norm_num extensions for inequalities.

def Mathlib.Meta.NormNum.inferOrderedSemiring {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(OrderedSemiring «$α»)

Helper function to synthesize a typed OrderedSemiring α expression.

Equations

• Mathlib.Meta.NormNum.inferOrderedSemiring α = do let __do_lift ← Qq.synthInstanceQ q(OrderedSemiring «$α») <|> Lean.throwError (Lean.toMessageData "not an ordered semiring") pure __do_lift

def Mathlib.Meta.NormNum.inferOrderedRing {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(OrderedRing «$α»)

Helper function to synthesize a typed OrderedRing α expression.

Equations

• Mathlib.Meta.NormNum.inferOrderedRing α = do let __do_lift ← Qq.synthInstanceQ q(OrderedRing «$α») <|> Lean.throwError (Lean.toMessageData "not an ordered ring") pure __do_lift

def Mathlib.Meta.NormNum.inferLinearOrderedField {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM Q(LinearOrderedField «$α»)

Helper function to synthesize a typed LinearOrderedField α expression.

Equations

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

theorem Mathlib.Meta.NormNum.isNat_le_true {α : Type u_1} [OrderedSemiring α] {a b : α} {a' b' : ℕ} : IsNat a a' → IsNat b b' → a'.ble b' = true → a ≤ b

theorem Mathlib.Meta.NormNum.isNat_lt_false {α : Type u_1} [OrderedSemiring α] {a b : α} {a' b' : ℕ} (ha : IsNat a a') (hb : IsNat b b') (h : b'.ble a' = true) : ¬a < b

theorem Mathlib.Meta.NormNum.isRat_le_true {α : Type u_1} [LinearOrderedRing α] {a b : α} {na nb : ℤ} {da db : ℕ} : IsRat a na da → IsRat b nb db → decide (na.mul (Int.ofNat db) ≤ nb.mul (Int.ofNat da)) = true → a ≤ b

theorem Mathlib.Meta.NormNum.isRat_lt_true {α : Type u_1} [LinearOrderedRing α] [Nontrivial α] {a b : α} {na nb : ℤ} {da db : ℕ} : IsRat a na da → IsRat b nb db → decide (na * ↑db < nb * ↑da) = true → a < b

theorem Mathlib.Meta.NormNum.isRat_le_false {α : Type u_1} [LinearOrderedRing α] [Nontrivial α] {a b : α} {na nb : ℤ} {da db : ℕ} (ha : IsRat a na da) (hb : IsRat b nb db) (h : decide (nb * ↑da < na * ↑db) = true) : ¬a ≤ b

theorem Mathlib.Meta.NormNum.isRat_lt_false {α : Type u_1} [LinearOrderedRing α] {a b : α} {na nb : ℤ} {da db : ℕ} (ha : IsRat a na da) (hb : IsRat b nb db) (h : decide (nb * ↑da ≤ na * ↑db) = true) : ¬a < b

(In)equalities

theorem Mathlib.Meta.NormNum.isNat_lt_true {α : Type u_1} [OrderedSemiring α] [CharZero α] {a b : α} {a' b' : ℕ} : IsNat a a' → IsNat b b' → b'.ble a' = false → a < b

theorem Mathlib.Meta.NormNum.isNat_le_false {α : Type u_1} [OrderedSemiring α] [CharZero α] {a b : α} {a' b' : ℕ} (ha : IsNat a a') (hb : IsNat b b') (h : a'.ble b' = false) : ¬a ≤ b

theorem Mathlib.Meta.NormNum.isInt_le_true {α : Type u_1} [OrderedRing α] {a b : α} {a' b' : ℤ} : IsInt a a' → IsInt b b' → decide (a' ≤ b') = true → a ≤ b

theorem Mathlib.Meta.NormNum.isInt_lt_true {α : Type u_1} [OrderedRing α] [Nontrivial α] {a b : α} {a' b' : ℤ} : IsInt a a' → IsInt b b' → decide (a' < b') = true → a < b

theorem Mathlib.Meta.NormNum.isInt_le_false {α : Type u_1} [OrderedRing α] [Nontrivial α] {a b : α} {a' b' : ℤ} (ha : IsInt a a') (hb : IsInt b b') (h : decide (b' < a') = true) : ¬a ≤ b

theorem Mathlib.Meta.NormNum.isInt_lt_false {α : Type u_1} [OrderedRing α] {a b : α} {a' b' : ℤ} (ha : IsInt a a') (hb : IsInt b b') (h : decide (b' ≤ a') = true) : ¬a < b

def Mathlib.Meta.NormNum.evalLE : NormNumExt

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

Equations

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

def Mathlib.Meta.NormNum.evalLE.core {u : Lean.Level} {α : Q(Type u)} (lα : Q(LE «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) : Lean.MetaM (Result q(«$a» ≤ «$b»))

Identify (as true or false) expressions of the form a ≤ b, where a and b are numeric expressions whose evaluations to NormNum.Result have already been computed.

def Mathlib.Meta.NormNum.evalLE.core.intArm {u : Lean.Level} {α : Q(Type u)} (lα : Q(LE «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) (e : Q(Prop)) : Lean.MetaM (Result e)

Equations

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def Mathlib.Meta.NormNum.evalLE.core.ratArm {u : Lean.Level} {α : Q(Type u)} (lα : Q(LE «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) (e : Q(Prop)) : Lean.MetaM (Result e)

Equations

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

def Mathlib.Meta.NormNum.evalLT : NormNumExt

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

Equations

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

def Mathlib.Meta.NormNum.evalLT.core {u : Lean.Level} {α : Q(Type u)} (lα : Q(LT «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) : Lean.MetaM (Result q(«$a» < «$b»))

Identify (as true or false) expressions of the form a < b, where a and b are numeric expressions whose evaluations to NormNum.Result have already been computed.

def Mathlib.Meta.NormNum.evalLT.core.intArm {u : Lean.Level} {α : Q(Type u)} (lα : Q(LT «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) (e : Q(Prop)) : Lean.MetaM (Result e)

Equations

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def Mathlib.Meta.NormNum.evalLT.core.ratArm {u : Lean.Level} {α : Q(Type u)} (lα : Q(LT «$α»)) {a b : Q(«$α»)} (ra : Result a) (rb : Result b) (e : Q(Prop)) : Lean.MetaM (Result e)

Equations

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