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Proof step 54 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⊥, m1 < # buf + n0, m1 < n0 + n, n4 < n, n0 < n1, n4 + n0 < n1, m1 < n1, n4 < # buf, afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1 ⊦ buf[n4] = ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n))[n4 + n0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 56:
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⊥, n4 < # buf, m1 < n1, n4 + n0 < n1, n0 < n1, n4 < n, m1 < n0 + n, m1 < # buf + n0, afs-cons(dirs, files1), ino ∈ files0, ino ∈ files1, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf ⊦ buf[n4] = ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n))[n4 + n0], ⌜n4 + n0⌝ < #(content0.stream to n0)

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [n4 + n0, (⌜buf⌝ to n) + (content0.stream from n0 + n), content0.stream to n0] 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 = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ ↔ 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: n4 + n0, (⌜buf⌝ to n) + (content0.stream from n0 + n), content0.stream to n0