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Proof step 39 from proof valid-wl-write-unmapped (previous step)

# apebs = # wlar, avhdr1.sqn = valid-sqns(VTBL_VOLID, VTBL_LNUM, apebs).max, avhdr0.sqn = valid-sqns(v, m, apebs).max, apebs[n] = mkpeb(aehdr0, avhdr2, buf2, m3, boolvar0), apebs[n1] = mkpeb(aechdr(m4), avhdr0, buf0, m1, false), wlar[n0] = wl-entry(m6, used), wlar[n] = wl-entry(m5, erasing), wlar[n1] = wl-entry(m4, used), vols[avhdr.vol][avhdr.leb] = unmapped, apebs[n0] = mkpeb(aechdr(m6), avhdr1, buf1, m2, false), n < # wlar, avhdr.leb < # vols[avhdr.vol], n0 < # wlar, n1 < # wlar, is-encoded(to-vtbl(vols), buf1), valid-peb(v, m, mkpeb(aehdr0, avhdr2, buf2, m3, boolvar0)), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m6), avhdr1, buf1, m2, false)), valid-peb(mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-peb(v, m, mkpeb(aechdr(m4), avhdr0, buf0, m1, false)), avhdr.vol ∈ vols, ∀ v, m, n2. vols[v][m] = ⌜n2⌝ ∧ v ∈ vols ∧ m < # vols[v] ↔ wlar[n2].status = used ∧ n2 < # wlar ∧ apebs[n2].vidhdr.valid? ∧ apebs[n2].vidhdr.vol = v ∧ apebs[n2].vidhdr.leb = m ∧ n2 ≠ n0 ⊦ avhdr0.sqn = (valid-sqns(v, m, apebs) -- avhdr2.sqn).max, n1 = n, VTBL_VOLID ∈ vols, v = avhdr.vol ∧ m = avhdr.leb

User applied rule apply lemma

And obtained the following new goals

Proof step 40:
⊦ n0 < # apebs ∧ n1 < # apebs ∧ n0 ≠ n1 ∧ valid-peb(v, m, apebs[n0]) ∧ valid-peb(v, m, apebs[n1]) → (∃ bag. valid-sqns(v, m, apebs) = bag ++ apebs[n0].vidhdr.sqn ++ apebs[n1].vidhdr.sqn)
Proof step 41:
# apebs = # wlar, avhdr0.sqn = valid-sqns(v, m, apebs).max, avhdr1.sqn = valid-sqns(VTBL_VOLID, VTBL_LNUM, apebs).max, apebs[n0] = mkpeb(aechdr(m6), avhdr1, buf1, m2, false), vols[avhdr.vol][avhdr.leb] = unmapped, wlar[n1] = wl-entry(m4, used), wlar[n] = wl-entry(m5, erasing), wlar[n0] = wl-entry(m6, used), apebs[n1] = mkpeb(aechdr(m4), avhdr0, buf0, m1, false), apebs[n] = mkpeb(aehdr0, avhdr2, buf2, m3, boolvar0), n1 < # wlar, n0 < # wlar, avhdr.leb < # vols[avhdr.vol], n < # wlar, is-encoded(to-vtbl(vols), buf1), valid-peb(v, m, mkpeb(aechdr(m4), avhdr0, buf0, m1, false)), valid-peb(mkpeb(aehdr, avhdr, buf, m0, boolvar)), valid-peb(VTBL_VOLID, VTBL_LNUM, mkpeb(aechdr(m6), avhdr1, buf1, m2, false)), valid-peb(v, m, mkpeb(aehdr0, avhdr2, buf2, m3, boolvar0)), avhdr.vol ∈ vols, n < # apebs ∧ n1 < # apebs ∧ n ≠ n1 ∧ valid-peb(v, m, apebs[n]) ∧ valid-peb(v, m, apebs[n1]) → (∃ bag. valid-sqns(v, m, apebs) = bag ++ apebs[n].vidhdr.sqn ++ apebs[n1].vidhdr.sqn), ∀ v, m, n2. vols[v][m] = ⌜n2⌝ ∧ v ∈ vols ∧ m < # vols[v] ↔ wlar[n2].status = used ∧ n2 < # wlar ∧ apebs[n2].vidhdr.valid? ∧ apebs[n2].vidhdr.vol = v ∧ apebs[n2].vidhdr.leb = m ∧ n2 ≠ n0 ⊦ avhdr0.sqn = (valid-sqns(v, m, apebs) -- avhdr2.sqn).max, n1 = n, VTBL_VOLID ∈ vols, v = avhdr.vol ∧ m = avhdr.leb

Rule argument

spec: sqns name: valid-sqns-extract2-sqn seq: ⊦ n0 < # apebs ∧ n1 < # apebs ∧ n0 ≠ n1 ∧ valid-peb(v, m, apebs[n0]) ∧ valid-peb(v, m, apebs[n1]) → (∃ bag. valid-sqns(v, m, apebs) = bag ++ apebs[n0].vidhdr.sqn ++ apebs[n1].vidhdr.sqn) subst: [n0, apebs, n1, v, m] → [n, apebs, n1, v, m]

Additional information

Used terms: n, apebs, n1, v, m