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Proof step 622 from proof refine-truncate-buff-some (previous step)

# buf-opt0.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n1 + m2 % VFS_PAGE_SIZE, cache_files[m4] = mkfile(m1, md2, # links(m4, cache_dirs), content1), cache_icache[m4] = mkientry(boolvar, mkinode(m0, md0, false, # links(m4, cache_dirs), m5, m6)), buf-opt0.some?, 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), pcache-cons(cache_pcache0, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m4 ∈ cache_icache, m4 ∈ cache_files, m2 ≤ m9, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data? ⊦ m2 % VFS_PAGE_SIZE = 0, ( n * VFS_PAGE_SIZE < m2 ∧ n ∈ (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)] ↔ n ∈ (content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)] upto m2) ⊕ mkcontent (m4, truncate(m4, m2, cache_pcache0)[datakey(m4, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1))])) ∧ ( n * VFS_PAGE_SIZE < m2 ∧ n ∈ (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)] → (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)][n] = ( (content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1)] upto m2) ⊕ mkcontent (m4, truncate(m4, m2, cache_pcache0)[datakey(m4, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 % VFS_PAGE_SIZE, n1))]) )[n ])

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 623:
n ≠ 0, m / n * n ≤ m ⊦ m % n = m - m / n * n
Proof step 624:
# buf-opt0.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n1 + (m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE), cache_icache[m4] = mkientry(boolvar, mkinode(m0, md0, false, # links(m4, cache_dirs), m5, m6)), cache_files[m4] = mkfile(m1, md2, # links(m4, cache_dirs), content1), buf-opt0.some?, 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), pcache-cons(cache_pcache0, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache0), m4 ∈ cache_files, m4 ∈ cache_icache, m2 ≤ m9, m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE ≤ m2, cache_sync → synced(cache_files, cache_icache, cache_pcache0), ∀ key. key ∈ cache_pcache0 → key.data? ⊦ m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE = 0, VFS_PAGE_SIZE = 0, ( n * VFS_PAGE_SIZE < m2 ∧ n ∈ (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)] ↔ n ∈ (content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)] upto m2) ⊕ mkcontent (m4, truncate(m4, m2, cache_pcache0) [datakey(m4, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)) ])) ∧ ( n * VFS_PAGE_SIZE < m2 ∧ n ∈ (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)] → (content1 ⊕ mkcontent(m4, cache_pcache0))[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)][n] = ( (content1[m2 / VFS_PAGE_SIZE, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)] upto m2) ⊕ mkcontent (m4, truncate(m4, m2, cache_pcache0) [datakey(m4, m2 / VFS_PAGE_SIZE), mkpentry(false, fill(buf-opt0.buf, zero, m2 - m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE, n1)) ]) )[n ])

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