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Proof step 16 from proof read-inv-rec (previous step)

# buf0 = # buf1, # buf1 = n4 + n + min(n3, n2, n1), VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), afs-cons(dirs, files), ino ∈ files, (n0 + n) % VFS_PAGE_SIZE + min(n3, n2, n1) ≤ VFS_PAGE_SIZE, n + min(n3, n2, n1) ≤ # buf1, n3 + n ≤ # buf1 ⊦ (((content.stream from n0) to n) + (⌜buf0⌝ from n) to n) + ((((content.stream from (n0 + n) / VFS_PAGE_SIZE * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from (n0 + n) % VFS_PAGE_SIZE) to min(n3, n2, n1)) + (((content.stream from n0) to n) + (⌜buf0⌝ from n) from n + min(n3, n2, n1)) = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ ¬ ⌜m⌝ < # σ1 → σ1 + σ2 from m = σ2 from (m - (# σ1).n)
Proof step 18:
# buf1 = n4 + n + min(n3, n2, n1), # buf0 = # buf1, VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), afs-cons(dirs, files), ino ∈ files, n3 + n ≤ # buf1, n + min(n3, n2, n1) ≤ # buf1, (n0 + n) % VFS_PAGE_SIZE + min(n3, n2, n1) ≤ VFS_PAGE_SIZE ⊦ (((content.stream from n0) to n) + (⌜buf0⌝ from n) to n) + ((((content.stream from (n0 + n) / VFS_PAGE_SIZE * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from (n0 + n) % VFS_PAGE_SIZE) to min(n3, n2, n1)) + (⌜buf0⌝ from n from (n + min(n3, n2, n1) - (#((content.stream from n0) to n)).n)) = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n), ⌜n + min(n3, n2, n1)⌝ < #((content.stream from n0) to n)

Rule argument

spec: stream, inst: buffer-inf name: append-from-ge-02 seq: ⊦ ¬ ⌜m⌝ < # σ1 → σ1 + σ2 from m = σ2 from (m - (# σ1).n) subst: [m, σ2, σ1] → [n + min(n3, n2, n1), ⌜buf0⌝ from n, (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)% ⊦ ⌜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: n + min(n3, n2, n1), ⌜buf0⌝ from n, (content.stream from n0) to n