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Proof step 154 from proof refine-write-buf (previous step)

# 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) = n0 + m2, m4 < # avol1, m2 < EB_PAGE_SIZE, n1 < n0, n1 < # buf3, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(n1 + m2, EB_PAGE_SIZE), is-aligned(n0 + m2, EB_PAGE_SIZE), is-aligned(# buf2, EB_PAGE_SIZE), n3 ≤ # buf3, EB_PAGE_SIZE ≤ m2 + n3, # buf2 + m2 + n3 ≤ LEB_SIZE, n0 ≤ # buf3, # buf2 + n0 + m2 ≤ LEB_SIZE, (# buf2 + m2) + n3 ≤ LEB_SIZE, ⌊err1⌋ ⊦ subarray(buf1, 0, m2) + subarray(buf3, 0, n1) = subarray(copy(buf3, 0, copy(buf1, 0, mkbuf(n0 + m2), 0, m2), m2, n0), 0, n1 + m2), n1 + m2 = 0, n1 = 0, n3 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 155:
⊦ 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 156:
# 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) = n0 + m2, n1 < # buf3, n1 < n0, m2 < EB_PAGE_SIZE, m4 < # avol1, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(# buf2, EB_PAGE_SIZE), is-aligned(n0 + m2, EB_PAGE_SIZE), is-aligned(n1 + m2, EB_PAGE_SIZE), (# buf2 + m2) + n3 ≤ LEB_SIZE, # buf2 + n0 + m2 ≤ LEB_SIZE, n0 ≤ # buf3, # buf2 + m2 + n3 ≤ LEB_SIZE, EB_PAGE_SIZE ≤ m2 + n3, n3 ≤ # buf3, 0 + m2 ≤ # buf1, 0 + m2 ≤ # mkbuf(n0 + m2), ⌊err1⌋ ⊦ n1 + m2 = 0, subarray(buf1, 0, m2) + subarray(buf3, 0, n1) = subarray (copy(buf3, 0, subarray(mkbuf(n0 + m2), 0, 0) + subarray(buf1, 0, m2) + subarray(mkbuf(n0 + m2), 0 + m2, # mkbuf(n0 + m2) - 0 - m2), m2, n0), 0, n1 + m2), n1 = 0, n3 = 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] → [m2, 0, mkbuf(n0 + m2), 0, buf1] 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% ⊦ # mkbuf(n) = n; (s)% ⊦ m ≤ m + n; (s)% ⊦ EB_PAGE_SIZE ≠ 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ m < n → n ≠ 0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m < n → ¬ n < m; (s)%n ≠ 0, m % n = 0 ⊦ align↓(m, n) = m; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m ≤ 0 ↔ m = 0; (s)% ⊦ # copy(buf, m, buf0, m0, n) = # buf0; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m < n → m ≤ n; (s)%n1 + n2 ≤ m0, m0 + m1 ≤ # buf ⊦ subarray(copy(buf0, n0, buf, n1, n2), m0, m1) = subarray(buf, m0, m1); (s)%m ≠ 0, is-aligned(n1, m) ⊦ is-aligned(n0 + n1, m) ↔ is-aligned(n0, m); (s)% ⊦ ¬ m + n < m; (s) Used terms: m2, 0, mkbuf(n0 + m2), 0, buf1