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Proof step 2 from proof valid-wl-next-valid-update (previous step)

# apebs = # wlar, apebs[m].echdr.ec = m2, apebs[m0].vidhdr.sqn = valid-sqns(v, n, apebs, m).max, m0 < m, m < # apebs, apebs[m].vidhdr.sqn > apebs[m0].vidhdr.sqn, unique-sqns(apebs), valid-peb(v, n, apebs[m]), valid-peb(v, n, apebs[m0]), valid-wl(subarray(apebs, 0, m), subarray(wlar, 0, m)) ⊦ valid-wl(subarray(apebs, 0, m + 1), subarray(wlar[m, wl-entry(m2, used)][m0, wl-entry(m1, erasing)], 0, m + 1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
# apebs = # wlar, apebs[m0].vidhdr.sqn = valid-sqns(v, n, apebs, m).max, m0 + 1 < # wlar, n0 < # wlar, m0 < m, valid-sqns(v, n, apebs, m).max < apebs[m].vidhdr.sqn, m < # wlar, m0 < # wlar, valid-peb(v, n, apebs[m0]), valid-peb(v, n, apebs[m]), n0 ≤ m, ∀ n0. n0 < m → (wlar[n0].status = erroneous ↔ apebs[n0].bad) ∧ (wlar[n0].status = free → apebs[n0].vidhdr = empty) ∧ (wlar[n0].status ≠ erroneous ∧ wlar[n0].status ≠ erasing → apebs[n0].echdr = aechdr(wlar[n0].ec)) ∧ (wlar[n0].status = used → (∃ v0, m2. valid-peb(v0, m2, apebs[n0]) ∧ apebs[n0].vidhdr.sqn = valid-sqns(v0, m2, apebs, m).max)), ∀ v, m. unique-elems(valid-sqns(v, m, apebs)) ⊦ (wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].status = erroneous ↔ apebs[n0].bad) ∧ (wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].status = free → apebs[n0].vidhdr = empty) ∧ ( wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].status ≠ erroneous ∧ wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].status ≠ erasing → apebs[n0].echdr = aechdr(wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].ec)) ∧ ( wlar[m, wl-entry(apebs[m].echdr.ec, used)][m0, wl-entry(m1, erasing)][n0].status = used → (∃ v0, m2. valid-peb(v0, m2, apebs[n0]) ∧ apebs[n0].vidhdr.sqn = (valid-sqns(v0, m2, apebs, m) ∪ (v = v0 ∧ n = m2 ⊃ {apebs[m].vidhdr.sqn} ;∅)).max))

Rule argument

Additional information

Used simplifier rules: valid-wl: ⊦ valid-wl(apebs, wlar) ↔ (∀ n. n < # wlar → (wlar[n].status = erroneous ↔ apebs[n].bad) ∧ (wlar[n].status = free → apebs[n].vidhdr = empty) ∧ ( wlar[n].status ≠ erroneous ∧ wlar[n].status ≠ erasing → apebs[n].echdr = aechdr(wlar[n].ec)) ∧ ( wlar[n].status = used → (∃ v, m. valid-peb(v, m, apebs[n]) ∧ apebs[n].vidhdr.sqn = valid-sqns(v, m, apebs).max)));%unique-sqns: ⊦ unique-sqns(apebs) ↔ (∀ v, m. unique-elems(valid-sqns(v, m, apebs)));%ls-at-subarray-wlar: n0 ≤ # wlar, n1 < n0 ⊦ subarray(wlar, 0, n0)[n1] = wlar[n1];%ls-at-subarray-pebs: n0 ≤ # apebs, n1 < n0 ⊦ subarray(apebs, 0, n0)[n1] = apebs[n1];%⊦ 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)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ # wlar[n, wle] = # wlar; (s)%n ≤ # apebs ⊦ valid-sqns(v, m, subarray(apebs, 0, n)) = valid-sqns(v, m, apebs, n); (s)% ⊦ n0 < n → subarray(wlar, m, n)[n0] = wlar[m + n0]; (s)% ⊦ n + 0 = n; (s)% ⊦ n0 < n → subarray(apebs, m, n)[n0] = apebs[m + n0]; (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m > n ↔ n < m; (s)% ⊦ # subarray(wlar, m, n) = n; (s)