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Proof step 11 from proof okarray-put-badfield (previous step)

st[r -- _type] = typeval(mkarraytype(ty0)), st[r -- _length] = intval(i), 0 ≤ i, r -- _length ∈ st, is_arrayref(r, i, st), r -- _type ∈ st, ty0 ≤ ty, r ∈ st, ∀ j. ¬ j < 0 ∧ ¬ i ≤ j → okval(st[r -- j '], ty) ⊦ fs ' = _length, fs ' = _type, r = jvmref, okreftype(refval(r), mkarraytype(ty), st[r -- fs ', val]) ∧ (0 ≤ j ∧ j < i → okval(st[r -- j '], ty))

User applied rule insert rewrite lemma

And obtained the following new goals

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)
Proof step 10:
st[r -- _type] = typeval(mkarraytype(ty0)), st[r -- _length] = intval(i), 0 ≤ i, r -- _length ∈ st, is_arrayref(r, i, st), r -- _type ∈ st, ty0 ≤ ty, r ∈ st, ∀ j. ¬ j < 0 ∧ ¬ i ≤ j → okval(st[r -- j '], ty) ⊦ fs ' = _length, fs ' = _type, r = jvmref, is_referencevalue(refval(r)) ∧ ( refval(r) .refval ≠ jvmref → refval(r) .refval -- _type ∈ st[r -- fs ', val] ∧ is_typevalue(st[r -- fs ', val][refval(r) .refval -- _type]) ∧ (st[r -- fs ', val][refval(r) .refval -- _type]) .type ≤ mkarraytype(ty)) ∧ (0 ≤ j ∧ j < i → okval(st[r -- j '], ty))

Rule argument

[crewrite-arg: okreftype , F , [st, ty, val] → [st[r -- fs ', val], mkarraytype(ty), refval(r)] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st[r -- fs ', val], mkarraytype(ty), refval(r))]]