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Proof step 811 from proof refine-truncate-buff-none-uncached (previous step)

VFS_PAGE_SIZE = n0 + m2 % VFS_PAGE_SIZE, cache_files[m1] = mkfile(m3, md1, # links(m1, cache_dirs), content1), (∅)[m1] = mkientry(boolvar0, mkinode(m4, md, boolvar, m5, m6, m7)), # (content1 ⊕ mkcontent(m1, cache_pcache0))[m2 / VFS_PAGE_SIZE] < VFS_PAGE_SIZE, afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache0) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m2 / VFS_PAGE_SIZE ∈ content1, m1 ∈ cache_files, m2 ≤ m, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data? ∧ # cache_pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ cache_files ∧ (¬ cache_pcache0[key].dirty → key.part ∈ cache_files[key.ino].content ∧ cache_pcache0[key].page = cache_files[key.ino].content[key.part]), ∀ key. key ∈ cache_pcache0 → key.data? ⊦ m2 % VFS_PAGE_SIZE = 0, m1 ∈ cache_icache, datakey(m1, m2 / VFS_PAGE_SIZE) ∈ cache_pcache0

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 812:
n ≠ 0, m / n * n ≤ m ⊦ m % n = m - m / n * n
Proof step 813:
VFS_PAGE_SIZE = n0 + (m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE), (∅)[m1] = mkientry(boolvar0, mkinode(m4, md, boolvar, m5, m6, m7)), cache_files[m1] = mkfile(m3, md1, # links(m1, cache_dirs), content1), # (content1 ⊕ mkcontent(m1, cache_pcache0))[m2 / VFS_PAGE_SIZE] < VFS_PAGE_SIZE, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache0) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m1 ∈ cache_files, m2 / VFS_PAGE_SIZE ∈ content1, m2 ≤ m, m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE ≤ m2, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data?, ∀ key. key ∈ cache_pcache0 → key.data? ∧ # cache_pcache0[key].page = VFS_PAGE_SIZE ∧ key.ino ∈ cache_files ∧ (¬ cache_pcache0[key].dirty → key.part ∈ cache_files[key.ino].content ∧ cache_pcache0[key].page = cache_files[key.ino].content[key.part]) ⊦ m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE = 0, VFS_PAGE_SIZE = 0, datakey(m1, m2 / VFS_PAGE_SIZE) ∈ cache_pcache0, m1 ∈ cache_icache

Rule argument

spec: nat-div name: rax seq: n ≠ 0, m / n * n ≤ m ⊦ m % n = m - m / n * n subst: [n, m] → [VFS_PAGE_SIZE, m2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ VFS_PAGE_SIZE ≠ 0; (s)%n ≠ 0 ⊦ m / n * n ≤ m; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: VFS_PAGE_SIZE, m2