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Proof step 63 from proof truncate-inv (previous step)

⟨if n0 ≠ 0 ∧ m2 ∈ content0 then {pageno := m2; if buf-opt.some? then buf := buf-opt.buf else buf := files2[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), files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), buf-opt = none, content = content0, (content2 upto (m2 * VFS_PAGE_SIZE + n0)).stream from m2 * VFS_PAGE_SIZE + n0 ≠ σ⊥, n0 < VFS_PAGE_SIZE, afs-cons(dirs0, files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))]), afs-cons(dirs0, files2), files2 ~ files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))], ino4 ∈ files2 ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 64:
⊦ n = n0 * VFS_PAGE_SIZE + n1 ∧ n1 ≠ 0 ∧ n1 < VFS_PAGE_SIZE → (content upto n).stream = (content.stream to (n0 + 1) * VFS_PAGE_SIZE) + σ⊥
Proof step 65:
⟨if n0 ≠ 0 ∧ m2 ∈ content0 then {pageno := m2; if buf-opt.some? then buf := buf-opt.buf else buf := files2[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), files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), buf-opt = none, content = content0, n0 < VFS_PAGE_SIZE, afs-cons(dirs0, files2), afs-cons(dirs0, files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))]), files2 ~ files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))], ino4 ∈ files2 ⊦ (content2 upto (m2 * VFS_PAGE_SIZE + n0)).stream from m2 * VFS_PAGE_SIZE + n0 = σ⊥, m2 * VFS_PAGE_SIZE + n0 = m2 * VFS_PAGE_SIZE + n0 ∧ n0 ≠ 0 ∧ n0 < VFS_PAGE_SIZE
Proof step 73:
⟨if n0 ≠ 0 ∧ m2 ∈ content0 then {pageno := m2; if buf-opt.some? then buf := buf-opt.buf else buf := files2[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), m2 * VFS_PAGE_SIZE + n0 = m2 * VFS_PAGE_SIZE + n0, files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), buf-opt = none, content = content0, n0 < VFS_PAGE_SIZE, afs-cons(dirs0, files2), afs-cons(dirs0, files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))]), files2 ~ files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content2 upto (m2 * VFS_PAGE_SIZE + n0))], ino4 ∈ files2 ⊦ (content2.stream to (m2 + 1) * VFS_PAGE_SIZE) + σ⊥ from m2 * VFS_PAGE_SIZE + n0 = σ⊥, n0 = 0

Rule argument

spec: afs-stream name: content-trunc seq: ⊦ n = n0 * VFS_PAGE_SIZE + n1 ∧ n1 ≠ 0 ∧ n1 < VFS_PAGE_SIZE → (content upto n).stream = (content.stream to (n0 + 1) * VFS_PAGE_SIZE) + σ⊥ subst: [n0, n1, n, content] → [m2, n0, m2 * VFS_PAGE_SIZE + n0, content2] flags: <all paths>

Additional information

Used terms: m2, n0, m2 * VFS_PAGE_SIZE + n0, content2