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Proof step 44 from proof junkrefs-put-nopathex-equiv (previous step)

st[r1 -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r1 -- _type] .type .type, st), i < st[r1 -- _length] .intval, is_valid_ref(str, r0, st), st[r1 -- _type] .type ≤ ty, is_array(str, r1, st), 0 < st[r1 -- _length] .intval, 0 ≤ i, ∀ str0, r, st0, sk0, ty0, val0. ¬ path∃(str0, r2, r, st0) ∧ is_valid_ref(str0, r, st0) ∧ sk0 ≠ _type → (junkrefs(str0, r2 ' + refs, ty0, st0[r -- sk0, val0]) ↔ junkrefs(str0, r2 ' + refs, ty0, st0)) ⊦ sk = _type, r1 = jvmref, r0 = r1, junkrefs(str, r1 ' + r2 ' + refs, ty, st[r0 -- sk, val]), path∃(str, r1, r0, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ 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 43:
is_array(str, r1, st[r0 -- sk, val]) ∧ (st[r0 -- sk, val][r1 -- _type]) .type ≤ ty, st[r1 -- i '] = refval(r2), junkrefs(str, r2 ' + refs, st[r1 -- _type] .type .type, st), i < st[r1 -- _length] .intval, is_valid_ref(str, r0, st), st[r1 -- _type] .type ≤ ty, is_array(str, r1, st), 0 < st[r1 -- _length] .intval, 0 ≤ i, ∀ str0, r, st0, sk0, ty0, val0. ¬ path∃(str0, r2, r, st0) ∧ is_valid_ref(str0, r, st0) ∧ sk0 ≠ _type → (junkrefs(str0, r2 ' + refs, ty0, st0[r -- sk0, val0]) ↔ junkrefs(str0, r2 ' + refs, ty0, st0)) ⊦ sk = _type, r1 = jvmref, r0 = r1, r2 ' + refs ≠ [] ∧ (∃ i. 0 ≤ i ∧ i < (st[r0 -- sk, val][r1 -- _length]) .intval ∧ is_referencevalue(st[r0 -- sk, val][r1 -- i ']) ∧ (r2 ' + refs) .first = (st[r0 -- sk, val][r1 -- i ']) .refval ∧ junkrefs(str, r2 ' + refs, (st[r0 -- sk, val][r1 -- _type]) .type .type, st[r0 -- sk, val])), path∃(str, r1, r0, st)

Rule argument

[speclemmaarg: , 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))) ,[st, ty, refs, r, str] → [st[r0 -- sk, val], ty, r2 ' + refs, r1, str] ,F ,junkrefs , ,junkrefs-array ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ≠ r0 → ( is_array (str, r0, st[r -- sk, val]) ↔ is_array(str, r0, st))], [CSIMPSEQ : ⊦ sk ≠ sk0 → (st[r -- sk, val][r0 -- sk0]) = st[r0 -- sk0]])], [EXTRATERMS : (st[r0 -- sk, val], ty, r2 ' + refs, r1, str)]]