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Proof step 152 from proof inv-truncate-buff-some (previous step)

# buf-opt.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m2 % VFS_PAGE_SIZE, files0[m3] = mkfile(m1, md1, # links(m3, dirs), content0), buf-opt.some?, afs-cons(dirs, files0), afs-cons (dirs, files0[m3, mkfile(m, md, # links(m3, dirs), content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)]), synced(dirs, icache, pcache0), m3 ∈ files0, m2 ≤ m, asynced1 → synced(files0, icache, pcache0), ∀ key. key ∈ pcache0 → key.data?, ∀ key0. key0 ∈ pcache0 → key0.data? ∧ # pcache0[key0].page = VFS_PAGE_SIZE ∧ key0.ino ∈ files0 ∧ (¬ pcache0[key0].dirty → key0.part ∈ files0[key0.ino].content ∧ pcache0[key0].page = files0[key0.ino].content[key0.part]) ⊦ m2 % VFS_PAGE_SIZE = 0, # (content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)[m2 / VFS_PAGE_SIZE] = VFS_PAGE_SIZE ∧ m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE < m2 ∧ fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0) = (content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)[m2 / VFS_PAGE_SIZE]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 153:
⊦ m * VFS_PAGE_SIZE < n ∧ m ∈ content → (content upto n)[m] = content[m]
Proof step 154:
# buf-opt.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m2 % VFS_PAGE_SIZE, files0[m3] = mkfile(m1, md1, # links(m3, dirs), content0), buf-opt.some?, afs-cons (dirs, files0[m3, mkfile(m, md, # links(m3, dirs), content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)]), afs-cons(dirs, files0), synced(dirs, icache, pcache0), m3 ∈ files0, m2 ≤ m, asynced1 → synced(files0, icache, pcache0), ∀ key0. key0 ∈ pcache0 → key0.data? ∧ # pcache0[key0].page = VFS_PAGE_SIZE ∧ key0.ino ∈ files0 ∧ (¬ pcache0[key0].dirty → key0.part ∈ files0[key0.ino].content ∧ pcache0[key0].page = files0[key0.ino].content[key0.part]), ∀ key. key ∈ pcache0 → key.data? ⊦ m2 % VFS_PAGE_SIZE = 0, # (content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)[m2 / VFS_PAGE_SIZE] = VFS_PAGE_SIZE ∧ m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE < m2 ∧ fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0) = (content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)] upto m2)[m2 / VFS_PAGE_SIZE], m2 / VFS_PAGE_SIZE * VFS_PAGE_SIZE < m2 ∧ m2 / VFS_PAGE_SIZE ∈ content0[m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0)]

Rule argument

spec: afs-base name: at-upto seq: ⊦ m * VFS_PAGE_SIZE < n ∧ m ∈ content → (content upto n)[m] = content[m] subst: [m, n, content] → [m2 / VFS_PAGE_SIZE, m2, content0 [m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0) ]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ content[n, buf]; (s)% ⊦ content[n, buf][n] = buf; (s)% ⊦ # fill(buf, d, m, n) = # buf; (s) Used terms: m2 / VFS_PAGE_SIZE, m2, content0 [m2 / VFS_PAGE_SIZE, fill(buf-opt.buf, zero, m2 % VFS_PAGE_SIZE, n0) ]