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Proof step 33 from proof union-okarray (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
⊦ 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 5:
⊦ oktype(val, ty, st) ↔ okval(val, ty)
Proof step 32:
r -- _length ∈ st, is_integervalue(st[r -- _length]), is_arrayref(r, st[r -- _length] .intval, st), r -- _type ∈ st, is_typevalue(st[r -- _type]), is_arraytype(st[r -- _type] .type), st[r -- _type] .type .type ≤ ty, 0 ≤ st[r -- _length] .intval, r ∈ st, ∀ j. ¬ j < 0 ∧ ¬ st[r -- _length] .intval ≤ j → okval(st[r -- j '], ty) ⊦ r = jvmref, okreftype(refval(r), mkarraytype(ty), st0 ∪ st) ∧ is_arrayref(r, st[r -- _length] .intval, st0 ∪ st) ∧ (0 ≤ j ∧ j < st[r -- _length] .intval → okval((st0 ∪ st)[r -- j '], ty))

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPFORWARD : ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j], [CSIMPFORWARD : ⊦ r -- sk ∈ st → r ∈ st], [CSIMPSEQ : ⊦ i ≤ j → (¬ ¬ j < i ↔ false)], [CSIMPSEQ : ⊦ rk ∈ st0 → (st ∪ st0)[rk] = st0[rk]], [CSIMPSEQ : ⊦ rk ∈ st ∪ st0 ↔ rk ∈ st ∨ rk ∈ st0], [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))])]]