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Proof step 69 from proof inv-write (previous step)

LEB_SIZE = n0 + m, LEB_SIZE = n4 + m1 + n1, avols[v][m2] = mapped(buf0, m), n1 < n, m2 < # avols[v], n3 < n4, n3 < LEB_SIZE, ebm-cons(avols), is-aligned(m, EB_PAGE_SIZE), is-aligned(n1, EB_PAGE_SIZE), is-aligned(m1, EB_PAGE_SIZE), is-aligned(n, EB_PAGE_SIZE), range=(buf0, m, emptyleb, 0, n0), v ∈ avols, n1 + m1 ≤ LEB_SIZE, m ≤ m1, m ≤ LEB_SIZE, n2 + n1 ≤ # buf, m1 + n ≤ LEB_SIZE, n + n2 ≤ # buf, m1 ≤ LEB_SIZE, m1 + n1 ≤ LEB_SIZE, n4 ≤ LEB_SIZE, ⌊err0⌋ ⊦ (subarray(buf0, 0, m1) + subarray(buf, n2, n1) + mkempbuf(n4))[(m1 + n1) + n3] = empty, n1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 71:
LEB_SIZE = n4 + m1 + n1, LEB_SIZE = n0 + m, avols[v][m2] = mapped(buf0, m), n3 < LEB_SIZE, n3 < n4, m2 < # avols[v], n1 < n, (m1 + n1) + n3 < # subarray(buf0, 0, m1) + #(subarray(buf, n2, n1) + mkempbuf(n4)), ebm-cons(avols), is-aligned(n, EB_PAGE_SIZE), is-aligned(m1, EB_PAGE_SIZE), is-aligned(n1, EB_PAGE_SIZE), is-aligned(m, EB_PAGE_SIZE), range=(buf0, m, emptyleb, 0, n0), v ∈ avols, n4 ≤ LEB_SIZE, m1 + n1 ≤ LEB_SIZE, m1 ≤ LEB_SIZE, n + n2 ≤ # buf, m1 + n ≤ LEB_SIZE, n2 + n1 ≤ # buf, m ≤ LEB_SIZE, m ≤ m1, n1 + m1 ≤ LEB_SIZE, ⌊err0⌋ ⊦ ((m1 + n1) + n3 < # subarray(buf0, 0, m1) ⊃ subarray(buf0, 0, m1)[(m1 + n1) + n3] ;(subarray(buf, n2, n1) + mkempbuf(n4))[(m1 + n1) + n3 - # subarray(buf0, 0, m1)] ) = empty, 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] → [(m1 + n1) + n3, subarray(buf, n2, n1) + mkempbuf(n4), subarray(buf0, 0, m1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ v ∈ avols ∧ m < # avols[v] ∧ avols[v][m] = mapped(buf, n) ∧ ebm-cons(avols) → # buf = LEB_SIZE; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # mkempbuf(n) = n; (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 < m + n0 ↔ n < n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%c: ⊦ m + n = n + m; from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: (m1 + n1) + n3, subarray(buf, n2, n1) + mkempbuf(n4), subarray(buf0, 0, m1)