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Proof step 51 from proof junkrefs-put-badfield (previous step)

st[r -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r -- _type] .type .type, st[r0 -- fs ', val]), 0 ≤ i, i < st[r -- _length] .intval, is_array(str, r, st), is_array(str, r, st[r0 -- fs ', val]), st[r -- _type] .type ≤ ty, 0 < st[r -- _length] .intval, ∀ str0, ty0, st0, r1, fs0, val0. junkrefs(str0, r2 ' + refs, ty0, st0[r1 -- fs0 ', val0]) → junkrefs(str0, r2 ' + refs, ty0, st0) ∨ is_typevalue(st0[r1 -- _type]) ∧ is_reftype(st0[r1 -- _type] .type), r ≠ r0 ∨ _type ≠ fs ', r ≠ r0 ∨ _length ≠ fs ' ⊦ r = jvmref, junkrefs(str, r ' + r2 ' + refs, ty, st), is_typevalue(st[r0 -- _type]) ∧ is_reftype(st[r0 -- _type] .type)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 46:
⊦ 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 50:
is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty, st[r -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r -- _type] .type .type, st[r0 -- fs ', val]), 0 ≤ i, i < st[r -- _length] .intval, is_array(str, r, st), is_array(str, r, st[r0 -- fs ', val]), st[r -- _type] .type ≤ ty, 0 < st[r -- _length] .intval, ∀ str0, ty0, st0, r1, fs0, val0. junkrefs(str0, r2 ' + refs, ty0, st0[r1 -- fs0 ', val0]) → junkrefs(str0, r2 ' + refs, ty0, st0) ∨ is_typevalue(st0[r1 -- _type]) ∧ is_reftype(st0[r1 -- _type] .type), r ≠ r0 ∨ _type ≠ fs ', r ≠ r0 ∨ _length ≠ fs ' ⊦ r = jvmref, r2 ' + refs ≠ [] ∧ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ (r2 ' + refs) .first = st[r -- i '] .refval ∧ junkrefs(str, r2 ' + refs, st[r -- _type] .type .type, st)), is_typevalue(st[r0 -- _type]) ∧ is_reftype(st[r0 -- _type] .type)

Rule argument

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

Additional information

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