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Proof step 41 from proof isarray-update (previous step)

st[r -- _type] = typeval(mkarraytype(ty)), r -- _type ∈ st, r ∈ st ⊦ r = r0, okreftype(refval(r), mkarraytype(ty), update(r0, fis, st)) ∧ r ≠ jvmref ∧ r -- _length ∈ update(r0, fis, st) ∧ is_integervalue(update(r0, fis, st)[r -- _length]) ∧ 0 ≤ (update(r0, fis, st)[r -- _length]) .intval ∧ is_arrayref(r, (update(r0, fis, st)[r -- _length]) .intval, update(r0, fis, st)) ∧ (∀ j. 0 ≤ j ∧ j < (update(r0, fis, st)[r -- _length]) .intval → oktype(update(r0, fis, st)[r -- j '], ty, update(r0, fis, st))) ↔ okarray(r, ty, st), r = jvmref

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
⊦ 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 7:
⊦ r ≠ r0 → (r -- sk ∈ update(r0, fis, st) ↔ r -- sk ∈ st)
Proof step 8:
⊦ r ≠ r0 → (update(r0, fis, st)[r -- sk]) = st[r -- sk]
Proof step 9:
⊦ oktype(val, ty, st) ↔ okval(val, ty)
Proof step 40:
st[r -- _type] = typeval(mkarraytype(ty)), r -- _type ∈ st, r ∈ st ⊦ r = jvmref, r = r0, is_typevalue(typeval(mkarraytype(ty))) ∧ is_arraytype(typeval(mkarraytype(ty)) .type) ∧ typeval(mkarraytype(ty)) .type .type ≤ ty ∧ r -- _length ∈ st ∧ is_integervalue(st[r -- _length]) ∧ 0 ≤ st[r -- _length] .intval ∧ is_arrayref(r, st[r -- _length] .intval, update(r0, fis, st)) ∧ (∀ j. ¬ j < 0 ∧ ¬ st[r -- _length] .intval ≤ j → okval(st[r -- j '], ty)) ↔ okarray(r, ty, st)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ ty0 ≤ mkarraytype(ty) ↔ is_arraytype(ty0) ∧ ty0 .type ≤ ty], [CSIMPSEQ : ⊦ refval(r) .refval = r], [CSIMPSEQ : ⊦ is_referencevalue(refval(r))])]]