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Proof step 53 from proof junkrefs-put-refval-02 (previous step)

is_array(str, r1, st), junkrefs(str, r1 ' + acc, ty, st[r -- i ', refval(r0)]), mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type, st[r1 -- _type] .type ≤ ty, ∀ st, r0, r, str, ty, i. mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type ∧ junkrefs(str, acc, ty, st[r -- i ', refval(r0)]) → r = jvmref ∨ r0 = jvmref ∨ junkrefs(str, acc, ty, st) ∨ junkref(str, r0, st) ⊦ r1 = jvmref, r0 = jvmref, r = jvmref, junkref(str, r0, st), acc ≠ [] ∧ (∃ i0. ¬ i0 < 0 ∧ ¬ st[r1 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r1 -- i0 ']) ∧ acc .first = st[r1 -- i0 '] .refval ∧ junkrefs(str, acc, st[r1 -- _type] .type .type, st))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( junkrefs(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 ∧ junkrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 31:
¬ (is_array(str, r1, st[r -- i ', refval(r0)]) ∧ (st[r -- i ', refval(r0)][r1 -- _type]) .type ≤ ty), is_array(str, r1, st), junkrefs(str, r1 ' + acc, ty, st[r -- i ', refval(r0)]), mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type, st[r1 -- _type] .type ≤ ty, ∀ st, r0, r, str, ty, i. mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type ∧ junkrefs(str, acc, ty, st[r -- i ', refval(r0)]) → r = jvmref ∨ r0 = jvmref ∨ junkrefs(str, acc, ty, st) ∨ junkref(str, r0, st) ⊦ r1 = jvmref, r0 = jvmref, r = jvmref, junkref(str, r0, st), acc ≠ [] ∧ (∃ i0. ¬ i0 < 0 ∧ ¬ st[r1 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r1 -- i0 ']) ∧ acc .first = st[r1 -- i0 '] .refval ∧ junkrefs(str, acc, st[r1 -- _type] .type .type, st))
Proof step 52:
is_array(str, r1, st[r -- i ', refval(r0)]) ∧ (st[r -- i ', refval(r0)][r1 -- _type]) .type ≤ ty, is_array(str, r1, st), acc ≠ [] ∧ (∃ i0. 0 ≤ i0 ∧ i0 < (st[r -- i ', refval(r0)][r1 -- _length]) .intval ∧ is_referencevalue(st[r -- i ', refval(r0)][r1 -- i0 ']) ∧ acc .first = (st[r -- i ', refval(r0)][r1 -- i0 ']) .refval ∧ junkrefs(str, acc, (st[r -- i ', refval(r0)][r1 -- _type]) .type .type, st[r -- i ', refval(r0)])), mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type, st[r1 -- _type] .type ≤ ty, ∀ st, r0, r, str, ty, i. mkarraytype(st[r0 -- _type] .type) ≤ st[r -- _type] .type ∧ junkrefs(str, acc, ty, st[r -- i ', refval(r0)]) → r = jvmref ∨ r0 = jvmref ∨ junkrefs(str, acc, ty, st) ∨ junkref(str, r0, st) ⊦ r1 = jvmref, r0 = jvmref, r = jvmref, junkref(str, r0, st), acc ≠ [] ∧ (∃ i0. ¬ i0 < 0 ∧ ¬ st[r1 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r1 -- i0 ']) ∧ acc .first = st[r1 -- i0 '] .refval ∧ junkrefs(str, acc, st[r1 -- _type] .type .type, st))

Rule argument

[crewrite-arg: junkrefs-array , F , [st, ty, refs, r, str] → [st[r -- i ', refval(r0)], ty, acc, r1, str] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st[r -- i ', refval(r0)], ty, acc, r1, str)]]