160 lines
4.4 KiB
Text
160 lines
4.4 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, Leonardo de Moura
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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.basic init.data.list init.coe init.data.repr init.data.to_string
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open nat
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/- the type, coercions, and notation -/
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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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attribute [extern cpp "lean::nat2int"] int.of_nat
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attribute [extern cpp "lean::int_neg_succ_of_nat"] int.neg_succ_of_nat
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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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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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private def nonneg : int → Prop
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| (of_nat _) := true
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| -[1+ _] := false
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def neg_of_nat : nat → int
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| 0 := 0
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| (succ m) := -[1+ m]
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@[extern cpp "lean::int_neg"]
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protected def neg (n : @& int) : int :=
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match n with
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| (of_nat n) := neg_of_nat n
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| -[1+ n] := succ n
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def sub_nat_nat (m n : nat) : 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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@[extern cpp "lean::int_add"]
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protected def add (m n : @& int) : int :=
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match m, n with
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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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@[extern cpp "lean::int_mul"]
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protected def mul (m n : @& int) : int :=
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match m, n with
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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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@[extern cpp "lean::int_sub"]
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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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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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@[extern cpp "lean::int_dec_eq"]
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protected def dec_eq (a b : @& int) : decidable (a = b) :=
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match a, b with
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| (of_nat a), (of_nat b) :=
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if h : a = b then is_true (h ▸ rfl)
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else is_false (λ h', int.no_confusion h' (λ h', absurd h' h))
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| (neg_succ_of_nat a), (neg_succ_of_nat b) :=
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if h : a = b then is_true (h ▸ rfl)
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else is_false (λ h', int.no_confusion h' (λ h', absurd h' h))
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| (of_nat a), (int.neg_succ_of_nat b) := is_false (λ h, int.no_confusion h)
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| (neg_succ_of_nat a), (of_nat b) := is_false (λ h, int.no_confusion h)
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instance int.decidable_eq : decidable_eq int :=
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{dec_eq := int.dec_eq}
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@[extern cpp "lean::int_dec_nonneg"]
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private def dec_nonneg (m : @& int) : decidable (nonneg m) :=
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match m with
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| (of_nat m) := show decidable true, from infer_instance
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| -[1+ m] := show decidable false, from infer_instance
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@[extern cpp "lean::int_dec_le"]
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instance dec_le (a b : @& int) : decidable (a ≤ b) :=
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dec_nonneg _
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@[extern cpp "lean::int_dec_lt"]
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instance dec_lt (a b : @& int) : decidable (a < b) :=
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dec_nonneg _
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@[extern cpp "lean::nat_abs"]
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def nat_abs (m : @& int) : nat :=
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match m with
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| (of_nat m) := m
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| (neg_succ_of_nat m) := nat.succ m
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protected def repr : int → string
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| (of_nat n) := nat.repr n
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| (neg_succ_of_nat n) := "-" ++ nat.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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def sign : int → int
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| (n+1:nat) := 1
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| 0 := 0
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| -[1+ n] := -1
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@[extern cpp "lean::int_div"]
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def div : (@& 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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@[extern cpp "lean::int_mod"]
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def mod : (@& 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 to_nat : int → nat
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| (n : nat) := n
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| -[1+ n] := 0
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def nat_mod (m n : int) : nat := (m % n).to_nat
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end int
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