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833ef4a954
Simplify tnum_step() from a 10-variable algorithm into a straight
line sequence of bitwise operations.
Problem Reduction:
tnum_step(): Given a tnum `(tval, tmask)` where `tval & tmask == 0`,
and a value `z` with `tval ≤ z < (tval | tmask)`, find the smallest
`r > z`, a tnum-satisfying value, i.e., `r & ~tmask == tval`.
Every tnum-satisfying value has the form tval | s where s is a subset
of tmask bits (s & ~tmask == 0). Since tval and tmask are disjoint:
tval | s = tval + s
Similarly z = tval + d where d = z - tval, so r > z becomes:
tval + s > tval + d
s > d
The problem reduces to: find the smallest s, a subset of tmask, such
that s > d.
Notice that `s` must be a subset of tmask, the problem now is simplified.
Algorithm:
The mask bits of `d` form a "counter" that we want to increment by one,
but the counter has gaps at the fixed-bit positions. A normal +1 would
stop at the first 0-bit it meets; we need it to skip over fixed-bit
gaps and land on the next mask bit.
Step 1 -- plug the gaps:
d | carry_mask | ~tmask
- ~tmask fills all fixed-bit positions with 1.
- carry_mask = (1 << fls64(d & ~tmask)) - 1 fills all positions
(including mask positions) below the highest non-mask bit of d.
After this, the only remaining 0s are mask bits above the highest
non-mask bit of d where d is also 0 -- exactly the positions where
the carry can validly land.
Step 2 -- increment:
(d | carry_mask | ~tmask) + 1
Adding 1 flips all trailing 1s to 0 and sets the first 0 to 1. Since
every gap has been plugged, that first 0 is guaranteed to be a mask bit
above all non-mask bits of d.
Step 3 -- mask:
((d | carry_mask | ~tmask) + 1) & tmask
Strip the scaffolding, keeping only mask bits. Call the result inc.
Step 4 -- result:
tval | inc
Reattach the fixed bits.
A simple 8-bit example:
tmask: 1 1 0 1 0 1 1 0
d: 1 0 1 0 0 0 1 0 (d = 162)
^
non-mask 1 at bit 5
With carry_mask = 0b00111111 (smeared from bit 5):
d|carry|~tm 1 0 1 1 1 1 1 1
+ 1 1 1 0 0 0 0 0 0
& tmask 1 1 0 0 0 0 0 0
The patch passes my local test: test_verifier, test_progs for
`-t verifier` and `-t reg_bounds`.
CBMC shows the new code is equiv to original one[1], and
a lean4 proof of correctness is available[2]:
theorem tnumStep_correct (tval tmask z : BitVec 64)
-- Precondition: valid tnum and input z
(h_consistent : (tval &&& tmask) = 0)
(h_lo : tval ≤ z)
(h_hi : z < (tval ||| tmask)) :
-- Postcondition: r must be:
-- (1) tnum member
-- (2) z < r
-- (3) for any other member w > z, r <= w
let r := tnumStep tval tmask z
satisfiesTnum64 r tval tmask ∧
tval ≤ r ∧ r ≤ (tval ||| tmask) ∧
z < r ∧
∀ w, satisfiesTnum64 w tval tmask → z < w → r ≤ w := by
-- unfold definition
unfold tnumStep satisfiesTnum64
simp only []
refine ⟨?_, ?_, ?_, ?_, ?_⟩
-- the solver proves each conjunct
· bv_decide
· bv_decide
· bv_decide
· bv_decide
· intro w hw1 hw2; bv_decide
[1] https://github.com/eddyz87/tnum-step-verif/blob/master/main.c
[2] https://pastebin.com/raw/czHKiyY0
Signed-off-by: Hao Sun <hao.sun@inf.ethz.ch>
Acked-by: Eduard Zingerman <eddyz87@gmail.com>
Acked-by: Shung-Hsi Yu <shung-hsi.yu@suse.com>
Reviewed-by: Harishankar Vishwanathan <harishankar.vishwanathan@gmail.com>
Link: https://lore.kernel.org/r/20260320162336.166542-1-hao.sun@inf.ethz.ch
Signed-off-by: Alexei Starovoitov <ast@kernel.org>
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