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
x : AdicCompletion I M
a : (n : ℕ) → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
m n : ℕ
hmn : m ≤ n
refine_1 :
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) (Submodule.Quotient.mk (a m)) (Submodule.Quotient.mk (a n))
refine_1₁ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = Submodule.Quotient.mk (a n)
refine_1₂ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = ((HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤).mkQ : (a : M) → M ⧸ I ^ m • ⊤) (a n)
refine_1₃ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
    (Submodule.factor ⋯ : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤)
      (((HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤).mkQ : (a : M) → M ⧸ I ^ n • ⊤) (a n))
refine_1₄ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
    (Submodule.factor ⋯ : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (Submodule.Quotient.mk (a n))
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1₅ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
  (Submodule.factor sorry : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
Tactic: rw [SModEq.def, ha m, ← mkQ_apply,
      ← factor_mk (Submodule.smul_mono_left (Ideal.pow_le_pow_right hmn)) (a n),
      mkQ_apply, ha n, x.property hmn]
-- 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
x : AdicCompletion I M
a : (n : ℕ) → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
m n : ℕ
hmn : m ≤ n
refine_1 :
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) (Submodule.Quotient.mk (a m)) (Submodule.Quotient.mk (a n))
refine_1₁ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = Submodule.Quotient.mk (a n)
refine_1₂ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = ((HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤).mkQ : (a : M) → M ⧸ I ^ m • ⊤) (a n)
refine_1₃ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
    (Submodule.factor ⋯ : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤)
      (((HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤).mkQ : (a : M) → M ⧸ I ^ n • ⊤) (a n))
refine_1₄ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
    (Submodule.factor ⋯ : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (Submodule.Quotient.mk (a n))
refine_1₅ :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
    (Submodule.factor ⋯ : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1₆ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m
Tactic: rw [SModEq.def, ha m, ← mkQ_apply,
      ← factor_mk (Submodule.smul_mono_left (Ideal.pow_le_pow_right hmn)) (a n),
      mkQ_apply, ha n, x.property hmn]
-- 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
x : AdicCompletion I M
a : (n : ℕ) → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
⊢ (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M)
      (Subtype.mk (p := fun (f : ℕ → M) ↦ IsAdicCauchy I M f) a ⋯) =
    x
have refine_2 : ∀ (n : ℕ),
  (↑((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M)
            (Subtype.mk (p := fun (f : ℕ → M) ↦ IsAdicCauchy I M f) a sorry)) :
        (n : ℕ) → M ⧸ I ^ n • ⊤)
      n =
    (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: ext n
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
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
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), transitionMap I N hle ∘ₗ f n = f m
x y : M
⊢ Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
      ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
      (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) (x + y)) ⋯ =
    HAdd.hAdd (α := AdicCompletion I N)
      (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
        ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
        (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) x) ⋯)
      (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
        ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
        (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) y) ⋯)
have simp : Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
    ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
    (fun (n : ℕ) ↦
      HAdd.hAdd (α := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) (β :=
        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) ⊤)
        ((f n : (a : M) → N ⧸ I ^ n • ⊤) y : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤))
    sorry =
  HAdd.hAdd (α := AdicCompletion I N)
    (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
      ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
      (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) x) sorry)
    (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
      ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
      (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) y) sorry)
Tactic: simp only [map_add]
-- Lean 4 proof step from Mathlib.RingTheory.AdicCompletion.Basic (goal)
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
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
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), transitionMap I N hle ∘ₗ f n = f m
r : R
x : M
⊢ Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
      ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
      (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) (r • x)) ⋯ =
    HSMul.hSMul (α := R) (β := AdicCompletion I N) ((RingHom.id R : (a : R) → R) r : R)
      (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
        ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
        (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) x) ⋯)
have simp : Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
    ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
    (fun (n : ℕ) ↦
      HSMul.hSMul (β := N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) r
        ((f n : (a : M) → N ⧸ I ^ n • ⊤) x : N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤))
    sorry =
  HSMul.hSMul (β := AdicCompletion I N) r
    (Subtype.mk (p := fun (f : (n : ℕ) → N ⧸ HSMul.hSMul (β := Submodule R N) (I ^ n) ⊤) ↦
      ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I N hmn : (a : N ⧸ I ^ n • ⊤) → N ⧸ I ^ m • ⊤) (f n) = f m)
      (fun (n : ℕ) ↦ (f n : (a : M) → N ⧸ I ^ n • ⊤) x) sorry)
Tactic: simp only [LinearMapClass.map_smul, RingHom.id_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
⊢ Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) ↔ IsHausdorff I M
have mp : Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsHausdorff I M
Tactic: constructor
-- 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
mp : Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsHausdorff I M
⊢ Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) ↔ IsHausdorff I M
have mpr : IsHausdorff I M → Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
Tactic: constructor
-- 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
⊢ Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsHausdorff I M
have mp : ∀ (h : Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)) (x : M)
  (hx : ∀ (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]),
  Eq (α := AdicCompletion I M) ((of I M : (a : M) → AdicCompletion I M) x : AdicCompletion I M)
    ((of I M : (a : M) → AdicCompletion I M) 0 : AdicCompletion I M)
Tactic: refine fun h ↦ ⟨fun x hx ↦ h ?_⟩
-- 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
h : Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
x : M
hx : ∀ (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
⊢ Eq (α := AdicCompletion I M) ((of I M : (a : M) → AdicCompletion I M) x : AdicCompletion I M)
    ((of I M : (a : M) → AdicCompletion I M) 0 : AdicCompletion I M)
have mp : ∀ (n : ℕ),
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    ((↑((of I M : (a : M) → AdicCompletion I M) x) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
    ((↑((of I M : (a : M) → AdicCompletion I M) 0) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
Tactic: ext n
-- 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
h : IsHausdorff I M
⊢ Function.Injective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
have mpr : Eq (α := Submodule R M) (LinearMap.ker (of I M)) ⊥
Tactic: rw [← LinearMap.ker_eq_bot]
-- 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
h : IsHausdorff I M
⊢ Eq (α := Submodule R M) (LinearMap.ker (of I M)) ⊥
have mpr : ∀ (x : M), Membership.mem (γ := Submodule R M) (LinearMap.ker (of I M)) x ↔ Membership.mem (γ := Submodule R M) ⊥ x
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
h : IsHausdorff I M
x : M
⊢ Eq (α := AdicCompletion I M) ((of I M : (a : M) → AdicCompletion I M) x : AdicCompletion I M)
      (0 : AdicCompletion I M) ↔
    x = 0
have mpr : ∀
  (hx :
    Eq (α := AdicCompletion I M) ((of I M : (a : M) → AdicCompletion I M) x : AdicCompletion I M)
      (0 : AdicCompletion I M))
  (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
Tactic: refine ⟨fun hx ↦ h.haus x fun n ↦ ?_, fun hx ↦ by simp [hx]⟩
-- 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
h : IsHausdorff I M
x : M
hx :
  Eq (α := AdicCompletion I M) ((of I M : (a : M) → AdicCompletion I M) x : AdicCompletion I M) (0 : AdicCompletion I M)
n : ℕ
⊢ x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
have mpr : ∀
  (hx :
    Eq.{u_4 + 1} (α := (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
      (↑((of I M : (a : M) → AdicCompletion I M) x) : (n : ℕ) → M ⧸ I ^ n • ⊤) (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤))
  (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
Tactic: rw [Subtype.ext_iff] at hx
Cleared hypotheses: hx, n
-- 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
⊢ Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) ↔ IsPrecomplete I M
have mp : Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsPrecomplete I M
Tactic: constructor
-- 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
mp :
  Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsPrecomplete I M
⊢ Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) ↔ IsPrecomplete I M
have mpr : IsPrecomplete I M → Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
Tactic: constructor
-- 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
⊢ Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M) → IsPrecomplete I M
have mp : ∀ (h : Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)) (f : ℕ → M)
  (hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]),
  ∃ (L : M), ∀ (n : ℕ), f n ≡ L [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
Tactic: refine fun h ↦ ⟨fun f hmn ↦ ?_⟩
-- 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
h : Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
f : ℕ → M
hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
u : AdicCompletion I M :=
  Subtype.mk (p := fun (f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) ↦
    ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (f n) = f m)
    (fun (n : ℕ) ↦ Submodule.Quotient.mk (f n)) ⋯
⊢ ∃ (L : M), ∀ (n : ℕ), f n ≡ L [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
have mp : ∀ (x : M) (hx : (of I M : (a : M) → AdicCompletion I M) x = u),
  ∃ (L : M), ∀ (n : ℕ), f n ≡ L [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
Tactic: obtain ⟨x, hx⟩ := h u
-- 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
h : Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
f : ℕ → M
hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
u : AdicCompletion I M :=
  Subtype.mk (p := fun (f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) ↦
    ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (f n) = f m)
    (fun (n : ℕ) ↦ Submodule.Quotient.mk (f n)) ⋯
x : M
hx : (of I M : (a : M) → AdicCompletion I M) x = u
⊢ ∃ (L : M), ∀ (n : ℕ), f n ≡ L [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
have mp : ∀ (n : ℕ), f n ≡ x [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
Tactic: refine ⟨x, fun n ↦ ?_⟩
-- 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
h : Function.Surjective (α := M) (β := AdicCompletion I M) (⇑(of I M) : (a : M) → AdicCompletion I M)
f : ℕ → M
hmn : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
u : AdicCompletion I M :=
  Subtype.mk (p := fun (f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) ↦
    ∀ {m n : ℕ} (hmn : m ≤ n), (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (f n) = f m)
    (fun (n : ℕ) ↦ Submodule.Quotient.mk (f n)) ⋯
x : M
hx : (of I M : (a : M) → AdicCompletion I M) x = u
n : ℕ
⊢ Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) (Submodule.Quotient.mk (f n)) (Submodule.Quotient.mk x)
have mp : Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) (Submodule.Quotient.mk (f n))
  ((eval I M n : (a : AdicCompletion I M) → M ⧸ I ^ n • ⊤) u)
Tactic: rw [← mkQ_apply _ x, ← eval_of, hx]
-- 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
h : IsPrecomplete I M
u : AdicCompletion I M
⊢ ∃ (a : M), (of I M : (a : M) → AdicCompletion I M) a = u
have mpr : ∀ (x : (n : ℕ) → M) (hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n),
  ∃ (a : M), (of I M : (a : M) → AdicCompletion I M) a = u
Tactic: choose x hx using (fun n ↦ Submodule.Quotient.mk_surjective (I ^ n • ⊤ : Submodule R M) (u.1 n))
-- 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
h : IsPrecomplete I M
u : AdicCompletion I M
x : (n : ℕ) → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n
m n : ℕ
hmn : m ≤ n
⊢ x m ≡ x n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have rw : Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
  ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (Submodule.Quotient.mk (x n)) :
    M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
  (Submodule.Quotient.mk (x n) : M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
Tactic: rw [SModEq, hx, ← u.2 hmn, ← hx]
-- 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
h : IsPrecomplete I M
u : AdicCompletion I M
x : (n : ℕ) → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n
⊢ ∃ (a : M), (of I M : (a : M) → AdicCompletion I M) a = u
have mpr : ∀ (a : M) (ha : ∀ (n : ℕ), x n ≡ a [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]),
  ∃ (a : M), (of I M : (a : M) → AdicCompletion I M) a = u
Tactic: obtain ⟨a, ha⟩ := h.prec (f := x) (fun hmn ↦ by rw [SModEq, hx, ← u.2 hmn, ← hx]; simp)
-- 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
h : IsPrecomplete I M
u : AdicCompletion I M
x : (n : ℕ) → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n
a : M
ha : ∀ (n : ℕ), x n ≡ a [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
⊢ ∃ (a : M), (of I M : (a : M) → AdicCompletion I M) a = u
have h₁ : (of I M : (a : M) → AdicCompletion I M) a = u
Tactic: use a
-- 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
h : IsPrecomplete I M
u : AdicCompletion I M
x : (n : ℕ) → M
hx : ∀ (n : ℕ), Submodule.Quotient.mk (x n) = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n
a : M
ha : ∀ (n : ℕ), x n ≡ a [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
⊢ (of I M : (a : M) → AdicCompletion I M) a = u
have h₁ : ∀ (n : ℕ), (↑((of I M : (a : M) → AdicCompletion I M) a) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑u : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: ext n
-- 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 : ℕ
⊢ HSMul.hSMul (β := Submodule R (AdicCompletion I M)) (I ^ n) ⊤ ≤ LinearMap.ker (eval I M n)
have simp : ∀ r ∈ I ^ n, ∀ (n_1 : AdicCompletion I M), r • (↑n_1 : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
Tactic: simp only [smul_le, mem_top, LinearMap.mem_ker, map_smul, coe_eval, forall_const]
-- 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 : ℕ
r : R
r_in : r ∈ I ^ n
x : AdicCompletion I M
⊢ r • (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
have rw : r • out ((↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n) ∈ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤
Tactic: rw [← Submodule.Quotient.mk_out (x.val n), ← Quotient.mk_smul, Quotient.mk_eq_zero]
-- 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
m n : ℕ
x : AdicCompletion I M
m_ge : n ≤ m
⊢ (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
      HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤ ↔
    (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
have refine_1 : ∀
  (h :
    (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
      HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤),
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
-- 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
m n : ℕ
x : AdicCompletion I M
m_ge : n ≤ m
refine_1 :
  ∀
    (h :
      (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
        HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤),
    (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
⊢ (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
      HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤ ↔
    (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
have refine_2 : ∀ (h : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0),
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
    HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤
Tactic: refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
-- 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
m n : ℕ
x : AdicCompletion I M
m_ge : n ≤ m
h :
  (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
    HSMul.hSMul (β := Submodule R (M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)) (I ^ n) ⊤
⊢ (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0
have refine_1 : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m ∈
  ((HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤).liftQ ((HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤).mkQ ∘ₗ LinearMap.id)
      sorry).ker
Tactic: rw [← x.prop m_ge, transitionMap, Submodule.factorPow, Submodule.factor, mapQ,
      ← LinearMap.mem_ker]
-- 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
⊢ of I N ∘ₗ lift I f ⋯ = AdicCompletion.lift I f ⋯
have h₁ : ∀ (x : M),
  Eq.{u_5 + 1} (α := AdicCompletion I N) ((of I N ∘ₗ lift I f sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N)
    ((AdicCompletion.lift I f sorry : (a : M) → AdicCompletion I N) x : AdicCompletion I N)
Tactic: ext1