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Proof step 64 from proof inv-replay (previous step)

replay-log(log0, fs0, fi0 ⊖ fo0) = (ri \ ro) \ datakeys(ro, ri), fo0.inodes?, ro.inodes?, file-node-cons(fs0, ri), log-cons(log0, fs0), orphans-cons(fs0, ri, ro), replay-safe(log0, fs0, fi0 ⊖ fo0), ri-closed(ri, fs0, ro), ri-cons(ri, fs0), ri-inj(ri), ri-inj(fi0), key ∈ ri, ro ⊆ dom ri, fo0 ⊆ dom fi0 ⊦ key.data?, dups(log0), inodekey(key.ino) ∈ ri, key ∈ ro, inodekey(0) ∈ ri, file-node-cons(fs0, replay-log(log0, fs0, fi0 ⊖ fo0)) ∧ orphans-cons(fs0, replay-log(log0, fs0, fi0 ⊖ fo0), ∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-base name: dels-with-data seq: ⊦ ri ⊖ ks = (ri \ ks) \ datakeys(ks, ri) subst: [ks, ri] → [fo0, fi0] flags: <all paths>

Additional information

Used simplifier rules: at-diff-ri: ⊦ (ri \ ks)[key] = (key ∈ ks ⊃ (∅)[key];ri[key]);%⊦ file-node-cons(fs, ri) → ( file-node-cons(fs0, ri0) ↔ (∀ ino. ( inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → fs[ri[inodekey(ino)]].nsubdirs = 0) → inodekey(ino) ∈ ri0 ∧ ¬ fs0[ri0[inodekey(ino)]].directory → fs0[ri0[inodekey(ino)]].nsubdirs = 0)); (ss)% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)%⊦ key ∈ datakeys(ks, ri) ↔ key ∈ ri ∧ key.data? ∧ inodekey(key.ino) ∈ ks; (s)% ⊦ ¬ inodekey(m).data?; (s)% ⊦ ¬ key ∈ ks → (ri \ ks)[key] = ri[key]; (s)%⊦ orphans-cons(fs, ri, ro) → ( orphans-cons(fs0, ri0, ro0) ↔ (∀ ino. ( inodekey(ino) ∈ ri → ( fs[ri[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro)) → inodekey(ino) ∈ ri0 → ( fs0[ri0[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro0))); (ss)% ⊦ ¬ key ∈ ∅; (s) Used terms: fo0, fi0