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

# apebs = # wlar, apebs[m] = mkpeb(aechdr(m0), empty, buf, m1, false), m < # apebs, valid-wl(subarray(apebs, 0, m), subarray(wlar, 0, m)) ⊦ valid-wl(subarray(apebs, 0, m + 1), subarray(wlar[m, wl-entry(m0, free)], 0, m + 1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
# apebs = # wlar, apebs[m] = mkpeb(aechdr(m0), empty, buf, m1, false), m < # wlar, n < # wlar, n ≤ m, ∀ n. n < m → (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, m2. valid-peb(v, m2, apebs[n]) ∧ apebs[n].vidhdr.sqn = valid-sqns(v, m2, apebs, m).max)) ⊦ (wlar[m, wl-entry(m0, free)][n].status = erroneous ↔ apebs[n].bad) ∧ (wlar[m, wl-entry(m0, free)][n].status = free → apebs[n].vidhdr = empty) ∧ ( wlar[m, wl-entry(m0, free)][n].status ≠ erroneous ∧ wlar[m, wl-entry(m0, free)][n].status ≠ erasing → apebs[n].echdr = aechdr(wlar[m, wl-entry(m0, free)][n].ec)) ∧ (wlar[m, wl-entry(m0, free)][n].status = used → (∃ v, m2. valid-peb(v, m2, apebs[n]) ∧ apebs[n].vidhdr.sqn = valid-sqns(v, m2, apebs, m).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)));%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];%union-com: ⊦ bag ∪ bag2 = bag2 ∪ bag; from specification bag-union[nat-bag]%union-ass: ⊦ bag ∪ bag3 ∪ bag4 = (bag ∪ bag3) ∪ bag4; from specification bag-union[nat-bag]% ⊦ ∅ ∪ bag0 = bag0; (s)% ⊦ ¬ valid-peb(v, m, mkpeb(aehdr, empty, buf, m0, boolvar)); (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[encoding-nat]%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[encoding-nat]%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic[encoding-nat]%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic[encoding-nat]%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat[encoding-nat]% ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic[encoding-nat]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic[encoding-nat]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ # subarray(wlar, m, n) = n; (s)