awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
m n : ℕ
hmn : m ≤ n
rw :
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) 0) (0 m)
rw₁ :
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) 0) 0
⊢ Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (0 n)) (0 m)
have rw₂ : Eq.{u_4 + 1} (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) 0 0
Tactic: rw [Pi.zero_apply, Pi.zero_apply, map_zero]
-- 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
c : R
f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤
hf :
  Membership.mem (γ := Set ((n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤))
    {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}
    f
m n : ℕ
hmn : m ≤ n
⊢ (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((c • f) n) = (c • f) m
have rw : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = (c • f) m
Tactic: rw [Pi.smul_apply, Pi.smul_apply, map_smul, hf hmn]
-- 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
c : R
f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤
hf :
  Membership.mem (γ := Set ((n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤))
    {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}
    f
m n : ℕ
hmn : m ≤ n
rw : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = (c • f) m
⊢ (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((c • f) n) = (c • f) m
have rw₁ : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = c • f m
Tactic: rw [Pi.smul_apply, Pi.smul_apply, map_smul, hf hmn]
-- 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
c : R
f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤
hf :
  Membership.mem (γ := Set ((n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤))
    {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}
    f
m n : ℕ
hmn : m ≤ n
rw : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = (c • f) m
rw₁ : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = c • f m
⊢ (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((c • f) n) = (c • f) m
have rw₂ : HSMul.hSMul (β := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) c
    ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (f n) :
      M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) =
  c • f m
Tactic: rw [Pi.smul_apply, Pi.smul_apply, map_smul, hf hmn]
-- 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
c : R
f : (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤
hf :
  Membership.mem (γ := Set ((n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤))
    {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}
    f
m n : ℕ
hmn : m ≤ n
rw : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = (c • f) m
rw₁ : (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (c • f n) = c • f m
rw₂ :
  HSMul.hSMul (β := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) c
      ((transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (f n) :
        M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) =
    c • f m
⊢ (transitionMap I M hmn : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) ((c • f) n) = (c • f) m
have rw₃ : c • f m = c • f m
Tactic: rw [Pi.smul_apply, Pi.smul_apply, map_smul, hf hmn]
-- 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
inst_5 : SMul S T
inst_6 : SMul S R
inst_7 : SMul T R
inst_8 : SMul S M
inst_9 : SMul T M
inst_10 : IsScalarTower S R M
inst_11 : IsScalarTower T R M
inst_12 : IsScalarTower S T M
s : S
t : T
f : AdicCompletion I M
⊢ (s • t) • f = s • t • f
have h : ∀ (n : ℕ), (↑((s • t) • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑(s • t • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: ext
-- 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
inst_5 : SMul S R
inst_6 : SMul T R
inst_7 : SMul S M
inst_8 : SMul T M
inst_9 : IsScalarTower S R M
inst_10 : IsScalarTower T R M
inst_11 : SMulCommClass S T M
s : S
t : T
f : AdicCompletion I M
⊢ s • t • f = t • s • f
have h : ∀ (n : ℕ), (↑(s • t • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑(t • s • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: ext
-- 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
inst_5 : SMul S R
inst_6 : SMul.{u_2, u_1} Sᵐᵒᵖ R
inst_7 : SMul S M
inst_8 : SMul.{u_2, u_4} Sᵐᵒᵖ M
inst_9 : IsScalarTower S R M
inst_10 : IsScalarTower.{u_2, u_1, u_4} Sᵐᵒᵖ R M
inst_11 : IsCentralScalar S M
s : S
f : AdicCompletion I M
⊢ MulOpposite.op s • f = s • f
have h : ∀ (n : ℕ), (↑(MulOpposite.op s • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑(s • f) : (n : ℕ) → M ⧸ I ^ n • ⊤) 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
ι : Type u_6
s : Finset ι
f : ι → AdicCompletion I M
⊢ Eq.{u_4 + 1} (α := (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    (↑(∑ i ∈ s, f i) : (n : ℕ) → M ⧸ I ^ n • ⊤) (∑ i ∈ s, (↑(f i) : (n : ℕ) → M ⧸ I ^ n • ⊤))
have simp_rw : Eq (α := (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
  ((incl I M : (a : AdicCompletion I M) → (n : ℕ) → M ⧸ I ^ n • ⊤) (∑ i ∈ s, f i) :
    (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
  (∑ x ∈ s, (incl I M : (a : AdicCompletion I M) → (n : ℕ) → M ⧸ I ^ n • ⊤) (f x) :
    (n : ℕ) → M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
Tactic: simp_rw [← funext (incl_apply _ _ _), map_sum]
-- 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
x : AdicCompletion I M
h : ∀ (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R (AdicCompletion I M)) (I ^ n) ⊤]
n : ℕ
⊢ (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n
have refine_1 : ∀ (r : R) (hr : r ∈ I ^ n) (x : AdicCompletion I M),
  Membership.mem (γ := Submodule R (AdicCompletion I M)) ⊤ x →
    (↑(r • x) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: refine smul_induction_on (SModEq.zero.1 <| h n) (fun r hr x _ ↦ ?_) (fun x y hx hy ↦ ?_)
-- 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
x : AdicCompletion I M
h : ∀ (n : ℕ), x ≡ 0 [SMOD HSMul.hSMul (β := Submodule R (AdicCompletion I M)) (I ^ n) ⊤]
n : ℕ
refine_1 :
  ∀ (r : R) (hr : r ∈ I ^ n) (x : AdicCompletion I M),
    Membership.mem (γ := Submodule R (AdicCompletion I M)) ⊤ x →
      (↑(r • x) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n
⊢ (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n
have refine_2 : ∀ (x y : AdicCompletion I M) (hx : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
  (hy : (↑y : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n),
  (↑(x + y) : (n : ℕ) → M ⧸ I ^ n • ⊤) n = (↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n
Tactic: refine smul_induction_on (SModEq.zero.1 <| h n) (fun r hr x _ ↦ ?_) (fun x y hx hy ↦ ?_)
-- 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
x_1 : AdicCompletion I M
h : ∀ (n : ℕ), x_1 ≡ 0 [SMOD HSMul.hSMul (β := Submodule R (AdicCompletion I M)) (I ^ n) ⊤]
n : ℕ
r : R
hr : r ∈ I ^ n
x : AdicCompletion I M
h_1 : Membership.mem (γ := Submodule R (AdicCompletion I M)) ⊤ x
⊢ r • (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n = 0 n
have refine_1 : ∀ (a : M), HSMul.hSMul (β := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤) r (Quotient.mk'' a) = 0 n
Tactic: induction x.val n using Quotient.inductionOn' with | _ 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
m n : ℕ
hmn : m ≤ n
⊢ Eq (α := AdicCompletion I M →ₗ[R] M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    (transitionMap I M hmn ∘ₗ eval I M n) (eval I M m)
have h : ∀ (x : AdicCompletion I M),
  Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    ((transitionMap I M hmn ∘ₗ eval I M n : (a : AdicCompletion I M) → M ⧸ I ^ m • ⊤) x :
      M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤)
    ((eval I M m : (a : AdicCompletion I M) → M ⧸ I ^ m • ⊤) x : M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ 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
f : ℕ → M
⊢ IsAdicCauchy I M f ↔ ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
have mp : IsAdicCauchy I M f → ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
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
f : ℕ → M
mp : IsAdicCauchy I M f → ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
⊢ IsAdicCauchy I M f ↔ ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
have mpr : (∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]) → IsAdicCauchy I M f
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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
m n_1 n : ℕ
hmn : m ≤ n
ih trans : f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
⊢ f m ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have trans₁ : f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
Tactic: trans
-- 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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
m n_1 n : ℕ
hmn : m ≤ n
ih trans : f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
trans₁ : f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
⊢ f m ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have trans₂ : M
Tactic: trans
-- 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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
m n : ℕ
hmn : m ≤ n
⊢ f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have mpr : f m ≡ f m [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
Tactic: induction n, hmn using Nat.le_induction with
    | base => rfl
    | succ n hmn ih =>
        trans
        · exact ih
        · refine SModEq.mono (smul_mono (Ideal.pow_le_pow_right hmn) (by rfl)) (h n)
Cleared hypotheses: 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
f : ℕ → M
h : ∀ (n : ℕ), f n ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤]
m n : ℕ
hmn : m ≤ n
mpr : f m ≡ f m [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
⊢ f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have mpr₁ : ∀ (n : ℕ) (hmn : m ≤ n) (ih : f m ≡ f n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]),
  f m ≡ f (n + 1) [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
Tactic: induction n, hmn using Nat.le_induction with
    | base => rfl
    | succ n hmn ih =>
        trans
        · exact ih
        · refine SModEq.mono (smul_mono (Ideal.pow_le_pow_right hmn) (by rfl)) (h n)
Cleared hypotheses: 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
f : AdicCauchySequence I M
h : ∃ (k : ℕ), ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤
⊢ Eq (α := AdicCompletion I M) ((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f : AdicCompletion I M)
    (0 : AdicCompletion I M)
have obtain : ∀ (k : ℕ) (h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤),
  Eq (α := AdicCompletion I M) ((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f : AdicCompletion I M)
    (0 : AdicCompletion I M)
Tactic: obtain ⟨k, h⟩ := h
Cleared hypotheses: 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
f : AdicCauchySequence I M
k : ℕ
h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤
⊢ Eq (α := AdicCompletion I M) ((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f : AdicCompletion I M)
    (0 : AdicCompletion I M)
have h₁ : ∀ (n : ℕ),
  Eq.{u_4 + 1} (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    ((↑((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
    ((↑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
f : AdicCauchySequence I M
k : ℕ
h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤
n : ℕ
⊢ Eq.{u_4 + 1} (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    ((↑((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
    ((↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
have h₁ : ∀ (m : ℕ) (hnm : m ≥ n + k) (l : ℕ) (hnl : l ≥ n + k)
  (hl : (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤),
  Eq.{u_4 + 1} (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    ((↑((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
    ((↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
Tactic: obtain ⟨m, hnm, l, hnl, hl⟩ := h (n + k) (by lia)
-- 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
f : AdicCauchySequence I M
k : ℕ
h : ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤
n m : ℕ
hnm : m ≥ n + k
l : ℕ
hnl : l ≥ n + k
hl : (↑f : ℕ → M) m ∈ HSMul.hSMul (β := Submodule R M) (I ^ l) ⊤
⊢ Eq.{u_4 + 1} (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
    ((↑((mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) f) : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
    ((↑0 : (n : ℕ) → M ⧸ I ^ n • ⊤) n)
have h₁ : Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
  (Submodule.Quotient.mk ((↑f : ℕ → M) m) : M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
  (0 n : M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤)
Tactic: rw [mk_apply_coe, Submodule.mkQ_apply, val_zero,
    ← AdicCauchySequence.mk_eq_mk (show n ≤ m by lia)]
-- 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 : AdicCauchySequence I M), (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) a = x
have choose : ∀ (a : (n : ℕ) → M) (ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n),
  ∃ (a : AdicCauchySequence I M), (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) a = x
Tactic: choose a ha using fun n ↦ Submodule.Quotient.mk_surjective _ (x.val 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
x : AdicCompletion I M
a : (n : ℕ) → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
⊢ ∃ (a : AdicCauchySequence I M), (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) a = x
have refine_1 : IsAdicCauchy I M a
Tactic: refine ⟨⟨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
x : AdicCompletion I M
a : (n : ℕ) → M
ha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) n
refine_1 : IsAdicCauchy I M a
⊢ ∃ (a : AdicCauchySequence I M), (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M) a = x
have refine_2 : (mk I M : (a : AdicCauchySequence I M) → AdicCompletion I M)
    (Subtype.mk (p := fun (f : ℕ → M) ↦ IsAdicCauchy I M f) a sorry) =
  x
Tactic: refine ⟨⟨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
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
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1 : Eq (α := M ⧸ HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤) (Submodule.Quotient.mk (a m)) (Submodule.Quotient.mk (a 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))
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1₁ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m = Submodule.Quotient.mk (a 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)
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have 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)
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)
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1₃ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
  (Submodule.factor sorry : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤)
    (((HSMul.hSMul (β := Submodule R M) (I ^ n) ⊤).mkQ : (a : M) → M ⧸ I ^ n • ⊤) (a 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))
⊢ a m ≡ a n [SMOD HSMul.hSMul (β := Submodule R M) (I ^ m) ⊤]
have refine_1₄ : (↑x : (n : ℕ) → M ⧸ I ^ n • ⊤) m =
  (Submodule.factor sorry : (a : M ⧸ I ^ n • ⊤) → M ⧸ I ^ m • ⊤) (Submodule.Quotient.mk (a 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]