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Proof step 34 from proof validrefs-putbyte-field (previous step)

st[r0 -- i '] = refval(r1), validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), i < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, st[r0 -- _type] .type ≤ ty, is_array(str, r0, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), 0 < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, 0 ≤ i, ∀ by, fieldvar, class, r, st0, ty, str, refs0. # refs0 < # (r0 + r1 + refs) → ( validrefs(str, refs0, ty, st0[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]) → refs0 = [] ∨ junkref(str, refs0 .first, st0) ∨ validrefs(str, refs0, ty, st0)) ⊦ r0 = jvmref, validrefs(str, r0 ' + r1 ' + refs, ty, st), junkref(str, r0, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ 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 16:
¬ (is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ ty), st[r0 -- i '] = refval(r1), validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), i < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, st[r0 -- _type] .type ≤ ty, is_array(str, r0, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), 0 < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, 0 ≤ i, ∀ by, fieldvar, class, r, st0, ty, str, refs0. # refs0 < # (r0 + r1 + refs) → ( validrefs(str, refs0, ty, st0[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]) → refs0 = [] ∨ junkref(str, refs0 .first, st0) ∨ validrefs(str, refs0, ty, st0)) ⊦ r0 = jvmref, validrefs(str, r0 ' + r1 ' + refs, ty, st), junkref(str, r0, st)
Proof step 33:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ ty, st[r0 -- i '] = refval(r1), validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), i < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, st[r0 -- _type] .type ≤ ty, is_array(str, r0, st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]), 0 < (st[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)][r0 -- _length]) .intval, 0 ≤ i, ∀ by, fieldvar, class, r, st0, ty, str, refs0. # refs0 < # (r0 + r1 + refs) → ( validrefs(str, refs0, ty, st0[r -- mkfs(class, byte_type, fieldvar) ', byteval(by)]) → refs0 = [] ∨ junkref(str, refs0 .first, st0) ∨ validrefs(str, refs0, ty, st0)) ⊦ r0 = jvmref, r1 ' + refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r0 -- _length] .intval ∧ is_referencevalue(st[r0 -- i ']) ∧ (r1 ' + refs) .first = st[r0 -- i '] .refval ∧ validrefs(str, r1 ' + refs, st[r0 -- _type] .type .type, st)), junkref(str, r0, st)

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st, ty, r1 ' + refs, r0, str] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, ty, r1 ' + refs, r0, str)]]