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Proof step 17 from proof onlymod-is-update (previous step)

st1 -- (r0 -- sk) = update(r0, fis, st0 -- (r0 -- sk)), # st0 = m + 1, st1 = (update(r0, fis, st0)[r0 -- sk, st1[r0 -- sk]]), sk ∈ fis, validFields(r0, fis, st0), hasKeys(r0, fis .keys, st0 -- (r0 -- sk)), unique_al(fis), r0 ∈ st0, r0 ∈ st1, r0 -- sk ∈ st0, r0 -- sk ∈ st1, r0 -- _length ∈ st0 → st0[r0 -- _length] = st1[r0 -- _length], r0 -- _type ∈ st0 → st0[r0 -- _type] = st1[r0 -- _type], ∀ r1, sk. r1 ≠ r0 → st0[r1 -- sk] = st1[r1 -- sk], ∀ rk. rk ∈ st0 ↔ rk ∈ st1 ⊦ st0 = ∅, st0[r0 -- sk] = st1[r0 -- sk], _type ∈ fis, _length ∈ fis

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ hasKeys(r, sks, st) ↔ (∀ sk. sk ∈ sks → r -- sk ∈ st)
Proof step 16:
st1 -- (r0 -- sk) = update(r0, fis, st0 -- (r0 -- sk)), # st0 = m + 1, st1 = (update(r0, fis, st0)[r0 -- sk, st1[r0 -- sk]]), sk ∈ fis, validFields(r0, fis, st0), ∀ sk0. sk0 ∈ fis .keys → r0 -- sk0 ∈ st0 -- (r0 -- sk), unique_al(fis), r0 ∈ st0, r0 ∈ st1, r0 -- sk ∈ st0, r0 -- sk ∈ st1, r0 -- _length ∈ st0 → st0[r0 -- _length] = st1[r0 -- _length], r0 -- _type ∈ st0 → st0[r0 -- _type] = st1[r0 -- _type], ∀ r1, sk. r1 ≠ r0 → st0[r1 -- sk] = st1[r1 -- sk], ∀ rk. rk ∈ st0 ↔ rk ∈ st1 ⊦ st0 = ∅, st0[r0 -- sk] = st1[r0 -- sk], _type ∈ fis, _length ∈ fis

Rule argument

[crewrite-arg: hasKeys-all , F , [st, sks, r] → [st0 -- (r0 -- sk), fis .keys, r0] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st0 -- (r0 -- sk), fis .keys, r0)]]