157 lines
4.8 KiB
Text
157 lines
4.8 KiB
Text
/-
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Copyright (c) 2015 Joe Hendrix. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Joe Hendrix
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Basic operations on bitvectors.
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This is a work-in-progress, and contains additions to other theories.
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-/
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import data.tuple
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definition bitvec (n : ℕ) := tuple bool n
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namespace bitvec
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open nat
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open tuple
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-- Create a zero bitvector
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definition zero { n : ℕ } : bitvec n := tuple.repeat ff
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-- Create a bitvector with the constant one.
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definition one {n : ℕ} : bitvec (succ n) := tuple.append (tuple.repeat ff : bitvec n) [ tt ]
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section shift
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-- shift left
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definition shl {n:ℕ} : bitvec n → ℕ → bitvec n
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| x i :=
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if le : i ≤ n then
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let eq := nat.sub_add_cancel le in
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cast (congr_arg bitvec eq) (tuple.append (dropn i x) (zero : bitvec i))
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else
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zero
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-- unsigned shift right
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definition ushr {n:ℕ} : bitvec n → ℕ → bitvec n
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| x i :=
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if le : i ≤ n then
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let y : bitvec (n-i) := @firstn _ _ _ (sub_le n i) x in
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let eq := calc (i+(n-i)) = (n - i) + i : nat.add_comm i (n - i)
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... = n : nat.sub_add_cancel le in
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cast (congr_arg bitvec eq) (tuple.append zero y)
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else
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zero
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-- signed shift right
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definition sshr {m:ℕ} : bitvec (succ m) → ℕ → bitvec (succ m)
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| x i :=
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let n := succ m in
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if le : i ≤ n then
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let z : bitvec i := repeat (head x) in
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let y : bitvec (n-i) := @firstn _ _ (n - i) (sub_le n i) x in
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let eq := calc (i+(n-i)) = (n-i) + i : nat.add_comm i (n - i)
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... = n : nat.sub_add_cancel le in
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cast (congr_arg bitvec eq) (tuple.append z y)
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else
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zero
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end shift
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section bitwise
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variable { n : ℕ }
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definition not : bitvec n → bitvec n := map bnot
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definition and : bitvec n → bitvec n → bitvec n := map₂ band
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definition or : bitvec n → bitvec n → bitvec n := map₂ bor
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definition xor : bitvec n → bitvec n → bitvec n := map₂ bxor
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end bitwise
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section arith
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variable { n : ℕ }
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protected definition xor3 (x:bool) (y:bool) (c:bool) := bxor (bxor x y) c
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protected definition carry (x:bool) (y:bool) (c:bool) :=
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x && y || x && c || y && c
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definition neg : bitvec n → bitvec n
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| x :=
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let f := λy c, (y || c, bxor y c) in
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prod.snd (mapAccumR f x ff)
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-- Add with carry (no overflow)
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definition adc : bitvec n → bitvec n → bool → bitvec (n+1)
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| x y c :=
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let f := λx y c, (bitvec.carry x y c, bitvec.xor3 x y c) in
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let z := tuple.mapAccumR₂ f x y c in
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prod.fst z :: prod.snd z
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definition add : bitvec n → bitvec n → bitvec n
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| x y := tail (adc x y ff)
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protected definition borrow (x:bool) (y:bool) (b:bool) :=
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bnot x && y || bnot x && b || y && b
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-- Subtract with borrow
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definition sbb : bitvec n → bitvec n → bool → bool × bitvec n
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| x y b :=
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let f := λx y c, (bitvec.borrow x y c, bitvec.xor3 x y c) in
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tuple.mapAccumR₂ f x y b
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definition sub : bitvec n → bitvec n → bitvec n
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| x y := prod.snd (sbb x y ff)
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instance : has_zero (bitvec n) := has_zero.mk zero
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instance : has_one (bitvec (succ n)) := has_one.mk one
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instance : has_add (bitvec n) := has_add.mk add
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instance : has_sub (bitvec n) := has_sub.mk sub
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instance : has_neg (bitvec n) := has_neg.mk neg
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definition mul : bitvec n → bitvec n → bitvec n
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| x y :=
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let f := λr b, cond b (r + r + y) (r + r) in
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list.foldl f 0 (to_list x)
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instance : has_mul (bitvec n) := has_mul.mk mul
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end arith
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section comparison
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variable {n : ℕ}
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definition uborrow : bitvec n → bitvec n → bool := λx y, prod.fst (sbb x y ff)
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definition ult (x y : bitvec n) : Prop := uborrow x y = tt
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definition ugt (x y : bitvec n) : Prop := ult y x
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definition ule (x y : bitvec n) : Prop := ¬ (ult y x)
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definition uge : bitvec n → bitvec n → Prop := λx y, ule y x
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definition sborrow : bitvec (succ n) → bitvec (succ n) → bool := λx y,
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bool.cases_on
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(head x)
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(bool.cases_on (head y) (uborrow (tail x) (tail y)) tt)
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(bool.cases_on (head y) ff (uborrow (tail x) (tail y)))
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definition slt : bitvec (succ n) → bitvec (succ n) → Prop := λx y,
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sborrow x y = tt
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definition sgt : bitvec (succ n) → bitvec (succ n) → Prop := λx y, slt y x
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definition sle : bitvec (succ n) → bitvec (succ n) → Prop := λx y, ¬ (slt y x)
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definition sge : bitvec (succ n) → bitvec (succ n) → Prop := λx y, sle y x
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end comparison
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section from_bitvec
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variable {A : Type}
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-- Convert a bitvector to another number.
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definition from_bitvec [p : has_add A] [q0 : has_zero A] [q1 : has_one A] {n:nat} (v:bitvec n) : A :=
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let f : A → bool → A := λr b, cond b (r + r + 1) (r + r) in
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list.foldl f (0 : A) (to_list v)
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end from_bitvec
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end bitvec
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