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Mathlib.Data.PNat.Defs

The positive natural numbers #

This file contains the definitions, and basic results. Most algebraic facts are deferred to Data.PNat.Basic, as they need more imports.

def PNat :

ℕ+ is the type of positive natural numbers. It is defined as a subtype, and the VM representation of ℕ+ is the same as because the proof is not stored.

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    ℕ+ is the type of positive natural numbers. It is defined as a subtype, and the VM representation of ℕ+ is the same as because the proof is not stored.

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      instance instOnePNat :
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      def PNat.val :
      ℕ+

      The underlying natural number

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        @[simp]
        theorem PNat.mk_coe (n : ) (h : 0 < n) :
        { val := n, property := h } = n
        def PNat.natPred (i : ℕ+) :

        Predecessor of a ℕ+, as a .

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          @[simp]
          theorem PNat.natPred_eq_pred {n : } (h : 0 < n) :
          PNat.natPred { val := n, property := h } = Nat.pred n
          def Nat.toPNat (n : ) (h : autoParam (0 < n) _auto✝) :

          Convert a natural number to a positive natural number. The positivity assumption is inferred by dec_trivial.

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            def Nat.succPNat (n : ) :

            Write a successor as an element of ℕ+.

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              @[simp]
              theorem Nat.succPNat_coe (n : ) :
              def Nat.toPNat' (n : ) :

              Convert a natural number to a PNat. n+1 is mapped to itself, and 0 becomes 1.

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                @[simp]
                theorem Nat.toPNat'_coe (n : ) :
                (Nat.toPNat' n) = if 0 < n then n else 1
                theorem PNat.mk_le_mk (n : ) (k : ) (hn : 0 < n) (hk : 0 < k) :
                { val := n, property := hn } { val := k, property := hk } n k

                We now define a long list of structures on ℕ+ induced by similar structures on ℕ. Most of these behave in a completely obvious way, but there are a few things to be said about subtraction, division and powers.

                theorem PNat.mk_lt_mk (n : ) (k : ) (hn : 0 < n) (hk : 0 < k) :
                { val := n, property := hn } < { val := k, property := hk } n < k
                @[simp]
                theorem PNat.coe_le_coe (n : ℕ+) (k : ℕ+) :
                n k n k
                @[simp]
                theorem PNat.coe_lt_coe (n : ℕ+) (k : ℕ+) :
                n < k n < k
                @[simp]
                theorem PNat.pos (n : ℕ+) :
                0 < n
                theorem PNat.eq {m : ℕ+} {n : ℕ+} :
                m = nm = n
                @[simp]
                theorem PNat.ne_zero (n : ℕ+) :
                n 0
                instance NeZero.pnat {a : ℕ+} :
                NeZero a
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                theorem PNat.toPNat'_coe {n : } :
                0 < n(Nat.toPNat' n) = n
                @[simp]
                theorem PNat.coe_toPNat' (n : ℕ+) :
                Nat.toPNat' n = n
                @[simp]
                theorem PNat.one_le (n : ℕ+) :
                1 n
                @[simp]
                theorem PNat.not_lt_one (n : ℕ+) :
                ¬n < 1
                @[simp]
                theorem PNat.mk_one {h : 0 < 1} :
                { val := 1, property := h } = 1
                @[simp]
                theorem PNat.one_coe :
                1 = 1
                @[simp]
                theorem PNat.coe_eq_one_iff {m : ℕ+} :
                m = 1 m = 1
                def PNat.strongInductionOn {p : ℕ+Sort u_1} (n : ℕ+) :
                ((k : ℕ+) → ((m : ℕ+) → m < kp m)p k)p n

                Strong induction on ℕ+.

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                  def PNat.modDivAux :
                  ℕ+ℕ+ ×

                  We define m % k and m / k in the same way as for except that when m = n * k we take m % k = k and m / k = n - 1. This ensures that m % k is always positive and m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

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                    def PNat.modDiv (m : ℕ+) (k : ℕ+) :

                    mod_div m k = (m % k, m / k). We define m % k and m / k in the same way as for except that when m = n * k we take m % k = k and m / k = n - 1. This ensures that m % k is always positive and m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

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                      def PNat.mod (m : ℕ+) (k : ℕ+) :

                      We define m % k in the same way as for except that when m = n * k we take m % k = k This ensures that m % k is always positive.

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                        def PNat.div (m : ℕ+) (k : ℕ+) :

                        We define m / k in the same way as for except that when m = n * k we take m / k = n - 1. This ensures that m = (m % k) + k * (m / k) in all cases. Later we define a function div_exact which gives the usual m / k in the case where k divides m.

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                          theorem PNat.mod_coe (m : ℕ+) (k : ℕ+) :
                          (PNat.mod m k) = if m % k = 0 then k else m % k
                          theorem PNat.div_coe (m : ℕ+) (k : ℕ+) :
                          PNat.div m k = if m % k = 0 then Nat.pred (m / k) else m / k
                          def PNat.divExact (m : ℕ+) (k : ℕ+) :

                          If h : k | m, then k * (div_exact m k) = m. Note that this is not equal to m / k.

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                            instance Nat.canLiftPNat :
                            CanLift ℕ+ PNat.val fun (n : ) => 0 < n
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                            instance Int.canLiftPNat :
                            CanLift ℕ+ (fun (x : ℕ+) => x) fun (x : ) => 0 < x
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