Documentation

Mathlib.Tactic.NormNum.Ineq

norm_num extensions for inequalities. #

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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 IsOrderedSemiring 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 IsOrderedSemiring 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
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        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
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        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
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        theorem Mathlib.Meta.NormNum.isRat_lt_true {α : Type u_1} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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
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        theorem Mathlib.Meta.NormNum.isRat_le_false {α : Type u_1} [Ring α] [LinearOrder α] [IsStrictOrderedRing 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
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        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 #

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        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
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        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
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        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
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        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
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        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
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        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
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        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.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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            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) :
            let e := q(«$a» ≤ «$b»); Lean.MetaM (Result e)
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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) :
              let e := q(«$a» ≤ «$b»); Lean.MetaM (Result e)
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                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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                  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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                    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) :
                    let e := q(«$a» < «$b»); Lean.MetaM (Result e)
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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) :
                      let e := q(«$a» < «$b»); Lean.MetaM (Result e)
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