152 lines
4.3 KiB
Text
152 lines
4.3 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.list
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import data.tuple
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namespace bv
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open algebra
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open bool
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open eq.ops
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open list
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open nat
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open prod
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open subtype
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open tuple
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definition bv [reducible] (n : ℕ) := tuple bool n
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-- Create a zero bitvector
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definition bv_zero (n : ℕ) : bv n := replicate ff
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-- Create a bitvector with the constant one.
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definition bv_one : Π (n : ℕ), bv n
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| 0 := replicate ff
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| (succ n) := (replicate ff : bv n) ++ (tt :: nil)
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definition bv_cong {a b : ℕ} : (a = b) → bv a → bv b
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| c (tag x p) := tag x (c ▸ p)
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section shift
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-- shift left
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definition bv_shl {n:ℕ} : bv n → ℕ → bv n
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| x i :=
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dite (i ≤ n)
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(λle,
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let r := dropn i x ++ replicate ff in
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let eq := calc (n-i) + i = n : nat.sub_add_cancel le in
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bv_cong eq r)
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(λp, bv_zero n)
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-- unsigned shift right
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definition bv_ushr {n:ℕ} : bv n → ℕ → bv n
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| x i :=
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dite (i ≤ n)
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(λle,
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let y : bv (n-i) := @firstn _ _ (n - i) (sub_le n i) x in
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let eq := calc (i+(n-i)) = (n - i) + i : add.comm
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... = n : nat.sub_add_cancel le in
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bv_cong eq (replicate ff ++ y))
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(λgt, bv_zero n)
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-- signed shift right
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definition bv_sshr {m:ℕ} : bv (succ m) → ℕ → bv (succ m)
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| x i :=
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let n := succ m in
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dite (i ≤ n)
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(λle,
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let z : bv i := replicate (head x) in
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let y : bv (n-i) := @firstn _ _ (n - i) (sub_le n i) x in
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let eq := calc (i+(n-i)) = (n-i) + i : add.comm
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... = n : nat.sub_add_cancel le in
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bv_cong eq (z ++ y))
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(λgt, bv_zero n)
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end shift
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section bitwise
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variable { n : ℕ }
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-- | Bitwise and
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definition bv_and : bv n → bv n → bv n := map₂ band
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-- | Bitwise or
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definition bv_or : bv n → bv n → bv n := map₂ bor
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-- | Bitwise xor
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definition bv_xor : bv n → bv n → bv n := map₂ bxor
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end bitwise
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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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-- Add with carry (no overflow)
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definition bv_adc {n:ℕ} : bv n → bv n → bool → bv (n+1)
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| x y c :=
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let f := λx y c, (bv.carry x y c, bv.xor3 x y c) in
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let z := tuple.mapAccumR₂ f x y c in
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(pr₁ z) :: (pr₂ z)
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definition bv_add {n:ℕ} : bv n → bv n → bv n
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| x y := tail (bv_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 bv_sbb {n:ℕ} : bv n → bv n → bool → bool × bv n
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| x y b :=
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let f := λx y c, (bv.borrow x y c, bv.xor3 x y c) in
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tuple.mapAccumR₂ f x y b
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definition bv_sub {n:ℕ} (x y: bv n) := pr₂ (bv_sbb x y ff)
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definition bv_neg {n:ℕ} : bv n → bv n
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| x :=
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let f := λy c, (y || c, bxor y c) in
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pr₂ (mapAccumR f x ff)
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protected definition mulc {n:ℕ} : list bool → bv n → bv n → bv n
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| [] y r := r
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| (tt::x) y r := mulc x y (bv_add r (bv_shl y (length x)))
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| (ff::x) y r := mulc x y r
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definition bv_mul {n:ℕ} : bv n → bv n → bv n
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| (tag x px) y := bv.mulc x y (bv_zero n)
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definition bv_has_zero [instance] {n : ℕ} : has_zero (bv n) :=
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has_zero.mk (bv_zero n)
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definition bv_has_one [instance] {n : ℕ} : has_one (bv n) :=
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has_one.mk (bv_one n)
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definition bv_has_add [instance] {n : ℕ} : has_add (bv n) :=
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has_add.mk bv_add
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definition bv_has_sub [instance] {n : ℕ} : has_sub (bv n) :=
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has_sub.mk bv_sub
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definition bv_has_neg [instance] {n : ℕ} : has_neg (bv n) :=
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has_neg.mk bv_neg
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definition bv_has_mul [instance] {n : ℕ} : has_mul (bv n) :=
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has_mul.mk bv_mul
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section from_bv
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variable {A : Type}
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protected definition fold_list_bits [p : has_add A] [q : has_one A]
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: list bool → A → A
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| [] r := r
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| (tt::l) r := fold_list_bits l (r+r+1)
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| (ff::l) r := fold_list_bits l (r+r)
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-- Convert a bitvector to another number.
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definition from_bv [p : has_add A] [q0 : has_zero A] [q1 : has_one A] {w:nat} (v:bv w) : A :=
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bv.fold_list_bits (to_list v) 0
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end from_bv
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end bv
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