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Proof step 258 from proof refine-write-buf (previous step)

# avol0 = # avol1, # lpt0 = # avol1, # buf1 = EB_PAGE_SIZE, avol0[m4] = mapped(buf2), avol1[m4] = mapped(buf2 + subarray(buf1, 0, m2)), m2 < EB_PAGE_SIZE, m2 + n < EB_PAGE_SIZE, m4 < # avol1, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(# buf2, EB_PAGE_SIZE), n ≤ # buf, # buf2 + m2 + n ≤ LEB_SIZE, (# buf2 + m2) + n ≤ LEB_SIZE ⊦ subarray(buf1, 0, m2) + subarray(buf, 0, n) = subarray(copy(buf, 0, buf1, m2, n), 0, m2 + n)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-splice, inst: buffer name: copy2append0 seq: ⊦ n1 ≤ # buf ∧ n0 + n1 ≤ # buf0 → copy(buf, 0, buf0, n0, n1) = subarray(buf0, 0, n0) + subarray(buf, 0, n1) + subarray(buf0, n0 + n1, # buf0 - n0 - n1) subst: [n1, n0, buf0, buf] → [n, m2, buf1, buf] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # subarray(buf, m, n) = n; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n = # buf ⊦ subarray(buf + buf0, 0, n) = buf; (s)% ⊦ buf + buf0 = buf + buf1 ↔ buf0 = buf1; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → ¬ n < m; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ n + 0 = n; (s)% ⊦ m + m0 ≤ m ↔ m0 = 0; (s)% ⊦ m < n → m ≤ n; (s)%m0 + m1 ≤ n1, m0 + m1 ≤ # buf ⊦ subarray(copy(buf0, n0, buf, n1, n2), m0, m1) = subarray(buf, m0, m1); (s)% ⊦ buf + buf0 = buf ↔ # buf0 = 0; (s)% ⊦ ¬ m + n < m; (s) Used terms: n, m2, buf1, buf