[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 15 from proof write-inv-post (previous step)

files1 -- ino = files0 -- ino, files0[ino] = mkfile(m1, md1, n2, content0), files1[ino] = mkfile(m1, md1, # links(ino, dirs), content1), content0.stream from m1 = σ⊥, afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1 ⊦ # splice(buf, 0, (content0.stream to m1).buf, n0, n) = # ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n) to (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1)).buf ∧ (∀ n1. n1 < # splice(buf, 0, (content0.stream to m1).buf, n0, n) → splice(buf, 0, (content0.stream to m1).buf, n0, n)[n1] = ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n) to (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1)).buf[n1])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
files1 -- ino = files0 -- ino, files1[ino] = mkfile(m1, md1, # links(ino, dirs), content1), files0[ino] = mkfile(m1, md1, n2, content0), content0.stream from m1 = σ⊥, afs-cons(dirs, files1), ino ∈ files0, ino ∈ files1, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf ⊦ # splice(buf, 0, (content0.stream to m1).buf, n0, n) = (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1) ∧ ( n3 < # splice(buf, 0, (content0.stream to m1).buf, n0, n) → splice(buf, 0, (content0.stream to m1).buf, n0, n)[n3] = ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n))[n3])

Rule argument

Additional information

Used simplifier rules: m < n, ¬ # σ0 < ⌜n⌝ ⊦ (σ0 to n)[m] = σ0[m]; (s)%finp(σ), ⌜n⌝ < # σ ⊦ σ.buf[n] = σ[n]; (s)% ⊦ m < n → n ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ ⌜m⌝ .n = m; (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf% ⊦ ⌜m⌝ + ∞ = ∞; (s)% ⊦ #(content.stream from n) = ∞; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ ¬ ∞ < n∞; (s)% ⊦ # content.stream = ∞; (s)% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)%finp(σ) ⊦ # σ.buf = (# σ).n; (s)% ⊦ finp(σ0 to m); (s)