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Proof step 123 from proof truncate-inv (previous step)

files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), n0 < VFS_PAGE_SIZE, afs-cons(dirs0, files2), afs-cons(dirs0, files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0 upto (m2 * VFS_PAGE_SIZE + n0))]), files2 ~ files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0 upto (m2 * VFS_PAGE_SIZE + n0))], ino4 ∈ files2 ⊦ (((content0.stream to VFS_PAGE_SIZE * m2) + (σ⊥ to VFS_PAGE_SIZE)) + σ⊥)[(m2 * VFS_PAGE_SIZE + n0) + n] = zero, n0 = 0, m2 ∈ content0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 124:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 125:
files2[ino4] = mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0), n0 < VFS_PAGE_SIZE, ⌜(m2 * VFS_PAGE_SIZE + n0) + n⌝ < #((content0.stream to VFS_PAGE_SIZE * m2) + (σ⊥ to VFS_PAGE_SIZE)), afs-cons(dirs0, files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0 upto (m2 * VFS_PAGE_SIZE + n0))]), afs-cons(dirs0, files2), files2 ~ files2[ino4, mkfile(m2 * VFS_PAGE_SIZE + n0, md, # links(ino4, dirs0), content0 upto (m2 * VFS_PAGE_SIZE + n0))], ino4 ∈ files2 ⊦ ((content0.stream to VFS_PAGE_SIZE * m2) + (σ⊥ to VFS_PAGE_SIZE))[(m2 * VFS_PAGE_SIZE + n0) + n] = zero, n0 = 0, m2 ∈ content0

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [(m2 * VFS_PAGE_SIZE + n0) + n, σ⊥, (content0.stream to VFS_PAGE_SIZE * m2) + (σ⊥ to VFS_PAGE_SIZE)] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ2[m - (# σ0).n]; (s)% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ # σ⊥ = ∞; (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]% ⊦ ⌜m⌝ ≠ ∞; (s)%n∞ ≠ ∞ ⊦ (⌜m⌝ + n∞).n = m + n∞.n; (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%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m + n - n + n1 = m - n1; (s)% ⊦ (σ⊥)[n] = zero; (s) Used terms: (m2 * VFS_PAGE_SIZE + n0) + n, σ⊥, (content0.stream to VFS_PAGE_SIZE * m2) + (σ⊥ to VFS_PAGE_SIZE)