162 lines
4.3 KiB
Text
162 lines
4.3 KiB
Text
/-
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Copyright (c) 2016 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad
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The integers, with addition, multiplication, and subtraction.
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-/
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prelude
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import init.data.nat.gcd init.meta.transfer init.data.list
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open nat
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/- the type, coercions, and notation -/
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@[derive decidable_eq]
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inductive int : Type
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| of_nat : nat → int
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| neg_succ_of_nat : nat → int
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notation `ℤ` := int
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instance : has_coe nat int := ⟨int.of_nat⟩
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notation `-[1+ ` n `]` := int.neg_succ_of_nat n
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protected def int.repr : int → string
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| (int.of_nat n) := repr n
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| (int.neg_succ_of_nat n) := "-" ++ repr (succ n)
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instance : has_repr int :=
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⟨int.repr⟩
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instance : has_to_string int :=
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⟨int.repr⟩
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namespace int
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protected def zero : int := of_nat 0
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protected def one : int := of_nat 1
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instance : has_zero int := ⟨int.zero⟩
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instance : has_one int := ⟨int.one⟩
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/- definitions of basic functions -/
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def neg_of_nat : ℕ → int
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| 0 := 0
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| (succ m) := -[1+ m]
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def sub_nat_nat (m n : ℕ) : int :=
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match (n - m : nat) with
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| 0 := of_nat (m - n) -- m ≥ n
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| (succ k) := -[1+ k] -- m < n, and n - m = succ k
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end
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protected def neg : int → int
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| (of_nat n) := neg_of_nat n
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| -[1+ n] := succ n
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protected def add : int → int → int
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| (of_nat m) (of_nat n) := of_nat (m + n)
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| (of_nat m) -[1+ n] := sub_nat_nat m (succ n)
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| -[1+ m] (of_nat n) := sub_nat_nat n (succ m)
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| -[1+ m] -[1+ n] := -[1+ succ (m + n)]
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protected def mul : int → int → int
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| (of_nat m) (of_nat n) := of_nat (m * n)
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| (of_nat m) -[1+ n] := neg_of_nat (m * succ n)
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| -[1+ m] (of_nat n) := neg_of_nat (succ m * n)
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| -[1+ m] -[1+ n] := of_nat (succ m * succ n)
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instance : has_neg int := ⟨int.neg⟩
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instance : has_add int := ⟨int.add⟩
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instance : has_mul int := ⟨int.mul⟩
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protected def sub (m n : int) : int :=
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m + -n
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instance : has_sub int := ⟨int.sub⟩
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@[simp] def nat_abs : int → ℕ
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| (of_nat m) := m
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| -[1+ m] := succ m
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def sign : int → int
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| (n+1:ℕ) := 1
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| 0 := 0
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| -[1+ n] := -1
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protected def div : int → int → int
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| (m : ℕ) (n : ℕ) := of_nat (m / n)
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| (m : ℕ) -[1+ n] := -of_nat (m / succ n)
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| -[1+ m] 0 := 0
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| -[1+ m] (n+1:ℕ) := -[1+ m / succ n]
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| -[1+ m] -[1+ n] := of_nat (succ (m / succ n))
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protected def mod : int → int → int
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| (m : ℕ) n := (m % nat_abs n : ℕ)
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| -[1+ m] n := sub_nat_nat (nat_abs n) (succ (m % nat_abs n))
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-- F-rounding: This pair satisfies fdiv x y = floor (x / y)
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def fdiv : int → int → int
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| 0 _ := 0
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| (m : ℕ) (n : ℕ) := of_nat (m / n)
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| (m+1:ℕ) -[1+ n] := -[1+ m / succ n]
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| -[1+ m] 0 := 0
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| -[1+ m] (n+1:ℕ) := -[1+ m / succ n]
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| -[1+ m] -[1+ n] := of_nat (succ m / succ n)
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def fmod : int → int → int
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| 0 _ := 0
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| (m : ℕ) (n : ℕ) := of_nat (m % n)
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| (m+1:ℕ) -[1+ n] := sub_nat_nat (m % succ n) n
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| -[1+ m] (n : ℕ) := sub_nat_nat n (succ (m % n))
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| -[1+ m] -[1+ n] := -of_nat (succ m % succ n)
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-- T-rounding: This pair satisfies quot x y = round_to_zero (x / y)
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def quot : int → int → int
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| (of_nat m) (of_nat n) := of_nat (m / n)
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| (of_nat m) -[1+ n] := -of_nat (m / succ n)
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| -[1+ m] (of_nat n) := -of_nat (succ m / n)
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| -[1+ m] -[1+ n] := of_nat (succ m / succ n)
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def rem : int → int → int
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| (of_nat m) (of_nat n) := of_nat (m % n)
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| (of_nat m) -[1+ n] := of_nat (m % succ n)
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| -[1+ m] (of_nat n) := -of_nat (succ m % n)
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| -[1+ m] -[1+ n] := -of_nat (succ m % succ n)
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instance : has_div int := ⟨int.div⟩
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instance : has_mod int := ⟨int.mod⟩
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def gcd (m n : int) : ℕ := gcd (nat_abs m) (nat_abs n)
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def to_nat : int → ℕ
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| (n : ℕ) := n
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| -[1+ n] := 0
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def nat_mod (m n : int) : ℕ := (m % n).to_nat
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private def nonneg : int → Prop
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| (of_nat _) := true
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| -[1+ _] := false
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protected def le (a b : int) : Prop := nonneg (b - a)
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instance : has_le int := ⟨int.le⟩
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protected def lt (a b : int) : Prop := (a + 1) ≤ b
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instance : has_lt int := ⟨int.lt⟩
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private def decidable_nonneg : Π (a : int), decidable (nonneg a)
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| (of_nat m) := decidable.true
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| -[1+ m] := decidable.false
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instance decidable_le (a b : int) : decidable (a ≤ b) :=
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decidable_nonneg _
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instance decidable_lt (a b : int) : decidable (a < b) :=
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decidable_nonneg _
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end int
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