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Proof step 157 from proof write-prepost (previous step)

⟨afs_truncate#(n0 + m0, (n0 + m0) / VFS_PAGE_SIZE; buf-opt, inode1, isdir, err, dirs1, files1, asynced1)⟩ ( buf-opt = buf-opt0 ∧ inode1 = mkinode(m, md, false, m3, 0, n0 + m0) ∧ (isdir ↔ isdir0) ∧ err = ESUCCESS ∧ dirs1 = dirs0 ∧ files1 = files0 ∧ (asynced1 ↔ asynced0)), # links(m, dirs1) = m3, files1 -- m = files0 -- m, files[m] = mkfile(n0 + m0, md, # links(m, dirs1), 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, err = ESUCCESS, content.stream from n0 + m0 = σ⊥, inode1 = 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 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, 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, m ∈ files1, m ∈ files, 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 158:
⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1)
Proof step 159:
⟨afs_truncate#(n0 + m0, (n0 + m0) / VFS_PAGE_SIZE; buf-opt, inode1, isdir, err, dirs1, files1, asynced1)⟩ ( buf-opt = buf-opt0 ∧ inode1 = mkinode(m, md, false, m3, 0, n0 + m0) ∧ (isdir ↔ isdir0) ∧ err = ESUCCESS ∧ dirs1 = dirs0 ∧ files1 = files0 ∧ (asynced1 ↔ asynced0)), # links(m, dirs1) = m3, files1 -- m = files0 -- 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, # links(m, dirs1), content1), buf-opt = none, err = ESUCCESS, content.stream from n0 + m0 = σ⊥, inode1 = 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, 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, fd ∈ of1, m ∈ files, m ∈ files1, n0 ≤ # buf, n1 ≤ # buf, ⌊err0⌋ , err1 = ESUCCESS ∨ ⌊err1⌋ ⊦ (content.stream to m0) + ((⌜buf⌝ to ino) + (content.stream from m0 + ino) to n0) = (content.stream to m0) + (⌜buf⌝ to n0), n0 = 0, n1 = 0, #((⌜buf⌝ to ino) + (content.stream from m0 + ino)) < ⌜n0⌝ , # content.stream < ⌜m0⌝ , mode ≠ MODE_W ∧ mode ≠ MODE_RW

Rule argument

spec: stream, inst: buffer-inf name: to-append-add seq: ⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1) subst: [n1, σ2, n, σ1] → [n0, (⌜buf⌝ to ino) + (content.stream from m0 + ino), m0, content.stream] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)% ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ #(content.stream from n) = ∞; (s)% ⊦ n∞ + ∞ = ∞; (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] Used terms: n0, (⌜buf⌝ to ino) + (content.stream from m0 + ino), m0, content.stream