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Proof step 74 from proof write-inv-rec (previous step)

n0 + n = m0 * VFS_PAGE_SIZE + n2, VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE = n3 + n0 + n, files1 -- ino = files0 -- ino, VFS_PAGE_SIZE = n4 + n2 + m, files0[ino] = mkfile(m2, md0, m4, content), files1[ino] = mkfile(m2, md0, # links(ino, dirs), content0), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino ∈ files0, ino ∈ files1, m0 * VFS_PAGE_SIZE ≤ n3 + n0 + n, m0 * VFS_PAGE_SIZE ≤ # buf + n0, m + n2 ≤ VFS_PAGE_SIZE, n0 + n ≤ n1, n1 ≤ # buf + n0, n ≤ # buf, n + m ≤ # buf, m0 * VFS_PAGE_SIZE ≤ n + n0, m0 * VFS_PAGE_SIZE ≤ n1, n2 + m ≤ VFS_PAGE_SIZE ⊦ n4 + m = 0, ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + (((content.stream to n0) + (⌜buf⌝ to n) to n0 + n) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) + ((content.stream from (n0 + n) + m) to n4) + (content.stream from (n0 + n) + n3) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m), #(((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n) from m0 * VFS_PAGE_SIZE) < ⌜VFS_PAGE_SIZE⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ # σ0 = ∞ → #(σ0 from n) = ∞
Proof step 76:
#(((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n)) = ∞, VFS_PAGE_SIZE * m0 + VFS_PAGE_SIZE = n3 + n0 + n, n0 + n = m0 * VFS_PAGE_SIZE + n2, files1 -- ino = files0 -- ino, VFS_PAGE_SIZE = n4 + n2 + m, files1[ino] = mkfile(m2, md0, # links(ino, dirs), content0), files0[ino] = mkfile(m2, md0, m4, content), n2 < VFS_PAGE_SIZE, afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, n2 + m ≤ VFS_PAGE_SIZE, m0 * VFS_PAGE_SIZE ≤ n1, m0 * VFS_PAGE_SIZE ≤ n + n0, n + m ≤ # buf, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1, m + n2 ≤ VFS_PAGE_SIZE, m0 * VFS_PAGE_SIZE ≤ # buf + n0, m0 * VFS_PAGE_SIZE ≤ n3 + n0 + n ⊦ ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + (((content.stream to n0) + (⌜buf⌝ to n) to n0 + n) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) + ((content.stream from (n0 + n) + m) to n4) + (content.stream from (n0 + n) + n3) = (content.stream to n0) + (⌜buf⌝ to n + m) + (content.stream from n0 + n + m), n4 + m = 0, ∞ < ⌜VFS_PAGE_SIZE⌝

Rule argument

spec: stream, inst: buffer-inf name: inf-size-from seq: ⊦ # σ0 = ∞ → #(σ0 from n) = ∞ subst: [n, σ0] → [m0 * VFS_PAGE_SIZE, ((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ #(content.stream from n) = ∞; (s)% ⊦ ⌜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%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf] Used terms: m0 * VFS_PAGE_SIZE, ((content.stream to n0) + (⌜buf⌝ to n)) + (content.stream from n0 + n)