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Proof step 99 from proof validrefs-junkrefs (previous step)

junkrefs(str, refs, ty, st) ⊦ ∃ refs0, r, refs1. refs = refs0 + r ' + refs1 ∧ validrefs(str, refs0, ty, st) ∧ r ≠ jvmref ∧ ( ¬ is_valid_ref(str, r, st) ∨ refs0 = [] ∧ ¬ st[r -- _type] .type ≤ ty ∨ refs0 ≠ [] ∧ ( is_obj(refs0 .last, st) ∧ (∃ fs. fs ' ∈ instfields(st[refs0 .last -- _type] .type .class) ∧ is_referencevalue(st[refs0 .last -- fs ']) ∧ r = st[refs0 .last -- fs '] .refval ∧ ¬ st[st[refs0 .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs0 .last, st) ∧ (∃ i. ¬ i < 0 ∧ ¬ st[refs0 .last -- _length] .intval ≤ i ∧ is_referencevalue(st[refs0 .last -- i ']) ∧ r = st[refs0 .last -- i '] .refval ∧ ¬ st[st[refs0 .last -- i '] .refval -- _type] .type ≤ st[refs0 .last -- _type] .type .type)))

User applied rule structural induction

And obtained the following new goals

Proof step 98:
∀ str, ty, st. junkrefs(str, acc, ty, st) → (∃ refs0, r, refs1. acc = refs0 + r ' + refs1 ∧ validrefs(str, refs0, ty, st) ∧ r ≠ jvmref ∧ ( ¬ is_valid_ref(str, r, st) ∨ refs0 = [] ∧ ¬ st[r -- _type] .type ≤ ty ∨ refs0 ≠ [] ∧ ( is_obj(refs0 .last, st) ∧ (∃ fs. fs ' ∈ instfields(st[refs0 .last -- _type] .type .class) ∧ is_referencevalue(st[refs0 .last -- fs ']) ∧ r = st[refs0 .last -- fs '] .refval ∧ ¬ st[st[refs0 .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs0 .last, st) ∧ (∃ i. ¬ i < 0 ∧ ¬ st[refs0 .last -- _length] .intval ≤ i ∧ is_referencevalue(st[refs0 .last -- i ']) ∧ r = st[refs0 .last -- i '] .refval ∧ ¬ st[st[refs0 .last -- i '] .refval -- _type] .type ≤ st[refs0 .last -- _type] .type .type)))), junkrefs(str, r + acc, ty, st) ⊦ ∃ refs0, r0, refs1. r + acc = refs0 + r0 ' + refs1 ∧ validrefs(str, refs0, ty, st) ∧ r0 ≠ jvmref ∧ ( ¬ is_valid_ref(str, r0, st) ∨ refs0 = [] ∧ ¬ st[r0 -- _type] .type ≤ ty ∨ refs0 ≠ [] ∧ ( is_obj(refs0 .last, st) ∧ (∃ fs. fs ' ∈ instfields(st[refs0 .last -- _type] .type .class) ∧ is_referencevalue(st[refs0 .last -- fs ']) ∧ r0 = st[refs0 .last -- fs '] .refval ∧ ¬ st[st[refs0 .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs0 .last, st) ∧ (∃ i. ¬ i < 0 ∧ ¬ st[refs0 .last -- _length] .intval ≤ i ∧ is_referencevalue(st[refs0 .last -- i ']) ∧ r0 = st[refs0 .last -- i '] .refval ∧ ¬ st[st[refs0 .last -- i '] .refval -- _type] .type ≤ st[refs0 .last -- _type] .type .type)))

Rule argument

[VARWITHVARSEQSARG : refs, ()]

Additional information

[[CONCRETESIMPRULES : ([CSIMPGEN : references generated by [], +;], [CSIMPSEQ : ⊦ ¬ junkrefs(str, [], ty, st)])]]