Init.Data.Nat.Linear

Nat.Linear.Poly.fuse [] = []
Nat.Linear.Poly.fuse ((k, v) :: p_2) = match Nat.Linear.Poly.fuse p_2 with | [] => [(k, v)] | (k', v') :: p' => bif v == v' then (Nat.add k k', v) :: p' else (k, v) :: (k', v') :: p'

source

def Nat.Linear.Poly.mul (k : Nat) (p : Nat.Linear.Poly) :

Nat.Linear.Poly

Equations

Nat.Linear.Poly.mul k p = bif k == 0 then [] else bif k == 1 then p else Nat.Linear.Poly.mul.go k p

source

def Nat.Linear.Poly.mul.go (k : Nat) :

Nat.Linear.Poly → Nat.Linear.Poly

Equations

Nat.Linear.Poly.mul.go k [] = []
Nat.Linear.Poly.mul.go k ((k_1, v) :: p_1) = (Nat.mul k k_1, v) :: Nat.Linear.Poly.mul.go k p_1

source

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

Nat.Linear.Poly × Nat.Linear.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)

source

def Nat.Linear.hugeFuel:

Nat

Equations

Nat.Linear.hugeFuel = 1000000

source

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

Nat.Linear.Poly × Nat.Linear.Poly

Equations

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

source

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

Option Nat

Equations

Nat.Linear.Poly.isNum? p = match p with | [] => some 0 | [(k, v)] => bif v == Nat.Linear.fixedVar then some k else none | x => none

source

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

Bool

Equations

Nat.Linear.Poly.isZero p = match p with | [] => true | x => false

source

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

source

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

Prop

Equations

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

source

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

Prop

Equations

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

source

def Nat.Linear.Poly.combineAux (fuel : Nat) (p₁ : Nat.Linear.Poly) (p₂ : Nat.Linear.Poly) :

Nat.Linear.Poly

Equations

One or more equations did not get rendered due to their size.
Nat.Linear.Poly.combineAux 0 p₁ p₂ = p₁ ++ p₂

source

def Nat.Linear.Poly.combine (p₁ : Nat.Linear.Poly) (p₂ : Nat.Linear.Poly) :

Nat.Linear.Poly

Equations

Nat.Linear.Poly.combine p₁ p₂ = Nat.Linear.Poly.combineAux Nat.Linear.hugeFuel p₁ p₂

source

def Nat.Linear.Expr.toPoly:

Nat.Linear.Expr → Nat.Linear.Poly

Equations

Nat.Linear.Expr.toPoly (Nat.Linear.Expr.num k) = bif k == 0 then [] else [(k, Nat.Linear.fixedVar)]
Nat.Linear.Expr.toPoly (Nat.Linear.Expr.var i) = [(1, i)]
Nat.Linear.Expr.toPoly (Nat.Linear.Expr.add a b) = Nat.Linear.Expr.toPoly a ++ Nat.Linear.Expr.toPoly b
Nat.Linear.Expr.toPoly (Nat.Linear.Expr.mulL k a) = Nat.Linear.Poly.mul k (Nat.Linear.Expr.toPoly a)
Nat.Linear.Expr.toPoly (Nat.Linear.Expr.mulR a k) = Nat.Linear.Poly.mul k (Nat.Linear.Expr.toPoly a)

source

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

Nat.Linear.Poly

Equations

Nat.Linear.Poly.norm p = Nat.Linear.Poly.fuse (Nat.Linear.Poly.sort p)

source

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

Nat.Linear.Poly

Equations

Nat.Linear.Expr.toNormPoly e = Nat.Linear.Poly.norm (Nat.Linear.Expr.toPoly e)

source

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

Nat.Linear.Expr

Equations

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

source

structure Nat.Linear.PolyCnstr:

Type

source

instance Nat.Linear.instBEqPolyCnstr:

BEq Nat.Linear.PolyCnstr

Equations

Nat.Linear.instBEqPolyCnstr = { beq := [anonymous] }

source

instance Nat.Linear.instLawfulBEqPolyCnstrInstBEqPolyCnstr:

LawfulBEq Nat.Linear.PolyCnstr

Equations

Nat.Linear.instLawfulBEqPolyCnstrInstBEqPolyCnstr = Nat.Linear.instLawfulBEqPolyCnstrInstBEqPolyCnstr.proof_1

source

def Nat.Linear.PolyCnstr.mul (k : Nat) (c : Nat.Linear.PolyCnstr) :

Nat.Linear.PolyCnstr

Equations

Nat.Linear.PolyCnstr.mul k c = { eq := c.eq, lhs := Nat.Linear.Poly.mul k c.lhs, rhs := Nat.Linear.Poly.mul k c.rhs }

source

def Nat.Linear.PolyCnstr.combine (c₁ : Nat.Linear.PolyCnstr) (c₂ : Nat.Linear.PolyCnstr) :

Nat.Linear.PolyCnstr

Equations

One or more equations did not get rendered due to their size.

source

structure Nat.Linear.ExprCnstr:

Type

source

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

source

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

Nat.Linear.PolyCnstr

Equations

One or more equations did not get rendered due to their size.

source

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

Bool

Equations

One or more equations did not get rendered due to their size.

source

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

Bool

Equations

Nat.Linear.PolyCnstr.isValid c = bif c.eq then Nat.Linear.Poly.isZero c.lhs && Nat.Linear.Poly.isZero c.rhs else Nat.Linear.Poly.isZero c.lhs

source

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

source

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

Nat.Linear.PolyCnstr

Equations

Nat.Linear.ExprCnstr.toPoly c = { eq := c.eq, lhs := Nat.Linear.Expr.toPoly c.lhs, rhs := Nat.Linear.Expr.toPoly c.rhs }

source

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

Nat.Linear.PolyCnstr

Equations

One or more equations did not get rendered due to their size.

source

@[inline]

abbrev Nat.Linear.Certificate:

Type

Equations

Nat.Linear.Certificate = List (Nat × Nat.Linear.ExprCnstr)

source

def Nat.Linear.Certificate.combineHyps (c : Nat.Linear.PolyCnstr) (hs : Nat.Linear.Certificate) :

Nat.Linear.PolyCnstr

Equations

One or more equations did not get rendered due to their size.
Nat.Linear.Certificate.combineHyps c [] = c

source

def Nat.Linear.Certificate.combine (hs : Nat.Linear.Certificate) :

Nat.Linear.PolyCnstr

Equations

One or more equations did not get rendered due to their size.

source

def Nat.Linear.Certificate.denote (ctx : Nat.Linear.Context) (c : Nat.Linear.Certificate) :

Prop

Equations

Nat.Linear.Certificate.denote ctx [] = False
Nat.Linear.Certificate.denote ctx ((k, c') :: hs_1) = (Nat.Linear.ExprCnstr.denote ctx c' → Nat.Linear.Certificate.denote ctx hs_1)

source

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

Nat.Linear.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)

source

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

Nat.Linear.Expr

Equations

Nat.Linear.Poly.toExpr p = match p with | [] => Nat.Linear.Expr.num 0 | (k, v) :: p => Nat.Linear.Poly.toExpr.go (Nat.Linear.monomialToExpr k v) p

source

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

Nat.Linear.Expr

Equations

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

source

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

Nat.Linear.ExprCnstr

Equations

Nat.Linear.PolyCnstr.toExpr c = { eq := c.eq, lhs := Nat.Linear.Poly.toExpr c.lhs, rhs := Nat.Linear.Poly.toExpr c.rhs }

source

theorem Nat.Linear.Poly.denote_insertSorted (ctx : Nat.Linear.Context) (k : Nat) (v : Nat.Linear.Var) (p : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.insertSorted k v p) = Nat.Linear.Poly.denote ctx p + k * Nat.Linear.Var.denote ctx v

source

theorem Nat.Linear.Poly.denote_sort_go (ctx : Nat.Linear.Context) (p : Nat.Linear.Poly) (r : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.sort.go p r) = Nat.Linear.Poly.denote ctx p + Nat.Linear.Poly.denote ctx r

source

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

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.sort m) = Nat.Linear.Poly.denote ctx m

source

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

Nat.Linear.Poly.denote ctx (p ++ q) = Nat.Linear.Poly.denote ctx p + Nat.Linear.Poly.denote ctx q

source

theorem Nat.Linear.Poly.denote_cons (ctx : Nat.Linear.Context) (k : Nat) (v : Nat.Linear.Var) (p : Nat.Linear.Poly) :

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

source

theorem Nat.Linear.Poly.denote_reverseAux (ctx : Nat.Linear.Context) (p : Nat.Linear.Poly) (q : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (List.reverseAux p q) = Nat.Linear.Poly.denote ctx (p ++ q)

source

theorem Nat.Linear.Poly.denote_reverse (ctx : Nat.Linear.Context) (p : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (List.reverse p) = Nat.Linear.Poly.denote ctx p

source

theorem Nat.Linear.Poly.denote_fuse (ctx : Nat.Linear.Context) (p : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.fuse p) = Nat.Linear.Poly.denote ctx p

source

theorem Nat.Linear.Poly.denote_mul (ctx : Nat.Linear.Context) (k : Nat) (p : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.mul k p) = k * Nat.Linear.Poly.denote ctx p

source

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

Nat.Linear.Poly.denote_eq ctx (Nat.Linear.Poly.cancelAux fuel m₁ m₂ r₁ r₂)

source

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

Nat.Linear.Poly.denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)

source

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

Nat.Linear.Poly.denote_eq ctx (Nat.Linear.Poly.cancel m₁ m₂)

source

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

Nat.Linear.Poly.denote_eq ctx (m₁, m₂)

source

theorem Nat.Linear.Poly.denote_eq_cancel_eq (ctx : Nat.Linear.Context) (m₁ : Nat.Linear.Poly) (m₂ : Nat.Linear.Poly) :

Nat.Linear.Poly.denote_eq ctx (Nat.Linear.Poly.cancel m₁ m₂) = Nat.Linear.Poly.denote_eq ctx (m₁, m₂)

source

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

Nat.Linear.Poly.denote_le ctx (Nat.Linear.Poly.cancelAux fuel m₁ m₂ r₁ r₂)

source

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

Nat.Linear.Poly.denote_le ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂)

source

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

Nat.Linear.Poly.denote_le ctx (Nat.Linear.Poly.cancel m₁ m₂)

source

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

Nat.Linear.Poly.denote_le ctx (m₁, m₂)

source

theorem Nat.Linear.Poly.denote_le_cancel_eq (ctx : Nat.Linear.Context) (m₁ : Nat.Linear.Poly) (m₂ : Nat.Linear.Poly) :

Nat.Linear.Poly.denote_le ctx (Nat.Linear.Poly.cancel m₁ m₂) = Nat.Linear.Poly.denote_le ctx (m₁, m₂)

source

theorem Nat.Linear.Poly.denote_combineAux (ctx : Nat.Linear.Context) (fuel : Nat) (p₁ : Nat.Linear.Poly) (p₂ : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.combineAux fuel p₁ p₂) = Nat.Linear.Poly.denote ctx p₁ + Nat.Linear.Poly.denote ctx p₂

source

theorem Nat.Linear.Poly.denote_combine (ctx : Nat.Linear.Context) (p₁ : Nat.Linear.Poly) (p₂ : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.combine p₁ p₂) = Nat.Linear.Poly.denote ctx p₁ + Nat.Linear.Poly.denote ctx p₂

source

theorem Nat.Linear.Expr.denote_toPoly (ctx : Nat.Linear.Context) (e : Nat.Linear.Expr) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Expr.toPoly e) = Nat.Linear.Expr.denote ctx e

source

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

Nat.Linear.Expr.denote ctx a = Nat.Linear.Expr.denote ctx b

source

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

(Nat.Linear.Expr.denote ctx a = Nat.Linear.Expr.denote ctx b) = (Nat.Linear.Expr.denote ctx c = Nat.Linear.Expr.denote ctx d)

source

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

(Nat.Linear.Expr.denote ctx a ≤ Nat.Linear.Expr.denote ctx b) = (Nat.Linear.Expr.denote ctx c ≤ Nat.Linear.Expr.denote ctx d)

source

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

(Nat.Linear.Expr.denote ctx a < Nat.Linear.Expr.denote ctx b) = (Nat.Linear.Expr.denote ctx c < Nat.Linear.Expr.denote ctx d)

source

theorem Nat.Linear.ExprCnstr.toPoly_norm_eq (c : Nat.Linear.ExprCnstr) :

Nat.Linear.PolyCnstr.norm (Nat.Linear.ExprCnstr.toPoly c) = Nat.Linear.ExprCnstr.toNormPoly c

source

theorem Nat.Linear.ExprCnstr.denote_toPoly (ctx : Nat.Linear.Context) (c : Nat.Linear.ExprCnstr) :

Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.ExprCnstr.toPoly c) = Nat.Linear.ExprCnstr.denote ctx c

source

theorem Nat.Linear.ExprCnstr.denote_toNormPoly (ctx : Nat.Linear.Context) (c : Nat.Linear.ExprCnstr) :

Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.ExprCnstr.toNormPoly c) = Nat.Linear.ExprCnstr.denote ctx c

source

theorem Nat.Linear.Poly.mul.go_denote (ctx : Nat.Linear.Context) (k : Nat) (p : Nat.Linear.Poly) :

Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.mul.go k p) = k * Nat.Linear.Poly.denote ctx p

source

theorem Nat.Linear.PolyCnstr.denote_mul (ctx : Nat.Linear.Context) (k : Nat) (c : Nat.Linear.PolyCnstr) :

Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.PolyCnstr.mul (k + 1) c) = Nat.Linear.PolyCnstr.denote ctx c

source

theorem Nat.Linear.PolyCnstr.denote_combine {ctx : Nat.Linear.Context} {c₁ : Nat.Linear.PolyCnstr} {c₂ : Nat.Linear.PolyCnstr} (h₁ : Nat.Linear.PolyCnstr.denote ctx c₁) (h₂ : Nat.Linear.PolyCnstr.denote ctx c₂) :

Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.PolyCnstr.combine c₁ c₂)

source

theorem Nat.Linear.Poly.isNum?_eq_some (ctx : Nat.Linear.Context) {p : Nat.Linear.Poly} {k : Nat} :

Nat.Linear.Poly.isNum? p = some k → Nat.Linear.Poly.denote ctx p = k

source

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

Nat.Linear.Poly.denote ctx p = 0

source

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

Nat.Linear.Poly.denote ctx p > 0

source

theorem Nat.Linear.PolyCnstr.eq_false_of_isUnsat (ctx : Nat.Linear.Context) {c : Nat.Linear.PolyCnstr} :

Nat.Linear.PolyCnstr.isUnsat c = true → Nat.Linear.PolyCnstr.denote ctx c = False

source

theorem Nat.Linear.PolyCnstr.eq_true_of_isValid (ctx : Nat.Linear.Context) {c : Nat.Linear.PolyCnstr} :

Nat.Linear.PolyCnstr.isValid c = true → Nat.Linear.PolyCnstr.denote ctx c = True

source

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

Nat.Linear.ExprCnstr.denote ctx c = False

source

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

Nat.Linear.ExprCnstr.denote ctx c = True

source

theorem Nat.Linear.Certificate.of_combineHyps (ctx : Nat.Linear.Context) (c : Nat.Linear.PolyCnstr) (cs : Nat.Linear.Certificate) (h : Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.Certificate.combineHyps c cs) → False) :

Nat.Linear.PolyCnstr.denote ctx c → Nat.Linear.Certificate.denote ctx cs

source

theorem Nat.Linear.Certificate.of_combine (ctx : Nat.Linear.Context) (cs : Nat.Linear.Certificate) (h : Nat.Linear.PolyCnstr.denote ctx (Nat.Linear.Certificate.combine cs) → False) :

Nat.Linear.Certificate.denote ctx cs

source

theorem Nat.Linear.Certificate.of_combine_isUnsat (ctx : Nat.Linear.Context) (cs : Nat.Linear.Certificate) (h : Nat.Linear.PolyCnstr.isUnsat (Nat.Linear.Certificate.combine cs) = true) :

Nat.Linear.Certificate.denote ctx cs

source

theorem Nat.Linear.denote_monomialToExpr (ctx : Nat.Linear.Context) (k : Nat) (v : Nat.Linear.Var) :

Nat.Linear.Expr.denote ctx (Nat.Linear.monomialToExpr k v) = k * Nat.Linear.Var.denote ctx v

source

theorem Nat.Linear.Poly.denote_toExpr_go (ctx : Nat.Linear.Context) (e : Nat.Linear.Expr) (p : Nat.Linear.Poly) :

Nat.Linear.Expr.denote ctx (Nat.Linear.Poly.toExpr.go e p) = Nat.Linear.Expr.denote ctx e + Nat.Linear.Poly.denote ctx p

source

theorem Nat.Linear.Poly.denote_toExpr (ctx : Nat.Linear.Context) (p : Nat.Linear.Poly) :

Nat.Linear.Expr.denote ctx (Nat.Linear.Poly.toExpr p) = Nat.Linear.Poly.denote ctx p

source

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

Nat.Linear.ExprCnstr.denote ctx c = Nat.Linear.ExprCnstr.denote ctx d

source

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

Nat.Linear.Expr.denote ctx e = Nat.Linear.Expr.denote ctx e'