-- Mathlib/RingTheory/Adjoin/Basic.lean
theorem adjoin_eq_span : Subalgebra.toSubmodule (adjoin R s) = span R (Submonoid.closure s)

Proof state:
case a.intro.intro.cons.cons.intro.intro.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
⊢ ∃ z_1 r_1, ∃ (_ : r_1 ∈ Submonoid.closure s), z_1 • r_1 = hd * z • r

Tactic:
rcases HL.1 with (⟨hd, rfl⟩ | hs)

Result:
case a.intro.intro.cons.cons.intro.intro.intro.inl.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
hd : R
HL : (algebraMap R A) hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ ∃ z_1 r_1, ∃ (_ : r_1 ∈ Submonoid.closure s), z_1 • r_1 = (algebraMap R A) hd * z • r

case a.intro.intro.cons.cons.intro.intro.intro.inr
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
hs : hd ∈ s
⊢ ∃ z_1 r_1, ∃ (_ : r_1 ∈ Submonoid.closure s), z_1 • r_1 = hd * z • r

Proof state:
case a.intro.intro.cons.cons.intro.intro.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
⊢ ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = (hd :: tl).prod

Tactic:
rw [List.prod_cons, ← hzr]

Result:
case a.intro.intro.cons.cons.intro.intro.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
⊢ ∃ z_1 r_1, ∃ (_ : r_1 ∈ Submonoid.closure s), z_1 • r_1 = hd * z • r

Proof state:
case a.intro.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
L : List (List A)
HL : ∀ t ∈ L, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ (List.map List.prod L).sum ∈ span R ↑(Submonoid.closure s)

Tactic:
induction' L with hd tl ih

Result:
case a.intro.intro.nil
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
HL : ∀ t ∈ [], ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ (List.map List.prod []).sum ∈ span R ↑(Submonoid.closure s)

case a.intro.intro.cons
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : List A
tl : List (List A)
ih :
  (∀ t ∈ tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → (List.map List.prod tl).sum ∈ span R ↑(Submonoid.closure s)
HL : ∀ t ∈ hd :: tl, ∀ y ∈ t, y ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ (List.map List.prod (hd :: tl)).sum ∈ span R ↑(Submonoid.closure s)

Proof state:
case a.intro.intro.cons.cons
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
⊢ ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = (hd :: tl).prod

Tactic:
rcases ih HL.2 with ⟨z, r, hr, hzr⟩

Result:
case a.intro.intro.cons.cons.intro.intro.intro
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
hd : A
tl : List A
ih : (∀ y ∈ tl, y ∈ Set.range ⇑(algebraMap R A) ∪ s) → ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = tl.prod
HL : hd ∈ Set.range ⇑(algebraMap R A) ∪ s ∧ ∀ x ∈ tl, x ∈ Set.range ⇑(algebraMap R A) ∪ s
z : R
r : A
hr : r ∈ Submonoid.closure s
hzr : z • r = tl.prod
⊢ ∃ z r, ∃ (_ : r ∈ Submonoid.closure s), z • r = (hd :: tl).prod

Proof state:
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ Subalgebra.toSubmodule (adjoin R s) = span R ↑(Submonoid.closure s)

Tactic:
apply le_antisymm

Result:
case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ Subalgebra.toSubmodule (adjoin R s) ≤ span R ↑(Submonoid.closure s)

case a
R : Type uR
A : Type uA
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
⊢ span R ↑(Submonoid.closure s) ≤ Subalgebra.toSubmodule (adjoin R s)