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Proof step 125 from proof pathex-put-nopathex (previous step)

st[r -- mkfs(class, ty, fieldvar) '] = refval(r1), is_obj(str, r, st), mkfs(class, ty, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r1 ' + refs1, ty, 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 = r0, path∃(str, r, (r1 ' + refs1) .last, st[r0 -- sk, val]), (is_reftype(ty) ∨ ty = abstract_type) ∧ r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ ty ∨ junkref(str, r1, st)), r1 ≠ jvmref ∧ st[r1 -- _type] .type ≤ ty ∧ path∃(str, r1, r0, st), r1 = jvmref ∧ r0 = jvmref

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 114:
⊦ is_obj(str, r, st) ∧ r ≠ r0 → ( path∃(str, r, r0, st) ↔ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ ( st[r -- fs '] .refval ≠ jvmref ∧ st[st[r -- fs '] .refval -- _type] .type ≤ fs .type ∧ path∃(str, st[r -- fs '] .refval, r0, st) ∨ st[r -- fs '] .refval = jvmref ∧ r0 = jvmref)))
Proof step 115:
⊦ path∃(str, r, r, st) ↔ r = jvmref ∨ r ≠ jvmref ∧ is_valid_ref(str, r, st)
Proof step 124:
is_obj(str, r, st[r0 -- sk, val]) ∧ r ≠ (r1 ' + refs1) .last, st[r -- mkfs(class, ty, fieldvar) '] = refval(r1), is_obj(str, r, st), mkfs(class, ty, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, r1 ' + refs1, ty, 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 = r0, ∃ fs. fs ' ∈ instfields((st[r0 -- sk, val][r -- _type]) .type .class) ∧ is_referencevalue(st[r0 -- sk, val][r -- fs ']) ∧ ( (st[r0 -- sk, val][r -- fs ']) .refval ≠ jvmref ∧ (st[r0 -- sk, val][(st[r0 -- sk, val][r -- fs ']) .refval -- _type]) .type ≤ fs .type ∧ path∃(str, (st[r0 -- sk, val][r -- fs ']) .refval, (r1 ' + refs1) .last, st[r0 -- sk, val]) ∨ (st[r0 -- sk, val][r -- fs ']) .refval = jvmref ∧ (r1 ' + refs1) .last = jvmref), (is_reftype(ty) ∨ ty = abstract_type) ∧ r1 ≠ jvmref ∧ (¬ st[r1 -- _type] .type ≤ ty ∨ junkref(str, r1, st)), r1 ≠ jvmref ∧ st[r1 -- _type] .type ≤ ty ∧ path∃(str, r1, r0, st), r1 = jvmref ∧ r0 = jvmref

Rule argument

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

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ≠ r0 → ( is_obj (str, r0, st[r -- sk, val]) ↔ is_obj(str, r0, st))], [CSIMPSEQ : ⊦ r ≠ r0 → ( is_valid_ref (str, r0, st[r -- sk, val]) ↔ is_valid_ref(str, r0, st))], [CSIMPSEQ : ⊦ is_obj(str, r, st) → is_valid_ref(str, r, st)])], [EXTRATERMS : (st[r0 -- sk, val], (r1 ' + refs1) .last, r, str)]]