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Proof step 66 from proof prepost-init-vols-mappings (previous step)

# apebs = # wlar, # wlar0 = # wlar, wlar[n0] = wl-entry(m7, used), vols[v][m0] = ⌜n0⌝ , apebs[n] = mkpeb(aehdr, avhdr, buf, m, boolvar), apebs[n0] = mkpeb(aehdr0, avidhdr(v, m0, m1, m5, m6), buf0, m2, false), recs1[v × m0] = recovery-entry(m3, m4), n0 < # wlar, m0 < # vols[v], bflipset-cons(bflipset, wlar), bflipset-cons(bflipset0, wlar0), is-encoded(vtbl, buf), unique-sqns(apebs), unmapped-vols(vols0), valid-eraseq(wlar, q), valid-eraseq(wlar0, q1), valid-eraseq-strong(apebs, vtbl, q), valid-eraseq-stronger(apebs, q1), valid-peb(mkpeb(aehdr0, avidhdr(v, m0, m1, m5, m6), buf0, m2, false)), valid-recs(apebs, wlar0, recs0), valid-vtbl(apebs, vols, wlar, n), valid-vtbl(apebs, vols0, wlar0, n), valid-wl(apebs, wlar), valid-wl(apebs, wlar0), v ∈ vols, v × m0 ∈ recs1, q1 ⊆ q, recs1 ⊥ vols ⊦ m3 = n0, n0 = n, recs1 = ∅, n ∈ q1, VTBL_VOLID ∈ vols, VTBL_VOLID × VTBL_LNUM ∈ recs1, VTBL_VOLID ∈ vols0, n0 ∈ q

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 67:
⊦ recs ⊥ vols ↔ (∀ v, m. ¬ v × m ∈ recs ∨ ¬ v ∈ vols ∨ ¬ m < # vols[v] ∨ vols[v][m] = unmapped)
Proof step 68:
# wlar0 = # wlar, # apebs = # wlar, recs1[v × m0] = recovery-entry(m3, m4), apebs[n0] = mkpeb(aehdr0, avidhdr(v, m0, m1, m5, m6), buf0, m2, false), apebs[n] = mkpeb(aehdr, avhdr, buf, m, boolvar), vols[v][m0] = ⌜n0⌝ , wlar[n0] = wl-entry(m7, used), m0 < # vols[v], n0 < # wlar, bflipset-cons(bflipset0, wlar0), bflipset-cons(bflipset, wlar), is-encoded(vtbl, buf), unique-sqns(apebs), unmapped-vols(vols0), valid-eraseq(wlar0, q1), valid-eraseq(wlar, q), valid-eraseq-strong(apebs, vtbl, q), valid-eraseq-stronger(apebs, q1), valid-peb(mkpeb(aehdr0, avidhdr(v, m0, m1, m5, m6), buf0, m2, false)), valid-recs(apebs, wlar0, recs0), valid-vtbl(apebs, vols0, wlar0, n), valid-vtbl(apebs, vols, wlar, n), valid-wl(apebs, wlar0), valid-wl(apebs, wlar), v × m0 ∈ recs1, v ∈ vols, q1 ⊆ q, ∀ v, m. ¬ v × m ∈ recs1 ∨ ¬ v ∈ vols ∨ ¬ m < # vols[v] ∨ vols[v][m] = unmapped ⊦ m3 = n0, n0 = n, recs1 = ∅, n0 ∈ q, VTBL_VOLID ∈ vols0, VTBL_VOLID × VTBL_LNUM ∈ recs1, VTBL_VOLID ∈ vols, n ∈ q1

Rule argument

spec: ubi-crash-recovery-base name: disj seq: ⊦ recs ⊥ vols ↔ (∀ v, m. ¬ v × m ∈ recs ∨ ¬ v ∈ vols ∨ ¬ m < # vols[v] ∨ vols[v][m] = unmapped) subst: [vols, recs] → [vols, recs1] flags: <all paths>

Additional information

Used terms: vols, recs1