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Proof step 43 from proof validrefs-putshort-field (previous step)

st[r0 -- _type] .type ≤ ty, is_shortvalue(st[r -- mkfs(class, short_type, fieldvar) ']), validrefs(str, r0 ' + refs1, ty, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]), is_array(str, r0, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]), ∀ sho, fieldvar, class, r, st, ty, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, refs, ty, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]) ∧ is_shortvalue(st[r -- mkfs(class, short_type, fieldvar) ']) → refs = [] ∨ junkref(str, refs .first, st) ∨ validrefs(str, refs, ty, st)) ⊦ r0 = jvmref, junkref(str, r0, st), validrefs(str, r0 ' + refs1, ty, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ 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, r0, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]) ∧ (st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)][r0 -- _type]) .type ≤ ty, st[r0 -- _type] .type ≤ ty, is_shortvalue(st[r -- mkfs(class, short_type, fieldvar) ']), refs1 = [] ∨ (∃ i. 0 ≤ i ∧ i < (st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)][r0 -- _length]) .intval ∧ is_referencevalue(st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)][r0 -- i ']) ∧ refs1 .first = (st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)][r0 -- i ']) .refval ∧ validrefs (str, refs1, (st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)][r0 -- _type]) .type .type, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)])), is_array(str, r0, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]), ∀ sho, fieldvar, class, r, st, ty, str, refs. # refs < # (r0 + refs1) → ( validrefs(str, refs, ty, st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)]) ∧ is_shortvalue(st[r -- mkfs(class, short_type, fieldvar) ']) → refs = [] ∨ junkref(str, refs .first, st) ∨ validrefs(str, refs, ty, st)) ⊦ r0 = jvmref, junkref(str, r0, st), validrefs(str, r0 ' + refs1, ty, st)

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)], ty, refs1, r0, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ (st[ r0 -- mkfs (class, short_type, fieldvar) ', val] [r -- _type ] ) = st[r -- _type]])], [EXTRATERMS : (st[r -- mkfs(class, short_type, fieldvar) ', shortval(sho)], ty, refs1, r0, str)]]