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Proof step 9 from proof okarray-union-03 (previous step)

okarray(refval(r), ty, noval .intval, st) ⊦ r = jvmref, r -- _length ∈ st, (0 ≤ j ∧ j < noval .intval → oktype((st0 ∪ st)[r -- j '], ty, st0 ∪ st)) ∧ is_arrayref(r, noval .intval, st0 ∪ st) ∧ ¬ noval .intval < 0 ∧ st0[r -- _length] = intval(noval .intval) ∧ r -- _length ∈ st0 ∧ okreftype(refval(r), mkarraytype(ty), st0 ∪ st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ okarray(val, ty, i, st) ↔ okreftype(val, mkarraytype(ty), st) ∧ val .refval ≠ jvmref ∧ (val .refval -- _length) × intval(i) ∈ st ∧ 0 ≤ i ∧ is_arrayref(val .refval, i, st) ∧ (∀ j. 0 ≤ j ∧ j < i → oktype(st[val .refval -- j '], ty, st))
Proof step 8:
⊦ okreftype(val, ty, st) ↔ is_referencevalue(val) ∧ (val .refval ≠ jvmref → val .refval -- _type ∈ st ∧ is_typevalue(st[val .refval -- _type]) ∧ st[val .refval -- _type] .type ≤ ty)

Rule argument

[crewrite-arg: okarrayOfLength , F , [st, i, ty, val] → [st, noval .intval, ty, refval(r)] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ refval(r) .refval = r], [CSIMPSEQ : ⊦ is_referencevalue(refval(r))], [CSIMPSEQ : ⊦ ty0 ≤ mkarraytype(ty) ↔ is_arraytype(ty0) ∧ ty0 .type ≤ ty], [CSIMPSEQ : ⊦ rk × val ∈ st ↔ rk ∈ st ∧ st[rk] = val])], [EXTRATERMS : (st, noval .intval, ty, refval(r))]]