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Proof step 33 from proof junkrefs-put-null-02 (previous step)

st[r0 -- i0 '] = refval(r2), junkrefs(str, r2 ' + refs, st[r0 -- _type] .type .type, st[r -- i ', refval(jvmref)]), i0 < st[r0 -- _length] .intval, is_array(str, r0, st), st[r0 -- _type] .type ≤ ty, is_array(str, r0, st[r -- i ', refval(jvmref)]), is_reftype(st[r -- _type] .type .type), 0 < st[r0 -- _length] .intval, 0 ≤ i0, ∀ st0, r1, str0, ty0, i1. is_reftype(st0[r1 -- _type] .type .type) ∧ junkrefs(str0, r2 ' + refs, ty0, st0[r1 -- i1 ', refval(jvmref)]) → r1 = jvmref ∨ junkrefs(str0, r2 ' + refs, ty0, st0), r0 ≠ r ∨ i0 ≠ i ⊦ r = jvmref, r0 = jvmref, junkrefs(str, r0 ' + r2 ' + refs, ty, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( junkrefs(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 ∧ junkrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 32:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ ty, st[r0 -- i0 '] = refval(r2), junkrefs(str, r2 ' + refs, st[r0 -- _type] .type .type, st[r -- i ', refval(jvmref)]), i0 < st[r0 -- _length] .intval, is_array(str, r0, st), st[r0 -- _type] .type ≤ ty, is_array(str, r0, st[r -- i ', refval(jvmref)]), is_reftype(st[r -- _type] .type .type), 0 < st[r0 -- _length] .intval, 0 ≤ i0, ∀ st0, r1, str0, ty0, i1. is_reftype(st0[r1 -- _type] .type .type) ∧ junkrefs(str0, r2 ' + refs, ty0, st0[r1 -- i1 ', refval(jvmref)]) → r1 = jvmref ∨ junkrefs(str0, r2 ' + refs, ty0, st0), r0 ≠ r ∨ i0 ≠ i ⊦ r = jvmref, r0 = jvmref, r2 ' + refs ≠ [] ∧ (∃ i. 0 ≤ i ∧ i < st[r0 -- _length] .intval ∧ is_referencevalue(st[r0 -- i ']) ∧ (r2 ' + refs) .first = st[r0 -- i '] .refval ∧ junkrefs(str, r2 ' + refs, st[r0 -- _type] .type .type, st))

Rule argument

[crewrite-arg: junkrefs-array , F , [st, ty, refs, r, str] → [st, ty, r2 ' + refs, r0, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st, ty, r2 ' + refs, r0, str)]]