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Proof step 41 from proof refine-read (previous step)

# buf2 = (n4 + n2) + n3, LEB_SIZE = n0 + n1, avols[default-volid][m2] = mapped(buf0, n1), n < # buf2, m2 < # avols[default-volid], ebm-cons(avols), ebm-vol-inv(abs(avols[default-volid])), default-volid ∈ avols, n2 + n3 ≤ # buf2, m1 + n2 ≤ LEB_SIZE, n3 + n2 ≤ # buf2, n1 ≤ LEB_SIZE, n3 ≤ # buf2 ⊦ (n < n3 ∨ n3 + n2 ≤ n ⊃ buf2[n];buf0[n + m1 - n3]) = (subarray(buf2, 0, n3) + subarray(subarray(buf0, 0, n1) + mkempbuf(n0), m1, n2) + subarray(buf2, n3 + n2, n4))[n]

User applied rule insert rewrite lemma

And obtained the following new goals

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(buf0, 0, n1) + mkempbuf(n0), m1, n2) + subarray(buf2, n3 + n2, n4), subarray(buf2, 0, n3)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ #(buf + buf0) = # buf + # buf0; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ ¬ n < n; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)%⊦ v ∈ avols ∧ m < # avols[v] ∧ avols[v][m] = mapped(buf, m0) ∧ ebm-cons(avols) → # buf = LEB_SIZE; (s)% ⊦ # mkempbuf(n) = n; (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)% ⊦ n < # buf → (buf + buf0)[n] = buf[n]; (s) Used terms: n, subarray(subarray(buf0, 0, n1) + mkempbuf(n0), m1, n2) + subarray(buf2, n3 + n2, n4), subarray(buf2, 0, n3)