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Proof step 48 from proof truncate-same-iso (previous step)

VFS_PAGE_SIZE = n0 + m % VFS_PAGE_SIZE, files0[ino0] = mkfile(m, md, # links(ino0, dirs), content), afs-cons(dirs, files0), m / VFS_PAGE_SIZE ∈ content, ino0 ∈ files0 ⊦ m % VFS_PAGE_SIZE = 0, content.stream to m = (content.stream to m / VFS_PAGE_SIZE * VFS_PAGE_SIZE) + ⌜fill(content[m / VFS_PAGE_SIZE], zero, m % VFS_PAGE_SIZE, n0)⌝ + (content.stream from VFS_PAGE_SIZE * m / VFS_PAGE_SIZE + VFS_PAGE_SIZE) to m

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 49:
⊦ n + m = # buf → ⌜fill(buf, zero, n, m)⌝ = (⌜buf⌝ to n) + (σ⊥ to m)
Proof step 50:
m % VFS_PAGE_SIZE + n0 = # content[m / VFS_PAGE_SIZE], VFS_PAGE_SIZE = n0 + m % VFS_PAGE_SIZE, files0[ino0] = mkfile(m, md, # links(ino0, dirs), content), afs-cons(dirs, files0), ino0 ∈ files0, m / VFS_PAGE_SIZE ∈ content ⊦ m % VFS_PAGE_SIZE = 0, content.stream to m = (content.stream to m / VFS_PAGE_SIZE * VFS_PAGE_SIZE) + ((⌜content[m / VFS_PAGE_SIZE]⌝ to m % VFS_PAGE_SIZE) + (σ⊥ to n0)) + (content.stream from VFS_PAGE_SIZE * m / VFS_PAGE_SIZE + VFS_PAGE_SIZE) to m

Rule argument

spec: afs-stream name: fill-stream-append seq: ⊦ n + m = # buf → ⌜fill(buf, zero, n, m)⌝ = (⌜buf⌝ to n) + (σ⊥ to m) subst: [m, n, buf] → [n0, m % VFS_PAGE_SIZE, content[m / VFS_PAGE_SIZE]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ files ∧ files[ino] = mkfile(m, md, n0, content) ∧ n ∈ content → # content[n] = VFS_PAGE_SIZE; (s)%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n0, m % VFS_PAGE_SIZE, content[m / VFS_PAGE_SIZE]