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Proof step 65 from proof fsync-inode-others-unchanged (previous step)

icache0[m] = mkientry(true, mkinode(m0, md, false, m2, 0, m3)), files0[m0] = mkfile(m1, md0, m2, content), icache-cons(icache0, dirs, files0), m ∈ icache0, m0 ∈ files0 ⊦ m0 = m ∧ (ino ≠ m ∧ ino ∈ files0 → (files0[m, (files0[m].meta:= md).size:= m3] ⊕ icache0)[ino] = (files0 ⊕ icache0)[ino])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 66:
⊦ icache-cons(icache, dirs, files) ↔ (∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ valid-file-inode(icache[ino].inode, dirs, files) ;valid-ino(ino, dirs, files) ∧ valid-inode(icache[ino].inode, dirs, files) ))
Proof step 67:
files0[m0] = mkfile(m1, md0, m2, content), icache0[m] = mkientry(true, mkinode(m0, md, false, m2, 0, m3)), m0 ∈ files0, m ∈ icache0, ∀ ino. ino ∈ icache0 → icache0[ino].inode.ino = ino ∧ (icache0[ino].dirty ⊃ valid-file-inode(icache0[ino].inode, dirs, files0) ;valid-ino(ino, dirs, files0) ∧ valid-inode(icache0[ino].inode, dirs, files0) ) ⊦ m0 = m ∧ (ino ≠ m ∧ ino ∈ files0 → (files0[m, (files0[m].meta:= md).size:= m3] ⊕ icache0)[ino] = (files0 ⊕ icache0)[ino])

Rule argument

spec: cache-base name: icache-cons seq: ⊦ icache-cons(icache, dirs, files) ↔ (∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ valid-file-inode(icache[ino].inode, dirs, files) ; valid-ino(ino, dirs, files) ∧ valid-inode(icache[ino].inode, dirs, files) )) subst: [files, dirs, icache] → [files0, dirs, icache0] flags: <all paths>

Additional information

Used terms: files0, dirs, icache0