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Proof step 55 from proof maxpath-object (previous step)

maxpath(str, r0, st) = 1 + # refs1, st[r -- fs '] = refval(r0), maxpath(str, r, st) = 2 + # refs, validrefs(str, r0 ' + refs1, 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 < # refs1, is_valid_ref(str, r0, st) ⊦ r ' + r0 ' + refs1 ∈ validrefset(str, r, st), validrefset(str, r, st) = ∅, # validrefset(str, r, st) = 0, r = r1, validrefset(str, r0, st) = ∅, # validrefset(str, r0, st) = 0, refs1 = [], junkref(str, r, st), cyclic(str, r, st), has_reffield(r1, st), r ∈ refs, r1 ∈ refs, dups(refs), r0 ∈ refs1, dups(refs1), has_reffield(refs1 .last, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ 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))
Proof step 54:
is_valid_ref(str, r, st), maxpath(str, r0, st) = 1 + # refs1, st[r -- fs '] = refval(r0), maxpath(str, r, st) = 2 + # refs, validrefs(str, r0 ' + refs1, 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 < # refs1, is_valid_ref(str, r0, st) ⊦ r ' + r0 ' + refs1 ≠ [] ∧ (r ' + r0 ' + refs1) .first = r ∧ ¬ has_reffield((r ' + r0 ' + refs1) .last, st) ∧ ¬ dups(r ' + r0 ' + refs1) ∧ validrefs(str, r ' + r0 ' + refs1, st[r -- _type] .type, st), validrefset(str, r, st) = ∅, # validrefset(str, r, st) = 0, r = r1, validrefset(str, r0, st) = ∅, # validrefset(str, r0, st) = 0, refs1 = [], junkref(str, r, st), cyclic(str, r, st), has_reffield(r1, st), r ∈ refs, r1 ∈ refs, dups(refs), r0 ∈ refs1, dups(refs1), has_reffield(refs1 .last, st)

Rule argument

[crewrite-arg: validrefset-def , F , [st, r, str, refs] → [st, r, str, r ' + r0 ' + refs1] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ is_valid_ref(str, r, st) → ( validrefs (str, r ' + refs, ty, st) ↔ r = jvmref ∧ refs = [])], [CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ r ' ≠ []], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st, r, str, r ' + r0 ' + refs1)]]