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Proof step 440 from proof prepost-scan-all (previous step)

# apebs = n0 + vtblpnum, # wlar0 = n0 + vtblpnum, avhdr.sqn = valid-sqns(v, m1, apebs, vtblpnum).max, apebs[m3] = mkpeb(aehdr, avhdr, buf, m5, boolvar), recs[v × m1] = recovery-entry(m3, avhdr.sqn), apebs[vtblpnum] = mkpeb(aechdr(m), avidhdr(v, m1, m2, 0, m4), buf1, m0, false), nx < vtblpnum, m3 < vtblpnum, m2 < avhdr.sqn, bflipset-cons(bflipset, subarray(wlar0, 0, vtblpnum)), indices-erasing(wlar0, nx), ubi-io-inv(apebs), ubi-io-peb-inv(mkpeb(aechdr(m), avidhdr(v, m1, m2, 0, m4), buf1, m0, false)), ubi-io-peb-inv(mkpeb(aehdr, avhdr, buf, m5, boolvar)), 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(mkpeb(aechdr(m), avidhdr(v, m1, m2, 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, valid-eraseq-stronger(subarray(apebs, 0, vtblpnum + 1), q + eq-entry(vtblpnum, v × m1) ') ∧ valid-wl(subarray(apebs, 0, vtblpnum + 1), subarray(wlar0[vtblpnum, wl-entry(m, erasing)], 0, vtblpnum + 1))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: ubi-base name: valid-wl-next-valid-erase seq: m < # apebs, # apebs = # wlar ⊦ valid-wl(subarray(apebs, 0, m), subarray(wlar, 0, m)) ∧ valid-peb(v, n, apebs[m]) ∧ ( valid-sqns(v, n, apebs, m) ≠ ∅ → apebs[m].vidhdr.sqn < valid-sqns(v, n, apebs, m).max) → valid-wl (subarray(apebs, 0, m + 1), subarray(wlar[m, wl-entry(m0, erasing)], 0, m + 1)) subst: [v, n, m0, wlar, m, apebs] → [v, m1, m, wlar0, vtblpnum, apebs] flags: <all paths>

Additional information

Used simplifier rules: ls-max: ⊦ (bag ++ n).max = (bag = ∅ ⊃ n;(bag.max > n ⊃ bag.max;n));% ⊦ 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%m < # apebs, # wlar = # apebs ⊦ valid-recs(subarray(apebs, 0, m), subarray(wlar, 0, m), recs) ∧ valid-wl(subarray(apebs, 0, m), subarray(wlar, 0, m)) ∧ v × n ∈ recs → valid-sqns(v, n, apebs, m) ≠ ∅; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)%m < # apebs ⊦ valid-eraseq-stronger(subarray(apebs, 0, m), q) → ( valid-eraseq-stronger (subarray(apebs, 0, m + 1), q + eq-entry(m, v × n) ') ↔ valid-peb(v, n, apebs[m]) ∧ apebs[m].vidhdr.sqn ≠ valid-sqns(v, n, apebs, m + 1).max); (ss)%⊦ valid-peb(v0, m0, apeb) → (valid-peb(v1, m1, apeb) ↔ v0 = v1 ∧ m0 = m1); (s)%⊦ n < # apebs → valid-sqns(v, m, apebs, n + 1) = valid-sqns(v, m, apebs, n) ∪ (valid-peb(v, m, apebs[n]) ⊃ {apebs[n].vidhdr.sqn} ;∅); (s)% ⊦ bag ∪ {n} = bag ++ n; (s)%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]% ⊦ m > n ↔ n < m; (s)%a: ⊦ (q + q0) + q1 = q + q0 + q1; from specification list[erasequeue-base]%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%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ mkpeb(aehdr, avhdr, buf, m, boolvar).vidhdr = avhdr; (s)% ⊦ avidhdr(v, m, m0, m1, m2).sqn = m0; (s)% ⊦ m < n → ¬ n < m; (s)%⊦ valid-peb(apeb) → (valid-peb(v, m, apeb) ↔ v = apeb.vidhdr.vol ∧ m = apeb.vidhdr.leb); (ss)% ⊦ avidhdr(v, m, m0, m1, m2).vol = v; (s)% ⊦ avidhdr(v, m, m0, m1, m2).leb = m; (s) Used terms: v, m1, m, wlar0, vtblpnum, apebs