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Proof step 55 from proof junkrefs-put-null-01 (previous step)

st[r0 -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, ty0, fieldvar) ', refval(jvmref)]), 0 ≤ i, i < st[r0 -- _length] .intval, is_array(str, r0, st), is_array(str, r0, st[r -- mkfs(class, ty0, fieldvar) ', refval(jvmref)]), st[r0 -- _type] .type ≤ ty, is_reftype(ty0), 0 < st[r0 -- _length] .intval, ∀ str0, ty1, st0, fs, r1. ¬ junkrefs(str0, r2 ' + refs, ty1, st0) ∧ is_reftype(fs .type) → ¬ junkrefs(str0, r2 ' + refs, ty1, st0[r1 -- fs ', refval(jvmref)]), r0 ≠ r ∨ _type ≠ mkfs(class, ty0, fieldvar) ', r0 ≠ r ∨ _length ≠ mkfs(class, ty0, fieldvar) ' ⊦ r0 = jvmref, junkrefs(str, r0 ' + r2 ' + refs, ty, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 50:
⊦ 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 54:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ ty, st[r0 -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, ty0, fieldvar) ', refval(jvmref)]), 0 ≤ i, i < st[r0 -- _length] .intval, is_array(str, r0, st), is_array(str, r0, st[r -- mkfs(class, ty0, fieldvar) ', refval(jvmref)]), st[r0 -- _type] .type ≤ ty, is_reftype(ty0), 0 < st[r0 -- _length] .intval, ∀ str0, ty1, st0, fs, r1. ¬ junkrefs(str0, r2 ' + refs, ty1, st0) ∧ is_reftype(fs .type) → ¬ junkrefs(str0, r2 ' + refs, ty1, st0[r1 -- fs ', refval(jvmref)]), r0 ≠ r ∨ _type ≠ mkfs(class, ty0, fieldvar) ', r0 ≠ r ∨ _length ≠ mkfs(class, ty0, fieldvar) ' ⊦ 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)]]