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Proof step 310 from proof prepost-scan-all (previous step)

# wlar0 = n1 + vtblpnum1, # apebs = n1 + vtblpnum1, apebs[vtblpnum1] = mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false), valid-sqns(v, m1, apebs, vtblpnum1) = ∅, nx < vtblpnum1, bflipset-cons(bflipset, subarray(wlar0, 0, vtblpnum1)), indices-erasing(wlar0, nx), ubi-io-inv(apebs), ubi-io-peb-inv(mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false)), unique-sqns(apebs), valid-eraseq(subarray(wlar0, 0, vtblpnum1), q), valid-eraseq-stronger(subarray(apebs, 0, vtblpnum1), q), valid-global-sqn(subarray(apebs, 0, vtblpnum1), sqnum), valid-recs(subarray(apebs, 0, vtblpnum1), subarray(wlar0, 0, vtblpnum1), recs), valid-recs-sqns(apebs, recs), valid-wl(subarray(apebs, 0, vtblpnum1), subarray(wlar0, 0, vtblpnum1)) ⊦ m3 = 0, n1 = 0, dups(nx), valid-peb(v, m1, mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false)), v × m1 ∈ recs, m3 ≤ datasize(buf1) ∧ m4 = checksum(buf1, m3), valid-global-sqn(subarray(apebs, 0, vtblpnum1 + 1), sqnum) ∧ valid-eraseq-stronger(subarray(apebs, 0, vtblpnum1 + 1), q + eq-entry(vtblpnum1, none) ') ∧ valid-wl(subarray(apebs, 0, vtblpnum1 + 1), subarray(wlar0[vtblpnum1, wl-entry(m, erasing)], 0, vtblpnum1 + 1))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 311:
m < # apebs ⊦ valid-eraseq-stronger(subarray(apebs, 0, m), q) → (valid-eraseq-stronger(subarray(apebs, 0, m + 1), q + eq-entry(m, none) ') ↔ ¬ valid-peb(apebs[m]))
Proof step 312:
# apebs = n1 + vtblpnum1, # wlar0 = n1 + vtblpnum1, apebs[vtblpnum1] = mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false), valid-sqns(v, m1, apebs, vtblpnum1) = ∅, nx < vtblpnum1, vtblpnum1 < # apebs, bflipset-cons(bflipset, subarray(wlar0, 0, vtblpnum1)), indices-erasing(wlar0, nx), ubi-io-inv(apebs), ubi-io-peb-inv(mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false)), unique-sqns(apebs), valid-eraseq(subarray(wlar0, 0, vtblpnum1), q), valid-eraseq-stronger(subarray(apebs, 0, vtblpnum1), q), valid-global-sqn(subarray(apebs, 0, vtblpnum1), sqnum), valid-recs(subarray(apebs, 0, vtblpnum1), subarray(wlar0, 0, vtblpnum1), recs), valid-recs-sqns(apebs, recs), valid-wl(subarray(apebs, 0, vtblpnum1), subarray(wlar0, 0, vtblpnum1)) ⊦ m3 = 0, n1 = 0, dups(nx), valid-peb(v, m1, mkpeb(aechdr(m), avidhdr(v, m1, m2, m3, m4), buf1, m0, false)), v × m1 ∈ recs, valid-global-sqn(subarray(apebs, 0, vtblpnum1 + 1), sqnum) ∧ ¬ valid-peb(apebs[vtblpnum1]) ∧ valid-wl(subarray(apebs, 0, vtblpnum1 + 1), subarray(wlar0[vtblpnum1, wl-entry(m, erasing)], 0, vtblpnum1 + 1)), m3 ≤ datasize(buf1) ∧ m4 = checksum(buf1, m3)

Rule argument

spec: ubi-crash-recovery-base name: valid-eraseq-stronger-next-append-none seq: m < # apebs ⊦ valid-eraseq-stronger(subarray(apebs, 0, m), q) → ( valid-eraseq-stronger (subarray(apebs, 0, m + 1), q + eq-entry(m, none) ') ↔ ¬ valid-peb(apebs[m])) subst: [q, m, apebs] → [q, vtblpnum1, apebs] flags: <all paths>

Additional information

Used simplifier rules: nat-sum-leq: ⊦ m = n0 + n1 → (m ≤ n1 ↔ n0 = 0);% ⊦ m < n ↔ ¬ n ≤ m; (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: q, vtblpnum1, apebs