theorem AlgebraicCombinatorics.FPS.coeff_derivative_eq.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) (nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
) :
  (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  nℕ)
      ((PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.) fPowerSeries R *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
      (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.↑nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.

theorem AlgebraicCombinatorics.FPS.derivative_eq_mk.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    PowerSeries.mkPowerSeries.mk.{u_2} {R : Type u_2} (f : ℕ → R) : PowerSeries RConstructor for formal power series.  fun nℕ ↦
      (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.↑nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
        (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.) fPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_eq_derivativeFun.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    fPowerSeries R.derivativeFunPowerSeries.derivativeFun.{u_1} {R : Type u_1} [CommSemiring R] (f : PowerSeries R) : PowerSeries RThe formal derivative of a power series in one variable.
This is defined here as a function, but will be packaged as a
derivation `derivative` on `R⟦X⟧`.

theorem AlgebraicCombinatorics.FPS.derivative_coeff_formula.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) (nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
) :
  (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  nℕ)
      ((PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.↑nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
      (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.) fPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_zero.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1] :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) 0 =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`. 0

theorem AlgebraicCombinatorics.FPS.derivative_one.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1] :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) 1 =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`. 0

theorem AlgebraicCombinatorics.FPS.derivative_C.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (cR : RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1)
      (PowerSeries.CPowerSeries.C.{u_1} {R : Type u_1} [Semiring R] : R →+* PowerSeries RThe constant formal power series.  cR) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    0

theorem AlgebraicCombinatorics.FPS.derivative_X.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1] :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1)
      PowerSeries.XPowerSeries.X.{u_1} {R : Type u_1} [Semiring R] : PowerSeries RThe variable of the formal power series ring.  =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    1

theorem AlgebraicCombinatorics.FPS.derivative_X_pow_succ.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1)
      (HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`.PowerSeries.XPowerSeries.X.{u_1} {R : Type u_1} [Semiring R] : PowerSeries RThe variable of the formal power series ring.  ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.)HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.↑nℕ +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. 1)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. •HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`. PowerSeries.XPowerSeries.X.{u_1} {R : Type u_1} [Semiring R] : PowerSeries RThe variable of the formal power series ring.  ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. nℕ

theorem AlgebraicCombinatorics.FPS.derivative_X_pow_of_pos.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1] {nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
}
  (hn0 < n : 0 <LT.lt.{u} {α : Type u} [self : LT α] : α → α → PropThe less-than relation: `x < y` 

Conventions for notations in identifiers:

 * The recommended spelling of `<` in identifiers is `lt`. nℕ) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1)
      (HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`.PowerSeries.XPowerSeries.X.{u_1} {R : Type u_1} [Semiring R] : PowerSeries RThe variable of the formal power series ring.  ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. nℕ)HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    ↑nℕ •HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`. PowerSeries.XPowerSeries.X.{u_1} {R : Type u_1} [Semiring R] : PowerSeries RThe variable of the formal power series ring.  ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. (HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).nℕ -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). 1)HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).

theorem AlgebraicCombinatorics.FPS.derivative_coe_polynomial.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (pPolynomial R : PolynomialPolynomial.{u_1} (R : Type u_1) [Semiring R] : Type u_1`Polynomial R` is the type of univariate polynomials over `R`,
denoted as `R[X]` within the `Polynomial` namespace.

Polynomials should be seen as (semi-)rings with the additional constructor `X`.
The embedding from `R` is called `C`.  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) ↑pPolynomial R =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    ↑(Polynomial.derivativePolynomial.derivative.{u} {R : Type u} [Semiring R] : Polynomial R →ₗ[R] Polynomial R`derivative p` is the formal derivative of the polynomial `p`  pPolynomial R)

theorem AlgebraicCombinatorics.FPS.derivative_add.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R gPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.fPowerSeries R +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. gPowerSeries R)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) gPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_sum.{u_1,
    u_2}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  {ιType u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } (sFinset ι : FinsetFinset.{u_4} (α : Type u_4) : Type u_4`Finset α` is the type of finite sets of elements of `α`. It is implemented
as a multiset (a list up to permutation) which has no duplicate elements.  ιType u_2)
  (fι → PowerSeries R : ιType u_2 → PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1)
      (∑ iι ∈ sFinset ι, fι → PowerSeries R iι) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    ∑ iι ∈ sFinset ι,
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) (fι → PowerSeries R iι)

theorem AlgebraicCombinatorics.FPS.derivative_summableFPSSum.{u_2,
    u_3}
  {R'Type u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  R'Type u_2]
  {ιType u_3 : Type u_3A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } (fι → PowerSeries R' : ιType u_3 → PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  R'Type u_2)
  (hfAlgebraicCombinatorics.FPS.SummableFPS f :
    AlgebraicCombinatorics.FPS.SummableFPSAlgebraicCombinatorics.FPS.SummableFPS.{u_1, u_2} {R : Type u_1} [CommRing R] {ι : Type u_2} (f : ι → PowerSeries R) :
  PropA family of FPS is summable if for each coefficient index n,
all but finitely many family members have that coefficient equal to zero.
(Definition def.fps.summable) 
      fι → PowerSeries R') :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  R'Type u_2)
      (AlgebraicCombinatorics.FPS.summableFPSSumAlgebraicCombinatorics.FPS.summableFPSSum.{u_1, u_2} {R : Type u_1} [CommRing R] {ι : Type u_2} (f : ι → PowerSeries R)
  (_hf : AlgebraicCombinatorics.FPS.SummableFPS f) : PowerSeries RThe sum of a summable family of FPS.
(Definition def.fps.summable, eq. eq.def.fps.summable.sum)

For a summable family (fᵢ)_{i ∈ I}, the sum ∑_{i ∈ I} fᵢ is the FPS whose
n-th coefficient is ∑_{i ∈ I} [x^n] fᵢ (an essentially finite sum). 
        fι → PowerSeries R' hfAlgebraicCombinatorics.FPS.SummableFPS f) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    AlgebraicCombinatorics.FPS.summableFPSSumAlgebraicCombinatorics.FPS.summableFPSSum.{u_1, u_2} {R : Type u_1} [CommRing R] {ι : Type u_2} (f : ι → PowerSeries R)
  (_hf : AlgebraicCombinatorics.FPS.SummableFPS f) : PowerSeries RThe sum of a summable family of FPS.
(Definition def.fps.summable, eq. eq.def.fps.summable.sum)

For a summable family (fᵢ)_{i ∈ I}, the sum ∑_{i ∈ I} fᵢ is the FPS whose
n-th coefficient is ∑_{i ∈ I} [x^n] fᵢ (an essentially finite sum). 
      (fun iι ↦
        (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  R'Type u_2) (fι → PowerSeries R' iι))
      ⋯

theorem AlgebraicCombinatorics.FPS.derivative_smul.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1] (cR : RType u_1)
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) (HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`.cR •HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`. fPowerSeries R)HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    cR •HSMul.hSMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSMul α β γ] : α → β → γ`a • b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent, but it is intended to be used for left actions. 

Conventions for notations in identifiers:

 * The recommended spelling of `•` in identifiers is `smul`. (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_mul.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R gPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) (HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.fPowerSeries R *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. gPowerSeries R)HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. gPowerSeries R +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.
      fPowerSeries R *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) gPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_div.{u_3}
  {KType u_3 : Type u_3A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [FieldField.{u} (K : Type u) : Type uA `Field` is a `CommRing` with multiplicative inverses for nonzero elements.

An instance of `Field K` includes maps `ratCast : ℚ → K` and `qsmul : ℚ → K → K`.
Those two fields are needed to implement the `DivisionRing K → Algebra ℚ K` instance since we need
to control the specific definitions for some special cases of `K` (in particular `K = ℚ` itself).
See also note [forgetful inheritance].

If the field has positive characteristic `p`, our division by zero convention forces
`ratCast (1 / p) = 1 / 0 = 0`.  KType u_3]
  (fPowerSeries K gPowerSeries K : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  KType u_3)
  (hgPowerSeries.constantCoeff g ≠ 0 : PowerSeries.constantCoeffPowerSeries.constantCoeff.{u_1} {R : Type u_1} [Semiring R] : PowerSeries R →+* RThe constant coefficient of a formal power series.  gPowerSeries K ≠Ne.{u} {α : Sort u} (a b : α) : Prop`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`. 0) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  KType u_3) (HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.fPowerSeries K *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. gPowerSeries K⁻¹Inv.inv.{u} {α : Type u} [self : Inv α] : α → α`a⁻¹` computes the inverse of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `⁻¹` in identifiers is `inv`.)HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).(PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  KType u_3) fPowerSeries K *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. gPowerSeries K -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).
        fPowerSeries K *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
          (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  KType u_3) gPowerSeries K)HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
      gPowerSeries K⁻¹Inv.inv.{u} {α : Type u} [self : Inv α] : α → α`a⁻¹` computes the inverse of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `⁻¹` in identifiers is `inv`. ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. 2

theorem AlgebraicCombinatorics.FPS.derivative_pow'.{u_4}
  {AType u_4 : Type u_4A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  AType u_4]
  (gPowerSeries A : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  AType u_4) (nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  AType u_4) (HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`.gPowerSeries A ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. nℕ)HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    ↑nℕ *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. gPowerSeries A ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. (HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).nℕ -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). 1)HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  AType u_4) gPowerSeries A

theorem AlgebraicCombinatorics.FPS.derivative_comp.{u_4}
  {AType u_4 : Type u_4A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  AType u_4]
  (fPowerSeries A gPowerSeries A : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  AType u_4)
  (hgPowerSeries.HasSubst g : PowerSeries.HasSubstPowerSeries.HasSubst.{u_3, u_4} {τ : Type u_3} {S : Type u_4} [CommRing S] (a : MvPowerSeries τ S) : Prop(Possibly multivariate) power series which can be substituted in a `PowerSeries`.  gPowerSeries A) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  AType u_4)
      (PowerSeries.substPowerSeries.subst.{u_2, u_3, u_4} {R : Type u_2} [CommRing R] {τ : Type u_3} {S : Type u_4} [CommRing S] [Algebra R S]
  (a : MvPowerSeries τ S) (f : PowerSeries R) : MvPowerSeries τ SSubstitution of power series into a power series.  gPowerSeries A fPowerSeries A) =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    PowerSeries.substPowerSeries.subst.{u_2, u_3, u_4} {R : Type u_2} [CommRing R] {τ : Type u_3} {S : Type u_4} [CommRing S] [Algebra R S]
  (a : MvPowerSeries τ S) (f : PowerSeries R) : MvPowerSeries τ SSubstitution of power series into a power series.  gPowerSeries A
        ((PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  AType u_4) fPowerSeries A) *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  AType u_4) gPowerSeries A

theorem AlgebraicCombinatorics.FPS.derivative_eq_imp_diff_const.{u_5}
  {RType u_5 : Type u_5A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  RType u_5]
  [IsAddTorsionFreeIsAddTorsionFree.{u_2} (M : Type u_2) [AddMonoid M] : PropAn additive monoid is torsion-free if scalar multiplication by every non-zero element `n : ℕ` is
injective.  RType u_5]
  {fPowerSeries R gPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_5}
  (h(PowerSeries.derivative R) f = (PowerSeries.derivative R) g :
    (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_5) fPowerSeries R =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_5) gPowerSeries R)
  (nℕ : ℕNat : TypeThe natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.
) :
  nℕ ≠Ne.{u} {α : Sort u} (a b : α) : Prop`a ≠ b`, or `Ne a b` is defined as `¬ (a = b)` or `a = b → False`,
and asserts that `a` and `b` are not equal.


Conventions for notations in identifiers:

 * The recommended spelling of `≠` in identifiers is `ne`. 0 →
    (PowerSeries.coeffPowerSeries.coeff.{u_1} {R : Type u_1} [Semiring R] (n : ℕ) : PowerSeries R →ₗ[R] RThe `n`th coefficient of a formal power series.  nℕ) (HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).fPowerSeries R -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). gPowerSeries R)HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`. 0

theorem AlgebraicCombinatorics.FPS.derivative_add.{u_1}
  {RType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommSemiringCommSemiring.{u} (R : Type u) : Type uA commutative semiring is a semiring with commutative multiplication.  RType u_1]
  (fPowerSeries R gPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_1) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) (HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.fPowerSeries R +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. gPowerSeries R)HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`. =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) fPowerSeries R +HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `+` in identifiers is `add`.
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_1) gPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_neg.{u_2}
  {RType u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  RType u_2]
  (fPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_2) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_2) (Neg.neg.{u} {α : Type u} [self : Neg α] : α → α`-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).-Neg.neg.{u} {α : Type u} [self : Neg α] : α → α`-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).fPowerSeries R)Neg.neg.{u} {α : Type u} [self : Neg α] : α → α`-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator). =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    -Neg.neg.{u} {α : Type u} [self : Neg α] : α → α`-a` computes the negative or opposite of `a`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).(PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_2) fPowerSeries R

theorem AlgebraicCombinatorics.FPS.derivative_sub.{u_2}
  {RType u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [CommRingCommRing.{u} (α : Type u) : Type uA commutative ring is a ring with commutative multiplication.  RType u_2]
  (fPowerSeries R gPowerSeries R : PowerSeriesPowerSeries.{u_1} (R : Type u_1) : Type u_1Formal power series over a coefficient type `R`  RType u_2) :
  (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_2) (HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).fPowerSeries R -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). gPowerSeries R)HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator). =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.
    (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_2) fPowerSeries R -HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 

Conventions for notations in identifiers:

 * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).
      (PowerSeries.derivativePowerSeries.derivative.{u_1} (R : Type u_1) [CommSemiring R] : Derivation R (PowerSeries R) (PowerSeries R)The formal derivative of a formal power series  RType u_2) gPowerSeries R