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Proof step 6 from proof update-put-same (previous step)

sk ∈ (sk0 × val0) ' + fis0, ∀ sk, r, st, val. sk ∈ fis0 → update(r, fis0, st[r -- sk, val]) = update(r, fis0, st) ⊦ (update(r, fis0, st[r -- sk, val])[r -- sk0, val0]) = (update(r, fis0, st)[r -- sk0, val0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ ¬ sk ∈ fis → update(r, fis, st[r -- sk, val]) = (update(r, fis, st)[r -- sk, val])
Proof step 5:
¬ ¬ sk ∈ fis0, sk ∈ (sk0 × val0) ' + fis0, ∀ sk, r, st, val. sk ∈ fis0 → update(r, fis0, st[r -- sk, val]) = update(r, fis0, st) ⊦ (update(r, fis0, st[r -- sk, val])[r -- sk0, val0]) = (update(r, fis0, st)[r -- sk0, val0])

Rule argument

[crewrite-arg: update-put-rew , F , [val, sk, st, fis, r] → [val, sk, st, fis0, r] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ sk ∈ fis + fis0 ↔ sk ∈ fis ∨ sk ∈ fis0], [CSIMPSEQ : ⊦ sk ∈ (sk0 × val) ' ↔ sk = sk0], [CSIMPSEQ : ⊦ (fis + fis0) + fis1 = fis + fis0 + fis1], [CSIMPSEQ : ⊦ (st[rk, val][rk, val0]) = st[rk, val0]])], [EXTRATERMS : (val, sk, st, fis0, r)]]