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Proof step 13 from proof write-inv-post (previous step)

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) = ((content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n) to (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1)).buf

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ buf = buf0 ↔ # buf = # buf0 ∧ (∀ n. n < # buf → buf[n] = buf0[n])
Proof step 15:
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])

Rule argument

spec: array, inst: buffer name: Extension seq: ⊦ buf = buf0 ↔ # buf = # buf0 ∧ (∀ n. n < # buf → buf[n] = buf0[n]) subst: [buf0, buf] → [( (content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n) to (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1) ).buf, splice(buf, 0, (content0.stream to m1).buf, n0, n)] flags: <all paths>

Additional information

Used terms: ( (content0.stream to n0) + (⌜buf⌝ to n) + (content0.stream from n0 + n) to (n ≠ 0 ∧ m1 < n0 + n ⊃ n0 + n;m1) ).buf, splice(buf, 0, (content0.stream to m1).buf, n0, n)