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Proof step 9 from proof Sep-forall (previous step)

((aP0 ⊗_ s))(H), ∀ a, H. a ∈ s ∧ aP0(a)(H) → aP(a)(H), ∀ s0, aP0, H, aP. # s0 < # s → ((aP0 ⊗_ s0))(H) ∧ (∀ a, H. a ∈ s0 ∧ aP0(a)(H) → aP(a)(H)) → ((aP ⊗_ s0))(H) ⊦ # s = 0, ((aP ⊗_ s))(H), ∀ a. a ∈ s ↔ a ∈ ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
a ∈ s, ((aP0 ⊗_ s))(H), ∀ a, H. a ∈ s ∧ aP0(a)(H) → aP(a)(H), ∀ s0, aP0, H, aP. # s0 < # s → ((aP0 ⊗_ s0))(H) ∧ (∀ a, H. a ∈ s0 ∧ aP0(a)(H) → aP(a)(H)) → ((aP ⊗_ s0))(H) ⊦ # s = 0, ((aP ⊗_ s))(H)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ a ∈ ∅; (s)