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Proof step 325 from proof refine-read-buf (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 326:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 327:
# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, LEB_SIZE = n0 + # buf, LEB_SIZE = n1 + # buf1, avol0[n2] = mapped(buf), avol1[n2] = mapped(buf1), n2 < # avol1, m4 < EB_PAGE_SIZE, n3 < # buf3, n3 + m0 < # buf + # mkempbuf(n0), awbuf-inv(avol1, buffered(n2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), m0 + n ≤ LEB_SIZE, # buf ≤ 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 ⊦ (n ≤ n3 ⊃ buf3[n3];(n3 + m0 < # buf ⊃ buf[n3 + m0];mkempbuf(n0)[n3 + m0 - # buf])) = (n ≤ n3 ⊃ buf3[n3];(buf1 + mkempbuf(n1))[n3 + m0]), m0 ≤ m6 + m4 ∧ m6 ≤ m0 + n

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] → [n3 + m0, mkempbuf(n0), buf] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # mkempbuf(n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: n3 + m0, mkempbuf(n0), buf