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Proof step 208 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, # buf3 = n5 + n1, n1 + n5 = n4 + min(n1, n2), LEB_SIZE = n0 + # buf0 + # buf, avol1[m2] = mapped(buf0 + buf), avol0[m2] = mapped(buf0 + buf), subarray(buf2, 0, n2 + n3) = subarray(buf, 0, n2 + n3), n < n5 + n1, n < n1 + n5, n3 + n2 < # buf, # buf0 + 1 < LEB_SIZE, # buf0 < LEB_SIZE, n3 < # buf, n2 + n3 < EB_PAGE_SIZE, m2 < # avol1, n3 < EB_PAGE_SIZE, (n3 + # buf0) + n2 < LEB_SIZE, n3 < n0 + # buf, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0, EB_PAGE_SIZE), min(n1, n2) ≤ n1 + n5, n3 + min(n1, n2) ≤ EB_PAGE_SIZE, n3 + n1 ≤ # buf + n0 ⊦ # buf = 1, EB_PAGE_SIZE = 1, (subarray(buf2, n3, min(n1, n2)) + subarray(subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5), min(n1, n2), n4))[n] = (subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5))[n], n3 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 209:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 210:
# buf3 = n5 + n1, # avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, n1 + n5 = n4 + min(n1, n2), LEB_SIZE = n0 + # buf0 + # buf, avol0[m2] = mapped(buf0 + buf), avol1[m2] = mapped(buf0 + buf), subarray(buf2, 0, n2 + n3) = subarray(buf, 0, n2 + n3), n3 < n0 + # buf, (n3 + # buf0) + n2 < LEB_SIZE, n3 < EB_PAGE_SIZE, m2 < # avol1, n2 + n3 < EB_PAGE_SIZE, n3 < # buf, # buf0 < LEB_SIZE, # buf0 + 1 < LEB_SIZE, n3 + n2 < # buf, n < n1 + n5, n < n5 + n1, n < # subarray(buf2, n3, min(n1, n2)) + # subarray(subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5), min(n1, n2), n4), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0, EB_PAGE_SIZE), n3 + n1 ≤ # buf + n0, n3 + min(n1, n2) ≤ EB_PAGE_SIZE, min(n1, n2) ≤ n1 + n5 ⊦ # buf = 1, (n < # subarray(buf2, n3, min(n1, n2)) ⊃ subarray(buf2, n3, min(n1, n2))[n] ;subarray(subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5), min(n1, n2), n4)[n - # subarray(buf2, n3, min(n1, n2))] ) = (subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5))[n], EB_PAGE_SIZE = 1, n3 = 0

Rule argument

spec: array-splice, inst: buffer name: append-at seq: ⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf]) subst: [n, buf0, buf] → [n, subarray ( subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5), min(n1, n2), n4), subarray(buf2, n3, min(n1, n2))] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]%c: ⊦ m + n = n + m; from specification nat-basic%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < m + n ↔ n ≠ 0; (s)% ⊦ m0 + m1 < n → min(n, m0) = m0; (s)% ⊦ m ≤ m + n; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%n + m ≤ # buf ⊦ subarray(buf + buf0, n, m) = subarray(buf, n, m); (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m ≤ m; (s)% ⊦ min(n, m) < n ↔ min(n, m) = m ∧ m < n; (ws)% ⊦ n1 < n0 → min(n0, n1) = n1; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)% ⊦ ¬ n < n; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n, subarray (subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5), min(n1, n2), n4), subarray(buf2, n3, min(n1, n2))