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Proof step 2 from proof refine-fsync (previous step)

afs_asynced ↔ cache_asynced, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, afs-cons(cache_dirs, cache_files), afs-cons(afs_dirs, afs_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), valid-file-inode(inode, afs_dirs, afs_files), cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ⟪cache_fsync#(inode, asynced, cache_sync; cache_dirs, cache_files, cache_icache, cache_pcache, cache_asynced; err)⟫ ⟨afs_fsync#(inode, asynced, afs_dirs, afs_files; ; afs_asynced, err0)⟩ ((afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)) ∧ err = err0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
inode.nlink = ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[inode.ino].nlink, inode.nsubdirs = 0, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ 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_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), inode.ino ∈ cache_files, ¬ inode.directory, cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ⟪cache_fsync#(inode, asynced, cache_sync; cache_dirs, cache_files, cache_icache, cache_pcache, cache_asynced; err)⟫ ⟨afs_fsync#(inode, asynced, afs_dirs, afs_files; ; afs_asynced, err0)⟩ (afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ err = err0)

Rule argument

Additional information

Used simplifier rules: ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ valid-file-ino(ino, dirs, files) ↔ ino ∈ files; (s)%⊦ valid-file-inode(inode, dirs, files) ↔ valid-file-ino(inode.ino, dirs, files) ∧ inode.nlink = files[inode.ino].nlink ∧ inode.nsubdirs = 0 ∧ ¬ inode.directory; (s)