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Proof step 45 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, LEB_SIZE = n3 + # buf1, LEB_SIZE = n0 + # buf4, avol1[m2] = mapped(buf1), avol0[m2] = mapped(buf4), n2 + (m0 ≤ m6 ⊃ 0;m0 - m6) < EB_PAGE_SIZE, m2 < # avol1, (m0 ≤ m6 ⊃ 0;m0 - m6) < EB_PAGE_SIZE, abs(avol1, buffered(m2), mkwbuf(buf2, m6, n2 + (m0 ≤ m6 ⊃ 0;m0 - m6)), avol0, rofs), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), # buf1 ≤ LEB_SIZE, m6 ≤ LEB_SIZE, (m0 ≤ m6 ⊃ m6 - m0;0) ≤ # buf3, (m0 ≤ m6 ⊃ 0;m0 - m6) + min(n1, n2) ≤ EB_PAGE_SIZE, (m0 ≤ m6 ⊃ m6 - m0;0) + min(n1, n2) ≤ # buf3, m0 ≤ m6 + n2 + (m0 ≤ m6 ⊃ 0;m0 - m6), m6 ≤ m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0), n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ # buf3, # buf4 ≤ LEB_SIZE, m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ LEB_SIZE, ¬ m0 ≤ m6 → m6 ≤ m0, rofs → m6 ≤ # buf4, ¬ rofs → # buf4 = m6 ⊦ copy(buf2, (m0 ≤ m6 ⊃ 0;m0 - m6), copy(buf4 + mkempbuf(n0), m0, buf3, 0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0)), (m0 ≤ m6 ⊃ m6 - m0;0), min(n1, n2)) = copy(buf1 + mkempbuf(n3), m0, buf3, 0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ 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 47:
# avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, LEB_SIZE = n0 + # buf4, LEB_SIZE = n3 + # buf1, avol0[m2] = mapped(buf4), avol1[m2] = mapped(buf1), (m0 ≤ m6 ⊃ 0;m0 - m6) < EB_PAGE_SIZE, m2 < # avol1, n2 + (m0 ≤ m6 ⊃ 0;m0 - m6) < EB_PAGE_SIZE, abs(avol1, buffered(m2), mkwbuf(buf2, m6, n2 + (m0 ≤ m6 ⊃ 0;m0 - m6)), avol0, rofs), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ LEB_SIZE, # buf4 ≤ LEB_SIZE, n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ # buf3, m6 ≤ m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0), m0 ≤ m6 + n2 + (m0 ≤ m6 ⊃ 0;m0 - m6), (m0 ≤ m6 ⊃ m6 - m0;0) + min(n1, n2) ≤ # buf3, (m0 ≤ m6 ⊃ 0;m0 - m6) + min(n1, n2) ≤ EB_PAGE_SIZE, (m0 ≤ m6 ⊃ m6 - m0;0) ≤ # buf3, m6 ≤ LEB_SIZE, # buf1 ≤ LEB_SIZE, m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ #(buf4 + mkempbuf(n0)), 0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ # buf3, ¬ rofs → # buf4 = m6, rofs → m6 ≤ # buf4, ¬ m0 ≤ m6 → m6 ≤ m0 ⊦ copy (buf2, (m0 ≤ m6 ⊃ 0;m0 - m6), subarray(buf3, 0, 0) + subarray(buf4 + mkempbuf(n0), m0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0)) + subarray(buf3, 0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0), # buf3 - 0 - n1 + (m0 ≤ m6 ⊃ m6 - m0;0)), (m0 ≤ m6 ⊃ m6 - m0;0), min(n1, n2)) = copy(buf1 + mkempbuf(n3), m0, buf3, 0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0))

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] → [n1 + (m0 ≤ m6 ⊃ m6 - m0;0), 0, buf3, m0, buf4 + mkempbuf(n0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # mkempbuf(n) = n; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ 0 ≤ m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ n < m → (m - n) + n = m; (s)% ⊦ m ≤ m + n; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%# buf ≤ n, n + m1 ≤ # buf + # buf0 ⊦ copy(buf + buf0, n, buf1, m0, m1) = copy(buf0, n - # buf, buf1, m0, m1); (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ ¬ n < n; (s)% ⊦ n ≤ n0 → (m + (n0 - n)) + n = m + n0; (s) Used terms: n1 + (m0 ≤ m6 ⊃ m6 - m0;0), 0, buf3, m0, buf4 + mkempbuf(n0)