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Proof step 61 from proof validrefs-set-null (previous step)

is_array(str, r, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), validrefs(str, (r ' + acc) + refs0, ty, st), is_reftype(fs .type), st[r -- _type] .type ≤ ty, ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ r = jvmref, refs0 = [], r ∈ acc, dups(acc), validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- fs ', refval(jvmref)])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ 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 60:
is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty, is_array(str, r, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), acc + refs0 = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ (acc + refs0) .first = st[r -- i '] .refval ∧ validrefs(str, acc + refs0, st[r -- _type] .type .type, st)), is_reftype(fs .type), st[r -- _type] .type ≤ ty, ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ r = jvmref, refs0 = [], r ∈ acc, dups(acc), validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- fs ', refval(jvmref)])

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st, ty, acc + refs0, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st, ty, acc + refs0, r, str)]]