[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 124 from proof refine-read-buf (previous step)

# buf3 = n4 + n2, # avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, (n0 + # buf4) + n3 + n2 = LEB_SIZE, n1 + n5 = n4 + n2, avol0[m2] = mapped(buf4), avol1[m2] = mapped(buf4 + subarray(buf2, 0, n2 + n0)), n < n4 + n2, m2 < # avol1, n2 + n0 < EB_PAGE_SIZE, n0 < EB_PAGE_SIZE, n < n2 + n4, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), n1 ≤ n2 + n3, # buf4 ≤ LEB_SIZE, n2 ≤ n1 ⊦ (n < n2 ⊃ buf2[n0 + n];subarray(mkempbuf(n1) + subarray(buf3, n1, n5), n2, n4)[n - n2]) = (subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1) + subarray(buf3, n1, n5))[n], EB_PAGE_SIZE = 1, n0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 125:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 126:
# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, # buf3 = n4 + n2, n1 + n5 = n4 + n2, (n0 + # buf4) + n3 + n2 = LEB_SIZE, avol1[m2] = mapped(buf4 + subarray(buf2, 0, n2 + n0)), avol0[m2] = mapped(buf4), n < n2 + n4, n0 < EB_PAGE_SIZE, n2 + n0 < EB_PAGE_SIZE, m2 < # avol1, n < n4 + n2, n < # subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1) + # subarray(buf3, n1, n5), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), n2 ≤ n1, # buf4 ≤ LEB_SIZE, n1 ≤ n2 + n3 ⊦ (n < n2 ⊃ buf2[n0 + n];subarray(mkempbuf(n1) + subarray(buf3, n1, n5), n2, n4)[n - n2]) = (n < # subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1) ⊃ subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1)[n] ;subarray(buf3, n1, n5)[n - # subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1)] ), EB_PAGE_SIZE = 1, n0 = 0

Rule argument

spec: array-splice, inst: buffer name: append-at seq: ⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf]) subst: [n, buf0, buf] → [n, subarray(buf3, n1, n5), subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%c: ⊦ m + n = n + m; from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n, subarray(buf3, n1, n5), subarray(subarray(buf2, 0, n2 + n0) + mkempbuf(n3), n0, n1)