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Proof step 9 from proof lookup-append (previous step)

∀ sks0, st, r. sks0 = [] ∨ st[r -- (sks1 + sks0)] = st[st[r -- sks1] .refval -- sks0] ⊦ sks1 = [], st[r -- ((sk ' + sks1) + sks0)] = st[st[st[r -- sk] .refval -- sks1] .refval -- sks0], sks0 = []

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks]
Proof step 8:
sks1 + sks0 ≠ [], ∀ sks0, st, r. sks0 = [] ∨ st[r -- (sks1 + sks0)] = st[st[r -- sks1] .refval -- sks0] ⊦ sks1 = [], st[st[r -- sk] .refval -- (sks1 + sks0)] = st[st[st[r -- sk] .refval -- sks1] .refval -- sks0], sks0 = []

Rule argument

[crewrite-arg: lookup-rec-01 , F , [sks, sk, r, st] → [sks1 + sks0, sk, r, st] ]

Additional information

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