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Proof step 7 from proof stream-split-partial-lastpage (previous step)

n < VFS_PAGE_SIZE, n0 < n + m * VFS_PAGE_SIZE, n0 / VFS_PAGE_SIZE ∈ content ⊦ content[n0 / VFS_PAGE_SIZE][n0 % VFS_PAGE_SIZE] = (⌜(λ n0. (n0 / VFS_PAGE_SIZE ∈ content ⊃ content[n0 / VFS_PAGE_SIZE];zeropage)[n0 % VFS_PAGE_SIZE]) to m * VFS_PAGE_SIZE⌝ + (σ⊥ to n))[n0], m ∈ content

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m]
Proof step 9:
n0 < n + m * VFS_PAGE_SIZE, n < VFS_PAGE_SIZE, n0 / VFS_PAGE_SIZE ∈ content ⊦ content[n0 / VFS_PAGE_SIZE][n0 % VFS_PAGE_SIZE] = (⌜(λ n0. (n0 / VFS_PAGE_SIZE ∈ content ⊃ content[n0 / VFS_PAGE_SIZE];zeropage)[n0 % VFS_PAGE_SIZE]) to m * VFS_PAGE_SIZE⌝ + (σ⊥ to n))[n0], ⌜n0⌝ < # ⌜(λ n0. (n0 / VFS_PAGE_SIZE ∈ content ⊃ content[n0 / VFS_PAGE_SIZE];zeropage)[n0 % VFS_PAGE_SIZE]) to m * VFS_PAGE_SIZE⌝ , m ∈ content

Rule argument

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

Additional information

Used simplifier rules: ⊦ ⌜dx⌝ [n] = dx[n]; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ # ⌜nf0 to n⌝ = ⌜n⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%m < n ⊦ (nf0 to n)[m] = nf0(m); (s) Used terms: n0, σ⊥ to n, ⌜ (λ n0. (n0 / VFS_PAGE_SIZE ∈ content ⊃ content[n0 / VFS_PAGE_SIZE] ;zeropage ) [n0 % VFS_PAGE_SIZE ]) to m * VFS_PAGE_SIZE ⌝