Module: Mathlib.Algebra.FreeAlgebra

Goal state before:
pp: case left_distrib
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ b✝ c✝ : Pre R X
⊢ liftFun R X f (a✝ * (b✝ + c✝)) = liftFun R X f (a✝ * b✝ + a✝ * c✝)

pp: case right_distrib
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ b✝ c✝ : Pre R X
⊢ liftFun R X f ((a✝ + b✝) * c✝) = liftFun R X f (a✝ * c✝ + b✝ * c✝)

pp: case zero_mul
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ : Pre R X
⊢ liftFun R X f (0 * a✝) = liftFun R X f 0

pp: case mul_zero
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ : Pre R X
⊢ liftFun R X f (a✝ * 0) = liftFun R X f 0

pp: case add_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ + c✝) = liftFun R X f (b✝ + c✝)

pp: case add_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ + a✝¹) = liftFun R X f (c✝ + b✝)

pp: case mul_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ * c✝) = liftFun R X f (b✝ * c✝)

pp: case mul_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ * a✝¹) = liftFun R X f (c✝ * b✝)

Tactic:
· change _ * (_ + _) = _ * _ + _ * _
        rw [left_distrib]

Goal state after:
case right_distrib
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ b✝ c✝ : Pre R X
⊢ liftFun R X f ((a✝ + b✝) * c✝) = liftFun R X f (a✝ * c✝ + b✝ * c✝)

case zero_mul
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ : Pre R X
⊢ liftFun R X f (0 * a✝) = liftFun R X f 0

case mul_zero
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝¹ : FreeAlgebra R X
a b a✝ : Pre R X
⊢ liftFun R X f (a✝ * 0) = liftFun R X f 0

case add_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ + c✝) = liftFun R X f (b✝ + c✝)

case add_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ + a✝¹) = liftFun R X f (c✝ + b✝)

case mul_compat_left
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (a✝¹ * c✝) = liftFun R X f (b✝ * c✝)

case mul_compat_right
R : Type u_1
X : Type u_2
inst✝² : CommSemiring R
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : X → A
a✝² : FreeAlgebra R X
a b a✝¹ b✝ c✝ : Pre R X
a✝ : Rel R X a✝¹ b✝
a_ih✝ : liftFun R X f a✝¹ = liftFun R X f b✝
⊢ liftFun R X f (c✝ * a✝¹) = liftFun R X f (c✝ * b✝)