feat: lemmas about BitVec (#3593)
Basic API lemmas for BitVec, motivated by thinking about bitblasting.
This commit is contained in:
Scott Morrison
GitHub
parent
f48079eb90
commit
89ec60befe
1 changed files with 92 additions and 0 deletions
|
|
@ -89,6 +89,9 @@ theorem eq_of_toFin_eq : ∀ {x y : BitVec w}, x.toFin = y.toFin → x = y
|
|||
@[simp] theorem toNat_ofBool (b : Bool) : (ofBool b).toNat = b.toNat := by
|
||||
cases b <;> rfl
|
||||
|
||||
@[simp] theorem msb_ofBool (b : Bool) : (ofBool b).msb = b := by
|
||||
cases b <;> simp [BitVec.msb]
|
||||
|
||||
theorem ofNat_one (n : Nat) : BitVec.ofNat 1 n = BitVec.ofBool (n % 2 = 1) := by
|
||||
rcases (Nat.mod_two_eq_zero_or_one n) with h | h <;> simp [h, BitVec.ofNat, Fin.ofNat']
|
||||
|
||||
|
|
@ -116,6 +119,8 @@ theorem getLsb_ofNat (n : Nat) (x : Nat) (i : Nat) :
|
|||
|
||||
@[simp] theorem getLsb_zero : (0#w).getLsb i = false := by simp [getLsb]
|
||||
|
||||
@[simp] theorem getMsb_zero : (0#w).getMsb i = false := by simp [getMsb]
|
||||
|
||||
@[simp] theorem toNat_mod_cancel (x : BitVec n) : x.toNat % (2^n) = x.toNat :=
|
||||
Nat.mod_eq_of_lt x.isLt
|
||||
|
||||
|
|
@ -241,6 +246,12 @@ theorem toInt_ofNat {n : Nat} (x : Nat) :
|
|||
else
|
||||
simp [n_le_i, toNat_ofNat]
|
||||
|
||||
theorem zeroExtend'_eq {x : BitVec w} (h : w ≤ v) : x.zeroExtend' h = x.zeroExtend v := by
|
||||
apply eq_of_toNat_eq
|
||||
rw [toNat_zeroExtend, toNat_zeroExtend']
|
||||
rw [Nat.mod_eq_of_lt]
|
||||
exact Nat.lt_of_lt_of_le x.isLt (Nat.pow_le_pow_right (Nat.zero_lt_two) h)
|
||||
|
||||
@[simp, bv_toNat] theorem toNat_truncate (x : BitVec n) : (truncate i x).toNat = x.toNat % 2^i :=
|
||||
toNat_zeroExtend i x
|
||||
|
||||
|
|
@ -285,6 +296,18 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
|
|||
getLsb (zeroExtend m x) i = (decide (i < m) && getLsb x i) := by
|
||||
simp [getLsb, toNat_zeroExtend, Nat.testBit_mod_two_pow]
|
||||
|
||||
@[simp] theorem getMsb_zeroExtend_add {x : BitVec w} (h : k ≤ i) :
|
||||
(x.zeroExtend (w + k)).getMsb i = x.getMsb (i - k) := by
|
||||
by_cases h : w = 0
|
||||
· subst h; simp
|
||||
simp only [getMsb, getLsb_zeroExtend]
|
||||
by_cases h₁ : i < w + k <;> by_cases h₂ : i - k < w <;> by_cases h₃ : w + k - 1 - i < w + k
|
||||
<;> simp [h₁, h₂, h₃]
|
||||
· congr 1
|
||||
omega
|
||||
all_goals (first | apply getLsb_ge | apply Eq.symm; apply getLsb_ge)
|
||||
<;> omega
|
||||
|
||||
@[simp] theorem getLsb_truncate (m : Nat) (x : BitVec n) (i : Nat) :
|
||||
getLsb (truncate m x) i = (decide (i < m) && getLsb x i) :=
|
||||
getLsb_zeroExtend m x i
|
||||
|
|
@ -301,11 +324,18 @@ theorem nat_eq_toNat (x : BitVec w) (y : Nat)
|
|||
(x.truncate l).truncate k = x.truncate k :=
|
||||
zeroExtend_zeroExtend_of_le x h
|
||||
|
||||
@[simp] theorem truncate_cast {h : w = v} : (cast h x).truncate k = x.truncate k := by
|
||||
apply eq_of_getLsb_eq
|
||||
simp
|
||||
|
||||
theorem msb_zeroExtend (x : BitVec w) : (x.zeroExtend v).msb = (decide (0 < v) && x.getLsb (v - 1)) := by
|
||||
rw [msb_eq_getLsb_last]
|
||||
simp only [getLsb_zeroExtend]
|
||||
cases getLsb x (v - 1) <;> simp; omega
|
||||
|
||||
theorem msb_zeroExtend' (x : BitVec w) (h : w ≤ v) : (x.zeroExtend' h).msb = (decide (0 < v) && x.getLsb (v - 1)) := by
|
||||
rw [zeroExtend'_eq, msb_zeroExtend]
|
||||
|
||||
/-! ## extractLsb -/
|
||||
|
||||
@[simp]
|
||||
|
|
@ -353,6 +383,13 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
|
|||
rw [← testBit_toNat, getLsb, getLsb]
|
||||
simp
|
||||
|
||||
@[simp] theorem getMsb_or {x y : BitVec w} : (x ||| y).getMsb i = (x.getMsb i || y.getMsb i) := by
|
||||
simp only [getMsb]
|
||||
by_cases h : i < w <;> simp [h]
|
||||
|
||||
@[simp] theorem msb_or {x y : BitVec w} : (x ||| y).msb = (x.msb || y.msb) := by
|
||||
simp [BitVec.msb]
|
||||
|
||||
/-! ### and -/
|
||||
|
||||
@[simp] theorem toNat_and (x y : BitVec v) :
|
||||
|
|
@ -367,6 +404,13 @@ protected theorem extractLsb_ofNat (x n : Nat) (hi lo : Nat) :
|
|||
rw [← testBit_toNat, getLsb, getLsb]
|
||||
simp
|
||||
|
||||
@[simp] theorem getMsb_and {x y : BitVec w} : (x &&& y).getMsb i = (x.getMsb i && y.getMsb i) := by
|
||||
simp only [getMsb]
|
||||
by_cases h : i < w <;> simp [h]
|
||||
|
||||
@[simp] theorem msb_and {x y : BitVec w} : (x &&& y).msb = (x.msb && y.msb) := by
|
||||
simp [BitVec.msb]
|
||||
|
||||
/-! ### xor -/
|
||||
|
||||
@[simp] theorem toNat_xor (x y : BitVec v) :
|
||||
|
|
@ -431,6 +475,19 @@ theorem not_def {x : BitVec v} : ~~~x = allOnes v ^^^ x := rfl
|
|||
cases h₁ : decide (i < m) <;> cases h₂ : decide (n ≤ i) <;> cases h₃ : decide (i < n)
|
||||
all_goals { simp_all <;> omega }
|
||||
|
||||
@[simp] theorem getMsb_shiftLeft (x : BitVec w) (i) :
|
||||
(x <<< i).getMsb k = x.getMsb (k + i) := by
|
||||
simp only [getMsb, getLsb_shiftLeft]
|
||||
by_cases h : w = 0
|
||||
· subst h; simp
|
||||
have t : w - 1 - k < w := by omega
|
||||
simp only [t]
|
||||
simp only [decide_True, Nat.sub_sub, Bool.true_and, Nat.add_assoc]
|
||||
by_cases h₁ : k < w <;> by_cases h₂ : w - (1 + k) < i <;> by_cases h₃ : k + i < w
|
||||
<;> simp [h₁, h₂, h₃]
|
||||
<;> (first | apply getLsb_ge | apply Eq.symm; apply getLsb_ge)
|
||||
<;> omega
|
||||
|
||||
theorem shiftLeftZeroExtend_eq {x : BitVec w} :
|
||||
shiftLeftZeroExtend x n = zeroExtend (w+n) x <<< n := by
|
||||
apply eq_of_toNat_eq
|
||||
|
|
@ -450,6 +507,10 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
|
|||
<;> simp_all
|
||||
<;> (rw [getLsb_ge]; omega)
|
||||
|
||||
@[simp] theorem msb_shiftLeftZeroExtend (x : BitVec w) (i : Nat) :
|
||||
(shiftLeftZeroExtend x i).msb = x.msb := by
|
||||
simp [shiftLeftZeroExtend_eq, BitVec.msb]
|
||||
|
||||
/-! ### ushiftRight -/
|
||||
|
||||
@[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
|
||||
|
|
@ -475,6 +536,34 @@ theorem append_def (x : BitVec v) (y : BitVec w) :
|
|||
· simp [h]
|
||||
· simp [h]; simp_all
|
||||
|
||||
theorem msb_append {x : BitVec w} {y : BitVec v} :
|
||||
(x ++ y).msb = bif (w == 0) then (y.msb) else (x.msb) := by
|
||||
rw [← append_eq, append]
|
||||
simp [msb_zeroExtend']
|
||||
by_cases h : w = 0
|
||||
· subst h
|
||||
simp [BitVec.msb, getMsb]
|
||||
· rw [cond_eq_if]
|
||||
have q : 0 < w + v := by omega
|
||||
have t : y.getLsb (w + v - 1) = false := getLsb_ge _ _ (by omega)
|
||||
simp [h, q, t, BitVec.msb, getMsb]
|
||||
|
||||
@[simp] theorem truncate_append {x : BitVec w} {y : BitVec v} :
|
||||
(x ++ y).truncate k = if h : k ≤ v then y.truncate k else (x.truncate (k - v) ++ y).cast (by omega) := by
|
||||
apply eq_of_getLsb_eq
|
||||
intro i
|
||||
simp only [getLsb_zeroExtend, Fin.is_lt, decide_True, getLsb_append, Bool.true_and]
|
||||
split
|
||||
· have t : i < v := by omega
|
||||
simp [t]
|
||||
· by_cases t : i < v
|
||||
· simp [t]
|
||||
· have t' : i - v < k - v := by omega
|
||||
simp [t, t']
|
||||
|
||||
@[simp] theorem truncate_cons {x : BitVec w} : (cons a x).truncate w = x := by
|
||||
simp [cons]
|
||||
|
||||
/-! ### rev -/
|
||||
|
||||
theorem getLsb_rev (x : BitVec w) (i : Fin w) :
|
||||
|
|
@ -511,6 +600,9 @@ theorem getMsb_rev (x : BitVec w) (i : Fin w) :
|
|||
have p2 : i - n ≠ 0 := by omega
|
||||
simp [p1, p2, Nat.testBit_bool_to_nat]
|
||||
|
||||
@[simp] theorem msb_cons : (cons a x).msb = a := by
|
||||
simp [cons, msb_cast, msb_append]
|
||||
|
||||
theorem truncate_succ (x : BitVec w) :
|
||||
truncate (i+1) x = cons (getLsb x i) (truncate i x) := by
|
||||
apply eq_of_getLsb_eq
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue