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Proof step 341 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, n3 + m0 = n4 + # buf, LEB_SIZE = n0 + # buf, LEB_SIZE = n1 + # buf1, avol0[n2] = mapped(buf), avol1[n2] = mapped(buf1), # buf < # buf1, n4 < n0, # buf < LEB_SIZE, n2 < # avol1, m4 < EB_PAGE_SIZE, n3 < # buf3, n3 + m0 < LEB_SIZE, n3 < n, n3 + m0 < # buf1, awbuf-inv(avol1, buffered(n2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), # buf ≤ n3 + m0, m0 + n ≤ LEB_SIZE, n ≤ # buf3, # buf1 ≤ LEB_SIZE, rofs → m6 = # buf ∧ buf1 = buf + subarray(buf2, 0, m4) ∨ m6 < # buf ∧ m6 + m4 < # buf ∧ buf1 = buf ∧ buf2 = subarray(buf, m6, m4), ¬ rofs → buf1 = buf + subarray(buf2, 0, m4), rofs → m6 ≤ # buf, ¬ rofs → # buf = m6 ⊦ empty = buf1[n3 + m0], m0 ≤ m6 + m4 ∧ m6 ≤ m0 + n

User applied rule case distinction

And obtained the following new goals

Proof step 342:
# avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, n3 + m0 = n4 + # buf, LEB_SIZE = n1 + # buf1, LEB_SIZE = n0 + # buf, avol1[n2] = mapped(buf1), avol0[n2] = mapped(buf), n3 + m0 < # buf1, n3 < n, n3 + m0 < LEB_SIZE, n3 < # buf3, m4 < EB_PAGE_SIZE, n2 < # avol1, # buf < LEB_SIZE, n4 < n0, # buf < # buf1, awbuf-inv(avol1, buffered(n2)), ebm-vol-inv(avol0), is-aligned(m6, EB_PAGE_SIZE), # buf1 ≤ LEB_SIZE, n ≤ # buf3, m0 + n ≤ LEB_SIZE, # buf ≤ n3 + m0, ¬ rofs → # buf = m6, rofs → m6 ≤ # buf, ¬ rofs → buf1 = buf + subarray(buf2, 0, m4), m6 = # buf ∧ buf1 = buf + subarray(buf2, 0, m4) ∨ m6 < # buf ∧ m6 + m4 < # buf ∧ buf1 = buf ∧ buf2 = subarray(buf, m6, m4) ⊦ empty = buf1[n3 + m0], m0 ≤ m6 + m4 ∧ m6 ≤ m0 + n

Rule argument

pos: left 25

Additional information

Used simplifier rules: nat0: ⊦ n0 = n1 + n → (n1 + m ≤ n0 ↔ m ≤ n);%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # subarray(buf, m, 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 < m + n ↔ n ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ m + n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%transitivity: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification oelem[nat-set]% ⊦ m < n → n ≠ 0; (s)% ⊦ m + n < m0 → m + n < m0 + n0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m + m0 < n → ¬ n + n0 < m; (s)% ⊦ n0 + n2 < # buf ∧ n1 < n2 → (buf + buf0)[n0 + n1] = buf[n0 + n1]; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n → ¬ n < m; (s)