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

# buf0 = # buf1, # buf1 = n4 + n + min(n3, n6, n1), n0 + n = m0 * VFS_PAGE_SIZE + n5, VFS_PAGE_SIZE = n6 + n5, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), n5 < VFS_PAGE_SIZE, ⌜n7⌝ < #(⌜buf0⌝ from n + min(n3, n6, n1)), afs-cons(dirs, files), ino ∈ files, min(n3, n6, n1) ≤ n6, n5 + min(n3, n6, n1) ≤ VFS_PAGE_SIZE, n + min(n3, n6, n1) ≤ # buf1, n3 + n ≤ # buf1, min(n3, n6, n1) + n ≤ # buf1 ⊦ (((content.stream from n0) to n) + ((content.stream from n0 + n) to min(n3, n6, n1)) + (⌜buf0⌝ from n + min(n3, n6, n1)) )[n7 + min(n3, n6, n1) + n ] = (⌜buf0⌝ from min(n3, n6, n1) + n)[n7], n6 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 47:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 48:
# buf1 = n4 + n + min(n3, n6, n1), # buf0 = # buf1, n0 + n = m0 * VFS_PAGE_SIZE + n5, VFS_PAGE_SIZE = n6 + n5, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), ⌜n7⌝ < #(⌜buf0⌝ from n + min(n3, n6, n1)), n5 < VFS_PAGE_SIZE, afs-cons(dirs, files), ino ∈ files, min(n3, n6, n1) + n ≤ # buf1, n3 + n ≤ # buf1, n + min(n3, n6, n1) ≤ # buf1, n5 + min(n3, n6, n1) ≤ VFS_PAGE_SIZE, min(n3, n6, n1) ≤ n6 ⊦ (((content.stream from n0) to n) + ((content.stream from n0 + n) to min(n3, n6, n1)) + (⌜buf0⌝ from n + min(n3, n6, n1)) )[n7 + min(n3, n6, n1) + n ] = (⌜buf0⌝ from min(n3, n6, n1) + n)[n7], n6 = 0, ⌜n7 + min(n3, n6, n1) + n⌝ < #((content.stream from n0) to n)

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [n7 + min(n3, n6, n1) + n, ((content.stream from n0 + n) to min(n3, n6, n1)) + (⌜buf0⌝ from n + min(n3, n6, n1)), (content.stream from n0) to n] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ #(content.stream from n) = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ ⌜n1⌝ + ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ n1 + n + m < n0; (s)% ⊦ ¬ m + n < m; (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: n7 + min(n3, n6, n1) + n, ((content.stream from n0 + n) to min(n3, n6, n1)) + (⌜buf0⌝ from n + min(n3, n6, n1)), (content.stream from n0) to n