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Proof step 59 from proof refine-read-buf (previous step)

# buf3 = n5 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0), # buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, n1 + n5 = n4 + min(n1, n2), 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), min(n1, n2) ≤ n1 + n5, m0 ≤ m6 + n2 + (m0 ≤ m6 ⊃ 0;m0 - m6), m6 ≤ m0 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0), # buf4 ≤ LEB_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 + n1 + (m0 ≤ m6 ⊃ m6 - m0;0) ≤ LEB_SIZE, ¬ m0 ≤ m6 → m6 ≤ m0, rofs → m6 ≤ # buf4, ¬ rofs → # buf4 = m6 ⊦ subarray(subarray(buf4 + mkempbuf(n0), m0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0)), 0, (m0 ≤ m6 ⊃ m6 - m0;0)) + subarray(buf2, (m0 ≤ m6 ⊃ 0;m0 - m6), min(n1, n2)) + subarray (subarray(buf4 + mkempbuf(n0), m0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0)) + subarray(buf3, n1 + (m0 ≤ m6 ⊃ m6 - m0;0), n5), (m0 ≤ m6 ⊃ m6 - m0;0) + min(n1, n2), n4) = subarray(buf1 + mkempbuf(n3), m0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0)) + subarray(buf3, n1 + (m0 ≤ m6 ⊃ m6 - m0;0), n5)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-copy, inst: buffer name: subarray-subarray-elim seq: ⊦ n0 + n1 ≤ # buf ∧ m0 + m1 ≤ n1 → subarray(subarray(buf, n0, n1), m0, m1) = subarray(buf, n0 + m0, m1) subst: [m1, m0, n1, n0, buf] → [(m0 ≤ m6 ⊃ m6 - m0;0), 0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0), m0, buf4 + mkempbuf(n0)] flags: <all paths>

Additional information

Used simplifier rules: nat0: ⊦ n0 = n1 + n → (n1 + m ≤ n0 ↔ m ≤ n);% ⊦ #(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% ⊦ m ≤ m + n; (s)% ⊦ 0 ≤ m; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ n < m → (m - n) + n = m; (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 ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m = m + n ↔ n = 0; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m ≤ m; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ min(n, m) < n ↔ min(n, m) = m ∧ m < n; (ws)% ⊦ n1 < n0 → min(n0, n1) = n1; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ m + n - n + n1 = m - n1; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ n ≤ n0 → (m + (n0 - n)) + n = m + n0; (s)% ⊦ ¬ n < n; (s) Used terms: (m0 ≤ m6 ⊃ m6 - m0;0), 0, n1 + (m0 ≤ m6 ⊃ m6 - m0;0), m0, buf4 + mkempbuf(n0)