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Proof step 24 from proof prepost-init-vols-mappings (previous step)

# wlar = # apebs, wlar[n0] = wl-entry(m4, used), apebs[n0] = mkpeb(aehdr0, avhdr0, buf0, m1, false), apebs[n] = mkpeb(aehdr, avhdr, buf, m, boolvar), recs0[avhdr0.vol × avhdr0.leb] = recovery-entry(m2, m3), avhdr0.valid?, n0 < # apebs, bflipset-cons(bflipset, wlar), unique-sqns(apebs), unmapped-vols(vols), valid-eraseq(wlar, q), valid-eraseq-stronger(apebs, q), valid-recs(apebs, wlar, recs0), valid-vtbl(apebs, vols, wlar, n), valid-wl(apebs, wlar) ⊦ n0 = n, n ∈ q, VTBL_VOLID ∈ vols, avhdr0.vol × avhdr0.leb ≠ VTBL_VOLID × VTBL_LNUM ∧ avhdr0.vol × avhdr0.leb ∈ recs0 ∧ m2 = n0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 25:
⊦ valid-recs(apebs, wlar, recs) ↔ (∀ ladr, m. ladr ∈ recs ∧ recs[ladr].pnum = m ↔ wlar[m].status = used ∧ m < # apebs ∧ apebs[m].vidhdr.vol = ladr.vol ∧ apebs[m].vidhdr.leb = ladr.leb)
Proof step 26:
# wlar = # apebs, recs0[avhdr0.vol × avhdr0.leb] = recovery-entry(m2, m3), apebs[n] = mkpeb(aehdr, avhdr, buf, m, boolvar), apebs[n0] = mkpeb(aehdr0, avhdr0, buf0, m1, false), wlar[n0] = wl-entry(m4, used), avhdr0.valid?, n0 < # apebs, bflipset-cons(bflipset, wlar), unique-sqns(apebs), unmapped-vols(vols), valid-eraseq(wlar, q), valid-eraseq-stronger(apebs, q), valid-vtbl(apebs, vols, wlar, n), valid-wl(apebs, wlar), ∀ ladr, m. ladr ∈ recs0 ∧ recs0[ladr].pnum = m ↔ wlar[m].status = used ∧ m < # apebs ∧ apebs[m].vidhdr.vol = ladr.vol ∧ apebs[m].vidhdr.leb = ladr.leb ⊦ n0 = n, VTBL_VOLID ∈ vols, n ∈ q, avhdr0.vol × avhdr0.leb ≠ VTBL_VOLID × VTBL_LNUM ∧ avhdr0.vol × avhdr0.leb ∈ recs0 ∧ m2 = n0

Rule argument

spec: ubi-crash-recovery-base name: valid-recs seq: ⊦ valid-recs(apebs, wlar, recs) ↔ (∀ ladr, m. ladr ∈ recs ∧ recs[ladr].pnum = m ↔ wlar[m].status = used ∧ m < # apebs ∧ apebs[m].vidhdr.vol = ladr.vol ∧ apebs[m].vidhdr.leb = ladr.leb) subst: [recs, wlar, apebs] → [recs0, wlar, apebs] flags: <all paths>

Additional information

Used terms: recs0, wlar, apebs