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Proof step 14 from proof replay-cons-add-orphan (previous step)

log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo) ⊦ (key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro) ∧ (key ∈ ri ⊖ (ro ++ inodekey(ino)) → (ri ⊖ (ro ++ inodekey(ino)))[key] = ((ri ⊖ inodekey(ino)) ⊖ ro)[key])

User applied rule cut formula

And obtained the following new goals

Proof step 15:
key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro, log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo) ⊦ (key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro) ∧ (key ∈ ri ⊖ (ro ++ inodekey(ino)) → (ri ⊖ (ro ++ inodekey(ino)))[key] = ((ri ⊖ inodekey(ino)) ⊖ ro)[key])
Proof step 35:
log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo) ⊦ key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro, (key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro) ∧ (key ∈ ri ⊖ (ro ++ inodekey(ino)) → (ri ⊖ (ro ++ inodekey(ino)))[key] = ((ri ⊖ inodekey(ino)) ⊖ ro)[key])

Rule argument

fma: key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro

Additional information

Used formulas: key ∈ ri ⊖ (ro ++ inodekey(ino)) ↔ key ∈ (ri ⊖ inodekey(ino)) ⊖ ro