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Proof step 39 from proof cyclic-addrefarray (previous step)

newref(r, st), type∃(str, ty), is_reftype(ty), validrefs(str, r ' + refs0, mkarraytype(ty), addarray(r, ty, vals, st)), is_array(str, r, addarray(r, ty, vals, st)), ∀ val. val ∈ vals → ¬ junkref(str, val .refval, st) ∧ ¬ cyclic(str, val .refval, st) ∧ is_referencevalue(val) ∧ (val .refval ≠ jvmref → st[val .refval -- _type] .type ≤ ty) ⊦ ty .jtelemtype = abstract_type, ty .jtelemtype = void_type, r = jvmref, ¬ r ∈ refs0 ∧ ¬ dups(refs0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 38:
is_array(str, r, addarray(r, ty, vals, st)) ∧ (addarray(r, ty, vals, st)[r -- _type]) .type ≤ mkarraytype(ty), newref(r, st), type∃(str, ty), is_reftype(ty), refs0 = [] ∨ (∃ i. 0 ≤ i ∧ i < (addarray(r, ty, vals, st)[r -- _length]) .intval ∧ is_referencevalue(addarray(r, ty, vals, st)[r -- i ']) ∧ refs0 .first = (addarray(r, ty, vals, st)[r -- i ']) .refval ∧ validrefs(str, refs0, (addarray(r, ty, vals, st)[r -- _type]) .type .type, addarray(r, ty, vals, st))), is_array(str, r, addarray(r, ty, vals, st)), ∀ val. val ∈ vals → ¬ junkref(str, val .refval, st) ∧ ¬ cyclic(str, val .refval, st) ∧ is_referencevalue(val) ∧ (val .refval ≠ jvmref → st[val .refval -- _type] .type ≤ ty) ⊦ ty .jtelemtype = abstract_type, ty .jtelemtype = void_type, r = jvmref, ¬ r ∈ refs0 ∧ ¬ dups(refs0)

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [addarray(r, ty, vals, st), mkarraytype(ty), refs0, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (addarray(r, ty, vals, st) [r -- _type ] ) = typeval(mkarraytype(ty))], [CSIMPSEQ : ⊦ typeval(ty) .type = ty], [CSIMPSEQ : ⊦ mkarraytype(ty) ≤ mkarraytype(ty0) ↔ ty ≤ ty0], [CSIMPSEQ : ⊦ ty ≤ ty])], [EXTRATERMS : (addarray(r, ty, vals, st), mkarraytype(ty), refs0, r, str)]]