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Proof step 4 from proof replay-safe-gc-drop (previous step)

as ∩ (adr + log0).toSet = ∅, ∀ as, fs, ri. as ∩ log0.toSet = ∅ → (replay-safe(log0, fs \ as, ri) ↔ replay-safe(log0, fs, ri)) ⊦ replay-safe(adr + log0, fs \ as, ri) ↔ replay-safe(adr + log0, fs, ri)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: replay-safe-rec: ⊦ replay-safe(adr ' + log, fs, ri) ↔ adr ∈ fs ∧ ¬ adr ∈ ri ∧ node-cons(fs[adr].key, fs[adr]) ∧ replay-safe(log, fs, replay-one(adr, fs[adr], ri));% ⊦ ¬ adr ∈ as → (fs \ as)[adr] = fs[adr]; (s)% ⊦ adr ∈ fs \ as ↔ adr ∈ fs ∧ ¬ adr ∈ as; (ws)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ as ∩ (as0 ++ adr) = ∅ ↔ as ∩ as0 = ∅ ∧ ¬ adr ∈ as; (ws)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]% ⊦ as ∪ ∅ = as; (s)% ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ adr '.toSet = ∅ ++ adr; (s)% ⊦ (ax + ax0).toSet = ax.toSet ∪ ax0.toSet; (s)% ⊦ adr + ax = adr ' + ax; (s)