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Proof step 11 from proof precise-Sep (previous step)

precise(adP(a, d)), ∀ 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), precise(adP ⊗_ st0), a ∈ st0

User applied rule apply induction

And obtained the following new goals

Proof step 12:
precise(adP(a, d)), ∀ 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 ⊗_ st0), precise(adP(a, d) * adP ⊗_ st0), a ∈ st0, ∀ a. a ∈ st0 → precise(adP(a, st0[a]))

Rule argument

subst: [st1, adP] → [st0, adP]

Additional information

Used simplifier rules: ⊦ ¬ a ∈ st → # st[a, d] = # st + 1; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: st0, adP