Helper definitions and theorems for constructing linear arithmetic proofs.

@[reducible, inline]

abbrev Nat.Linear.Var : Type

Equations

• Nat.Linear.Var = Nat

@[reducible, inline]

abbrev Nat.Linear.Context : Type

Equations

• Nat.Linear.Context = Lean.RArray Nat

def Nat.Linear.fixedVar : Nat

When encoding polynomials. We use fixedVar for encoding numerals. The denotation of fixedVar is always 1.

Equations

• Nat.Linear.fixedVar = 100000000

def Nat.Linear.Var.denote (ctx : Context) (v : Var) : Nat

Equations

• Nat.Linear.Var.denote ctx v = bif v == Nat.Linear.fixedVar then 1 else Lean.RArray.get ctx v

inductive Nat.Linear.Expr : Type

• num (v : Nat) : Expr
• var (i : Var) : Expr
• add (a b : Expr) : Expr
• mulL (k : Nat) (a : Expr) : Expr
• mulR (a : Expr) (k : Nat) : Expr

instance Nat.Linear.instInhabitedExpr : Inhabited Expr

Equations

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

instance Nat.Linear.instBEqExpr : BEq Expr

Equations

• Nat.Linear.instBEqExpr = { beq := Nat.Linear.beqExpr✝ }

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

Equations

• Nat.Linear.Expr.denote ctx (a.add b) = (Nat.Linear.Expr.denote ctx a).add (Nat.Linear.Expr.denote ctx b)
• Nat.Linear.Expr.denote ctx (Nat.Linear.Expr.num k) = k
• Nat.Linear.Expr.denote ctx (Nat.Linear.Expr.var v) = Nat.Linear.Var.denote ctx v
• Nat.Linear.Expr.denote ctx (Nat.Linear.Expr.mulL k e) = k.mul (Nat.Linear.Expr.denote ctx e)
• Nat.Linear.Expr.denote ctx (e.mulR k) = (Nat.Linear.Expr.denote ctx e).mul k

@[reducible, inline]

abbrev Nat.Linear.Poly : Type

Equations

• Nat.Linear.Poly = List (Nat × Nat.Linear.Var)

def Nat.Linear.Poly.denote (ctx : Context) (p : Poly) : Nat

Equations

• Nat.Linear.Poly.denote ctx [] = 0
• Nat.Linear.Poly.denote ctx ((k, v) :: p_2) = (k.mul (Nat.Linear.Var.denote ctx v)).add (Nat.Linear.Poly.denote ctx p_2)

def Nat.Linear.Poly.insert (k : Nat) (v : Var) (p : Poly) : Poly

Equations

• One or more equations did not get rendered due to their size.
• Nat.Linear.Poly.insert k v [] = [(k, v)]

def Nat.Linear.Poly.norm (p : Poly) : Poly

Equations

• p.norm = Nat.Linear.Poly.norm.go p []

def Nat.Linear.Poly.norm.go (p r : Poly) : Poly

Equations

• Nat.Linear.Poly.norm.go [] r = r
• Nat.Linear.Poly.norm.go ((k, v) :: p_2) r = Nat.Linear.Poly.norm.go p_2 (Nat.Linear.Poly.insert k v r)

def Nat.Linear.Poly.cancelAux (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly) : Poly × Poly

Equations

• One or more equations did not get rendered due to their size.
• Nat.Linear.Poly.cancelAux 0 m₁ m₂ r₁ r₂ = (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)
• Nat.Linear.Poly.cancelAux fuel_2.succ m₁ [] r₁ r₂ = (List.reverse r₁ ++ m₁, List.reverse r₂)
• Nat.Linear.Poly.cancelAux fuel_2.succ [] m₂ r₁ r₂ = (List.reverse r₁, List.reverse r₂ ++ m₂)

def Nat.Linear.hugeFuel : Nat

Equations

• Nat.Linear.hugeFuel = 1000000

def Nat.Linear.Poly.cancel (p₁ p₂ : Poly) : Poly × Poly

Equations

• p₁.cancel p₂ = Nat.Linear.Poly.cancelAux Nat.Linear.hugeFuel p₁ p₂ [] []

def Nat.Linear.Poly.isNum? (p : Poly) : Option Nat

Equations

• Nat.Linear.Poly.isNum? [] = some 0
• Nat.Linear.Poly.isNum? [(k, v)] = bif v == Nat.Linear.fixedVar then some k else none
• p.isNum? = none

def Nat.Linear.Poly.isZero (p : Poly) : Bool

Equations

• Nat.Linear.Poly.isZero [] = true
• p.isZero = false

def Nat.Linear.Poly.isNonZero (p : Poly) : Bool

Equations

• Nat.Linear.Poly.isNonZero [] = false
• Nat.Linear.Poly.isNonZero ((k, v) :: p_2) = bif v == Nat.Linear.fixedVar then decide (k > 0) else Nat.Linear.Poly.isNonZero p_2

def Nat.Linear.Poly.denote_eq (ctx : Context) (mp : Poly × Poly) : Prop

Equations

• Nat.Linear.Poly.denote_eq ctx mp = (Nat.Linear.Poly.denote ctx mp.fst = Nat.Linear.Poly.denote ctx mp.snd)

def Nat.Linear.Poly.denote_le (ctx : Context) (mp : Poly × Poly) : Prop

Equations

• Nat.Linear.Poly.denote_le ctx mp = (Nat.Linear.Poly.denote ctx mp.fst ≤ Nat.Linear.Poly.denote ctx mp.snd)

def Nat.Linear.Expr.toPoly (e : Expr) : Poly

Equations

• e.toPoly = Nat.Linear.Expr.toPoly.go 1 e []

def Nat.Linear.Expr.toPoly.go (coeff : Nat) : Expr → Poly → Poly

Equations

• Nat.Linear.Expr.toPoly.go coeff (Nat.Linear.Expr.num k) = bif k == 0 then id else fun (x : Nat.Linear.Poly) => (coeff * k, Nat.Linear.fixedVar) :: x
• Nat.Linear.Expr.toPoly.go coeff (Nat.Linear.Expr.var i) = fun (x : Nat.Linear.Poly) => (coeff, i) :: x
• Nat.Linear.Expr.toPoly.go coeff (a.add b) = Nat.Linear.Expr.toPoly.go coeff a ∘ Nat.Linear.Expr.toPoly.go coeff b
• Nat.Linear.Expr.toPoly.go coeff (Nat.Linear.Expr.mulL k a) = bif k == 0 then id else Nat.Linear.Expr.toPoly.go (coeff * k) a
• Nat.Linear.Expr.toPoly.go coeff (a.mulR k) = bif k == 0 then id else Nat.Linear.Expr.toPoly.go (coeff * k) a

def Nat.Linear.Expr.toNormPoly (e : Expr) : Poly

Equations

• e.toNormPoly = e.toPoly.norm

def Nat.Linear.Expr.inc (e : Expr) : Expr

Equations

• e.inc = e.add (Nat.Linear.Expr.num 1)

structure Nat.Linear.PolyCnstr : Type

• eq : Bool
• lhs : Poly
• rhs : Poly

instance Nat.Linear.instBEqPolyCnstr : BEq PolyCnstr

Equations

• Nat.Linear.instBEqPolyCnstr = { beq := Nat.Linear.beqPolyCnstr✝ }

instance Nat.Linear.instLawfulBEqPolyCnstr : LawfulBEq PolyCnstr

structure Nat.Linear.ExprCnstr : Type

• eq : Bool
• lhs : Expr
• rhs : Expr

def Nat.Linear.PolyCnstr.denote (ctx : Context) (c : PolyCnstr) : Prop

Equations

• Nat.Linear.PolyCnstr.denote ctx c = bif c.eq then Nat.Linear.Poly.denote_eq ctx (c.lhs, c.rhs) else Nat.Linear.Poly.denote_le ctx (c.lhs, c.rhs)

def Nat.Linear.PolyCnstr.norm (c : PolyCnstr) : PolyCnstr

Equations

• c.norm = match c.lhs.norm.cancel c.rhs.norm with | (lhs, rhs) => { eq := c.eq, lhs := lhs, rhs := rhs }

def Nat.Linear.PolyCnstr.isUnsat (c : PolyCnstr) : Bool

Equations

• c.isUnsat = bif c.eq then c.lhs.isZero && c.rhs.isNonZero || c.lhs.isNonZero && c.rhs.isZero else c.lhs.isNonZero && c.rhs.isZero

def Nat.Linear.PolyCnstr.isValid (c : PolyCnstr) : Bool

Equations

• c.isValid = bif c.eq then c.lhs.isZero && c.rhs.isZero else c.lhs.isZero

def Nat.Linear.ExprCnstr.denote (ctx : Context) (c : ExprCnstr) : Prop

Equations

• Nat.Linear.ExprCnstr.denote ctx c = bif c.eq then Nat.Linear.Expr.denote ctx c.lhs = Nat.Linear.Expr.denote ctx c.rhs else Nat.Linear.Expr.denote ctx c.lhs ≤ Nat.Linear.Expr.denote ctx c.rhs

def Nat.Linear.ExprCnstr.toPoly (c : ExprCnstr) : PolyCnstr

Equations

• c.toPoly = { eq := c.eq, lhs := c.lhs.toPoly, rhs := c.rhs.toPoly }

def Nat.Linear.ExprCnstr.toNormPoly (c : ExprCnstr) : PolyCnstr

Equations

• c.toNormPoly = match c.lhs.toNormPoly.cancel c.rhs.toNormPoly with | (lhs, rhs) => { eq := c.eq, lhs := lhs, rhs := rhs }

def Nat.Linear.monomialToExpr (k : Nat) (v : Var) : Expr

Equations

• Nat.Linear.monomialToExpr k v = bif v == Nat.Linear.fixedVar then Nat.Linear.Expr.num k else bif k == 1 then Nat.Linear.Expr.var v else Nat.Linear.Expr.mulL k (Nat.Linear.Expr.var v)

def Nat.Linear.Poly.toExpr (p : Poly) : Expr

Equations

• Nat.Linear.Poly.toExpr [] = Nat.Linear.Expr.num 0
• Nat.Linear.Poly.toExpr ((k, v) :: p_2) = Nat.Linear.Poly.toExpr.go (Nat.Linear.monomialToExpr k v) p_2

def Nat.Linear.Poly.toExpr.go (e : Expr) (p : Poly) : Expr

Equations

• Nat.Linear.Poly.toExpr.go e [] = e
• Nat.Linear.Poly.toExpr.go e ((k, v) :: p_2) = Nat.Linear.Poly.toExpr.go (e.add (Nat.Linear.monomialToExpr k v)) p_2

def Nat.Linear.PolyCnstr.toExpr (c : PolyCnstr) : ExprCnstr

Equations

• c.toExpr = { eq := c.eq, lhs := c.lhs.toExpr, rhs := c.rhs.toExpr }

theorem Nat.Linear.Poly.denote_insert (ctx : Context) (k : Nat) (v : Var) (p : Poly) : denote ctx (insert k v p) = denote ctx p + k * Var.denote ctx v

theorem Nat.Linear.Poly.denote_norm_go (ctx : Context) (p r : Poly) : denote ctx (norm.go p r) = denote ctx p + denote ctx r

theorem Nat.Linear.Poly.denote_sort (ctx : Context) (m : Poly) : denote ctx m.norm = denote ctx m

theorem Nat.Linear.Poly.denote_append (ctx : Context) (p q : Poly) : denote ctx (p ++ q) = denote ctx p + denote ctx q

theorem Nat.Linear.Poly.denote_cons (ctx : Context) (k : Nat) (v : Var) (p : Poly) : denote ctx ((k, v) :: p) = k * Var.denote ctx v + denote ctx p

theorem Nat.Linear.Poly.denote_reverse (ctx : Context) (p : Poly) : denote ctx (List.reverse p) = denote ctx p

theorem Nat.Linear.Poly.denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly) (h : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)

theorem Nat.Linear.Poly.of_denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly) (h : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)) : denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)

theorem Nat.Linear.Poly.denote_eq_cancel {ctx : Context} {m₁ m₂ : Poly} (h : denote_eq ctx (m₁, m₂)) : denote_eq ctx (m₁.cancel m₂)

theorem Nat.Linear.Poly.of_denote_eq_cancel {ctx : Context} {m₁ m₂ : Poly} (h : denote_eq ctx (m₁.cancel m₂)) : denote_eq ctx (m₁, m₂)

theorem Nat.Linear.Poly.denote_eq_cancel_eq (ctx : Context) (m₁ m₂ : Poly) : denote_eq ctx (m₁.cancel m₂) = denote_eq ctx (m₁, m₂)

theorem Nat.Linear.Poly.denote_le_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly) (h : denote_le ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)) : denote_le ctx (cancelAux fuel m₁ m₂ r₁ r₂)

theorem Nat.Linear.Poly.of_denote_le_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly) (h : denote_le ctx (cancelAux fuel m₁ m₂ r₁ r₂)) : denote_le ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)

theorem Nat.Linear.Poly.denote_le_cancel {ctx : Context} {m₁ m₂ : Poly} (h : denote_le ctx (m₁, m₂)) : denote_le ctx (m₁.cancel m₂)

theorem Nat.Linear.Poly.of_denote_le_cancel {ctx : Context} {m₁ m₂ : Poly} (h : denote_le ctx (m₁.cancel m₂)) : denote_le ctx (m₁, m₂)

theorem Nat.Linear.Poly.denote_le_cancel_eq (ctx : Context) (m₁ m₂ : Poly) : denote_le ctx (m₁.cancel m₂) = denote_le ctx (m₁, m₂)

theorem Nat.Linear.Expr.denote_toPoly_go {k : Nat} {p : Poly} (ctx : Context) (e : Expr) : Poly.denote ctx (toPoly.go k e p) = k * denote ctx e + Poly.denote ctx p

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

theorem Nat.Linear.Expr.eq_of_toNormPoly (ctx : Context) (a b : Expr) (h : a.toNormPoly = b.toNormPoly) : denote ctx a = denote ctx b

theorem Nat.Linear.Expr.of_cancel_eq (ctx : Context) (a b c d : Expr) (h : a.toNormPoly.cancel b.toNormPoly = (c.toPoly, d.toPoly)) : (denote ctx a = denote ctx b) = (denote ctx c = denote ctx d)

theorem Nat.Linear.Expr.of_cancel_le (ctx : Context) (a b c d : Expr) (h : a.toNormPoly.cancel b.toNormPoly = (c.toPoly, d.toPoly)) : (denote ctx a ≤ denote ctx b) = (denote ctx c ≤ denote ctx d)

theorem Nat.Linear.Expr.of_cancel_lt (ctx : Context) (a b c d : Expr) (h : a.inc.toNormPoly.cancel b.toNormPoly = (c.inc.toPoly, d.toPoly)) : (denote ctx a < denote ctx b) = (denote ctx c < denote ctx d)

theorem Nat.Linear.ExprCnstr.toPoly_norm_eq (c : ExprCnstr) : c.toPoly.norm = c.toNormPoly

theorem Nat.Linear.ExprCnstr.denote_toPoly (ctx : Context) (c : ExprCnstr) : PolyCnstr.denote ctx c.toPoly = denote ctx c

theorem Nat.Linear.ExprCnstr.denote_toNormPoly (ctx : Context) (c : ExprCnstr) : PolyCnstr.denote ctx c.toNormPoly = denote ctx c

theorem Nat.Linear.Poly.of_isZero (ctx : Context) {p : Poly} (h : p.isZero = true) : denote ctx p = 0

theorem Nat.Linear.Poly.of_isNonZero (ctx : Context) {p : Poly} (h : p.isNonZero = true) : denote ctx p > 0

theorem Nat.Linear.PolyCnstr.eq_false_of_isUnsat (ctx : Context) {c : PolyCnstr} : c.isUnsat = true → denote ctx c = False

theorem Nat.Linear.PolyCnstr.eq_true_of_isValid (ctx : Context) {c : PolyCnstr} : c.isValid = true → denote ctx c = True

theorem Nat.Linear.ExprCnstr.eq_false_of_isUnsat (ctx : Context) (c : ExprCnstr) (h : c.toNormPoly.isUnsat = true) : denote ctx c = False

theorem Nat.Linear.ExprCnstr.eq_true_of_isValid (ctx : Context) (c : ExprCnstr) (h : c.toNormPoly.isValid = true) : denote ctx c = True

theorem Nat.Linear.ExprCnstr.eq_of_toNormPoly_eq (ctx : Context) (c d : ExprCnstr) (h : (c.toNormPoly == d.toPoly) = true) : denote ctx c = denote ctx d

theorem Nat.Linear.Expr.eq_of_toNormPoly_eq (ctx : Context) (e e' : Expr) (h : (e.toNormPoly == e'.toPoly) = true) : denote ctx e = denote ctx e'

def Nat.elimOffset {α : Sort u} (a b k : Nat) (h₁ : a + k = b + k) (h₂ : a = b → α) : α

Equations

• a.elimOffset b k h₁ h₂ = h₂ ⋯