-- Module: Mathlib.Algebra.Star.Subalgebra
Goal:
R : Type u_2
A : Type u_3
inst : CommSemiring R
inst_1 : StarRing R
inst_2 : Semiring A
inst_3 : Algebra R A
inst_4 : StarRing A
inst_5 : StarModule R A
s : Set A
p :
  (x y : A) →
    Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x →
      Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y → Prop
mem_mem : ∀ (x y : A) (hx : x ∈ s) (hy : y ∈ s), p x y ⋯ ⋯
algebraMap_both : ∀ (r₁ r₂ : R), p ((algebraMap R A : (a : R) → A) r₁) ((algebraMap R A : (a : R) → A) r₂) ⋯ ⋯
algebraMap_left : ∀ (r : R) (x : A) (hx : x ∈ s), p ((algebraMap R A : (a : R) → A) r) x ⋯ ⋯
algebraMap_right : ∀ (r : R) (x : A) (hx : x ∈ s), p x ((algebraMap R A : (a : R) → A) r) ⋯ ⋯
add_left :
  ∀ (x y z : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y)
    (hz : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) z), p x z hx hz → p y z hy hz → p (x + y) z ⋯ hz
add_right :
  ∀ (x y z : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y)
    (hz : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) z), p x y hx hy → p x z hx hz → p x (y + z) hx ⋯
mul_left :
  ∀ (x y z : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y)
    (hz : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) z), p x z hx hz → p y z hy hz → p (x * y) z ⋯ hz
mul_right :
  ∀ (x y z : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y)
    (hz : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) z), p x y hx hy → p x z hx hz → p x (y * z) hx ⋯
star_left :
  ∀ (x y : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y), p x y hx hy → p (star x) y ⋯ hy
star_right :
  ∀ (x y : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y), p x y hx hy → p x (star y) hx ⋯
a b : A
ha : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) a
z : A
hz : z ∈ s
mem : ∀ (x : A) (h : x ∈ s), p x z ⋯ ⋯
mem₁ : ∀ (r : R), p ((algebraMap R A : (a : R) → A) r) z ⋯ ⋯
mem₂ :
  ∀ (x y : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x * y) z ⋯ ⋯
mem₃ :
  ∀ (x y : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x)
    (hy : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) y) (h₁ : p x z hx ⋯) (h₂ : p y z hy ⋯), p (x + y) z ⋯ ⋯
mem₄ : ∀ (x : A) (hx : Membership.mem (γ := StarSubalgebra R A) (adjoin R s) x) (h : p x z hx ⋯), p (star x) z ⋯ ⋯
⊢ p a z ha ⋯

Tactic:
induction ha using adjoin_induction with
    | mem _ h => exact mem_mem _ _ h hz
    | algebraMap _ => exact algebraMap_left _ _ hz
    | mul _ _ _ _ h₁ h₂ => exact mul_left _ _ _ _ _ _ h₁ h₂
    | add _ _ _ _ h₁ h₂ => exact add_left _ _ _ _ _ _ h₁ h₂
    | star _ _ h => exact star_left _ _ _ _ h

Premises used: Algebra.algebraMap, Star.star