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Proof step 75 from proof refine-write-buf (previous step)

# buf1 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, avol1[m4] = mapped(buf2 + subarray(buf1, 0, m2)), avol0[m4] = mapped(buf2), align↓(m2 + n3, EB_PAGE_SIZE) = n + m2, m4 < # avol1, m2 < EB_PAGE_SIZE, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(# buf2, EB_PAGE_SIZE), is-aligned(n + m2, EB_PAGE_SIZE), (# buf2 + m2) + n3 ≤ LEB_SIZE, n + (m2 + n3) % EB_PAGE_SIZE ≤ # buf3, # buf2 + n + m2 ≤ LEB_SIZE, n ≤ # buf3, # buf2 + m2 + n3 ≤ LEB_SIZE, EB_PAGE_SIZE ≤ m2 + n3, n3 ≤ # buf3 ⊦ n + m2 = 0, n3 = 0, subarray(buf3, 0, n3) = copy(buf3, 0, subarray(mkbuf(n + m2), m2, n), 0, n) + subarray(buf3, n, (m2 + n3) % EB_PAGE_SIZE)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 76:
⊦ n0 + m ≤ # buf ∧ n1 + m ≤ # buf0 → copy(buf, n0, buf0, n1, m) = subarray(buf0, 0, n1) + subarray(buf, n0, m) + subarray(buf0, n1 + m, # buf0 - n1 - m)
Proof step 77:
# avol0 = # avol1, # lpt0 = # avol1, # buf1 = EB_PAGE_SIZE, avol0[m4] = mapped(buf2), avol1[m4] = mapped(buf2 + subarray(buf1, 0, m2)), align↓(m2 + n3, EB_PAGE_SIZE) = n + m2, m2 < EB_PAGE_SIZE, m4 < # avol1, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(n + m2, EB_PAGE_SIZE), is-aligned(# buf2, EB_PAGE_SIZE), n3 ≤ # buf3, EB_PAGE_SIZE ≤ m2 + n3, # buf2 + m2 + n3 ≤ LEB_SIZE, n ≤ # buf3, # buf2 + n + m2 ≤ LEB_SIZE, n + (m2 + n3) % EB_PAGE_SIZE ≤ # buf3, (# buf2 + m2) + n3 ≤ LEB_SIZE, 0 + n ≤ # buf3, 0 + n ≤ # subarray(mkbuf(n + m2), m2, n) ⊦ n + m2 = 0, n3 = 0, subarray(buf3, 0, n3) = ( subarray(subarray(mkbuf(n + m2), m2, n), 0, 0) + subarray(buf3, 0, n) + subarray(subarray(mkbuf(n + m2), m2, n), 0 + n, # subarray(mkbuf(n + m2), m2, n) - 0 - n)) + subarray(buf3, n, (m2 + n3) % EB_PAGE_SIZE)

Rule argument

spec: array-splice, inst: buffer name: copy2append seq: ⊦ n0 + m ≤ # buf ∧ n1 + m ≤ # buf0 → copy(buf, n0, buf0, n1, m) = subarray(buf0, 0, n1) + subarray(buf, n0, m) + subarray(buf0, n1 + m, # buf0 - n1 - m) subst: [m, n1, buf0, n0, buf] → [n, 0, subarray(mkbuf(n + m2), m2, n), 0, buf3] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m ≤ m; (s) Used terms: n, 0, subarray(mkbuf(n + m2), m2, n), 0, buf3