norm_num extension for integer div/mod and divides

This file adds support for the %, /, and ∣ (divisibility) operators on ℤ to the norm_num tactic.

theorem Mathlib.Meta.NormNum.isInt_ediv_zero {a b r : ℤ} : IsInt a r → IsNat b 0 → IsNat (a / b) 0

theorem Mathlib.Meta.NormNum.isInt_ediv {a b q m a' : ℤ} {b' r : ℕ} (ha : IsInt a a') (hb : IsNat b b') (hm : q * ↑b' = m) (h : ↑r + m = a') (h₂ : r.blt b' = true) : IsInt (a / b) q

theorem Mathlib.Meta.NormNum.isInt_ediv_neg {a b q q' : ℤ} (h : IsInt (a / -b) q) (hq : -q = q') : IsInt (a / b) q'

theorem Mathlib.Meta.NormNum.isNat_neg_of_isNegNat {a : ℤ} {b : ℕ} (h : IsInt a (Int.negOfNat b)) : IsNat (-a) b

def Mathlib.Meta.NormNum.evalIntDiv : NormNumExt

The norm_num extension which identifies expressions of the form Int.ediv a b, such that norm_num successfully recognises both a and b.

def Mathlib.Meta.NormNum.evalIntDiv.core (a na : Q(ℤ)) (za : ℤ) (pa : Q(IsInt «$a» «$na»)) (b : Q(ℤ)) (nb : Q(ℕ)) (pb : Q(IsNat «$b» «$nb»)) : ℤ × (q : Q(ℤ)) × Q(IsInt («$a» / «$b») «$q»)

Given a result for evaluating a b in ℤ where b > 0, evaluate a / b.

theorem Mathlib.Meta.NormNum.isInt_emod_zero {a b r : ℤ} : IsInt a r → IsNat b 0 → IsInt (a % b) r

theorem Mathlib.Meta.NormNum.isInt_emod {a b q m a' : ℤ} {b' r : ℕ} (ha : IsInt a a') (hb : IsNat b b') (hm : q * ↑b' = m) (h : ↑r + m = a') (h₂ : r.blt b' = true) : IsNat (a % b) r

theorem Mathlib.Meta.NormNum.isInt_emod_neg {a b : ℤ} {r : ℕ} (h : IsNat (a % -b) r) : IsNat (a % b) r

def Mathlib.Meta.NormNum.evalIntMod : NormNumExt

The norm_num extension which identifies expressions of the form Int.emod a b, such that norm_num successfully recognises both a and b.

def Mathlib.Meta.NormNum.evalIntMod.go (a na : Q(ℤ)) (za : ℤ) (pa : Q(IsInt «$a» «$na»)) (b : Q(ℤ)) : Result b → Option (Result q(«$a» % «$b»))

Given a result for evaluating a b in ℤ, evaluate a % b.

def Mathlib.Meta.NormNum.evalIntMod.core (a na : Q(ℤ)) (za : ℤ) (pa : Q(IsInt «$a» «$na»)) (b : Q(ℤ)) (nb : Q(ℕ)) (pb : Q(IsNat «$b» «$nb»)) : (r : Q(ℕ)) × Q(IsNat («$a» % «$b») «$r»)

Given a result for evaluating a b in ℤ where b > 0, evaluate a % b.

theorem Mathlib.Meta.NormNum.isInt_dvd_true {a b a' b' c : ℤ} : IsInt a a' → IsInt b b' → a'.mul c = b' → a ∣ b

theorem Mathlib.Meta.NormNum.isInt_dvd_false {a b a' b' : ℤ} : IsInt a a' → IsInt b b' → (b'.emod a' != 0) = true → ¬a ∣ b

def Mathlib.Meta.NormNum.evalIntDvd : NormNumExt

The norm_num extension which identifies expressions of the form (a : ℤ) ∣ b, such that norm_num successfully recognises both a and b.