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Proof step 107 from proof initial-ex-abstract-initial (previous step)

# buf = PEB_SIZE, m0 + 1 = # pebs, pebs[m0] = mkpeb(buf, EB_PAGE_SIZE, false), n0 < m0, abs(subarray(pebs, 0, m0), apebs), is-encoded(echdr(n), subarray(buf, 0, EB_PAGE_SIZE)), range=(buf, EB_PAGE_SIZE, emptypeb, EB_PAGE_SIZE, LEB_SIZE + EB_PAGE_SIZE), ubi-io-initial(apebs), ubi-io-initial(pebs), EB_PAGE_SIZE ≤ PEB_SIZE ⊦ n0 = m0, abs-peb(pebs[n0], copy(apebs, 0, mkapebs(m0 + 1), 0, m0)[n0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 108:
⊦ m0 < # apebs → copy(apebs0, n, apebs, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ apebs[m0];apebs0[m0 + n - n0])
Proof step 109:
# buf = PEB_SIZE, m0 + 1 = # pebs, pebs[m0] = mkpeb(buf, EB_PAGE_SIZE, false), n0 < m0, n0 < # mkapebs(m0 + 1), abs(subarray(pebs, 0, m0), apebs), is-encoded(echdr(n), subarray(buf, 0, EB_PAGE_SIZE)), range=(buf, EB_PAGE_SIZE, emptypeb, EB_PAGE_SIZE, LEB_SIZE + EB_PAGE_SIZE), ubi-io-initial(pebs), ubi-io-initial(apebs), EB_PAGE_SIZE ≤ PEB_SIZE ⊦ n0 = m0, abs-peb(pebs[n0], (n0 < 0 ∨ 0 + m0 ≤ n0 ⊃ mkapebs(m0 + 1)[n0];apebs[n0 + 0 - 0]))

Rule argument

spec: array-copy, inst: apebs name: at-copy seq: ⊦ m0 < # apebs → copy(apebs0, n, apebs, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ apebs[m0];apebs0[m0 + n - n0]) subst: [m0, m, n0, apebs, n, apebs0] → [n0, m0, 0, mkapebs(m0 + 1), 0, apebs] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # mkapebs(n) = n; (s)% ⊦ m ≠ n → (m < n + 1 ↔ m < n); (s)% ⊦ m ≠ n → (m < n ↔ ¬ n < m); (ss)% ⊦ m < n → ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n0, m0, 0, mkapebs(m0 + 1), 0, apebs