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Proof step 18 from proof validreflist-getarray-less-01 (previous step)

is_reftype(st[r -- _type] .type), 0 < st[r -- _length] .intval, j < st[r -- _length] .intval, 0 ≤ j, j + i1 + 1 < st[r -- _length] .intval, validref(str, r, mkarraytype(ty), st), is_reftype(ty), 0 ≤ i1, validreflist(str, getarray(r, j + 1, i1, st), ty, st), is_valid_ref(str, r, st), ∀ j, st, r, str, ty, i0. i→n(i0) < i→n(i1 + 1) → ( is_reftype(ty) ∧ validrefnotnull(str, r, mkarraytype(ty), st) ∧ j + i0 < st[r -- _length] .intval ∧ 0 ≤ i0 ∧ 0 ≤ j → r = jvmref ∨ validreflist(str, getarray(r, j, i0, st), ty, st)) ⊦ r = jvmref, junkref(str, r, st), is_referencevalue(st[r -- j '])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ is_array(str, r, st) → ( junkref(str, r, 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 ∨ junkref(str, st[r -- i '] .refval, st))))
Proof step 13:
¬ is_array(str, r, st), is_reftype(st[r -- _type] .type), 0 < st[r -- _length] .intval, j < st[r -- _length] .intval, 0 ≤ j, j + i1 + 1 < st[r -- _length] .intval, validref(str, r, mkarraytype(ty), st), is_reftype(ty), 0 ≤ i1, validreflist(str, getarray(r, j + 1, i1, st), ty, st), is_valid_ref(str, r, st), ∀ j, st, r, str, ty, i0. i→n(i0) < i→n(i1 + 1) → ( is_reftype(ty) ∧ validrefnotnull(str, r, mkarraytype(ty), st) ∧ j + i0 < st[r -- _length] .intval ∧ 0 ≤ i0 ∧ 0 ≤ j → r = jvmref ∨ validreflist(str, getarray(r, j, i0, st), ty, st)) ⊦ r = jvmref, junkref(str, r, st), is_referencevalue(st[r -- j '])
Proof step 17:
is_array(str, r, st), is_reftype(st[r -- _type] .type), 0 < st[r -- _length] .intval, j < st[r -- _length] .intval, 0 ≤ j, j + i1 + 1 < st[r -- _length] .intval, validref(str, r, mkarraytype(ty), st), is_reftype(ty), 0 ≤ i1, validreflist(str, getarray(r, j + 1, i1, st), ty, st), is_valid_ref(str, r, st), ∀ j, st, r, str, ty, i0. i→n(i0) < i→n(i1 + 1) → ( is_reftype(ty) ∧ validrefnotnull(str, r, mkarraytype(ty), st) ∧ j + i0 < st[r -- _length] .intval ∧ 0 ≤ i0 ∧ 0 ≤ j → r = jvmref ∨ validreflist(str, getarray(r, j, i0, st), ty, st)) ⊦ r = jvmref, ∃ 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 ∨ junkref(str, st[r -- i '] .refval, st)), is_referencevalue(st[r -- j '])

Rule argument

[crewrite-arg: junkref-array, F, [st, r, str] → [st, r, str]]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, r, str)]]