awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
inst_3 : IsDomain R
inst_4 : Module.IsTorsionFree R P
hS : 0 ∉ S
I : FractionalIdeal S P
hI : I.num = ⊥
x : P
hx : x ∈ (↑I : Submodule R P)
rw : Eq (α := Submodule R P) (Submodule.map (Algebra.linearMap R P) I.num) ⊥
rw₁ : Eq (α := Submodule R P) (Submodule.map (Algebra.linearMap R P) ⊥) ⊥
⊢ I.den • (↑I : Submodule R P) = ⊥
have rw₂ : Eq.{u_2 + 1} (α := Submodule R P) ⊥ ⊥
Tactic: rw [den_mul_self_eq_num, hI, Submodule.map_bot]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
inst_3 : IsDomain R
inst_4 : Module.IsTorsionFree R P
hS : 0 ∉ S
I : FractionalIdeal S P
hI : I.num = ⊥
x : P
hx : x ∈ (↑I : Submodule R P)
⊢ (↑I.den : R) • x = 0
have h_eq : I.den • (↑I : Submodule R P) = ⊥
Tactic: have h_eq : I.den • (I : Submodule R P) = ⊥ := by rw [den_mul_self_eq_num, hI, Submodule.map_bot]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
⊢ Eq.{u_2 + 1} (α := Submodule R P) (↑1 : Submodule R P) 1
have rw : Eq (α := Submodule R P) (coeSubmodule P ⊤ : Submodule R P) 1
Tactic: rw [coe_one_eq_coeSubmodule_top, coeSubmodule_top]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
rw : Eq (α := Submodule R P) (coeSubmodule P ⊤ : Submodule R P) 1
⊢ Eq.{u_2 + 1} (α := Submodule R P) (↑1 : Submodule R P) 1
have rw₁ : Eq.{u_2 + 1} (α := Submodule R P) 1 1
Tactic: rw [coe_one_eq_coeSubmodule_top, coeSubmodule_top]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : FractionalIdeal S P
x : P
hx : Membership.mem (γ := FractionalIdeal S P) 0 x
⊢ x ∈ I
have h : x = 0
Tactic: convert! zero_mem I
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : FractionalIdeal S P
x : P
hx : Membership.mem (γ := FractionalIdeal S P) 0 x
⊢ x = 0
have h : Eq.{u_2 + 1} (α := P) 0 0
Tactic: rw [(mem_zero_iff _).mp hx]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • b)
have rcases : ∀ (bI : P) (hbI : bI ∈ I) (bJ : P) (hbJ : bJ ∈ J) (hb : bI + bJ ∈ I ⊔ J), IsInteger R ((aI * aJ) • (bI + bJ))
Tactic: rcases mem_sup.mp hb with ⟨bI, hbI, bJ, hbJ, rfl⟩
Cleared hypotheses: b, hb
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • (bI + bJ))
have rw : IsInteger R ((aI * aJ) • bI + (aI * aJ) • bJ)
Tactic: rw [smul_add]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI + (aI * aJ) • bJ)
have ha : IsInteger R ((aI * aJ) • bI)
Tactic: apply isInteger_add
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
ha : IsInteger R ((aI * aJ) • bI)
⊢ IsInteger R ((aI * aJ) • bI + (aI * aJ) • bJ)
have hb₁ : IsInteger R ((aI * aJ) • bJ)
Tactic: apply isInteger_add
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bI)
have ha : IsInteger R (aJ • aI • bI)
Tactic: rw [mul_smul, smul_comm]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
bI : P
hbI : bI ∈ I
bJ : P
hbJ : bJ ∈ J
hb : bI + bJ ∈ I ⊔ J
⊢ IsInteger R ((aI * aJ) • bJ)
have hb₁ : IsInteger R (aI • aJ • bJ)
Tactic: rw [mul_smul]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
J : Submodule R P
b : P
hb : b ∈ I ⊓ J
⊢ IsInteger R (aI • b)
have hbI : b ∈ I
Tactic: rcases mem_inf.mp hb with ⟨hbI, _⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
J : Submodule R P
b : P
hb : b ∈ I ⊓ J
hbI : b ∈ I
⊢ IsInteger R (aI • b)
have right : b ∈ J
Tactic: rcases mem_inf.mp hb with ⟨hbI, _⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : FractionalIdeal S P
x : P
⊢ x ∈ I + J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = x
have rw : x ∈ (↑I : Submodule R P) ⊔ (↑J : Submodule R P) ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = x
Tactic: rw [← mem_coe, coe_add, Submodule.add_eq_sup]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
inst_3 : FaithfulSMul R P
I J : Ideal R
⊢ Eq.{u_2 + 1} (α := FractionalIdeal S P) (↑(I ⊓ J) : FractionalIdeal S P)
    ((↑I : FractionalIdeal S P) ⊓ (↑J : FractionalIdeal S P))
have a : Eq.{u_2 + 1} (α := Submodule R P)
  ((fun (I : FractionalIdeal S P) ↦ (↑I : Submodule R P)) (↑(I ⊓ J) : FractionalIdeal S P))
  ((fun (I : FractionalIdeal S P) ↦ (↑I : Submodule R P)) ((↑I : FractionalIdeal S P) ⊓ (↑J : FractionalIdeal S P)))
Tactic: apply coeToSubmodule_injective
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : Submodule R P
h : IsFractional S I
⊢ IsFractional.{u_1, u_2} S (P := P) 0
have h : Eq.{u_2 + 1} (α := Submodule R P) 0 (↑(↑0 : FractionalIdeal S P) : Submodule R P)
Tactic: convert! ((0 : Ideal R) : FractionalIdeal S P).isFractional
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I : Submodule R P
n : ℕ
h : IsFractional S I
⊢ IsFractional S ((n + 1) • I)
have rw : IsFractional S (n • I + I)
Tactic: rw [succ_nsmul]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
⊢ IsInteger R ((aI * aJ) • b)
have refine_1 : ∀ m ∈ I, ∀ n ∈ J, IsInteger R ((aI * aJ) • (m * n))
Tactic: refine Submodule.mul_induction_on hb ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
refine_1 : ∀ m ∈ I, ∀ n ∈ J, IsInteger R ((aI * aJ) • (m * n))
⊢ IsInteger R ((aI * aJ) • b)
have refine_2 : ∀ (x y : P), IsInteger R ((aI * aJ) • x) → IsInteger R ((aI * aJ) • y) → IsInteger R ((aI * aJ) • (x + y))
Tactic: refine Submodule.mul_induction_on hb ?_ ?_
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : n ∈ J
⊢ IsInteger R ((aI * aJ) • (m * n))
have refine_1 : ∀ (n' : R) (hn' : (algebraMap R P : (a : R) → P) n' = aJ • n), IsInteger R ((aI * aJ) • (m * n))
Tactic: obtain ⟨n', hn'⟩ := hJ n hn
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : n ∈ J
n' : R
hn' : (algebraMap R P : (a : R) → P) n' = aJ • n
⊢ IsInteger R ((aI * aJ) • (m * n))
have refine_1 : IsInteger R (aI • n' • m)
Tactic: rw [mul_smul, mul_comm m, ← smul_mul_assoc, ← hn', ← Algebra.smul_def]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
m : P
hm : m ∈ I
n : P
hn : n ∈ J
n' : R
hn' : (algebraMap R P : (a : R) → P) n' = aJ • n
⊢ IsInteger R (aI • n' • m)
have refine_1 : n' • m ∈ I
Tactic: apply hI
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Submodule R P
aI : R
haI : aI ∈ S
hI : ∀ b ∈ I, IsInteger R (aI • b)
aJ : R
haJ : aJ ∈ S
hJ : ∀ b ∈ J, IsInteger R (aJ • b)
b : P
hb : b ∈ I * J
x y : P
hx : IsInteger R ((aI * aJ) • x)
hy : IsInteger R ((aI * aJ) • y)
⊢ IsInteger R ((aI * aJ) • (x + y))
have refine_2 : IsInteger R ((aI * aJ) • x + (aI * aJ) • y)
Tactic: rw [smul_add]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : Ideal R
⊢ (↑(I * J) : FractionalIdeal S P) =
    HMul.hMul (α := FractionalIdeal S P) (↑I : FractionalIdeal S P) (↑J : FractionalIdeal S P)
have simp : (↑(I * J) : FractionalIdeal S P) =
  Subtype.mk (p := fun (I : Submodule R P) ↦ IsFractional S I)
    (HMul.hMul (α := Submodule R P) (↑(↑I : FractionalIdeal S P) : Submodule R P)
      (↑(↑J : FractionalIdeal S P) : Submodule R P))
    sorry
Tactic: simp only [mul_def]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : FractionalIdeal S P
i j : P
hi : i ∈ I
hj : j ∈ J
⊢ i * j ∈ I * J
have simp : i * j ∈ Subtype.mk (p := fun (I : Submodule R P) ↦ IsFractional S I) ((↑I : Submodule R P) * (↑J : Submodule R P)) sorry
Tactic: simp only [mul_def]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J K : FractionalIdeal S P
⊢ I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K
have simp : Subtype.mk (p := fun (I : Submodule R P) ↦ IsFractional S I) ((↑I : Submodule R P) * (↑J : Submodule R P)) sorry ≤ K ↔
  ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K
Tactic: simp only [mul_def]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
I J : FractionalIdeal S P
C : P → Prop
r : P
hr : r ∈ I * J
hm : ∀ i ∈ I, ∀ j ∈ J, C (i * j)
ha : ∀ (x y : P), C x → C y → C (x + y)
⊢ C r
have hr₁ : r ∈ Subtype.mk (p := fun (I : Submodule R P) ↦ IsFractional S I) ((↑I : Submodule R P) * (↑J : Submodule R P)) sorry
Tactic: simp only [mul_def] at hr
Cleared hypotheses: hr
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
n : ℕ
⊢ Eq.{u_2 + 1} (α := Submodule R P) (↑n.unaryCast : Submodule R P) (↑n : Submodule R P)
have zero : Eq.{u_2 + 1} (α := Submodule R P) (↑(Nat.unaryCast 0) : Submodule R P) (↑0 : Submodule R P)
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
n : ℕ
zero : Eq.{u_2 + 1} (α := Submodule R P) (↑(Nat.unaryCast 0) : Submodule R P) (↑0 : Submodule R P)
⊢ Eq.{u_2 + 1} (α := Submodule R P) (↑n.unaryCast : Submodule R P) (↑n : Submodule R P)
have succ : ∀ (n : ℕ),
  Eq.{u_2 + 1} (α := Submodule R P) (↑n.unaryCast : Submodule R P) (↑n : Submodule R P) →
    Eq.{u_2 + 1} (α := Submodule R P) (↑(n + 1).unaryCast : Submodule R P) (↑(n + 1) : Submodule R P)
Tactic: induction n
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
⊢ J ≤ 1 ↔ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mp : J ≤ 1 → ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
mp : J ≤ 1 → ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
⊢ J ≤ 1 ↔ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mpr : (∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J) → J ≤ 1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mp : ∀ {a b : R},
  Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} a →
    Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} b →
      Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} (a + b)
Tactic: refine ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, ?_⟩, ?_⟩, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
mp :
  ∀ {a b : R},
    Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} a →
      Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} b →
        Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} (a + b)
⊢ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mp₁ : Membership.mem (γ := Set R)
  (AddSubsemigroup.carrier
    { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry })
  0
Tactic: refine ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, ?_⟩, ?_⟩, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
mp :
  ∀ {a b : R},
    Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} a →
      Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} b →
        Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} (a + b)
mp₁ :
  Membership.mem (γ := Set R)
    (AddSubsemigroup.carrier
      { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯ })
    0
⊢ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mp₂ : ∀ (c : R) {x : R},
  x ∈
      AddSubsemigroup.carrier
        (AddSubmonoid.toAddSubsemigroup
          { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
            zero_mem' := sorry : AddSubmonoid R }) →
    c • x ∈
      AddSubsemigroup.carrier
        (AddSubmonoid.toAddSubsemigroup
          { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
            zero_mem' := sorry : AddSubmonoid R })
Tactic: refine ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, ?_⟩, ?_⟩, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
mp :
  ∀ {a b : R},
    Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} a →
      Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} b →
        Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} (a + b)
mp₁ :
  Membership.mem (γ := Set R)
    (AddSubsemigroup.carrier
      { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯ })
    0
mp₂ :
  ∀ (c : R) {x : R},
    x ∈
        AddSubsemigroup.carrier
          (AddSubmonoid.toAddSubsemigroup
            { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
              zero_mem' := ⋯ : AddSubmonoid R }) →
      c • x ∈
        AddSubsemigroup.carrier
          (AddSubmonoid.toAddSubsemigroup
            { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
              zero_mem' := ⋯ : AddSubmonoid R })
⊢ ∃ (I : Ideal R), (↑I : FractionalIdeal S P) = J
have mp₃ : (↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
        zero_mem' := sorry, smul_mem' := sorry } :
    FractionalIdeal S P) =
  J
Tactic: refine ⟨⟨⟨⟨{ x : R | algebraMap R P x ∈ J }, ?_⟩, ?_⟩, ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
a b : R
ha : Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} a
hb : Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} b
⊢ Membership.mem (γ := Set R) {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)} (a + b)
have mp : HAdd.hAdd (α := P) (β := P) ((algebraMap R P : (a : R) → P) a : P) ((algebraMap R P : (a : R) → P) b : P) ∈ J
Tactic: rw [mem_setOf, map_add]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ Membership.mem (γ := Set R)
    (AddSubsemigroup.carrier
      { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯ })
    0
have mp : 0 ∈ J
Tactic: rw [mem_setOf, map_zero]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
c x : R
hx :
  x ∈
    AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup
        { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
          zero_mem' := ⋯ : AddSubmonoid R })
⊢ c • x ∈
    AddSubsemigroup.carrier
      (AddSubmonoid.toAddSubsemigroup
        { carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
          zero_mem' := ⋯ : AddSubmonoid R })
have mp : HSMul.hSMul (β := P) c ((algebraMap R P : (a : R) → P) x : P) ∈ J
Tactic: rw [smul_eq_mul, mem_setOf, map_mul, ← Algebra.smul_def]
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
⊢ (↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
          zero_mem' := ⋯, smul_mem' := ⋯ } :
      FractionalIdeal S P) =
    J
have mp : ∀ (x : P),
  Membership.mem (γ := FractionalIdeal S P)
      ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
              zero_mem' := sorry, smul_mem' := sorry } :
          FractionalIdeal S P) :
        FractionalIdeal S P)
      x ↔
    x ∈ J
Tactic: ext x
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
⊢ Membership.mem (γ := FractionalIdeal S P)
      ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
              zero_mem' := ⋯, smul_mem' := ⋯ } :
          FractionalIdeal S P) :
        FractionalIdeal S P)
      x ↔
    x ∈ J
have mp : Membership.mem (γ := FractionalIdeal S P)
    ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
            zero_mem' := sorry, smul_mem' := sorry } :
        FractionalIdeal S P) :
      FractionalIdeal S P)
    x →
  x ∈ J
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.FractionalIdeal.Basic (goal)
Goal:
R : Type u_1
inst : CommRing R
S : Submonoid R
P : Type u_2
inst_1 : CommRing P
inst_2 : Algebra R P
J : FractionalIdeal S P
hJ : J ≤ 1
x : P
mp :
  Membership.mem (γ := FractionalIdeal S P)
      ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
              zero_mem' := ⋯, smul_mem' := ⋯ } :
          FractionalIdeal S P) :
        FractionalIdeal S P)
      x →
    x ∈ J
⊢ Membership.mem (γ := FractionalIdeal S P)
      ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := ⋯,
              zero_mem' := ⋯, smul_mem' := ⋯ } :
          FractionalIdeal S P) :
        FractionalIdeal S P)
      x ↔
    x ∈ J
have mp₁ : x ∈ J →
  Membership.mem (γ := FractionalIdeal S P)
    ((↑{ carrier := {x : R | Membership.mem (α := P) J ((algebraMap R P : (a : R) → P) x : P)}, add_mem' := sorry,
            zero_mem' := sorry, smul_mem' := sorry } :
        FractionalIdeal S P) :
      FractionalIdeal S P)
    x
Tactic: constructor