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Proof step 5 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) ⊦ precise(adP ⊗_ st), ∀ a, d, st0. ¬(st = st0[a, d] ∧ ¬ a ∈ st0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
∀ a0. a0 ∈ st0[a, d] → precise(adP(a0, st0[a, d][a0])), ∀ st1, adP. # st1 < # st0[a, d] → (∀ a. a ∈ st1 → precise(adP(a, st1[a]))) → precise(adP ⊗_ st1) ⊦ precise(adP(a, d) * adP ⊗_ st0), a ∈ st0

Rule argument

Additional information

Used simplifier rules: Sep-ins: ⊦ ¬ a ∈ st → adP ⊗_ st[a, d] = adP(a, d) * adP ⊗_ st;