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Proof step 182 from proof exvalidrefs-maxlength (previous step)

st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- 0 '] = refval(r0), ty .jtelemtype = mkclasstype(`java.lang.Object' .class), 0 < st[r0 -- _length] .intval, junkrefs(str, r0 ' + refs0, (st -- r0)[r0 -- _type] .type, st -- r0), mkarraytype(ty) ≤ ty, is_array(r0, st), is_array(str, r0, st), is_reftype(ty), ∀ r, str, st0. # st0 .refs < # st .refs → ( true → cyclic(str, r, st0) ∨ junkref(str, r, st0) ∨ (∃ refs. validrefs(str, refs, st0[refs .first -- _type] .type, st0) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r ∧ ¬ has_reffield(refs .last, st0))) ⊦ r0 = jvmref, ty = void_type, junkref(str, r0, st), validrefs(str, r0 ' + r0 ', mkarraytype(ty), st), ∃ refs. validrefs(str, refs, st[refs .first -- _type] .type, st) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r0 ∧ ¬ has_reffield(refs .last, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 176:
⊦ 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 181:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ mkarraytype(ty), st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- 0 '] = refval(r0), ty .jtelemtype = mkclasstype(`java.lang.Object' .class), 0 < st[r0 -- _length] .intval, junkrefs(str, r0 ' + refs0, (st -- r0)[r0 -- _type] .type, st -- r0), mkarraytype(ty) ≤ ty, is_array(r0, st), is_array(str, r0, st), is_reftype(ty), ∀ r, str, st0. # st0 .refs < # st .refs → ( true → cyclic(str, r, st0) ∨ junkref(str, r, st0) ∨ (∃ refs. validrefs(str, refs, st0[refs .first -- _type] .type, st0) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r ∧ ¬ has_reffield(refs .last, st0))) ⊦ r0 = jvmref, ty = void_type, junkref(str, r0, st), r0 ' = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r0 -- _length] .intval ∧ is_referencevalue(st[r0 -- i ']) ∧ r0 ' .first = st[r0 -- i '] .refval ∧ validrefs(str, r0 ', st[r0 -- _type] .type .type, st)), ∃ refs. validrefs(str, refs, st[refs .first -- _type] .type, st) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r0 ∧ ¬ has_reffield(refs .last, st)

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st, mkarraytype(ty), r0 ', r0, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ mkarraytype(ty) ≤ mkarraytype(ty0) ↔ ty ≤ ty0], [CSIMPSEQ : ⊦ ty ≤ ty])], [EXTRATERMS : (st, mkarraytype(ty), r0 ', r0, str)]]