norm_num extensions for inequalities.

def Mathlib.Meta.NormNum.inferOrderedSemiring {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM ((x : Q(Semiring «$α»)) × (x_1 : Q(PartialOrder «$α»)) × Q(IsOrderedRing «$α»))

Helper function to synthesize typed Semiring α PartialOrder α IsOrderedRing α expressions.

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def Mathlib.Meta.NormNum.inferOrderedRing {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM ((x : Q(Ring «$α»)) × (x_1 : Q(PartialOrder «$α»)) × Q(IsOrderedRing «$α»))

Helper function to synthesize typed Ring α PartialOrder α IsOrderedRing α expressions.

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def Mathlib.Meta.NormNum.inferLinearOrderedSemifield {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM ((x : Q(Semifield «$α»)) × (x_1 : Q(LinearOrder «$α»)) × Q(IsStrictOrderedRing «$α»))

Helper function to synthesize typed Semifield α LinearOrder α IsStrictOrderedRing α expressions.

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def Mathlib.Meta.NormNum.inferLinearOrderedField {u : Lean.Level} (α : Q(Type u)) : Lean.MetaM ((x : Q(Field «$α»)) × (x_1 : Q(LinearOrder «$α»)) × Q(IsStrictOrderedRing «$α»))

Helper function to synthesize typed Field α LinearOrder α IsStrictOrderedRing α expressions.

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theorem Mathlib.Meta.NormNum.isNat_le_true {α : Type u_1} [Semiring α] [PartialOrder α] [IsOrderedRing 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} [Semiring α] [PartialOrder α] [IsOrderedRing 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.isNNRat_le_true {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {na nb da db : ℕ} : IsNNRat a na da → IsNNRat b nb db → decide (na.mul db ≤ nb.mul da) = true → a ≤ b

theorem Mathlib.Meta.NormNum.isNNRat_lt_true {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {na nb da db : ℕ} : IsNNRat a na da → IsNNRat b nb db → decide (na * db < nb * da) = true → a < b

theorem Mathlib.Meta.NormNum.isNNRat_le_false {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {na nb da db : ℕ} (ha : IsNNRat a na da) (hb : IsNNRat b nb db) (h : decide (nb * da < na * db) = true) : ¬a ≤ b

theorem Mathlib.Meta.NormNum.isNNRat_lt_false {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a b : α} {na nb da db : ℕ} (ha : IsNNRat a na da) (hb : IsNNRat b nb db) (h : decide (nb * da ≤ na * db) = true) : ¬a < b

theorem Mathlib.Meta.NormNum.isRat_le_true {α : Type u_1} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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} [Semiring α] [PartialOrder α] [IsOrderedRing 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} [Semiring α] [PartialOrder α] [IsOrderedRing 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} [Ring α] [PartialOrder α] [IsOrderedRing 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} [Ring α] [PartialOrder α] [IsOrderedRing 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} [Ring α] [PartialOrder α] [IsOrderedRing 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} [Ring α] [PartialOrder α] [IsOrderedRing 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.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.

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• Mathlib.Meta.NormNum.evalLE.core lα (Mathlib.Meta.NormNum.Result'.isBool val proof) rb = failure
• Mathlib.Meta.NormNum.evalLE.core lα ra (Mathlib.Meta.NormNum.Result'.isBool val proof) = failure
• Mathlib.Meta.NormNum.evalLE.core lα (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) rb = Mathlib.Meta.NormNum.evalLE.core.intArm✝ lα (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) rb
• Mathlib.Meta.NormNum.evalLE.core lα ra (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) = Mathlib.Meta.NormNum.evalLE.core.intArm✝ lα ra (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof)

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.

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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.

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• Mathlib.Meta.NormNum.evalLT.core lα (Mathlib.Meta.NormNum.Result'.isBool val proof) rb = failure
• Mathlib.Meta.NormNum.evalLT.core lα ra (Mathlib.Meta.NormNum.Result'.isBool val proof) = failure
• Mathlib.Meta.NormNum.evalLT.core lα (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) rb = Mathlib.Meta.NormNum.evalLT.core.intArm✝ lα (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) rb
• Mathlib.Meta.NormNum.evalLT.core lα ra (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof) = Mathlib.Meta.NormNum.evalLT.core.intArm✝ lα ra (Mathlib.Meta.NormNum.Result'.isNegNat inst lit proof)

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.

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