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Proof step 227 from proof write-prepost-partial (previous step)

⟨afs_truncate#(n0 + m0, (n0 + m0) / VFS_PAGE_SIZE; buf-opt, inode, isdir, err2, dirs1, files1, asynced)⟩ (buf-opt = buf-opt0 ∧ inode = inode0 ∧ (isdir ↔ isdir0) ∧ err2 = ESUCCESS ∧ dirs1 = dirs0 ∧ files1 = files0 ∧ (asynced ↔ asynced0)), files0 -- m = files -- m, files1 -- m = files -- m, files1[m] = mkfile(n0 + m0, md, # links(m, dirs1), content0), of1[fd] = mkfile(m, mode, m0), files0[m] = mkfile(n0 + m0, md, # links(m, dirs1), content), files[m] = mkfile(n0 + m0, md, m3, content1), buf-opt = none, err2 = ESUCCESS, content.stream from n0 + m0 = σ⊥, inode = mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), inode0 = mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), content0.stream to n0 + m0 = content.stream to n0 + m0, n0 ≠ 0, n1 ≠ 0, (content.stream to m0) + (⌜buf⌝ to ino) + (content.stream from m0 + ino) to n0 + m0 ≠ (content.stream to n0 + m0 to m0) + (⌜buf⌝ to n0), n0 < ino, afs-cons(dirs1, files1), of-cons(of1, dirs1, files1), write-inv(m0, m0 + n1, mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), buf, 0, dirs0, files0, files0), write-inv(m0, m0 + n1, mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), buf, ino, dirs0, files0, files), files1 ~ files0, m ∈ files, m ∈ files1, fd ∈ of1, n1 ≤ # buf, n0 ≤ # buf, ⌊err0⌋ , err1 = ESUCCESS ∨ ⌊err1⌋ ⊦ mode ≠ MODE_W ∧ mode ≠ MODE_RW

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 228:
¬ # σ0 < ⌜n⌝ ⊦ m ≤ n → σ0 to n to m = σ0 to m
Proof step 229:
⟨afs_truncate#(n0 + m0, (n0 + m0) / VFS_PAGE_SIZE; buf-opt, inode, isdir, err2, dirs1, files1, asynced)⟩ (buf-opt = buf-opt0 ∧ inode = inode0 ∧ (isdir ↔ isdir0) ∧ err2 = ESUCCESS ∧ dirs1 = dirs0 ∧ files1 = files0 ∧ (asynced ↔ asynced0)), files1 -- m = files -- m, files0 -- m = files -- m, files[m] = mkfile(n0 + m0, md, m3, content1), files0[m] = mkfile(n0 + m0, md, # links(m, dirs1), content), of1[fd] = mkfile(m, mode, m0), files1[m] = mkfile(n0 + m0, md, # links(m, dirs1), content0), buf-opt = none, err2 = ESUCCESS, content.stream from n0 + m0 = σ⊥, inode = mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), inode0 = mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), content0.stream to n0 + m0 = content.stream to n0 + m0, n0 < ino, afs-cons(dirs1, files1), of-cons(of1, dirs1, files1), write-inv(m0, m0 + n1, mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), buf, ino, dirs0, files0, files), write-inv(m0, m0 + n1, mkinode(m, md, false, # links(m, dirs1), 0, n0 + m0), buf, 0, dirs0, files0, files0), files1 ~ files0, fd ∈ of1, m ∈ files1, m ∈ files, n0 ≤ # buf, n1 ≤ # buf, m0 ≤ n0 + m0, ⌊err0⌋ , err1 = ESUCCESS ∨ ⌊err1⌋ ⊦ n0 = 0, n1 = 0, (content.stream to m0) + (⌜buf⌝ to ino) + (content.stream from m0 + ino) to n0 + m0 = (content.stream to m0) + (⌜buf⌝ to n0), # content.stream < ⌜n0 + m0⌝ , mode ≠ MODE_W ∧ mode ≠ MODE_RW

Rule argument

spec: stream, inst: buffer-inf name: to-to-leq seq: ¬ # σ0 < ⌜n⌝ ⊦ m ≤ n → σ0 to n to m = σ0 to m subst: [m, n, σ0] → [m0, n0 + m0, content.stream] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # content.stream = ∞; (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% ⊦ ¬ ∞ < n∞; (s)% ⊦ m ≤ m + n; (s) Used terms: m0, n0 + m0, content.stream