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Proof step 8 from proof lookup-last-rec (previous step)

sks0 = [] ∨ (∀ st, r, sk. st[r -- (sks0 + sk ')] = st[st[r -- sks0] .refval -- sk]) ⊦ st[r -- ((sk0 ' + sks0) + sk ')] = st[st[r -- (sk0 ' + sks0)] .refval -- sk]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 1:
⊦ sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks]
Proof step 7:
sks0 + sk ' ≠ [], sks0 = [] ∨ (∀ st, r, sk. st[r -- (sks0 + sk ')] = st[st[r -- sks0] .refval -- sk]) ⊦ st[st[r -- sk0] .refval -- (sks0 + sk ')] = st[st[r -- (sk0 ' + sks0)] .refval -- sk]

Rule argument

[crewrite-arg: lookup-rec-01 , F , [sks, sk, r, st] → [sks0 + sk ', sk0, r, st] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ sks + sks0 ≠ [] ↔ sks = [] ∧ sks0 = []], [CSIMPSEQ : ⊦ sk ' ≠ []], [CSIMPSEQ : ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1])], [EXTRATERMS : (sks0 + sk ', sk0, r, st)]]