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Proof step 27 from proof unchNorec-isarray (previous step)

st[r -- _type] = typeval(mkarraytype(ty)), st0[r -- _type] = typeval(mkarraytype(ty)), st[r -- _length] = intval(i0), st0[r -- _length] = intval(i0), 0 ≤ i0, r -- _length ∈ st0, r ∈ st, okreftype(refval(r), mkarraytype(ty), st), r -- _length ∈ st, is_arrayref(r, i0, st), r -- _type ∈ st, unchangedNorec(r, st, st0), r -- _type ∈ st0, r ∈ st0, ∀ i. ¬ i < 0 ∧ ¬ i0 ≤ i → st[r -- i '] = st0[r -- i '], ∀ j0. ¬ j0 < 0 ∧ ¬ i0 ≤ j0 → okval(st[r -- j0 '], ty) ⊦ r = jvmref, (0 ≤ j ∧ j < i0 → okval(st0[r -- j '], ty)) ∧ is_arrayref(r, i0, st0) ∧ okreftype(refval(r), mkarraytype(ty), st0)

User applied rule insert rewrite lemma

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 26:
st[r -- _type] = typeval(mkarraytype(ty)), st0[r -- _type] = typeval(mkarraytype(ty)), st[r -- _length] = intval(i0), st0[r -- _length] = intval(i0), 0 ≤ i0, r -- _length ∈ st0, r ∈ st, okreftype(refval(r), mkarraytype(ty), st), r -- _length ∈ st, is_arrayref(r, i0, st), r -- _type ∈ st, unchangedNorec(r, st, st0), r -- _type ∈ st0, r ∈ st0, ∀ i. ¬ i < 0 ∧ ¬ i0 ≤ i → st[r -- i '] = st0[r -- i '], ∀ j0. ¬ j0 < 0 ∧ ¬ i0 ≤ j0 → okval(st[r -- j0 '], ty) ⊦ r = jvmref, (0 ≤ j ∧ j < i0 → okval(st0[r -- j '], ty)) ∧ is_arrayref(r, i0, st0) ∧ is_referencevalue(refval(r)) ∧ ( refval(r) .refval ≠ jvmref → refval(r) .refval -- _type ∈ st0 ∧ is_typevalue(st0[refval(r) .refval -- _type]) ∧ st0[refval(r) .refval -- _type] .type ≤ mkarraytype(ty))

Rule argument

[x0lemmaarg: spec-name: okfuns instance-name: lemma-name: okreftype antecedent: succedent: 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) substlist: [st, ty, val] → [st0, mkarraytype(ty), refval(r)] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 1, 1, 1, 1, 1)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st0, mkarraytype(ty), refval(r))]]