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Proof step 23 from proof prepost-read-orphans (previous step)

m4 + # lpt = # avol, LEB_SIZE = n0 + # buf, avol[m5] = mapped(buf1), avol[m1] = mapped(buf), avol[0] = mapped(buf2), m1 < # avol, disk-layout(mksb(adr, m6, m5, m1, m2, m3, m4), # lpt), ebm-vol-inv(avol), is-encoded(mksb(adr, m6, m5, m1, m2, m3, m4), buf2), is-encoded(lpt, lebs(avol, m3, lptlebs(# lpt, LEB_SIZE))), is-encoded(log, buf1), is-encoded-pad(orphans0, m2, buf), mapped-full?(avol, m3, lptlebs(# lpt, LEB_SIZE)), # buf ≤ LEB_SIZE, m2 * ENCODED_NAT_SIZE ≤ LEB_SIZE, logoff ≠ LEB_SIZE → # buf1 = logoff ⊦ ⟪persistence_io_decode_orphans# (m2, copy(avol[m1].data + mkempbuf(n0), 0, mkbuf(m2 * ENCODED_NAT_SIZE), 0, m2 * ENCODED_NAT_SIZE); ; orphans, err0)⟫ ((err0 = ESUCCESS ∨ ⌊err0⌋ ) ∧ (err0 = ESUCCESS → orphans = orphans0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ 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 25:
m4 + # lpt = # avol, LEB_SIZE = n0 + # buf, avol[0] = mapped(buf2), avol[m1] = mapped(buf), avol[m5] = mapped(buf1), m1 < # avol, disk-layout(mksb(adr, m6, m5, m1, m2, m3, m4), # lpt), ebm-vol-inv(avol), is-encoded(log, buf1), is-encoded(lpt, lebs(avol, m3, lptlebs(# lpt, LEB_SIZE))), is-encoded(mksb(adr, m6, m5, m1, m2, m3, m4), buf2), is-encoded-pad(orphans0, m2, buf), mapped-full?(avol, m3, lptlebs(# lpt, LEB_SIZE)), m2 * ENCODED_NAT_SIZE ≤ LEB_SIZE, # buf ≤ LEB_SIZE, 0 + m2 * ENCODED_NAT_SIZE ≤ #(avol[m1].data + mkempbuf(n0)), 0 + m2 * ENCODED_NAT_SIZE ≤ # mkbuf(m2 * ENCODED_NAT_SIZE), logoff ≠ LEB_SIZE → # buf1 = logoff ⊦ ⟪persistence_io_decode_orphans# (m2, subarray(mkbuf(m2 * ENCODED_NAT_SIZE), 0, 0) + subarray(avol[m1].data + mkempbuf(n0), 0, m2 * ENCODED_NAT_SIZE) + subarray(mkbuf(m2 * ENCODED_NAT_SIZE), 0 + m2 * ENCODED_NAT_SIZE, # mkbuf(m2 * ENCODED_NAT_SIZE) - 0 - m2 * ENCODED_NAT_SIZE) ; ; orphans, err0)⟫ ((err0 = ESUCCESS ∨ ⌊err0⌋ ) ∧ (err0 = ESUCCESS → orphans = orphans0))

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 * ENCODED_NAT_SIZE, 0, mkbuf(m2 * ENCODED_NAT_SIZE), 0, avol[m1].data + mkempbuf(n0)] 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% ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # mkempbuf(n) = n; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ # mkbuf(n) = n; (s)% ⊦ m ≤ m; (s)% ⊦ 0 ≤ m; (s)% ⊦ m - 0 = m; (s)% ⊦ m - m = 0; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m0 * m = 0 ↔ m0 = 0 ∨ m = 0; (ws)% ⊦ ENCODED_NAT_SIZE ≠ 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ mapped(buf).data = buf; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ LEB_SIZE ≠ 0; (s)% ⊦ ¬ n < n; (s)%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]%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat Used terms: m2 * ENCODED_NAT_SIZE, 0, mkbuf(m2 * ENCODED_NAT_SIZE), 0, avol[m1].data + mkempbuf(n0)