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feat: BitVec.shiftLeft_shiftLeft, BitVec.shiftRight_shiftRight #4148

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May 13, 2024
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feat: BitVec.shiftLeft_shiftLeft, BitVec.shiftRight_shiftRight #4148

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a07c20c
feat: shiftLeft_shiftLeft, shiftRight_shiftRight
bollu May 13, 2024
ba6ecb2
Update src/Init/Data/BitVec/Lemmas.lean
bollu May 13, 2024
0a67d3d
chore: golf shiftLeft_shiftLeft
bollu May 13, 2024
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31 changes: 31 additions & 0 deletions src/Init/Data/BitVec/Lemmas.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -608,6 +608,32 @@ theorem shiftLeftZeroExtend_eq {x : BitVec w} :
(shiftLeftZeroExtend x i).msb = x.msb := by
simp [shiftLeftZeroExtend_eq, BitVec.msb]

theorem BitVec.shiftLeft_shiftLeft (w : Nat) (x : BitVec w) (n m : Nat) :
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(x <<< n) <<< m = x <<< (n + m) := by
ext i
simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and]
-- omega claims that the system cannot be proven, so we case bash.
-- The entire proof below should be redundant once omega is complete.
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decide statements are out of scope for omega!

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If you want to case bash and then hand off to omega, use:

theorem BitVec.shiftLeft_shiftLeft (w : Nat) (x : BitVec w) (n m : Nat) :
    (x <<< n) <<< m = x <<< (n + m) := by
  ext i
  simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and]
  rw [show i - (n + m) = (i - m - n) by omega]
  cases h₂ : decide (i < m) <;> cases h₃ : decide (i - m < w) <;> cases h₄ : decide (i - m < n) <;> cases h₅ : decide (i < n + m) <;>
    simp at * <;> omega

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Sweet, thanks for the tutorial! rw [show _ by omega] is a neat one, I'm going to remember this golf trick :)

rw [Bool.eq_iff_iff]
have hgetLsbIndex : i - (n + m) = (i - m - n) := by omega
rw [hgetLsbIndex]
apply Iff.intro
. intro h
simp_all only [Bool.and_eq_true, Bool.not_eq_true', decide_eq_false_iff_not, Nat.not_lt,
decide_eq_true_eq]
rcases h with ⟨h₁, h₂, _⟩
constructor
. omega
. trivial
. intro h
simp_all only [Bool.and_eq_true, Bool.not_eq_true', decide_eq_false_iff_not, Nat.not_lt,
decide_eq_true_eq]
rcases h with ⟨h₁, h₂⟩
constructor
. omega
. simp only [h₂, and_true]
omega

/-! ### ushiftRight -/

@[simp, bv_toNat] theorem toNat_ushiftRight (x : BitVec n) (i : Nat) :
Expand Down Expand Up @@ -693,6 +719,11 @@ theorem msb_append {x : BitVec w} {y : BitVec v} :
simp only [getLsb_append, cond_eq_if]
split <;> simp [*]

theorem BitVec.shiftRight_shiftRight (w : Nat) (x : BitVec w) (n m : Nat) :
(x >>> n) >>> m = x >>> (n + m) := by
ext i
simp [Nat.add_assoc n m i]

/-! ### rev -/

theorem getLsb_rev (x : BitVec w) (i : Fin w) :
Expand Down
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