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Proof step 7 from proof precise-Sep (previous step)

a ∈ s, ∀ a. a ∈ s → precise(aP(a)), ∀ s0, aP. # s0 < # s → (∀ a. a ∈ s0 → precise(aP(a))) → precise(aP ⊗_ s0) ⊦ # s = 0, precise(aP ⊗_ s)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ a ∈ s ↔ (∃ s0. s = s0 ++ a ∧ ¬ a ∈ s0)
Proof step 9:
∀ a. a ∈ s → precise(aP(a)), ∃ s0. s = s0 ++ a ∧ ¬ a ∈ s0, ∀ s0, aP. # s0 < # s → (∀ a. a ∈ s0 → precise(aP(a))) → precise(aP ⊗_ s0) ⊦ # s = 0, precise(aP ⊗_ s)

Rule argument

spec: set-basic name: indef seq: ⊦ a ∈ s ↔ (∃ s0. s = s0 ++ a ∧ ¬ a ∈ s0) subst: [s, a] → [s, a] flags: <all paths>

Additional information

Used terms: s, a