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Proof step 9 from proof update-put-notin (previous step)

∀ r0, st, sk, val. (update(r0, fis0, st)[r0 -- sk, val]) = update(r0, fis0, st[r0 -- sk, val]) ∨ sk ∈ fis0 ⊦ (update(r0, fis0, st)[r0 -- fv .1, fv .2][r0 -- sk, val]) = (update(r0, fis0, st[r0 -- sk, val])[r0 -- fv .1, fv .2]), sk ∈ fv ', sk ∈ fis0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 8:
fv = sk0 × val0, ∀ r0, st, sk, val. (update(r0, fis0, st)[r0 -- sk, val]) = update(r0, fis0, st[r0 -- sk, val]) ∨ sk ∈ fis0 ⊦ (update(r0, fis0, st)[r0 -- sk0, val0][r0 -- sk, val]) = (update(r0, fis0, st[r0 -- sk, val])[r0 -- sk0, val0]), sk ∈ fv ', sk ∈ fis0

Rule argument

[speclemmaarg: ,sk = fv .1 ∧ val = fv .2 ↔ fv = sk × val ,[fv, sk, val] → [fv, sk0, val0] ,T ,pair ,fieldinit ,elim ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : ⊦ sk = fv .1 ∧ val = fv .2 ↔ fv = sk × val])], [EXTRATERMS : (fv, sk0, val0)]]