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Proof step 11 from proof ex-validrefs-fits-stronger (previous step)

st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), # sks = # refs1, is_obj(str, r, st), st[r -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r0 ' + refs1, ty0, st), fits(r0, sks, st), m ≤ # refs1, ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, st[r -- (mkfs(class, ty0, fieldvar) ' ' + firstn(m, sks))] .refval = (r0 ' + refs1)[m], is_arraytype(st[r -- _type] .type), ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks]
Proof step 10:
firstn(m, sks) ≠ [], st[r -- mkfs(class, ty0, fieldvar) '] = refval(r0), # sks = # refs1, is_obj(str, r, st), st[r -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r0 ' + refs1, ty0, st), fits(r0, sks, st), m ≤ # refs1, ∀ n. n ≠ 0 ∧ ¬ # refs1 < n → st[r0 -- firstn(n, sks)] .refval = (r0 ' + refs1)[n], ∀ st0, ty, r1, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, r1 ' + refs, ty, st0) → (∃ sks. fits(r1, sks, st0) ∧ # sks = # refs ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st0[r1 -- firstn(n, sks)] .refval = (r1 ' + refs)[n]))) ⊦ r = jvmref, st[st[r -- mkfs(class, ty0, fieldvar) '] .refval -- firstn(m, sks)] .refval = (r0 ' + refs1)[m], is_arraytype(st[r -- _type] .type), ∃ sks. fits(r, sks, st) ∧ # sks = 1 + # refs1 ∧ (∀ n. n ≠ 0 ∧ n ≤ # sks → st[r -- firstn(n, sks)] .refval = (r ' + r0 ' + refs1)[n])

Rule argument

[speclemmaarg: ,sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks] ,[sks, sk, r, st] → [firstn(m, sks), mkfs(class, ty0, fieldvar) ', r, st] ,F ,storefuns , ,lookup-rec-01 ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ firstn(n, sks) = [] ↔ n = 0], [CSIMPSEQ : ⊦ firstn(0, sks) = []], [CSIMPSEQ : ⊦ sks + [] = sks], [CSIMPSEQ : ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1], [CSIMPSEQ : ⊦ st[r -- sk '] = st[r -- sk]], [CSIMPSEQ : ⊦ (r ' + refs)[0] = r], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ 0 ≤ m], [CSIMPSEQ : ⊦ refval(r) .refval = r])], [EXTRATERMS : (firstn(m, sks), mkfs(class, ty0, fieldvar) ', r, st)]]