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Proof step 133 from proof refine-write (previous step)

LEB_SIZE = (n5 + n0) + n1 + n4, avols[default-volid][m2] = mapped(buf0, n4), n0 < n, m2 < # avols[default-volid], n2 < n4 + n1, n2 < LEB_SIZE, ebm-cons(avols), ebm-vol-inv(abs(avols[default-volid])), is-aligned(n0, EB_PAGE_SIZE), is-aligned(n, EB_PAGE_SIZE), is-aligned(n1, EB_PAGE_SIZE), default-volid ∈ avols, n0 + n1 + n4 ≤ LEB_SIZE, n3 + n0 ≤ # buf, n1 + n4 ≤ LEB_SIZE, n4 ≤ LEB_SIZE, n3 + n ≤ # buf, (n1 + n4) + n ≤ LEB_SIZE, ⌊err1⌋ ⊦ (subarray(buf0, 0, n4) + mkempbuf(n1))[n2] = buf0[n2], n0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 134:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 135:
LEB_SIZE = (n5 + n0) + n1 + n4, avols[default-volid][m2] = mapped(buf0, n4), n2 < LEB_SIZE, n2 < n4 + n1, m2 < # avols[default-volid], n0 < n, n2 < # subarray(buf0, 0, n4) + # mkempbuf(n1), ebm-cons(avols), ebm-vol-inv(abs(avols[default-volid])), is-aligned(n1, EB_PAGE_SIZE), is-aligned(n, EB_PAGE_SIZE), is-aligned(n0, EB_PAGE_SIZE), default-volid ∈ avols, (n1 + n4) + n ≤ LEB_SIZE, n3 + n ≤ # buf, n4 ≤ LEB_SIZE, n1 + n4 ≤ LEB_SIZE, n3 + n0 ≤ # buf, n0 + n1 + n4 ≤ LEB_SIZE, ⌊err1⌋ ⊦ (n2 < # subarray(buf0, 0, n4) ⊃ subarray(buf0, 0, n4)[n2];mkempbuf(n1)[n2 - # subarray(buf0, 0, n4)]) = buf0[n2], n0 = 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] → [n2, mkempbuf(n1), subarray(buf0, 0, n4)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ v ∈ avols ∧ m < # avols[v] ∧ avols[v][m] = mapped(buf, m0) ∧ ebm-cons(avols) → # buf = LEB_SIZE; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n2, mkempbuf(n1), subarray(buf0, 0, n4)