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Proof step 68 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⊥, n4 + n0 < m1, n0 < m1, afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, n0 + n ≤ m1, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1 ⊦ (n ≤ n4 ⊃ getbyte(n4 + n0, content0);buf[n4]) = ((⌜buf⌝ to n) + (content0.stream from n0 + n))[n4], n = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 69:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 70:
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⊥, n0 < m1, n4 + n0 < m1, afs-cons(dirs, files1), ino ∈ files0, ino ∈ files1, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf, n0 + n ≤ m1 ⊦ (n ≤ n4 ⊃ getbyte(n4 + n0, content0);buf[n4]) = ((⌜buf⌝ to n) + (content0.stream from n0 + n))[n4], n = 0, ⌜n4⌝ < #(⌜buf⌝ 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] → [n4, content0.stream from n0 + n, ⌜buf⌝ to n] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)%m < n, ¬ # σ0 < ⌜n⌝ ⊦ (σ0 to n)[m] = σ0[m]; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ n < # buf → ⌜buf⌝ [n] = buf[n]; (s) Used terms: n4, content0.stream from n0 + n, ⌜buf⌝ to n