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Proof step 110 from proof prepost-read-log (previous step)

# buf2 = EB_PAGE_SIZE, LEB_SIZE = n1 + logoff0, LEB_SIZE = n0 + # buf0, avol[m5] = mapped(buf0), n < EB_PAGE_SIZE, m5 < # avol, LEB_SIZE < n1 + logoff0 + EB_PAGE_SIZE, # buf0 < n1 + logoff0 + EB_PAGE_SIZE, # buf0 < logoff0 + EB_PAGE_SIZE, logoff0 < LEB_SIZE, # buf0 + 1 < n1 + logoff0 + EB_PAGE_SIZE, # buf0 < LEB_SIZE, ebm-vol-inv(avol), is-aligned(logoff0, EB_PAGE_SIZE), is-encoded(log0, buf0), is-encoded(log, subarray(buf0, 0, logoff0)), is-encoded(rnode(m), subarray(buf0 + mkempbuf(n0), logoff0, EB_PAGE_SIZE)), logoff0 + EB_PAGE_SIZE ≤ LEB_SIZE, logoff0 ≤ # buf0 ⊦ (buf0 + mkempbuf(n0))[logoff0 + n] = empty, n0 = 0, n1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 111:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 112:
# buf2 = EB_PAGE_SIZE, LEB_SIZE = n0 + # buf0, LEB_SIZE = n1 + logoff0, avol[m5] = mapped(buf0), # buf0 < LEB_SIZE, # buf0 + 1 < n1 + logoff0 + EB_PAGE_SIZE, logoff0 < LEB_SIZE, # buf0 < logoff0 + EB_PAGE_SIZE, # buf0 < n1 + logoff0 + EB_PAGE_SIZE, LEB_SIZE < n1 + logoff0 + EB_PAGE_SIZE, m5 < # avol, n < EB_PAGE_SIZE, logoff0 + n < # buf0 + # mkempbuf(n0), ebm-vol-inv(avol), is-aligned(logoff0, EB_PAGE_SIZE), is-encoded(rnode(m), subarray(buf0 + mkempbuf(n0), logoff0, EB_PAGE_SIZE)), is-encoded(log, subarray(buf0, 0, logoff0)), is-encoded(log0, buf0), logoff0 ≤ # buf0, logoff0 + EB_PAGE_SIZE ≤ LEB_SIZE ⊦ (logoff0 + n < # buf0 ⊃ buf0[logoff0 + n];mkempbuf(n0)[logoff0 + n - # buf0]) = empty, n0 = 0, n1 = 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] → [logoff0 + n, mkempbuf(n0), buf0] 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%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ m < m + n ↔ n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: logoff0 + n, mkempbuf(n0), buf0