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Proof step 336 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, n3 + m0 = n4 + # buf, LEB_SIZE = n0 + # buf, LEB_SIZE = n1 + # buf1, avol0[n2] = mapped(buf), avol1[n2] = mapped(buf1), # buf < LEB_SIZE, n2 < # avol1, m4 < EB_PAGE_SIZE, n3 < # buf3, n3 + m0 < LEB_SIZE, n3 < n, awbuf-inv(avol1, buffered(n2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), # buf ≤ n3 + m0, m0 + n ≤ LEB_SIZE, n ≤ # buf3, # buf1 ≤ LEB_SIZE, rofs → m6 = # buf ∧ buf1 = buf + subarray(buf2, 0, m4) ∨ m6 < # buf ∧ m6 + m4 < # buf ∧ buf1 = buf ∧ subarray(buf2, 0, m4) = subarray(buf, m6, m4), ¬ rofs → buf1 = buf + subarray(buf2, 0, m4), rofs → m6 ≤ # buf, ¬ rofs → # buf = m6 ⊦ mkempbuf(n0)[n4] = (n3 + m0 < # buf1 ⊃ buf1[n3 + m0];mkempbuf(n1)[n3 + m0 - # buf1]), n0 = 0, m0 ≤ m6 + m4 ∧ m6 ≤ m0 + n

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 337:
⊦ m < n → mkempbuf(n)[m] = empty
Proof step 338:
# avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, n3 + m0 = n4 + # buf, LEB_SIZE = n1 + # buf1, LEB_SIZE = n0 + # buf, avol1[n2] = mapped(buf1), avol0[n2] = mapped(buf), n3 < n, n3 + m0 < LEB_SIZE, n3 < # buf3, m4 < EB_PAGE_SIZE, n2 < # avol1, # buf < LEB_SIZE, n4 < n0, awbuf-inv(avol1, buffered(n2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), # buf1 ≤ LEB_SIZE, n ≤ # buf3, m0 + n ≤ LEB_SIZE, # buf ≤ n3 + m0, ¬ rofs → # buf = m6, rofs → m6 ≤ # buf, ¬ rofs → buf1 = buf + subarray(buf2, 0, m4), rofs → m6 = # buf ∧ buf1 = buf + subarray(buf2, 0, m4) ∨ m6 < # buf ∧ m6 + m4 < # buf ∧ buf1 = buf ∧ subarray(buf2, 0, m4) = subarray(buf, m6, m4) ⊦ empty = (n3 + m0 < # buf1 ⊃ buf1[n3 + m0];mkempbuf(n1)[n3 + m0 - # buf1]), n0 = 0, m0 ≤ m6 + m4 ∧ m6 ≤ m0 + n

Rule argument

spec: array-splice, inst: buffer name: emparray-at seq: ⊦ m < n → mkempbuf(n)[m] = empty subst: [m, n] → [n4, n0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m ≤ n → (n < m ↔ false); (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)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s) Used terms: n4, n0