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Proof step 3 from proof precise-Sep (previous step)

∀ a. a ∈ st → precise(adP(a, st[a])), ∀ st0, adP. # st0 < # st → (∀ a. a ∈ st0 → precise(adP(a, st0[a]))) → precise(adP ⊗_ st0) ⊦ st = ∅, precise(adP ⊗_ st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ st = ∅ ↔ ¬(∃ a, d, st0. st = st0[a, d] ∧ ¬ a ∈ st0)
Proof step 5:
∀ a. a ∈ st → precise(adP(a, st[a])), ∀ st0, adP. # st0 < # st → (∀ a. a ∈ st0 → precise(adP(a, st0[a]))) → precise(adP ⊗_ st0) ⊦ precise(adP ⊗_ st), ∀ a, d, st0. ¬(st = st0[a, d] ∧ ¬ a ∈ st0)

Rule argument

spec: store name: exnotempty seq: ⊦ st = ∅ ↔ ¬(∃ a, d, st0. st = st0[a, d] ∧ ¬ a ∈ st0) subst: [st] → [st] flags: <all paths>

Additional information

Used terms: st