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Proof step 55 from proof truncate-inv (previous step)

⟨afs_inode_sync#(mkinode(ino4, md, false, # links(ino4, dirs), 0, m0); dirs, files1)⟩ (dirs = dirs0 ∧ files1 = files0), ⟨if m0 % VFS_PAGE_SIZE ≠ 0 ∧ m0 / VFS_PAGE_SIZE ∈ content0 then {pageno := m0 / VFS_PAGE_SIZE; if buf-opt.some? then buf := buf-opt.buf else buf := files[ino4].content[pageno]; buf := fill(buf, zero, m0 % VFS_PAGE_SIZE, VFS_PAGE_SIZE - m0 % VFS_PAGE_SIZE); content[pageno] := buf; buf-opt := some(buf)}⟩ (pageno = pageno1 ∧ buf = buf0 ∧ content = content2 ∧ buf-opt = buf-opt0), files[ino4] = mkfile(m0, md, # links(ino4, dirs), content0), files0[ino4] = mkfile(m, md0, m1, content1), buf-opt = none, content = content0, files1 = files[ino4, mkfile(m0, md, # links(ino4, dirs), content2 upto m0)], content1.stream from m0 ≠ σ⊥, afs-cons(dirs0, files0), afs-cons(dirs, files), files ~ files0, ino4 ∈ files ⊦

User applied rule insert elim lemma

And obtained the following new goals

Proof step 56:
⟨afs_inode_sync#(mkinode(ino4, md, false, # links(ino4, dirs), 0, m0); dirs, files1)⟩ (dirs = dirs0 ∧ files1 = files0), ⟨if n0 ≠ 0 ∧ m2 ∈ content0 then {pageno := m2; if buf-opt.some? then buf := buf-opt.buf else buf := files[ino4].content[pageno]; buf := fill(buf, zero, n0, VFS_PAGE_SIZE - n0); content[pageno] := buf; buf-opt := some(buf)}⟩ (pageno = pageno1 ∧ buf = buf0 ∧ content = content2 ∧ buf-opt = buf-opt0), files0[ino4] = mkfile(m, md0, m1, content1), files[ino4] = mkfile(m0, md, # links(ino4, dirs), content0), buf-opt = none, content = content0, files1 = files[ino4, mkfile(m0, md, # links(ino4, dirs), content2 upto m0)], m0 = m2 * VFS_PAGE_SIZE + n0, n0 < VFS_PAGE_SIZE, afs-cons(dirs, files), afs-cons(dirs0, files0), files ~ files0, ino4 ∈ files ⊦ VFS_PAGE_SIZE = 0, content1.stream from m0 = σ⊥

Rule argument

spec: nat-div name: div-elim seq: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n subst: [n, m, m0, n0] → [VFS_PAGE_SIZE, m0, m8, n]

Additional information

Used simplifier rules: ⊦ VFS_PAGE_SIZE ≠ 0; (s)%n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n; (elim) Used terms: VFS_PAGE_SIZE, m0, m8, n