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Proof step 183 from proof pathex-put-nopathex (previous step)

st[r -- i '] = refval(r1), is_array(str, r, st[r0 -- sk, val]), 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, r = (r1 ' + refs1) .last, path∃(str, r, r0, st), r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ st[r -- _type] .type .type ∨ junkref(str, r1, st)), ¬ (st[r0 -- sk, val][r -- _length]) .intval ≤ i ∧ is_referencevalue(st[r0 -- sk, val][r -- i ']) ∧ ( (st[r0 -- sk, val][r -- i ']) .refval ≠ jvmref ∧ st[(st[r0 -- sk, val][r -- i ']) .refval -- _type] .type ≤ st[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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 168:
⊦ rk ≠ rk0 → (st[rk0, val][rk]) = st[rk]
Proof step 169:
⊦ path∃(str, r, r, st) ↔ r = jvmref ∨ r ≠ jvmref ∧ is_valid_ref(str, r, st)
Proof step 182:
r -- i ' ≠ r0 -- sk, st[r -- i '] = refval(r1), is_array(str, r, st[r0 -- sk, val]), 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, r = (r1 ' + refs1) .last, path∃(str, r, r0, st), r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ st[r -- _type] .type .type ∨ junkref(str, r1, st)), ¬ (st[r0 -- sk, val][r -- _length]) .intval ≤ i ∧ 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, (r1 ' + refs1) .last, st[r0 -- sk, val]) ∨ st[r -- i '] .refval = jvmref ∧ (r1 ' + refs1) .last = jvmref)

Rule argument

[speclemmaarg: ,rk ≠ rk0 → (st[rk0, val][rk]) = st[rk] ,[rk, val, rk0, st] → [r -- i ', val, r0 -- sk, st] ,F ,store ,javastore-ostorea ,At-other ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r -- sk = r0 -- sk0 ↔ r = r0 ∧ sk = sk0], [CSIMPSEQ : ⊦ _type ≠ i '], [CSIMPSEQ : ⊦ is_array(str, r, st) → is_valid_ref(str, r, st)])], [EXTRATERMS : (r -- i ', val, r0 -- sk, st)]]