awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_3
S : Type u_4
A : Type u_5
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
I : Ideal S
inst_3 : IsAdicComplete I S
inst_4 : CommRing A
inst_5 : Algebra R A
f g : A →ₐ[R] S
H :
  ∀ (n : ℕ),
    Eq (α := AlgHom.{u_3, u_5, u_4} R A (S ⧸ I ^ n)) (AlgHom.comp (Quotient.mkₐ R (I ^ n)) f)
      (AlgHom.comp (Quotient.mkₐ R (I ^ n)) g)
x : A
⊢ Eq.{u_4 + 1} (α := AdicCompletion I S)
    (((AlgHom.comp (↑(AlgEquiv.restrictScalars R (ofAlgEquiv I)) : S →ₐ[R] AdicCompletion I S) f :
            AlgHom.{u_3, u_5, u_4} R A (AdicCompletion I S)) :
          (a : A) → AdicCompletion I S)
        x :
      AdicCompletion I S)
    (((AlgHom.comp (↑(AlgEquiv.restrictScalars R (ofAlgEquiv I)) : S →ₐ[R] AdicCompletion I S) g :
            AlgHom.{u_3, u_5, u_4} R A (AdicCompletion I S)) :
          (a : A) → AdicCompletion I S)
        x :
      AdicCompletion I S)
have H₁ : ∀ (n : ℕ),
  Eq (α := S ⧸ I ^ n)
    ((evalₐ I n : (a : AdicCompletion I S) → S ⧸ I ^ n)
        (((AlgHom.comp (↑(AlgEquiv.restrictScalars R (ofAlgEquiv I)) : S →ₐ[R] AdicCompletion I S) f :
              AlgHom.{u_3, u_5, u_4} R A (AdicCompletion I S)) :
            (a : A) → AdicCompletion I S)
          x) :
      S ⧸ I ^ n)
    ((evalₐ I n : (a : AdicCompletion I S) → S ⧸ I ^ n)
        (((AlgHom.comp (↑(AlgEquiv.restrictScalars R (ofAlgEquiv I)) : S →ₐ[R] AdicCompletion I S) g :
              AlgHom.{u_3, u_5, u_4} R A (AdicCompletion I S)) :
            (a : A) → AdicCompletion I S)
          x) :
      S ⧸ I ^ n)
Tactic: refine AdicCompletion.ext_evalₐ fun n ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
m n : ℕ
hle : m ≤ n
⊢ RingHom.comp (factorPow I ⋯) (f n) = f m
have a₁ : ∀ (x : R),
  Eq.{u_2 + 1} (α := S ⧸ I ^ a m)
    (((RingHom.comp (factorPow I sorry) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) : (a_1 : R) → S ⧸ I ^ a m) x :
      S ⧸ I ^ a m)
    ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
m n : ℕ
hle : m ≤ n
x : R
⊢ Eq.{u_2 + 1} (α := S ⧸ I ^ a m)
    (((RingHom.comp (factorPow I ⋯) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) : (a_1 : R) → S ⧸ I ^ a m) x :
      S ⧸ I ^ a m)
    ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
have a₁ : Eq.{u_2 + 1} (α := S ⧸ I ^ a m) ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
  (((RingHom.comp (factorPow I sorry) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) : (a_1 : R) → S ⧸ I ^ a m) x :
    S ⧸ I ^ a m)
Tactic: symm
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
m n : ℕ
hle : m ≤ n
x : R
⊢ Eq.{u_2 + 1} (α := S ⧸ I ^ a m) ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
    (((RingHom.comp (factorPow I ⋯) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) : (a_1 : R) → S ⧸ I ^ a m) x :
      S ⧸ I ^ a m)
have a₁ : Antitone.{0, u_2} (β := Submodule S S) fun (n : ℕ) ↦ I ^ a n
Tactic: refine Submodule.eq_factor_of_eq_factor_succ ?_ (fun n ↦ f n x) ?_ hle
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
m n : ℕ
hle : m ≤ n
x : R
a₁ : Antitone.{0, u_2} (β := Submodule S S) fun (n : ℕ) ↦ I ^ a n
⊢ Eq.{u_2 + 1} (α := S ⧸ I ^ a m) ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
    (((RingHom.comp (factorPow I ⋯) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) : (a_1 : R) → S ⧸ I ^ a m) x :
      S ⧸ I ^ a m)
have a₂ : ∀ (m : ℕ),
  Eq.{u_2 + 1} (α := S ⧸ I ^ a m) ((fun (n : ℕ) ↦ ((f n : (a_1 : R) → S ⧸ I ^ a n) x : S ⧸ I ^ a n)) m)
    ((Submodule.factor sorry : (a_1 : S ⧸ I ^ a m.succ) → S ⧸ I ^ a m)
      ((fun (n : ℕ) ↦ ((f n : (a_1 : R) → S ⧸ I ^ a n) x : S ⧸ I ^ a n)) (m + 1)))
Tactic: refine Submodule.eq_factor_of_eq_factor_succ ?_ (fun n ↦ f n x) ?_ hle
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (hypothesis)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
m n : ℕ
hle : m ≤ n
x : R
s : ℕ
⊢ Eq.{u_2 + 1} (α := S ⧸ I ^ a s) ((fun (n : ℕ) ↦ ((f n : (a_1 : R) → S ⧸ I ^ a n) x : S ⧸ I ^ a n)) s)
    ((Submodule.factor ⋯ : (a_1 : S ⧸ I ^ a s.succ) → S ⧸ I ^ a s)
      ((fun (n : ℕ) ↦ ((f n : (a_1 : R) → S ⧸ I ^ a n) x : S ⧸ I ^ a n)) (s + 1)))
have hf₁ : ∀ {m : ℕ} (x : R),
  Eq.{u_2 + 1} (α := S ⧸ I ^ a m)
    (((@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I sorry : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
              (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) :
            RingHom.{u_1, u_2} R (S ⧸ I ^ a m)) :
          (a_1 : R) → S ⧸ I ^ a m)
        x :
      S ⧸ I ^ a m)
    ((f m : (a_1 : R) → S ⧸ I ^ a m) x : S ⧸ I ^ a m)
Tactic: simp only [RingHom.ext_iff] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
inst_2 : IsAdicComplete I S
m n : ℕ
hle : m ≤ n
⊢ Eq (α := RingHom.{u_1, u_2} R (S ⧸ I ^ m))
    (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I hle : S ⧸ I ^ n →+* S ⧸ I ^ m)
        ((fun (n : ℕ) ↦ (RingHom.comp (factorPow I ⋯) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ id n))) n :
          RingHom.{u_1, u_2} R (S ⧸ I ^ id n)) :
      RingHom.{u_1, u_2} R (S ⧸ I ^ m))
    ((fun (n : ℕ) ↦ (RingHom.comp (factorPow I ⋯) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ id n))) m :
      RingHom.{u_1, u_2} R (S ⧸ I ^ id m))
have a₁ : ∀ (x : R),
  Eq.{u_2 + 1} (α := S ⧸ I ^ m)
    (((@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I hle : S ⧸ I ^ n →+* S ⧸ I ^ m)
              ((fun (n : ℕ) ↦ (RingHom.comp (factorPow I sorry) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ id n))) n :
                RingHom.{u_1, u_2} R (S ⧸ I ^ id n)) :
            RingHom.{u_1, u_2} R (S ⧸ I ^ m)) :
          (a : R) → S ⧸ I ^ m)
        x :
      S ⧸ I ^ m)
    ((((fun (n : ℕ) ↦ (RingHom.comp (factorPow I sorry) (f n) : RingHom.{u_1, u_2} R (S ⧸ I ^ id n))) m :
            RingHom.{u_1, u_2} R (S ⧸ I ^ id m)) :
          (a : R) → S ⧸ I ^ m)
        x :
      S ⧸ I ^ m)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
inst_2 : IsAdicComplete I S
n : ℕ
⊢ RingHom.comp (Ideal.Quotient.mk (I ^ a n)) (liftRingHom I ha f ⋯) = f n
have a₁ : ∀ (x : R),
  Eq.{u_2 + 1} (α := S ⧸ I ^ a n)
    (((RingHom.comp (Ideal.Quotient.mk (I ^ a n)) (liftRingHom I ha f sorry) : RingHom.{u_1, u_2} R (S ⧸ I ^ a n)) :
          (a_1 : R) → S ⧸ I ^ a n)
        x :
      S ⧸ I ^ a n)
    ((f n : (a_1 : R) → S ⧸ I ^ a n) x : S ⧸ I ^ a n)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
inst_2 : IsAdicComplete I S
F : R →+* S
hF : ∀ (n : ℕ), RingHom.comp (Ideal.Quotient.mk (I ^ a n)) F = f n
⊢ F = liftRingHom I ha f ⋯
have a₁ : Eq (α := (a : R) → S) ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) F)
  ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) (liftRingHom I ha f sorry))
Tactic: apply DFunLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.RingHom (goal)
Goal:
R : Type u_1
S : Type u_2
inst : NonAssocSemiring R
inst_1 : CommRing S
I : Ideal S
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → RingHom.{u_1, u_2} R (S ⧸ I ^ a n)
hf :
  ∀ {m : ℕ},
    @RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (factorPow I ⋯ : S ⧸ I ^ a (m + 1) →+* S ⧸ I ^ a m)
        (f (m + 1) : RingHom.{u_1, u_2} R (S ⧸ I ^ a (m + 1))) =
      f m
inst_2 : IsAdicComplete I S
F : R →+* S
hF : ∀ (n : ℕ), RingHom.comp (Ideal.Quotient.mk (I ^ a n)) F = f n
⊢ Eq (α := (a : R) → S) ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) F)
    ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) (liftRingHom I ha f ⋯))
have a₁ : ∀ (n : ℕ) (r : R),
  Eq (α := S ⧸ I ^ a n)
    ((Ideal.Quotient.mk (I ^ a n) : (a_1 : S) → S ⧸ I ^ a n)
      ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) F r))
    ((Ideal.Quotient.mk (I ^ a n) : (a_1 : S) → S ⧸ I ^ a n)
      ((fun (f : R →+* S) ↦ ((⇑f : (a : R) → S) : (a : R) → S)) (liftRingHom I ha f sorry) r))
Tactic: apply IsHausdorff.StrictMono.funext' I ha
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : TopologicalSpace R
I : Ideal R
hI : IsAdic I
⊢ IsHausdorff I R ↔ T2Space R
have rw : (∀ (x : R), (∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) x 0) → x = 0) ↔
  Subset (α := Set R) (⋂₀ FilterBasis.sets (@AddGroupFilterBasis.toFilterBasis _ AddGroupWithOne.toAddGroup _)) {0}
Tactic: rw [I.ringFilterBasis.t2Space_iff_sInter_subset hI.symm, isHausdorff_iff]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
⊢ IsPrecomplete I R ↔ CompleteSpace R
have this : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
Tactic: have := hI.hasBasis_nhds_zero.isCountablyGenerated
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
⊢ IsPrecomplete I R ↔ CompleteSpace R
have this₁ : (𝓤 R).IsCountablyGenerated
Tactic: have : (𝓤 R).IsCountablyGenerated := IsUniformAddGroup.uniformity_countably_generated
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
⊢ IsPrecomplete I R ↔ CompleteSpace R
have simp : (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n) ↔ CompleteSpace R
Tactic: simp only [isPrecomplete_iff, smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
⊢ (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n) ↔ CompleteSpace R
have mp : (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n) → CompleteSpace R
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
mp : (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n) → CompleteSpace R
⊢ (∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n) ↔ CompleteSpace R
have mpr : CompleteSpace R → ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
⊢ CompleteSpace R
have mp : ∀ (u : ℕ → R) (hu : CauchySeq u), ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
Tactic: refine UniformSpace.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
⊢ ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
have this₁ : ∀ (i : ℕ), ∃ (N : ℕ), ∀ (m : ℕ), N ≤ m → ∀ (n : ℕ), N ≤ n → u n - u m ∈ I ^ i
Tactic: have : ∀ i, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → u n - u m ∈ I ^ i := by
      simpa using hI.hasBasis_nhds_zero.uniformity_of_nhds_zero.cauchySeq_iff.mp hu
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this_2 : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
this : ∀ (i : ℕ), ∃ (N : ℕ), ∀ (m : ℕ), N ≤ m → ∀ (n : ℕ), N ≤ n → u n - u m ∈ I ^ i
⊢ ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
have mp : ∀ (N : (i : ℕ) → ℕ) (hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i),
  ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
Tactic: choose N hN using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
⊢ ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
have mp : ∀ (L : R) (hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n), ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
Tactic: obtain ⟨L, hL⟩ := H (fun i ↦ u ((Finset.Iic i).sup N))
      fun _ ↦ hN _ _ (Finset.le_sup (by simpa)) _ (Finset.le_sup (by simp))
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
⊢ ∃ (a : R), Filter.Tendsto u Filter.atTop (𝓝 a)
have h : Filter.Tendsto u Filter.atTop (𝓝 L)
Tactic: use L
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
⊢ Filter.Tendsto u Filter.atTop (𝓝 L)
have h : ∀ (i : ℕ), ∃ (N : ℕ), ∀ (n : ℕ), N ≤ n → u n - L ∈ I ^ i
Tactic: suffices ∀ i, ∃ N, ∀ n, N ≤ n → u n - L ∈ I ^ i by
      simpa [(hI.hasBasis_nhds L).tendsto_right_iff, sub_eq_neg_add]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
⊢ ∀ (i : ℕ), ∃ (N : ℕ), ∀ (n : ℕ), N ≤ n → u n - L ∈ I ^ i
have h : ∀ (i n : ℕ) (hn : (Finset.Iic i).sup N ≤ n), u n - L ∈ I ^ i
Tactic: refine fun i ↦ ⟨(Finset.Iic i).sup N, fun n hn ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
i n : ℕ
hn : (Finset.Iic i).sup N ≤ n
⊢ u n - L ∈ I ^ i
have this₁ : u n - u ((Finset.Iic i).sup N) + (u ((Finset.Iic i).sup N) - L) ∈ I ^ i
Tactic: have := Ideal.add_mem _ (hN i ((Finset.Iic i).sup N) (Finset.le_sup (by simp))
      n (.trans (Finset.le_sup (by simp)) hn)) (hL i)
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this_2 : (𝓤 R).IsCountablyGenerated
H : ∀ (f : ℕ → R), (∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m) → ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
u : ℕ → R
hu : CauchySeq u
N : (i : ℕ) → ℕ
hN : ∀ (i m : ℕ), N i ≤ m → ∀ (n : ℕ), N i ≤ n → u n - u m ∈ I ^ i
L : R
hL : ∀ (n : ℕ), u ((Finset.Iic n).sup N) - L ∈ I ^ n
i n : ℕ
hn : (Finset.Iic i).sup N ≤ n
this : u n - u ((Finset.Iic i).sup N) + (u ((Finset.Iic i).sup N) - L) ∈ I ^ i
⊢ u n - L ∈ I ^ i
have h : ∀ (this : u n - L ∈ I ^ i), u n - L ∈ I ^ i
Tactic: rwa [sub_add_sub_cancel] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : CompleteSpace R
f : ℕ → R
hf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m
⊢ ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
have mpr : ∀ (L : R) (hL : Filter.map f Filter.atTop ≤ 𝓝 L), ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
Tactic: obtain ⟨L, hL⟩ := CompleteSpace.complete (f := Filter.atTop.map f)
      (hI.hasBasis_nhds_zero.uniformity_of_nhds_zero.cauchySeq_iff.mpr fun i _ ↦
        ⟨i, fun m hm n hn ↦ by simpa using Ideal.sub_mem _ (hf hm) (hf hn)⟩)
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : CompleteSpace R
f : ℕ → R
hf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m
L : R
hL : Filter.map f Filter.atTop ≤ 𝓝 L
⊢ ∃ (L : R), ∀ (n : ℕ), f n - L ∈ I ^ n
have mpr : ∀ (i : ℕ), f i - L ∈ I ^ i
Tactic: refine ⟨L, fun i ↦ ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
this_1 : Filter.IsCountablyGenerated.{u_1} (α := R) (𝓝 0)
this : (𝓤 R).IsCountablyGenerated
H : CompleteSpace R
f : ℕ → R
hf : ∀ {m n : ℕ}, m ≤ n → f m - f n ∈ I ^ m
L : R
hL : Filter.map f Filter.atTop ≤ 𝓝 L
i : ℕ
⊢ f i - L ∈ I ^ i
have mpr : ∀ (N : ℕ) (hN : ∀ (n : ℕ), N ≤ n → f n - L ∈ I ^ i), f i - L ∈ I ^ i
Tactic: obtain ⟨N, hN⟩ : ∃ N, ∀ n, N ≤ n → f n - L ∈ I ^ i := by
      simpa [sub_eq_neg_add] using (hI.hasBasis_nhds L).tendsto_right_iff.mp hL i
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
⊢ IsAdicComplete I R ↔ CompleteSpace R ∧ T2Space R
have rw : IsHausdorff I R ∧ IsPrecomplete I R ↔ CompleteSpace R ∧ T2Space R
Tactic: rw [isAdicComplete_iff, hI.isHausdorff_iff, hI.isPrecomplete_iff, and_comm]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
rw : IsHausdorff I R ∧ IsPrecomplete I R ↔ CompleteSpace R ∧ T2Space R
⊢ IsAdicComplete I R ↔ CompleteSpace R ∧ T2Space R
have rw₁ : T2Space R ∧ IsPrecomplete I R ↔ CompleteSpace R ∧ T2Space R
Tactic: rw [isAdicComplete_iff, hI.isHausdorff_iff, hI.isPrecomplete_iff, and_comm]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Topology (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : UniformSpace R
inst_2 : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
rw : IsHausdorff I R ∧ IsPrecomplete I R ↔ CompleteSpace R ∧ T2Space R
rw₁ : T2Space R ∧ IsPrecomplete I R ↔ CompleteSpace R ∧ T2Space R
⊢ IsAdicComplete I R ↔ CompleteSpace R ∧ T2Space R
have rw₂ : T2Space R ∧ CompleteSpace R ↔ CompleteSpace R ∧ T2Space R
Tactic: rw [isAdicComplete_iff, hI.isHausdorff_iff, hI.isPrecomplete_iff, and_comm]