[Imprint][Data Privacy Statement][E-Mail]

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Proof step 5 from proof content-trunc (previous step)

n1 < VFS_PAGE_SIZE ⊦ n1 = 0, # ⌜λ n. (n / VFS_PAGE_SIZE * VFS_PAGE_SIZE < n0 * VFS_PAGE_SIZE + n1 ∧ n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage) [n % VFS_PAGE_SIZE ] ⌝ = # (⌜(λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage)[n % VFS_PAGE_SIZE]) to VFS_PAGE_SIZE * n0 + VFS_PAGE_SIZE⌝ + σ⊥) ∧ (∀ n. ⌜n⌝ < # ⌜λ n. (n / VFS_PAGE_SIZE * VFS_PAGE_SIZE < n0 * VFS_PAGE_SIZE + n1 ∧ n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage) [n % VFS_PAGE_SIZE ] ⌝ → ⌜λ n. (n / VFS_PAGE_SIZE * VFS_PAGE_SIZE < n0 * VFS_PAGE_SIZE + n1 ∧ n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage) [n % VFS_PAGE_SIZE ] ⌝ [n ] = ( ⌜(λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage)[n % VFS_PAGE_SIZE]) to VFS_PAGE_SIZE * n0 + VFS_PAGE_SIZE⌝ + σ⊥ )[n ])

User applied rule case distinction

And obtained the following new goals

Proof step 6:
n1 < VFS_PAGE_SIZE ⊦ n1 = 0, ∀ n. ⌜n⌝ < # ⌜λ n. (n / VFS_PAGE_SIZE * VFS_PAGE_SIZE < n0 * VFS_PAGE_SIZE + n1 ∧ n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage) [n % VFS_PAGE_SIZE ] ⌝ → ⌜λ n. (n / VFS_PAGE_SIZE * VFS_PAGE_SIZE < n0 * VFS_PAGE_SIZE + n1 ∧ n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage) [n % VFS_PAGE_SIZE ] ⌝ [n ] = (⌜(λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage)[n % VFS_PAGE_SIZE]) to VFS_PAGE_SIZE * n0 + VFS_PAGE_SIZE⌝ + σ⊥ )[n ]

Rule argument

pos: right 2

Additional information

Used simplifier rules: zerostream-size: ⊦ # σ⊥ = ∞;% ⊦ # ⌜nf0⌝ = ∞; (s)% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # ⌜nf0 to n⌝ = ⌜n⌝ ; (s)% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ⌜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%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]