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Proof step 187 from proof pathex-put-nopathex (previous step)

st[r -- i '] = refval(r1), 0 ≤ i, 0 < st[r -- _length] .intval, is_array(str, r, st), i < st[r -- _length] .intval, validrefs(str, r1 ' + refs1, st[r -- _type] .type .type, st), path∃(str, r1, r0, st) ∨ junkref(str, r1, st) ∨ path∃(str, r1, (r1 ' + refs1) .last, st[r0 -- sk, val]), ∀ val, r0, sk, st0, r2, str, refs. # refs < # (r1 + refs1) → ( validrefs(str, r2 ' + refs, st0[r2 -- _type] .type, st0) → sk = _type ∨ path∃(str, r2, r0, st0) ∨ junkref(str, r2, st0) ∨ path∃(str, r2, (r2 ' + refs) .last, st0[r0 -- sk, val])) ⊦ sk = _type, r = jvmref, path∃(str, r, r0, st), path∃(str, r, (r1 ' + refs1) .last, st[r0 -- sk, val]), r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ st[r -- _type] .type .type ∨ junkref(str, r1, st))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 154:
⊦ is_array(str, r, st) ∧ r ≠ r0 → ( path∃(str, r, r0, 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 ∧ path∃(str, st[r -- i '] .refval, r0, st) ∨ st[r -- i '] .refval = jvmref ∧ r0 = jvmref)))
Proof step 167:
¬ (is_array(str, r, st[r0 -- sk, val]) ∧ r ≠ (r1 ' + refs1) .last), st[r -- i '] = refval(r1), 0 ≤ i, 0 < st[r -- _length] .intval, is_array(str, r, st), i < st[r -- _length] .intval, validrefs(str, r1 ' + refs1, st[r -- _type] .type .type, st), path∃(str, r1, r0, st) ∨ junkref(str, r1, st) ∨ path∃(str, r1, (r1 ' + refs1) .last, st[r0 -- sk, val]), ∀ val, r0, sk, st0, r2, str, refs. # refs < # (r1 + refs1) → ( validrefs(str, r2 ' + refs, st0[r2 -- _type] .type, st0) → sk = _type ∨ path∃(str, r2, r0, st0) ∨ junkref(str, r2, st0) ∨ path∃(str, r2, (r2 ' + refs) .last, st0[r0 -- sk, val])) ⊦ sk = _type, r = jvmref, path∃(str, r, r0, st), path∃(str, r, (r1 ' + refs1) .last, st[r0 -- sk, val]), r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ st[r -- _type] .type .type ∨ junkref(str, r1, st))
Proof step 186:
is_array(str, r, st[r0 -- sk, val]) ∧ r ≠ (r1 ' + refs1) .last, st[r -- i '] = refval(r1), 0 ≤ i, 0 < st[r -- _length] .intval, is_array(str, r, st), i < st[r -- _length] .intval, validrefs(str, r1 ' + refs1, st[r -- _type] .type .type, st), path∃(str, r1, r0, st) ∨ junkref(str, r1, st) ∨ path∃(str, r1, (r1 ' + refs1) .last, st[r0 -- sk, val]), ∀ val, r0, sk, st0, r2, str, refs. # refs < # (r1 + refs1) → ( validrefs(str, r2 ' + refs, st0[r2 -- _type] .type, st0) → sk = _type ∨ path∃(str, r2, r0, st0) ∨ junkref(str, r2, st0) ∨ path∃(str, r2, (r2 ' + refs) .last, st0[r0 -- sk, val])) ⊦ sk = _type, r = jvmref, path∃(str, r, r0, st), ∃ i. 0 ≤ i ∧ i < (st[r0 -- sk, val][r -- _length]) .intval ∧ is_referencevalue(st[r0 -- sk, val][r -- i ']) ∧ ( (st[r0 -- sk, val][r -- i ']) .refval ≠ jvmref ∧ (st[r0 -- sk, val][(st[r0 -- sk, val][r -- i ']) .refval -- _type]) .type ≤ (st[r0 -- sk, val][r -- _type]) .type .type ∧ path∃(str, (st[r0 -- sk, val][r -- i ']) .refval, (r1 ' + refs1) .last, st[r0 -- sk, val]) ∨ (st[r0 -- sk, val][r -- i ']) .refval = jvmref ∧ (r1 ' + refs1) .last = jvmref), r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ st[r -- _type] .type .type ∨ junkref(str, r1, st))

Rule argument

[crewrite-arg: pathex-array , F , [st, r0, r, str] → [st[r0 -- sk, val], (r1 ' + refs1) .last, r, str] ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st[r0 -- sk, val], (r1 ' + refs1) .last, r, str)]]