awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subsemiring R) (closure s) x
hy : Membership.mem (γ := Subsemiring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
⊢ p x y hx hy
have zero : p x 0 hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subsemiring R) (closure s) x
hy : Membership.mem (γ := Subsemiring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 hx ⋯
⊢ p x y hx hy
have one : p x 1 hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subsemiring R) (closure s) x
hy : Membership.mem (γ := Subsemiring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 hx ⋯
one : p x 1 hx ⋯
⊢ p x y hx hy
have mul : ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subsemiring R) (closure s) x_1)
  (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x x_1 hx hx_1) (h₂ : p x y hx hy), p x (x_1 * y) hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u
inst : NonAssocSemiring R
s : Set R
p :
  (x y : R) →
    Membership.mem (γ := Subsemiring R) (closure s) x → Membership.mem (γ := Subsemiring R) (closure s) y → Prop
mem_mem : ∀ (x y : R) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
zero_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 0 x ⋯ hx
zero_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 0 hx ⋯
one_left : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p 1 x ⋯ hx
one_right : ∀ (x : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x), p x 1 hx ⋯
add_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : R) (hx : Membership.mem (γ := Subsemiring R) (closure s) x)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (hz : Membership.mem (γ := Subsemiring R) (closure s) z),
    p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
x y : R
hx : Membership.mem (γ := Subsemiring R) (closure s) x
hy : Membership.mem (γ := Subsemiring R) (closure s) y
mem : ∀ (z : R) (hz : z ∈ s), p x z hx ⋯
zero : p x 0 hx ⋯
one : p x 1 hx ⋯
mul :
  ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subsemiring R) (closure s) x_1)
    (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x x_1 hx hx_1) (h₂ : p x y hx hy),
    p x (x_1 * y) hx ⋯
⊢ p x y hx hy
have add : ∀ (x_1 y : R) (hx_1 : Membership.mem (γ := Subsemiring R) (closure s) x_1)
  (hy : Membership.mem (γ := Subsemiring R) (closure s) y) (h₁ : p x x_1 hx hx_1) (h₂ : p x y hx hy), p x (x_1 + y) hx sorry
Tactic: induction hy using closure_induction with
  | mem z hz => induction hx using closure_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | zero => exact zero_left _ _
    | one => exact one_left _ _
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
  | zero => exact zero_right x hx
  | one => exact one_right x hx
  | mul _ _ _ _ h₁ h₂ => exact mul_right _ _ _ _ _ _ h₁ h₂
  | add _ _ _ _ h₁ h₂ => exact add_right _ _ _ _ _ _ h₁ h₂
Cleared hypotheses: hy
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x : R
⊢ Membership.mem (γ := Subsemiring R) (closure s) x ↔
    ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
have mp : Membership.mem (γ := Subsemiring R) (closure s) x →
  ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x : R
mp :
  Membership.mem (γ := Subsemiring R) (closure s) x →
    ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
⊢ Membership.mem (γ := Subsemiring R) (closure s) x ↔
    ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
have mpr : (∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x) →
  Membership.mem (γ := Subsemiring R) (closure s) x
Tactic: constructor
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (hypothesis)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x : R
hx : Membership.mem (γ := Subsemiring R) (closure s) x
⊢ ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
have hx₁ : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
Tactic: rw [mem_closure_iff] at hx
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
this : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
t : List R
ht1 : ∀ y ∈ t, y ∈ s
ht2 : t.prod = x
⊢ (List.map List.prod [t]).sum = x
have rw : [t.prod].sum = x
Tactic: rw [List.map_singleton, List.sum_singleton, ht2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
this : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
t : List R
ht1 : ∀ y ∈ t, y ∈ s
ht2 : t.prod = x
rw : [t.prod].sum = x
rw₁ : t.prod = x
⊢ (List.map List.prod [t]).sum = x
have rw₂ : x = x
Tactic: rw [List.map_singleton, List.sum_singleton, ht2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
⊢ ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
have mp : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
Tactic: suffices ∃ t : List R, (∀ y ∈ t, y ∈ s) ∧ t.prod = x from
        let ⟨t, ht1, ht2⟩ := this
        ⟨[t], List.forall_mem_singleton.2 ht1, by
          rw [List.map_singleton, List.sum_singleton, ht2]⟩
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x_2 x y : R
hx : Membership.mem (γ := Submonoid R) (Submonoid.closure s) x
hy : Membership.mem (γ := Submonoid R) (Submonoid.closure s) y
ht : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
hu : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = y
⊢ ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x * y
have mp : ∀ (t : List R) (ht1 : ∀ y ∈ t, y ∈ s) (ht2 : t.prod = x) (u : List R) (hu1 : ∀ y ∈ u, y ∈ s) (hu2 : u.prod = y),
  ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x * y
Tactic: obtain ⟨⟨t, ht1, ht2⟩, ⟨u, hu1, hu2⟩⟩ := And.intro ht hu
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x_2 x y : R
hx : Membership.mem (γ := Submonoid R) (Submonoid.closure s) x
hy : Membership.mem (γ := Submonoid R) (Submonoid.closure s) y
ht : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
hu : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = y
t : List R
ht1 : ∀ y ∈ t, y ∈ s
ht2 : t.prod = x
u : List R
hu1 : ∀ y ∈ u, y ∈ s
hu2 : u.prod = y
⊢ (t ++ u).prod = x * y
have rw : t.prod * u.prod = x * y
Tactic: rw [List.prod_append, ht2, hu2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x_2 x y : R
hx : Membership.mem (γ := Submonoid R) (Submonoid.closure s) x
hy : Membership.mem (γ := Submonoid R) (Submonoid.closure s) y
ht : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
hu : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = y
t : List R
ht1 : ∀ y ∈ t, y ∈ s
ht2 : t.prod = x
u : List R
hu1 : ∀ y ∈ u, y ∈ s
hu2 : u.prod = y
rw : t.prod * u.prod = x * y
⊢ (t ++ u).prod = x * y
have rw₁ : x * u.prod = x * y
Tactic: rw [List.prod_append, ht2, hu2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x_2 x y : R
hx : Membership.mem (γ := Submonoid R) (Submonoid.closure s) x
hy : Membership.mem (γ := Submonoid R) (Submonoid.closure s) y
ht : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
hu : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = y
t : List R
ht1 : ∀ y ∈ t, y ∈ s
ht2 : t.prod = x
u : List R
hu1 : ∀ y ∈ u, y ∈ s
hu2 : u.prod = y
rw : t.prod * u.prod = x * y
rw₁ : x * u.prod = x * y
⊢ (t ++ u).prod = x * y
have rw₂ : x * y = x * y
Tactic: rw [List.prod_append, ht2, hu2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
⊢ ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
have mp : ∀ (x : R) (hx : x ∈ s), ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
Tactic: induction hx using Submonoid.closure_induction with
      | mem x hx => exact ⟨[x], List.forall_mem_singleton.2 hx, List.prod_singleton⟩
      | one => exact ⟨[], List.forall_mem_nil _, rfl⟩
      | mul x y _ _ ht hu =>
        obtain ⟨⟨t, ht1, ht2⟩, ⟨u, hu1, hu2⟩⟩ := And.intro ht hu
        exact ⟨t ++ u, List.forall_mem_append.2 ⟨ht1, hu1⟩, by rw [List.prod_append, ht2, hu2]⟩
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
mp : ∀ (x : R) (hx : x ∈ s), ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
⊢ ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
have mp₁ : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = 1
Tactic: induction hx using Submonoid.closure_induction with
      | mem x hx => exact ⟨[x], List.forall_mem_singleton.2 hx, List.prod_singleton⟩
      | one => exact ⟨[], List.forall_mem_nil _, rfl⟩
      | mul x y _ _ ht hu =>
        obtain ⟨⟨t, ht1, ht2⟩, ⟨u, hu1, hu2⟩⟩ := And.intro ht hu
        exact ⟨t ++ u, List.forall_mem_append.2 ⟨ht1, hu1⟩, by rw [List.prod_append, ht2, hu2]⟩
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x : R
hx : x ∈ (↑(Submonoid.closure s) : Set R)
mp : ∀ (x : R) (hx : x ∈ s), ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
mp₁ : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = 1
⊢ ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x
have mp₂ : ∀ (x y : R),
  Membership.mem (γ := Submonoid R) (Submonoid.closure s) x →
    Membership.mem (γ := Submonoid R) (Submonoid.closure s) y →
      ∀ (ht : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x) (hu : ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = y),
        ∃ (t : List R), (∀ y ∈ t, y ∈ s) ∧ t.prod = x * y
Tactic: induction hx using Submonoid.closure_induction with
      | mem x hx => exact ⟨[x], List.forall_mem_singleton.2 hx, List.prod_singleton⟩
      | one => exact ⟨[], List.forall_mem_nil _, rfl⟩
      | mul x y _ _ ht hu =>
        obtain ⟨⟨t, ht1, ht2⟩, ⟨u, hu1, hu2⟩⟩ := And.intro ht hu
        exact ⟨t ++ u, List.forall_mem_append.2 ⟨ht1, hu1⟩, by rw [List.prod_append, ht2, hu2]⟩
Cleared hypotheses: hx
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x y : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
hy : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) y
hL : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
hM : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = y
⊢ ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x + y
have mp : ∀ (L : List (List R)) (HL1 : ∀ t ∈ L, ∀ y ∈ t, y ∈ s) (HL2 : (List.map List.prod L).sum = x) (M : List (List R))
  (HM1 : ∀ t ∈ M, ∀ y ∈ t, y ∈ s) (HM2 : (List.map List.prod M).sum = y),
  ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x + y
Tactic: obtain ⟨⟨L, HL1, HL2⟩, ⟨M, HM1, HM2⟩⟩ := And.intro hL hM
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x y : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
hy : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) y
hL : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
hM : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = y
L : List (List R)
HL1 : ∀ t ∈ L, ∀ y ∈ t, y ∈ s
HL2 : (List.map List.prod L).sum = x
M : List (List R)
HM1 : ∀ t ∈ M, ∀ y ∈ t, y ∈ s
HM2 : (List.map List.prod M).sum = y
⊢ (List.map List.prod (L ++ M)).sum = x + y
have rw : (List.map List.prod L ++ List.map List.prod M).sum = x + y
Tactic: rw [List.map_append, List.sum_append, HL2, HM2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x y : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
hy : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) y
hL : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
hM : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = y
L : List (List R)
HL1 : ∀ t ∈ L, ∀ y ∈ t, y ∈ s
HL2 : (List.map List.prod L).sum = x
M : List (List R)
HM1 : ∀ t ∈ M, ∀ y ∈ t, y ∈ s
HM2 : (List.map List.prod M).sum = y
rw : (List.map List.prod L ++ List.map List.prod M).sum = x + y
⊢ (List.map List.prod (L ++ M)).sum = x + y
have rw₁ : (List.map List.prod L).sum + (List.map List.prod M).sum = x + y
Tactic: rw [List.map_append, List.sum_append, HL2, HM2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x y : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
hy : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) y
hL : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
hM : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = y
L : List (List R)
HL1 : ∀ t ∈ L, ∀ y ∈ t, y ∈ s
HL2 : (List.map List.prod L).sum = x
M : List (List R)
HM1 : ∀ t ∈ M, ∀ y ∈ t, y ∈ s
HM2 : (List.map List.prod M).sum = y
rw : (List.map List.prod L ++ List.map List.prod M).sum = x + y
rw₁ : (List.map List.prod L).sum + (List.map List.prod M).sum = x + y
⊢ (List.map List.prod (L ++ M)).sum = x + y
have rw₂ : x + (List.map List.prod M).sum = x + y
Tactic: rw [List.map_append, List.sum_append, HL2, HM2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x_1 x y : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
hy : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) y
hL : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
hM : ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = y
L : List (List R)
HL1 : ∀ t ∈ L, ∀ y ∈ t, y ∈ s
HL2 : (List.map List.prod L).sum = x
M : List (List R)
HM1 : ∀ t ∈ M, ∀ y ∈ t, y ∈ s
HM2 : (List.map List.prod M).sum = y
rw : (List.map List.prod L ++ List.map List.prod M).sum = x + y
rw₁ : (List.map List.prod L).sum + (List.map List.prod M).sum = x + y
rw₂ : x + (List.map List.prod M).sum = x + y
⊢ (List.map List.prod (L ++ M)).sum = x + y
have rw₃ : x + y = x + y
Tactic: rw [List.map_append, List.sum_append, HL2, HM2]
-- Lean 4 proof step from Mathlib.Algebra.Ring.Subsemiring.Basic (goal)
Goal:
R : Type u_1
inst : Semiring R
s : Set R
x : R
hx : Membership.mem (γ := AddSubmonoid R) (AddSubmonoid.closure (↑(Submonoid.closure s) : Set R)) x
⊢ ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
have mp : ∀ (x : R) (hx : x ∈ (↑(Submonoid.closure s) : Set R)),
  ∃ (L : List (List R)), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x
Tactic: induction hx using AddSubmonoid.closure_induction with
    | mem x hx =>
      suffices ∃ t : List R, (∀ y ∈ t, y ∈ s) ∧ t.prod = x from
        let ⟨t, ht1, ht2⟩ := this
        ⟨[t], List.forall_mem_singleton.2 ht1, by
          rw [List.map_singleton, List.sum_singleton, ht2]⟩
      induction hx using Submonoid.closure_induction with
      | mem x hx => exact ⟨[x], List.forall_mem_singleton.2 hx, List.prod_singleton⟩
      | one => exact ⟨[], List.forall_mem_nil _, rfl⟩
      | mul x y _ _ ht hu =>
        obtain ⟨⟨t, ht1, ht2⟩, ⟨u, hu1, hu2⟩⟩ := And.intro ht hu
        exact ⟨t ++ u, List.forall_mem_append.2 ⟨ht1, hu1⟩, by rw [List.prod_append, ht2, hu2]⟩
    | zero => exact ⟨[], List.forall_mem_nil _, rfl⟩
    | add x y _ _ hL hM =>
      obtain ⟨⟨L, HL1, HL2⟩, ⟨M, HM1, HM2⟩⟩ := And.intro hL hM
      exact ⟨L ++ M, List.forall_mem_append.2 ⟨HL1, HM1⟩, by
        rw [List.map_append, List.sum_append, HL2, HM2]⟩
Cleared hypotheses: hx