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Proof step 14 from proof keys-put-in (previous step)

sk ∈ fis0, ∀ sk, val. sk ∈ fis0 → fis0[sk, val] .keys = fis0 .keys ⊦ ((sk0 × val0) ' + fis0)[sk, val] .keys = sk0 ' + fis0 .keys

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ sk ≠ sk0 → ((sk0 × val) ' + fis)[sk, val0] = (sk0 × val) ' + fis[sk, val0]
Proof step 8:
⊦ (fis + fis0) .keys = fis .keys + fis0 .keys
Proof step 9:
⊦ fv ' .keys = fv .1 '
Proof step 13:
sk ≠ sk0, sk ∈ fis0, ∀ sk, val. sk ∈ fis0 → fis0[sk, val] .keys = fis0 .keys ⊦ ((sk0 × val0) ' + fis0[sk, val]) .keys = sk0 ' + fis0 .keys

Rule argument

[speclemmaarg: , sk ≠ sk0 → ((sk0 × val) ' + fis)[sk, val0] = (sk0 × val) ' + fis[sk, val0] ,[val0, sk, fis, val, sk0] → [val, sk, fis0, val0, sk0] ,F ,assoclist ,fieldinit ,put-no ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ((sk × val) ' + fis)[sk, val0] = (sk × val0) ' + fis], [CSIMPSEQ : ⊦ (fis + fis0) + fis1 = fis + fis0 + fis1], [CSIMPSEQ : ⊦ (sk × val) .1 = sk], [CSIMPSEQ : ⊦ sks + sks0 = sks + sks1 ↔ sks0 = sks1], [CSIMPSEQ : ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1])], [EXTRATERMS : (val, sk, fis0, val0, sk0)]]