awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsAdicComplete I N
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
n : ℕ
x : M
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    (Submodule.Quotient.mk ((lift I f ⋯ : (a : M) → N) x) : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
have simp : Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
  (Submodule.Quotient.mk
      ((LinearEquiv.symm (ofLinearEquiv I N) : (a : AdicCompletion I N) → N)
        ((AdicCompletion.lift I f sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N)) :
    N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
  ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
Tactic: simp only [lift, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsAdicComplete I N
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
n : ℕ
x : M
⊢ Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    (Submodule.Quotient.mk
        ((LinearEquiv.symm (ofLinearEquiv I N) : (a : AdicCompletion I N) → N)
          ((AdicCompletion.lift I f ⋯ : (a : M) → AdicCompletion I N) x : AdicCompletion I N)) :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
have rw : Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
  ((AdicCompletion.eval I N n : (a : AdicCompletion I N) → N ⧸ I ^ n • ⊤)
      ((of I N : (a : N) → AdicCompletion I N)
        ((LinearEquiv.symm (ofLinearEquiv I N) : (a : AdicCompletion I N) → N)
          ((AdicCompletion.lift I f sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N))) :
    N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
  ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
Tactic: rw [← mkQ_apply, ← eval_of]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsAdicComplete I N
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
n : ℕ
⊢ (HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤).mkQ ∘ₗ lift I f ⋯ = f n
have h₁ : ∀ (x : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    (((HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤).mkQ ∘ₗ lift I f sorry : (a : M) → N ⧸ I ^ n • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsAdicComplete I N
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
F : M →ₗ[R] N
hF : ∀ (n : ℕ), (HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤).mkQ ∘ₗ F = f n
⊢ F = lift I f ⋯
have a : Eq (α := (a : M) → N) ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F)
  ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I f sorry))
Tactic: apply DFunLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
inst_5 : IsAdicComplete I N
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
h : ∀ {m n : ℕ} (hle : m ≤ n), Submodule.factorPow I N hle ∘ₗ f n = f m
F : M →ₗ[R] N
hF : ∀ (n : ℕ), (HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤).mkQ ∘ₗ F = f n
⊢ Eq (α := (a : M) → N) ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F)
    ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I f ⋯))
have a : ∀ (n : ℕ) (m : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤)
    (Submodule.Quotient.mk ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F m))
    (Submodule.Quotient.mk ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I f sorry) m))
Tactic: apply IsHausdorff.funext I
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
⊢ Submodule.factorPow I N ⋯ ∘ₗ f n = f m
have h : ∀ (x : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((Submodule.factorPow I N sorry ∘ₗ f n : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
x : M
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((Submodule.factorPow I N ⋯ ∘ₗ f n : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
have h : Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
  ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
  ((Submodule.factorPow I N sorry ∘ₗ f n : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
    N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
Tactic: symm
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
x : M
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((Submodule.factorPow I N ⋯ ∘ₗ f n : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
have h : Antitone.{0, u_5} (β := Submodule R N) fun (n : ℕ) ↦ HSMul.hSMul (β := Submodule R 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.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
x : M
h : Antitone.{0, u_5} (β := Submodule R N) fun (n : ℕ) ↦ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((Submodule.factorPow I N ⋯ ∘ₗ f n : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
have h₁ : ∀ (m : ℕ),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((fun (n : ℕ) ↦ ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)) m)
    ((Submodule.factor sorry : (a_1 : N ⧸ I ^ a m.succ • ⊤) → N ⧸ I ^ a m • ⊤)
      ((fun (n : ℕ) ↦ ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (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.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
x : M
s : ℕ
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a s) ⊤)
    ((fun (n : ℕ) ↦ ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)) s)
    ((Submodule.factor ⋯ : (a_1 : N ⧸ I ^ a s.succ • ⊤) → N ⧸ I ^ a s • ⊤)
      ((fun (n : ℕ) ↦ ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤))
        (s + 1)))
have hf₁ : ∀ {m : ℕ} (x : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((Submodule.factorPow I N sorry ∘ₗ f (m + 1) : (a_1 : M) → N ⧸ I ^ a m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
    ((f m : (a_1 : M) → N ⧸ I ^ a m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a m) ⊤)
Tactic: simp only [LinearMap.ext_iff] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
m n : ℕ
hle : m ≤ n
⊢ Submodule.factorPow I N hle ∘ₗ extend ha f n = extend ha f m
have h : ∀ (x : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ m) ⊤)
    ((Submodule.factorPow I N hle ∘ₗ extend ha f n : (a : M) → N ⧸ I ^ m • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ m) ⊤)
    ((extend ha f m : (a : M) → N ⧸ I ^ m • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ m) ⊤)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
inst_5 : IsAdicComplete I N
n : ℕ
x : M
⊢ Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    (Submodule.Quotient.mk ((lift I ha f ⋯ : (a : M) → N) x) : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
have simp : Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
  (Submodule.Quotient.mk
      ((LinearEquiv.symm (LinearEquiv.ofBijective (of I N) sorry) : (a : AdicCompletion I N) → N)
        ((AdicCompletion.lift I (extend ha f) sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N)) :
    N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
  ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
Tactic: simp only [lift, IsAdicComplete.lift, ofLinearEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe,
    Function.comp_apply]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
inst_5 : IsAdicComplete I N
n : ℕ
x : M
⊢ Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    (Submodule.Quotient.mk
        ((LinearEquiv.symm (LinearEquiv.ofBijective (of I N) ⋯) : (a : AdicCompletion I N) → N)
          ((AdicCompletion.lift I (extend ha f) ⋯ : (a : M) → AdicCompletion I N) x : AdicCompletion I N)) :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
have rw : Eq (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
  ((AdicCompletion.eval I N (a n) : (a_1 : AdicCompletion I N) → N ⧸ I ^ a n • ⊤)
      ((of I N : (a : N) → AdicCompletion I N)
        ((LinearEquiv.symm (LinearEquiv.ofBijective (of I N) sorry) : (a : AdicCompletion I N) → N)
          ((AdicCompletion.lift I (extend ha f) sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N))) :
    N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
  ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
Tactic: rw [← mkQ_apply, ← eval_of]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
inst_5 : IsAdicComplete I N
n : ℕ
⊢ (HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤).mkQ ∘ₗ lift I ha f ⋯ = f n
have h : ∀ (x : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    (((HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤).mkQ ∘ₗ lift I ha f sorry : (a_1 : M) → N ⧸ I ^ a n • ⊤) x :
      N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    ((f n : (a_1 : M) → N ⧸ I ^ a n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
inst_5 : IsAdicComplete I N
F : M →ₗ[R] N
hF : ∀ (n : ℕ), (HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤).mkQ ∘ₗ F = f n
⊢ F = lift I ha f ⋯
have a₁ : Eq (α := (a : M) → N) ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F)
  ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I ha f sorry))
Tactic: apply DFunLike.coe_injective
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
M : Type u_4
inst_1 : AddCommGroup M
inst_2 : Module R M
N : Type u_5
inst_3 : AddCommGroup N
inst_4 : Module R N
a : ℕ → ℕ
ha : StrictMono a
f : (n : ℕ) → LinearMap.{u_1, u_1, u_4, u_5} (RingHom.id R) M (N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
hf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m
inst_5 : IsAdicComplete I N
F : M →ₗ[R] N
hF : ∀ (n : ℕ), (HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤).mkQ ∘ₗ F = f n
⊢ Eq (α := (a : M) → N) ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F)
    ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I ha f ⋯))
have a₁ : ∀ (n : ℕ) (m : M),
  Eq.{u_5 + 1} (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ a n) ⊤)
    (Submodule.Quotient.mk ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) F m))
    (Submodule.Quotient.mk ((fun (f : M →ₗ[R] N) ↦ ((⇑f : (a : M) → N) : (a : M) → N)) (lift I ha f sorry) m))
Tactic: apply IsHausdorff.StrictMono.funext I ha
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
⊢ Membership.mem (γ := Submodule R R) (jacobson ⊥) x
have rw : ∀ (y : R), IsUnit (-x * y + 1)
Tactic: rw [← Ideal.neg_mem_iff, Ideal.mem_jacobson_bot]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
⊢ IsUnit (-x * y + 1)
have rw : IsUnit (1 + -x * y)
Tactic: rw [add_comm]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
have simp : HSub.hSub (α := R) (∑ i ∈ range m, (x * y) ^ i) (∑ i ∈ range n, (x * y) ^ i) ∈ I ^ m
Tactic: simp only [f, smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ HSub.hSub (α := R) (∑ i ∈ range m, (x * y) ^ i) (∑ i ∈ range n, (x * y) ^ i) ∈ I ^ m
have rw : ∑ x_1 ∈ range (n - m), (x * y) ^ (m + x_1) ∈ I ^ m
Tactic: rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
m n : ℕ
h : m ≤ n
⊢ ∑ x_1 ∈ range (n - m), (x * y) ^ (m + x_1) ∈ I ^ m
have a : ∀ c ∈ range (n - m), (x * y) ^ (m + c) ∈ I ^ m
Tactic: apply Submodule.sum_mem
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
m n_1 : ℕ
h : m ≤ n_1
n : ℕ
h_1 : n ∈ range (n_1 - m)
⊢ (x * y) ^ (m + n) ∈ I ^ m
have a : x ^ m * (x ^ n * y ^ (m + n)) ∈ I ^ m
Tactic: rw [mul_pow, pow_add, mul_assoc]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
⊢ IsUnit (1 + -x * y)
have hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
Tactic: have hf : ∀ m n, m ≤ n → f m ≡ f n [SMOD I ^ m • (⊤ : Submodule R R)] := by
    intro m n h
    simp only [f, smul_eq_mul, Ideal.mul_top, SModEq.sub_mem]
    rw [← add_tsub_cancel_of_le h, Finset.sum_range_add, ← sub_sub, sub_self, zero_sub,
      @neg_mem_iff]
    apply Submodule.sum_mem
    intro n _
    rw [mul_pow, pow_add, mul_assoc]
    exact Ideal.mul_mem_right _ (I ^ m) (Ideal.pow_mem_pow hx m)
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
⊢ IsUnit (1 + -x * y)
have obtain : ∀ (L : R) (hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L), IsUnit (1 + -x * y)
Tactic: obtain ⟨L, hL⟩ := IsPrecomplete.prec toIsPrecomplete @hf
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
⊢ IsUnit (1 + -x * y)
have rw : ∃ (b : R), (1 + -x * y) * b = 1
Tactic: rw [isUnit_iff_exists_inv]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
⊢ ∃ (b : R), (1 + -x * y) * b = 1
have h : (1 + -x * y) * L = 1
Tactic: use L
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
⊢ (1 + -x * y) * L = 1
have h : (1 + -(x * y)) * L - 1 = 0
Tactic: rw [← sub_eq_zero, neg_mul]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
⊢ (1 + -(x * y)) * L - 1 = 0
have h : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) ((1 + -(x * y)) * L - 1) 0
Tactic: apply IsHausdorff.haus (toIsHausdorff : IsHausdorff I R)
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
hL : ∀ (n : ℕ), SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
n : ℕ
⊢ SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) ((1 + -(x * y)) * L - 1) 0
have hL₁ : SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
Tactic: specialize hL n
Cleared hypotheses: hL
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
I : Ideal R
inst_1 : IsAdicComplete I R
x : R
hx : x ∈ I
y : R
f : ℕ → R := fun (n : ℕ) ↦ ∑ i ∈ range n, (x * y) ^ i
hf : ∀ (m n : ℕ), m ≤ n → SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ m) ⊤) (f m) (f n)
L : R
n : ℕ
hL : SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) (f n) L
⊢ SModEq (R := R) (HSMul.hSMul (β := Submodule R R) (I ^ n) ⊤) ((1 + -(x * y)) * L - 1) 0
have h : ∀ (hL : f n - L ∈ I ^ n), (1 + -(x * y)) * L - 1 - 0 ∈ I ^ n
Tactic: rw [SModEq.sub_mem, smul_eq_mul, Ideal.mul_top] at hL ⊢
Cleared hypotheses: hL