inductive Nat.SOM.Expr : Type

• num (i : Nat) : Expr
• var (v : Linear.Var) : Expr
• add (a b : Expr) : Expr
• mul (a b : Expr) : Expr

instance Nat.SOM.instInhabitedExpr : Inhabited Expr

Equations

• Nat.SOM.instInhabitedExpr = { default := Nat.SOM.Expr.num default }

def Nat.SOM.Expr.denote (ctx : Linear.Context) : Expr → Nat

Equations

• Nat.SOM.Expr.denote ctx (Nat.SOM.Expr.num n) = n
• Nat.SOM.Expr.denote ctx (Nat.SOM.Expr.var v) = Nat.Linear.Var.denote ctx v
• Nat.SOM.Expr.denote ctx (a.add b) = (Nat.SOM.Expr.denote ctx a).add (Nat.SOM.Expr.denote ctx b)
• Nat.SOM.Expr.denote ctx (a.mul b) = (Nat.SOM.Expr.denote ctx a).mul (Nat.SOM.Expr.denote ctx b)

@[reducible, inline]

abbrev Nat.SOM.Mon : Type

Equations

• Nat.SOM.Mon = List Nat.Linear.Var

def Nat.SOM.Mon.denote (ctx : Linear.Context) : Mon → Nat

Equations

• Nat.SOM.Mon.denote ctx [] = 1
• Nat.SOM.Mon.denote ctx (v :: vs) = (Nat.Linear.Var.denote ctx v).mul (Nat.SOM.Mon.denote ctx vs)

def Nat.SOM.Mon.mul (m₁ m₂ : Mon) : Mon

Equations

• m₁.mul m₂ = Nat.SOM.Mon.mul.go Nat.Linear.hugeFuel m₁ m₂

def Nat.SOM.Mon.mul.go (fuel : Nat) (m₁ m₂ : Mon) : Mon

Equations

• One or more equations did not get rendered due to their size.
• Nat.SOM.Mon.mul.go 0 m₁ m₂ = m₁ ++ m₂
• Nat.SOM.Mon.mul.go fuel_2.succ m₁ [] = m₁
• Nat.SOM.Mon.mul.go fuel_2.succ [] m₂ = m₂

@[reducible, inline]

abbrev Nat.SOM.Poly : Type

Equations

• Nat.SOM.Poly = List (Nat × Nat.SOM.Mon)

def Nat.SOM.Poly.denote (ctx : Linear.Context) : Poly → Nat

Equations

• Nat.SOM.Poly.denote ctx [] = 0
• Nat.SOM.Poly.denote ctx ((k, m) :: p) = (k.mul (Nat.SOM.Mon.denote ctx m)).add (Nat.SOM.Poly.denote ctx p)

def Nat.SOM.Poly.add (p₁ p₂ : Poly) : Poly

Equations

• p₁.add p₂ = Nat.SOM.Poly.add.go Nat.Linear.hugeFuel p₁ p₂

def Nat.SOM.Poly.add.go (fuel : Nat) (p₁ p₂ : Poly) : Poly

Equations

• One or more equations did not get rendered due to their size.
• Nat.SOM.Poly.add.go 0 p₁ p₂ = p₁ ++ p₂
• Nat.SOM.Poly.add.go fuel_2.succ p₁ [] = p₁
• Nat.SOM.Poly.add.go fuel_2.succ [] p₂ = p₂

def Nat.SOM.Poly.insertSorted (k : Nat) (m : Mon) (p : Poly) : Poly

Equations

• Nat.SOM.Poly.insertSorted k m [] = [(k, m)]
• Nat.SOM.Poly.insertSorted k m ((k_1, m_1) :: p_1) = bif decide (m < m_1) then (k, m) :: (k_1, m_1) :: p_1 else (k_1, m_1) :: Nat.SOM.Poly.insertSorted k m p_1

def Nat.SOM.Poly.mulMon (p : Poly) (k : Nat) (m : Mon) : Poly

Equations

• p.mulMon k m = Nat.SOM.Poly.mulMon.go k m p []

def Nat.SOM.Poly.mulMon.go (k : Nat) (m : Mon) (p acc : Poly) : Poly

Equations

• Nat.SOM.Poly.mulMon.go k m [] acc = acc
• Nat.SOM.Poly.mulMon.go k m ((k_1, m_1) :: p_1) acc = Nat.SOM.Poly.mulMon.go k m p_1 (Nat.SOM.Poly.insertSorted (k * k_1) (m.mul m_1) acc)

def Nat.SOM.Poly.mul (p₁ p₂ : Poly) : Poly

Equations

• p₁.mul p₂ = Nat.SOM.Poly.mul.go p₂ p₁ []

def Nat.SOM.Poly.mul.go (p₂ p₁ acc : Poly) : Poly

Equations

• Nat.SOM.Poly.mul.go p₂ [] acc = acc
• Nat.SOM.Poly.mul.go p₂ ((k, m) :: p) acc = Nat.SOM.Poly.mul.go p₂ p (acc.add (p₂.mulMon k m))

def Nat.SOM.Expr.toPoly : Expr → Poly

Equations

• (Nat.SOM.Expr.num n).toPoly = bif n == 0 then [] else [(n, [])]
• (Nat.SOM.Expr.var v).toPoly = [(1, [v])]
• (a.add b).toPoly = a.toPoly.add b.toPoly
• (a.mul b).toPoly = a.toPoly.mul b.toPoly

theorem Nat.SOM.Mon.append_denote (ctx : Linear.Context) (m₁ m₂ : Mon) : denote ctx (m₁ ++ m₂) = denote ctx m₁ * denote ctx m₂

theorem Nat.SOM.Mon.mul_denote (ctx : Linear.Context) (m₁ m₂ : Mon) : denote ctx (m₁.mul m₂) = denote ctx m₁ * denote ctx m₂

theorem Nat.SOM.Mon.mul_denote.go (ctx : Linear.Context) (fuel : Nat) (m₁ m₂ : Mon) : denote ctx (mul.go fuel m₁ m₂) = denote ctx m₁ * denote ctx m₂

theorem Nat.SOM.Poly.append_denote (ctx : Linear.Context) (p₁ p₂ : Poly) : denote ctx (p₁ ++ p₂) = denote ctx p₁ + denote ctx p₂

theorem Nat.SOM.Poly.add_denote (ctx : Linear.Context) (p₁ p₂ : Poly) : denote ctx (p₁.add p₂) = denote ctx p₁ + denote ctx p₂

theorem Nat.SOM.Poly.add_denote.go (ctx : Linear.Context) (fuel : Nat) (p₁ p₂ : Poly) : denote ctx (add.go fuel p₁ p₂) = denote ctx p₁ + denote ctx p₂

theorem Nat.SOM.Poly.denote_insertSorted (ctx : Linear.Context) (k : Nat) (m : Mon) (p : Poly) : denote ctx (insertSorted k m p) = denote ctx p + k * Mon.denote ctx m

theorem Nat.SOM.Poly.mulMon_denote (ctx : Linear.Context) (p : Poly) (k : Nat) (m : Mon) : denote ctx (p.mulMon k m) = denote ctx p * k * Mon.denote ctx m

theorem Nat.SOM.Poly.mulMon_denote.go (ctx : Linear.Context) (k : Nat) (m : Mon) (p acc : Poly) : denote ctx (mulMon.go k m p acc) = denote ctx acc + denote ctx p * k * Mon.denote ctx m

theorem Nat.SOM.Poly.mul_denote (ctx : Linear.Context) (p₁ p₂ : Poly) : denote ctx (p₁.mul p₂) = denote ctx p₁ * denote ctx p₂

theorem Nat.SOM.Poly.mul_denote.go (ctx : Linear.Context) (p₂ p₁ acc : Poly) : denote ctx (mul.go p₂ p₁ acc) = denote ctx acc + denote ctx p₁ * denote ctx p₂

theorem Nat.SOM.Expr.toPoly_denote (ctx : Linear.Context) (e : Expr) : Poly.denote ctx e.toPoly = denote ctx e

theorem Nat.SOM.Expr.eq_of_toPoly_eq (ctx : Linear.Context) (a b : Expr) (h : (a.toPoly == b.toPoly) = true) : denote ctx a = denote ctx b