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Proof step 43 from proof prepost-init-vols-mappings (previous step)

# wlar = # apebs, avhdr0.sqn = avhdr.sqn, avhdr.sqn = (avhdr.sqn < (bag = ∅ ⊃ avhdr.sqn;(avhdr.sqn < bag.max ⊃ bag.max;avhdr.sqn)) ⊃ (bag = ∅ ⊃ avhdr.sqn;bag.max);avhdr.sqn), recs0[VTBL_VOLID × VTBL_LNUM] = recovery-entry(n0, m3), apebs[n] = mkpeb(aechdr(m0), avhdr, buf, m, false), apebs[n0] = mkpeb(aechdr(m4), avhdr0, buf0, m1, false), wlar[n0] = wl-entry(m4, used), wlar[n] = wl-entry(m0, used), valid-sqns(VTBL_VOLID, VTBL_LNUM, apebs) = bag ++ avhdr.sqn ++ avhdr.sqn, avhdr0.valid?, n0 < # apebs, n < # apebs, bflipset-cons(bflipset, wlar), is-encoded(to-vtbl(vols), buf), unique-sqns(apebs), unmapped-vols(vols), valid-eraseq(wlar, q), valid-eraseq-stronger(apebs, q), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m0), avhdr, buf, m, false)), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m4), avhdr0, buf0, m1, false)), VTBL_VOLID × VTBL_LNUM ∈ recs0 ⊦ n0 = n, VTBL_VOLID ∈ vols, n ∈ q

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 44:
⊦ unique-sqns(apebs) ↔ (∀ v, m. unique-elems(valid-sqns(v, m, apebs)))
Proof step 45:
# wlar = # apebs, avhdr.sqn = (avhdr.sqn < (bag = ∅ ⊃ avhdr.sqn;(avhdr.sqn < bag.max ⊃ bag.max;avhdr.sqn)) ⊃ (bag = ∅ ⊃ avhdr.sqn;bag.max);avhdr.sqn), avhdr0.sqn = avhdr.sqn, wlar[n] = wl-entry(m0, used), wlar[n0] = wl-entry(m4, used), apebs[n0] = mkpeb(aechdr(m4), avhdr0, buf0, m1, false), apebs[n] = mkpeb(aechdr(m0), avhdr, buf, m, false), recs0[VTBL_VOLID × VTBL_LNUM] = recovery-entry(n0, m3), valid-sqns(VTBL_VOLID, VTBL_LNUM, apebs) = bag ++ avhdr.sqn ++ avhdr.sqn, avhdr0.valid?, n < # apebs, n0 < # apebs, bflipset-cons(bflipset, wlar), is-encoded(to-vtbl(vols), buf), unmapped-vols(vols), valid-eraseq(wlar, q), valid-eraseq-stronger(apebs, q), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m4), avhdr0, buf0, m1, false)), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m0), avhdr, buf, m, false)), VTBL_VOLID × VTBL_LNUM ∈ recs0, ∀ v, m. unique-elems(valid-sqns(v, m, apebs)) ⊦ n0 = n, n ∈ q, VTBL_VOLID ∈ vols

Rule argument

spec: ubi-base name: unique-sqns seq: ⊦ unique-sqns(apebs) ↔ (∀ v, m. unique-elems(valid-sqns(v, m, apebs))) subst: [apebs] → [apebs] flags: <all paths>

Additional information

Used terms: apebs