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Proof step 41 from proof junkrefs-put-null-02 (previous step)

is_array(str, r0, st), junkrefs(str, r0 ' + acc, ty, st[r -- i ', refval(jvmref)]), is_reftype(st[r -- _type] .type .type), st[r0 -- _type] .type ≤ ty, ∀ st, r, str, ty, i. is_reftype(st[r -- _type] .type .type) ∧ junkrefs(str, acc, ty, st[r -- i ', refval(jvmref)]) → r = jvmref ∨ junkrefs(str, acc, ty, st) ⊦ r0 = jvmref, r = jvmref, junkrefs(str, r0 ' + acc, ty, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ 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 26:
¬ (is_array(str, r0, st[r -- i ', refval(jvmref)]) ∧ (st[r -- i ', refval(jvmref)][r0 -- _type]) .type ≤ ty), is_array(str, r0, st), junkrefs(str, r0 ' + acc, ty, st[r -- i ', refval(jvmref)]), is_reftype(st[r -- _type] .type .type), st[r0 -- _type] .type ≤ ty, ∀ st, r, str, ty, i. is_reftype(st[r -- _type] .type .type) ∧ junkrefs(str, acc, ty, st[r -- i ', refval(jvmref)]) → r = jvmref ∨ junkrefs(str, acc, ty, st) ⊦ r0 = jvmref, r = jvmref, junkrefs(str, r0 ' + acc, ty, st)
Proof step 40:
is_array(str, r0, st[r -- i ', refval(jvmref)]) ∧ (st[r -- i ', refval(jvmref)][r0 -- _type]) .type ≤ ty, is_array(str, r0, st), acc ≠ [] ∧ (∃ i0. 0 ≤ i0 ∧ i0 < (st[r -- i ', refval(jvmref)][r0 -- _length]) .intval ∧ is_referencevalue(st[r -- i ', refval(jvmref)][r0 -- i0 ']) ∧ acc .first = (st[r -- i ', refval(jvmref)][r0 -- i0 ']) .refval ∧ junkrefs(str, acc, (st[r -- i ', refval(jvmref)][r0 -- _type]) .type .type, st[r -- i ', refval(jvmref)])), is_reftype(st[r -- _type] .type .type), st[r0 -- _type] .type ≤ ty, ∀ st, r, str, ty, i. is_reftype(st[r -- _type] .type .type) ∧ junkrefs(str, acc, ty, st[r -- i ', refval(jvmref)]) → r = jvmref ∨ junkrefs(str, acc, ty, st) ⊦ r0 = jvmref, r = jvmref, junkrefs(str, r0 ' + acc, ty, st)

Rule argument

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

Additional information

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