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Proof step 51 from proof junkrefs-put-nopathex-equiv (previous step)

st[r1 -- _type] .type ≤ ty, is_obj(str, r1, st) ∨ is_array(str, r1, st), is_valid_ref(str, r0, st), junkrefs(str, r1 ' + acc, ty, st), ∀ str0, r2, st0, sk0, ty0, val0. ¬ path∃(str0, acc .first, r2, st0) ∧ is_valid_ref(str0, r2, st0) ∧ sk0 ≠ _type ∧ acc ≠ [] → (junkrefs(str0, acc, ty0, st0[r2 -- sk0, val0]) ↔ junkrefs(str0, acc, ty0, st0)) ⊦ r0 = r1, r1 = jvmref, sk = _type, path∃(str, r1, r0, st), junkrefs(str, r1 ' + acc, ty, st[r0 -- sk, val])

User applied rule case distinction

And obtained the following new goals

Proof step 29:
is_obj(str, r1, st), st[r1 -- _type] .type ≤ ty, is_valid_ref(str, r0, st), junkrefs(str, r1 ' + acc, ty, st), ∀ str0, r2, st0, sk0, ty0, val0. ¬ path∃(str0, acc .first, r2, st0) ∧ is_valid_ref(str0, r2, st0) ∧ sk0 ≠ _type ∧ acc ≠ [] → (junkrefs(str0, acc, ty0, st0[r2 -- sk0, val0]) ↔ junkrefs(str0, acc, ty0, st0)) ⊦ r0 = r1, r1 = jvmref, sk = _type, path∃(str, r1, r0, st), junkrefs(str, r1 ' + acc, ty, st[r0 -- sk, val])
Proof step 50:
is_array(str, r1, st), st[r1 -- _type] .type ≤ ty, is_valid_ref(str, r0, st), junkrefs(str, r1 ' + acc, ty, st), ∀ str0, r2, st0, sk0, ty0, val0. ¬ path∃(str0, acc .first, r2, st0) ∧ is_valid_ref(str0, r2, st0) ∧ sk0 ≠ _type ∧ acc ≠ [] → (junkrefs(str0, acc, ty0, st0[r2 -- sk0, val0]) ↔ junkrefs(str0, acc, ty0, st0)) ⊦ r0 = r1, r1 = jvmref, sk = _type, path∃(str, r1, r0, st), junkrefs(str, r1 ' + acc, ty, st[r0 -- sk, val])

Rule argument

[CASEDARG : pos: left 2, ()]

Additional information

[[CONCRETESIMPRULES : ()]]