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Proof step 43 from proof validrefs-put-notin-02 (previous step)

acc + r0 ' = r2 ' + refs, st[r1 -- i '] = refval(r2), is_array(str, r1, st), okval(val, ty0), st[r1 -- _type] .type ≤ ty, i < st[r1 -- _length] .intval, validrefs(str, r2 ' + refs, st[r1 -- _type] .type .type, st), 0 ≤ i, 0 < st[r1 -- _length] .intval, ∀ str, r0, ty, st, val, ty0, class, fieldvar, r. validrefs(str, acc + r0 ', ty, st) ∧ okval(val, ty0) → mkfs(class, ty0, fieldvar) ' = _length ∨ mkfs(class, ty0, fieldvar) ' = _type ∨ validrefs(str, acc + r0 ', ty, st[r -- mkfs(class, ty0, fieldvar) ', val]) ∨ r ∈ acc ⊦ r1 = jvmref, r = r1, mkfs(class, ty0, fieldvar) ' = _type, mkfs(class, ty0, fieldvar) ' = _length, r ∈ acc, validrefs(str, (r1 ' + acc) + r0 ', ty, st[r -- mkfs(class, ty0, fieldvar) ', val]), r0 ≠ r2 ∧ ¬ r0 ∈ refs, ¬ r2 ∈ acc ∧ r2 ≠ r0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ 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 42:
is_array(str, r1, st[r -- mkfs(class, ty0, fieldvar) ', val]) ∧ (st[r -- mkfs(class, ty0, fieldvar) ', val][r1 -- _type]) .type ≤ ty, acc + r0 ' = r2 ' + refs, st[r1 -- i '] = refval(r2), is_array(str, r1, st), okval(val, ty0), st[r1 -- _type] .type ≤ ty, i < st[r1 -- _length] .intval, validrefs(str, r2 ' + refs, st[r1 -- _type] .type .type, st), 0 ≤ i, 0 < st[r1 -- _length] .intval, ∀ str, r0, ty, st, val, ty0, class, fieldvar, r. validrefs(str, acc + r0 ', ty, st) ∧ okval(val, ty0) → mkfs(class, ty0, fieldvar) ' = _length ∨ mkfs(class, ty0, fieldvar) ' = _type ∨ validrefs(str, acc + r0 ', ty, st[r -- mkfs(class, ty0, fieldvar) ', val]) ∨ r ∈ acc ⊦ r1 = jvmref, r = r1, mkfs(class, ty0, fieldvar) ' = _type, mkfs(class, ty0, fieldvar) ' = _length, r ∈ acc, acc + r0 ' = [] ∨ (∃ i. 0 ≤ i ∧ i < (st[r -- mkfs(class, ty0, fieldvar) ', val][r1 -- _length]) .intval ∧ is_referencevalue(st[r -- mkfs(class, ty0, fieldvar) ', val][r1 -- i ']) ∧ (acc + r0 ') .first = (st[r -- mkfs(class, ty0, fieldvar) ', val][r1 -- i ']) .refval ∧ validrefs(str, acc + r0 ', (st[r -- mkfs(class, ty0, fieldvar) ', val][r1 -- _type]) .type .type, st[r -- mkfs(class, ty0, fieldvar) ', val])), r0 ≠ r2 ∧ ¬ r0 ∈ refs, ¬ r2 ∈ acc ∧ r2 ≠ r0

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st[r -- mkfs(class, ty0, fieldvar) ', val], ty, acc + r0 ', r1, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ fs ' ≠ _length ∧ fs ' ≠ _type → ( is_array (str, r, st[r0 -- fs ', val]) ↔ is_array(str, r, st))], [CSIMPSEQ : ⊦ sk ≠ sk0 → (st[r -- sk, val][r0 -- sk0]) = st[r0 -- sk0]], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st[r -- mkfs(class, ty0, fieldvar) ', val], ty, acc + r0 ', r1, str)]]