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Proof step 67 from proof validrefs-set-null (previous step)

is_reftype(fs .type), validrefs(str, (r ' + acc) + refs0, ty, st), is_obj(str, (r ' + acc) .last, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ refs0 = [], validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- fs ', refval(jvmref)]), dups(acc), r ∈ acc

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 1:
⊦ r = jvmref → (validrefs(str, r ' + refs, ty, st) ↔ refs = [])
Proof step 66:
r ≠ jvmref, is_reftype(fs .type), validrefs(str, (r ' + acc) + refs0, ty, st), is_obj(str, (r ' + acc) .last, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), ∀ str0, refs, ty0, st0, fs0. validrefs(str0, acc + refs, ty0, st0) ∧ acc ≠ [] ∧ is_obj(str0, acc .last, st0) ∧ fs0 ' ∈ instfields(st0[acc .last -- _type] .type .class) ∧ is_reftype(fs0 .type) ∧ refs ≠ [] → validrefs(str0, acc + jvmref ', ty0, st0[acc .last -- fs0 ', refval(jvmref)]) ⊦ refs0 = [], validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- fs ', refval(jvmref)]), dups(acc), r ∈ acc

Rule argument

[crewrite-arg: validrefs-null , F , [st, ty, refs, r, str] → [st, ty, acc + refs0, r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1])], [EXTRATERMS : (st, ty, acc + refs0, r, str)]]