[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 7 from proof read-inv-rec (previous step)

# buf0 = # buf1, VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), ⌜buf1⌝ = ((content.stream from n0) to n) + (⌜buf0⌝ from n), afs-cons(dirs, files), ino ∈ files, n3 + n ≤ # buf1 ⊦ ⌜copy(getpage((n0 + n) / VFS_PAGE_SIZE, content), (n0 + n) % VFS_PAGE_SIZE, buf1, n, min(n3, n2, n1))⌝ = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ n0 + m ≤ # buf0 ∧ n1 + m ≤ # buf1 → ⌜copy(buf0, n0, buf1, n1, m)⌝ = (⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + ((⌜buf1⌝ from n1 + m) to (# buf1 - n1 + m))
Proof step 9:
# buf0 = # buf1, VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), ⌜buf1⌝ = ((content.stream from n0) to n) + (⌜buf0⌝ from n), afs-cons(dirs, files), ino ∈ files, n3 + n ≤ # buf1, (n0 + n) % VFS_PAGE_SIZE + min(n3, n2, n1) ≤ # getpage((n0 + n) / VFS_PAGE_SIZE, content), n + min(n3, n2, n1) ≤ # buf1 ⊦ (⌜buf1⌝ to n) + ((⌜getpage((n0 + n) / VFS_PAGE_SIZE, content)⌝ from (n0 + n) % VFS_PAGE_SIZE) to min(n3, n2, n1)) + ((⌜buf1⌝ from n + min(n3, n2, n1)) to (# buf1 - n + min(n3, n2, n1))) = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n)

Rule argument

spec: afs-stream name: copy-stream-append seq: ⊦ n0 + m ≤ # buf0 ∧ n1 + m ≤ # buf1 → ⌜copy(buf0, n0, buf1, n1, m)⌝ = (⌜buf1⌝ to n1) + ((⌜buf0⌝ from n0) to m) + ((⌜buf1⌝ from n1 + m) to (# buf1 - n1 + m)) subst: [m, n1, buf1, n0, buf0] → [min(n3, n2, n1), n, buf1, (n0 + n) % VFS_PAGE_SIZE, getpage((n0 + n) / VFS_PAGE_SIZE, content)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ files ∧ files[ino] = mkfile(m, md, m0, content) → # getpage(n, content) = VFS_PAGE_SIZE; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n1 < min(n0, n1, n2); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ ¬ n0 < min(n0, n1, n2); (s) Used terms: min(n3, n2, n1), n, buf1, (n0 + n) % VFS_PAGE_SIZE, getpage((n0 + n) / VFS_PAGE_SIZE, content)