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Proof step 10 from proof validref-forward-array (previous step)

st0[r -- i '] = refval(r0), st0[r -- _type] .type = mkarraytype(ty0), is_array(str, r, st0), validref(str, r, mkarraytype(ty), st0), i < st0[r -- _length] .intval, 0 ≤ i, 0 < st0[r -- _length] .intval, ty0 ≤ ty ⊦ r = jvmref, validref(str, r0, ty, st0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validref(str, r, ty, st) ↔ ¬ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ st[r -- i '] .refval ≠ jvmref ∧ ( ¬ st[st[r -- i '] .refval -- _type] .type ≤ st[r -- _type] .type .type ∨ ¬ validref(str, st[r -- i '] .refval, st[r -- _type] .type .type, st))))
Proof step 9:
is_array(str, r, st0) ∧ st0[r -- _type] .type ≤ mkarraytype(ty), st0[r -- i '] = refval(r0), st0[r -- _type] .type = mkarraytype(ty0), is_array(str, r, st0), ∀ i. ¬ ( 0 ≤ i ∧ i < st0[r -- _length] .intval ∧ is_referencevalue(st0[r -- i ']) ∧ st0[r -- i '] .refval ≠ jvmref ∧ ( ¬ st0[st0[r -- i '] .refval -- _type] .type ≤ st0[r -- _type] .type .type ∨ ¬ validref(str, st0[r -- i '] .refval, st0[r -- _type] .type .type, st0))), i < st0[r -- _length] .intval, 0 ≤ i, 0 < st0[r -- _length] .intval, ty0 ≤ ty ⊦ r = jvmref, validref(str, r0, ty, st0)

Rule argument

[crewrite-arg: validref-array-rec , F , [st, ty, r, str] → [st0, mkarraytype(ty), r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ mkarraytype(ty) ≤ mkarraytype(ty0) ↔ ty ≤ ty0])], [EXTRATERMS : (st0, mkarraytype(ty), r, str)]]