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Proof step 9 from proof cyclic-class (previous step)

st[r -- mkfs(class, ty, fieldvar) '] = refval(r0), st[r0 -- _type] .type ≤ ty, is_reftype(ty), mkfs(class, ty, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), validrefs(str, r0 ' + refs0, st[r0 -- _type] .type, st) ⊦ validrefs(str, r ' + r0 ' + refs0, st[r -- _type] .type, st), ¬ r0 ∈ refs0 ∧ ¬ dups(refs0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ is_obj(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ refs .first = st[r -- fs '] .refval ∧ validrefs(str, refs, fs .type, st)))
Proof step 8:
is_obj(str, r, st) ∧ st[r -- _type] .type ≤ st[r -- _type] .type, st[r -- mkfs(class, ty, fieldvar) '] = refval(r0), st[r0 -- _type] .type ≤ ty, is_reftype(ty), mkfs(class, ty, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), validrefs(str, r0 ' + refs0, st[r0 -- _type] .type, st) ⊦ r0 ' + refs0 = [] ∨ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ (r0 ' + refs0) .first = st[r -- fs '] .refval ∧ validrefs(str, r0 ' + refs0, fs .type, st)), ¬ r0 ∈ refs0 ∧ ¬ dups(refs0)

Rule argument

[crewrite-arg: validrefs-class , F , [st, ty, refs, r, str] → [st, st[r -- _type] .type, r0 ' + refs0, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ty ≤ ty])], [EXTRATERMS : (st, st[r -- _type] .type, r0 ' + refs0, r, str)]]