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Proof step 16 from proof okarray-put-08 (previous step)

⊦ fs ' = _length, fs ' = _type, is_referencevalue(val) ∧ val .refval -- _type ∈ st[r -- fs ', val0] ∧ is_typevalue(st[val .refval -- _type]) ∧ is_arraytype(st[val .refval -- _type] .type) ∧ st[val .refval -- _type] .type .type ≤ ty ∧ val .refval ≠ jvmref ∧ val .refval -- _length ∈ st[r -- fs ', val0] ∧ st[val .refval -- _length] = intval(i) ∧ 0 ≤ i ∧ is_arrayref(val .refval, i, st) ∧ (∀ j. ¬ j < 0 ∧ ¬ i ≤ j → okval(st[val .refval -- j '], ty)) ↔ 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))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
⊦ 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 8:
⊦ oktype(val, ty, st) ↔ okval(val, ty)
Proof step 15:
st[val .refval -- _length] = intval(i), is_referencevalue(val), val .refval -- _type ∈ st[r -- fs ', val0], is_typevalue(st[val .refval -- _type]), is_arraytype(st[val .refval -- _type] .type), st[val .refval -- _type] .type .type ≤ ty, val .refval -- _length ∈ st[r -- fs ', val0], is_arrayref(val .refval, i, st), j < i, val .refval -- _length ∈ st, val .refval -- _type ∈ st, val .refval ∈ st[r -- fs ', val0], 0 ≤ i, 0 ≤ j, 0 < i, val .refval ∈ st, ∀ j. ¬ j < 0 ∧ ¬ i ≤ j → okval(st[val .refval -- j '], ty) ⊦ fs ' = _length, fs ' = _type, val .refval = jvmref, okval(st[val .refval -- j '], ty)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPFORWARD : ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j], [CSIMPFORWARD : ⊦ i < j ∧ 0 ≤ i → 0 < j], [CSIMPFORWARD : ⊦ r -- sk ∈ st → r ∈ st], [CSIMPSEQ : ⊦ intval(i) ≠ noval], [CSIMPSEQ : ⊦ ¬ is_typevalue(noval)], [CSIMPSEQ : ⊦ ∅[rk] = noval], [CSIMPSEQ : ⊦ ¬ rk ∈ st → st[rk] = ∅[rk]], [CSIMPSEQ : ⊦ r -- sk = r0 -- sk0 ↔ ¬ ¬ (r = r0 ∧ sk = sk0)], [CSIMPSEQ : ⊦ rk ∈ st[rk0, val] ↔ rk = rk0 ∨ rk ∈ st], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ rk × val ∈ st ↔ rk ∈ st ∧ st[rk] = val], [CSIMPSEQ : ⊦ ty0 ≤ mkarraytype(ty) ↔ is_arraytype(ty0) ∧ ty0 .type ≤ ty])]]