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feat: add sshiftRight/shiftLeft_*_distrib #13

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feat: add sshiftRight/shiftLeft_*_distrib #13

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c94c835
feat: add sshiftRight/shiftLeft_*_distrib
tobiasgrosser Aug 8, 2024
35160b3
chore: drop unnecessary parenthesis
tobiasgrosser Aug 8, 2024
eb23e9c
chore: fix indentation of <;>
tobiasgrosser Aug 8, 2024
e3023b4
chore: squeeze simps
tobiasgrosser Aug 9, 2024
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54 changes: 54 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 @@ -559,6 +559,15 @@ theorem and_comm (x y : BitVec w) :
rw [← testBit_toNat, getLsb, getLsb]
simp

@[simp] theorem getMsb_xor {x y : BitVec w} :
(x ^^^ y).getMsb i = (xor (x.getMsb i) (y.getMsb i)) := by
simp only [getMsb]
by_cases h : i < w <;> simp [h]

@[simp] theorem msb_xor {x y : BitVec w} :
(x ^^^ y).msb = (xor x.msb y.msb) := by
simp [BitVec.msb]

@[simp] theorem truncate_xor {x y : BitVec w} :
(x ^^^ y).truncate k = x.truncate k ^^^ y.truncate k := by
ext
Expand Down Expand Up @@ -651,6 +660,27 @@ theorem zero_shiftLeft (n : Nat) : 0#w <<< n = 0#w := by
cases h₁ : decide (i < m) <;> cases h₂ : decide (n ≤ i) <;> cases h₃ : decide (i < n)
all_goals { simp_all <;> omega }

theorem shiftLeft_xor_distrib (x y : BitVec w) (n : Nat) :
(x ^^^ y) <<< n = (x <<< n) ^^^ (y <<< n) := by
ext i
simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_xor]
by_cases h : i < n
<;> simp [h]

theorem shiftLeft_and_distrib (x y : BitVec w) (n : Nat) :
(x &&& y) <<< n = (x <<< n) &&& (y <<< n) := by
ext i
simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_and]
by_cases h : i < n
<;> simp [h]

theorem shiftLeft_or_distrib (x y : BitVec w) (n : Nat) :
(x ||| y) <<< n = (x <<< n) ||| (y <<< n) := by
ext i
simp only [getLsb_shiftLeft, Fin.is_lt, decide_True, Bool.true_and, getLsb_or]
by_cases h : i < n
<;> simp [h]

@[simp] theorem getMsb_shiftLeft (x : BitVec w) (i) :
(x <<< i).getMsb k = x.getMsb (k + i) := by
simp only [getMsb, getLsb_shiftLeft]
Expand Down Expand Up @@ -822,6 +852,30 @@ theorem sshiftRight_eq_of_msb_true {x : BitVec w} {s : Nat} (h : x.msb = true) :
Nat.not_lt, decide_eq_true_eq]
omega

theorem sshiftRight_xor_distrib (x y : BitVec w) (n : Nat) :
(x ^^^ y).sshiftRight n = (x.sshiftRight n) ^^^ (y.sshiftRight n) := by
ext i
simp only [getLsb_sshiftRight, getLsb_xor, msb_xor]
split
<;> by_cases w ≤ i
<;> simp [*]

theorem sshiftRight_and_distrib (x y : BitVec w) (n : Nat) :
(x &&& y).sshiftRight n = (x.sshiftRight n) &&& (y.sshiftRight n) := by
ext i
simp only [getLsb_sshiftRight, getLsb_and, msb_and]
split
<;> by_cases w ≤ i
<;> simp [*]

theorem sshiftRight_or_distrib (x y : BitVec w) (n : Nat) :
(x ||| y).sshiftRight n = (x.sshiftRight n) ||| (y.sshiftRight n) := by
ext i
simp only [getLsb_sshiftRight, getLsb_or, msb_or]
split
<;> by_cases w ≤ i
<;> simp [*]

/-- The msb after arithmetic shifting right equals the original msb. -/
theorem sshiftRight_msb_eq_msb {n : Nat} {x : BitVec w} :
(x.sshiftRight n).msb = x.msb := by
Expand Down
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