt_finprod_cond`

```lean
theorem one_lt_finprod {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedCancelMonoid M]
    {f : ι → M}
    (h : ∀ i, 1 ≤ f i) (h' : ∃ i, 1 < f i) (hf : HasFiniteMulSupport f) : 1 < ∏ᶠ i, f i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Algebra.BigOperators.Finprod).

Complete the following proof.

## Imports
```lean
import Mathlib.Algebra.FiniteSupport.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
```

Open namespaces: Function, Set, Classical, in, Batteries.ExtendedBinder

## Surrounding context in Mathlib/Algebra/BigOperators/Finprod.lean
```lean
-- Mathlib.Algebra.BigOperators.Finprod.finprod_eq_prod_plift_of_mulSupport_toFinset_subset (L178-L184)
theorem finprod_eq_prod_plift_of_mulSupport_toFinset_subset {f : α → M}
    (hf : HasFiniteMulSupport (f ∘ PLift.down)) {s : Finset (PLift α)} (hs : hf.toFinset ⊆ s) :
    ∏ᶠ i, f i = ∏ i ∈ s, f i.down

-- Mathlib.Algebra.BigOperators.Finprod.finprod_eq_prod_plift_of_mulSupport_subset (L186-L191)
theorem finprod_eq_prod_plift_of_mulSupport_subset {f : α → M} {s : Finset (PLift α)}
    (hs : mulSupport (f ∘ PLift.down) ⊆ s) : ∏ᶠ i, f i = ∏ i ∈ s, f i.down

-- Mathlib.Algebra.BigOperators.Finprod.finprod_one (L193-L197)
theorem finprod_one : (∏ᶠ _ : α, (1 : M)) = 1
```

Relevant lemmas available in scope: `one_lt_finprod_cond`

```lean
theorem one_lt_finprod {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedCancelMonoid M]
    {f : ι → M}
    (h : ∀ i, 1 ≤ f i) (h' : ∃ i, 1 < f i) (hf : HasFiniteMulSupport f) : 1 < ∏ᶠ i, f i
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.


[System] You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Preorder.Finsupp).

[User]
The following Lean code fails to compile. Fix it.

```lean
lemma support_inf_union_support_sup : (f ⊓ g).support ∪ (f ⊔ g).support = f.support ∪ g.support
:= _unbalancedParenXYZ
coe_injective <| compl_injective <| by ext; simp [inf_eq_and_sup_eq_iff]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=199, messages=[Message(data='Invalid field notation: Type of\n  f ⊓ g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=53), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  f ⊔ g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=56), pos_end=Position(line=1, column=71), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  f\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=74), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=86), pos_end=Position(line=1, column=95), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.

<s>[SYSTEM_PROMPT]You are Leanstral, a Lean 4 proof assistant. Complete the task below using the provided Lean source context. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Order.Preorder.Finsupp).[/SYSTEM_PROMPT][MODEL_SETTINGS]{"reasoning_effort": "none"}[/MODEL_SETTINGS][INST]The following Lean code fails to compile. Fix it.

```lean
lemma support_inf_union_support_sup : (f ⊓ g).support ∪ (f ⊔ g).support = f.support ∪ g.support
:= _unbalancedParenXYZ
coe_injective <| compl_injective <| by ext; simp [inf_eq_and_sup_eq_iff]
```

## Compiler diagnostic
```
[CompilationUnit(i_begin=0, i_end=199, messages=[Message(data='Invalid field notation: Type of\n  f ⊓ g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=38), pos_end=Position(line=1, column=53), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  f ⊔ g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=56), pos_end=Position(line=1, column=71), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  f\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=74), pos_end=Position(line=1, column=83), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Invalid field notation: Type of\n  g\nis not known; cannot resolve field `support`\n\nHint: Consider replacing the field projection with a call to one of the following:\n  • `AddSubmonoid.support`\n  • `BoundingSieve.support`\n  • `ContDiffBumpBase.support`\n  • `DFinsupp.support`\n  • `Finsupp.support`\n  • `FreeAbelianGroup.support`\n  • `Function.support`\n  • `HahnSeries.support`\n  • `Module.support`\n  • `MvPolynomial.support`\n  • `PMF.support`\n  • `Polynomial.support`\n  • `RingPreordering.support`\n  • `SimpleGraph.support`\n  • `SkewMonoidAlgebra.support`\n  • `SkewPolynomial.support`\n  • `SummationFilter.support`\n  • `Equiv.Perm.support`\n  • `Filter.EventuallyEq.support`\n  • `Function.locallyFinsuppWithin.support`\n  • `MeasureTheory.Measure.support`\n  • `RootPairing.Base.support`\n  • `SimpleGraph.Subgraph.support`\n  • `SimpleGraph.Walk.support`\n  • `AlgebraicGeometry.Scheme.IdealSheafData.support`\n  • `Lean.Lsp.ShowDocumentClientCapabilities.support`', pos=Position(line=1, column=86), pos_end=Position(line=1, column=95), severity=<Severity.ERROR: 3>, kind='lean.invalidField._namedError'), Message(data='Unknown identifier `_unbalancedParenXYZ`', pos=Position(line=2, column=3), pos_end=Position(line=2, column=22), severity=<Severity.ERROR: 3>, kind='lean.unknownIdentifier._namedError')], invocations=None, goal_state=None, goal_src_boundaries=None, new_constants=None)]
```

Return a corrected, compiling version of the code.[/INST]


[System] You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Continuous).

[User]
Complete the following proof.

Open namespaces: FiniteDimensional, Metric, Set, List, Bornology, scoped, Topology

## Surrounding context in Mathlib/Analysis/Convex/Continuous.lean
```lean
-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L69-L73)
lemma ConcaveOn.lipschitzOnWith_of_abs_le (hf : ConcaveOn ℝ (ball x₀ r) f) (hε : 0 < ε)
    (hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
    LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))

-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L83-L87)
lemma ConcaveOn.exists_lipschitzOnWith_of_isBounded (hf : ConcaveOn ℝ (ball x₀ r) f) (hr : r' < r)
    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r')

-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L109-L112)
lemma ConcaveOn.isBoundedUnder_abs (hf : ConcaveOn ℝ C f) {x₀ : E} (hC : C ∈ 𝓝 x₀) :
    (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f| ↔ (𝓝 x₀).IsBoundedUnder (· ≥ ·) f
```

Relevant lemmas available in scope: `oneside`, `hx`, `hx'`, `hy'`, `hz`, `hab`, `hxyz`, `h`

```lean
lemma ConvexOn.exists_lipschitzOnWith_of_isBounded (hf : ConvexOn ℝ (ball x₀ r) f) (hr : r' < r)
    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r')
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.

You are Leanstral, a Lean 4 proof assistant. The target mathlib commit is 5e932f97dd25535344f80f9dd8da3aab83df0fe6 (module Mathlib.Analysis.Convex.Continuous).

Complete the following proof.

Open namespaces: FiniteDimensional, Metric, Set, List, Bornology, scoped, Topology

## Surrounding context in Mathlib/Analysis/Convex/Continuous.lean
```lean
-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L69-L73)
lemma ConcaveOn.lipschitzOnWith_of_abs_le (hf : ConcaveOn ℝ (ball x₀ r) f) (hε : 0 < ε)
    (hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
    LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε))

-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L83-L87)
lemma ConcaveOn.exists_lipschitzOnWith_of_isBounded (hf : ConcaveOn ℝ (ball x₀ r) f) (hr : r' < r)
    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r')

-- Mathlib.Analysis.Convex.Continuous.ConcaveOn (L109-L112)
lemma ConcaveOn.isBoundedUnder_abs (hf : ConcaveOn ℝ C f) {x₀ : E} (hC : C ∈ 𝓝 x₀) :
    (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f| ↔ (𝓝 x₀).IsBoundedUnder (· ≥ ·) f
```

Relevant lemmas available in scope: `oneside`, `hx`, `hx'`, `hy'`, `hz`, `hab`, `hxyz`, `h`

```lean
lemma ConvexOn.exists_lipschitzOnWith_of_isBounded (hf : ConvexOn ℝ (ball x₀ r) f) (hr : r' < r)
    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r')
```

Provide a complete Lean proof for the statement above. Do not include the statement in your answer.