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Proof step 52 from proof onlymod-is-update (previous step)

st1[r0 -- sk0] = (update(r0, fis, st0 -- (r0 -- sk))[r0 -- sk0]), r0 -- sk0 ∈ st1 ↔ r0 -- sk0 ∈ update(r0, fis, st0 -- (r0 -- sk)), # st0 = m + 1, r0 -- sk ∈ st1, r0 -- sk ∈ st0, onlyModified(r0, st0, st1), r0 ∈ st1, r0 ∈ st0, validFields(r0, fis, st0 -- (r0 -- sk)), ∀ rk. (rk ∈ st1 -- (r0 -- sk) ↔ rk ∈ update(r0, fis, st0 -- (r0 -- sk))) ∧ (st1 -- (r0 -- sk))[rk] = (update(r0, fis, st0 -- (r0 -- sk))[rk]) ⊦ sk0 = sk, st1[r0 -- sk0] = (update(r0, fis, st0)[r0 -- sk0]), st0[r0 -- sk] = st1[r0 -- sk], st0 = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ ¬ sk ∈ fis → (update(r, fis, st)[r0 -- sk]) = st[r0 -- sk]
Proof step 43:
⊦ sk ∈ fis → (update(r, fis, st)[r -- sk]) = fis[sk]
Proof step 51:
¬ sk0 ∈ fis, st1[r0 -- sk0] = (update(r0, fis, st0 -- (r0 -- sk))[r0 -- sk0]), r0 -- sk0 ∈ st1 ↔ r0 -- sk0 ∈ update(r0, fis, st0 -- (r0 -- sk)), # st0 = m + 1, r0 -- sk ∈ st1, r0 -- sk ∈ st0, onlyModified(r0, st0, st1), r0 ∈ st1, r0 ∈ st0, validFields(r0, fis, st0 -- (r0 -- sk)), ∀ rk. (rk ∈ st1 -- (r0 -- sk) ↔ rk ∈ update(r0, fis, st0 -- (r0 -- sk))) ∧ (st1 -- (r0 -- sk))[rk] = (update(r0, fis, st0 -- (r0 -- sk))[rk]) ⊦ sk0 = sk, st1[r0 -- sk0] = st0[r0 -- sk0], st0[r0 -- sk] = st1[r0 -- sk], st0 = ∅

Rule argument

[crewrite-arg: update-at-notin , F , [sk, r0, st, fis, r] → [sk0, r0, st0, fis, r0] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ rk ∈ st → st[rk] = ∅[rk]], [CSIMPSEQ : ⊦ ∅[rk] = noval])], [EXTRATERMS : (sk0, r0, st0, fis, r0)]]