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Proof step 406 from proof prepost-scan-all (previous step)

# apebs = n0 + vtblpnum, # wlar0 = n0 + vtblpnum, avhdr.sqn = valid-sqns(v, m1, apebs, vtblpnum).max, apebs[vtblpnum] = mkpeb(aechdr(m), avidhdr(v, m1, avhdr.sqn, 0, m4), buf1, m0, false), recs[v × m1] = recovery-entry(m3, avhdr.sqn), apebs[m3] = mkpeb(aehdr, avhdr, buf, m5, boolvar), wlar0[m3] = wl-entry(m6, stat), m3 < n0 + vtblpnum, m3 < vtblpnum, nx < vtblpnum, bflipset-cons(bflipset, subarray(wlar0, 0, vtblpnum)), indices-erasing(wlar0, nx), ubi-io-inv(apebs), ubi-io-peb-inv(mkpeb(aehdr, avhdr, buf, m5, boolvar)), ubi-io-peb-inv(mkpeb(aechdr(m), avidhdr(v, m1, avhdr.sqn, 0, m4), buf1, m0, false)), unique-sqns(apebs), valid-eraseq(subarray(wlar0, 0, vtblpnum), q), valid-eraseq-stronger(subarray(apebs, 0, vtblpnum), q), valid-global-sqn(subarray(apebs, 0, vtblpnum), sqnum), valid-peb(v, m1, mkpeb(aechdr(m), avidhdr(v, m1, avhdr.sqn, 0, m4), buf1, m0, false)), valid-peb(mkpeb(aechdr(m), avidhdr(v, m1, avhdr.sqn, 0, m4), buf1, m0, false)), valid-peb(v, m1, mkpeb(aehdr, avhdr, buf, m5, boolvar)), valid-recs(subarray(apebs, 0, vtblpnum), subarray(wlar0, 0, vtblpnum), recs), valid-recs-sqns(apebs, recs), valid-wl(subarray(apebs, 0, vtblpnum), subarray(wlar0, 0, vtblpnum)), v × m1 ∈ recs ⊦ n0 = 0, dups(nx), m3 ∈ q, indices-erasing(wlar0[vtblpnum, wl-entry(m, used)][m3, wl-entry(m6, erasing)], nx) ∧ valid-eraseq-stronger(subarray(apebs, 0, vtblpnum + 1), q + eq-entry(m3, v × m1) ') ∧ valid-wl(subarray(apebs, 0, vtblpnum + 1), subarray(wlar0[vtblpnum, wl-entry(m, used)][m3, wl-entry(m6, erasing)], 0, vtblpnum + 1)) ∧ valid-recs-sqns(apebs, recs[v × m1, recovery-entry(vtblpnum, avhdr.sqn)])

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: ubi-base name: unique-sqns-2-valid-pebs seq: ⊦ m0 < # apebs ∧ m1 < # apebs ∧ valid-peb(v, m, apebs[m0]) ∧ valid-peb(v, m, apebs[m1]) ∧ unique-sqns(apebs) ∧ m0 ≠ m1 → apebs[m0].vidhdr.sqn ≠ apebs[m1].vidhdr.sqn subst: [m0, apebs, m1, v, m] → [vtblpnum, apebs, m3, v, m1]

Additional information

Used simplifier rules: ⊦ m < m + n ↔ n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)% ⊦ avidhdr(v, m, m0, m1, m2).sqn = m0; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ ¬ n < n; (s)%# wlar = n ⊦ subarray(wlar, 0, n) = wlar; (s)%# apebs = n ⊦ subarray(apebs, 0, n) = apebs; (s)%succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic% ⊦ m < m0 ∧ n0 < n + m → n0 + 1 ≠ n + m0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)%m < # wlar ⊦ wlar[m, wle][m, wle0] = wlar[m, wle0]; (s)% ⊦ # wlar[n, wle] = # wlar; (s)%⊦ valid-recs-sqns(apebs, recs) → ( valid-recs-sqns(apebs, recs[v × m, recovery-entry(m0, m1)]) ↔ m0 < # apebs ∧ apebs[m0].vidhdr.sqn = m1); (ss) Used terms: vtblpnum, apebs, m3, v, m1