| maxpath(str, r0, st) = # refs0, st[r -- fs '] = refval(r0), maxpath(str, r, st) = 2 + # refs, refs0 ∈ validrefset(str, r0, st), maxpath(str, r, st) ≤ maxpath(str, r0, st), validrefs(str, r ' + refs + r1 ', st[r -- _type] .type, st), fs ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), # refs + 1 < maxpath(str, r0, st) ⊦ validrefset(str, r, st) = ∅, # validrefset(str, r, st) = 0, r = r1, validrefset(str, r0, st) = ∅, # validrefset(str, r0, st) = 0, junkref(str, r, st), cyclic(str, r, st), has_reffield(r1, st), r ∈ refs, r1 ∈ refs, dups(refs) |
⊦ is_valid_ref(str, r, st) → ( refs ∈ validrefset(str, r, st) ↔ refs ≠ [] ∧ refs .first = r ∧ ¬ has_reffield(refs .last, st) ∧ ¬ dups(refs) ∧ validrefs(str, refs, st[r -- _type] .type, st))
¬ is_valid_ref(str, r0, st), maxpath(str, r0, st) = # refs0, st[r -- fs '] = refval(r0), maxpath(str, r, st) = 2 + # refs, refs0 ∈ validrefset(str, r0, st), maxpath(str, r, st) ≤ maxpath(str, r0, st), validrefs(str, r ' + refs + r1 ', st[r -- _type] .type, st), fs ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), # refs + 1 < maxpath(str, r0, st) ⊦ validrefset(str, r, st) = ∅, # validrefset(str, r, st) = 0, r = r1, validrefset(str, r0, st) = ∅, # validrefset(str, r0, st) = 0, junkref(str, r, st), cyclic(str, r, st), has_reffield(r1, st), r ∈ refs, r1 ∈ refs, dups(refs)
is_valid_ref(str, r0, st), maxpath(str, r0, st) = # refs0, st[r -- fs '] = refval(r0), maxpath(str, r, st) = 2 + # refs, refs0 ≠ [] ∧ refs0 .first = r0 ∧ ¬ has_reffield(refs0 .last, st) ∧ ¬ dups(refs0) ∧ validrefs(str, refs0, st[r0 -- _type] .type, st), maxpath(str, r, st) ≤ maxpath(str, r0, st), validrefs(str, r ' + refs + r1 ', st[r -- _type] .type, st), fs ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), # refs + 1 < maxpath(str, r0, st) ⊦ validrefset(str, r, st) = ∅, # validrefset(str, r, st) = 0, r = r1, validrefset(str, r0, st) = ∅, # validrefset(str, r0, st) = 0, junkref(str, r, st), cyclic(str, r, st), has_reffield(r1, st), r ∈ refs, r1 ∈ refs, dups(refs)
[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, r0, str, refs0)]]