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Proof step 45 from proof validrefs-set-null (previous step)

acc + refs0 = st[r -- i '] .refval ' + refs1, is_array(str, r, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), is_reftype(ty1), st[r -- _type] .type ≤ ty, i < st[r -- _length] .intval, is_referencevalue(st[r -- i ']), validrefs(str, acc + refs0, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ 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)]) ⊦ acc + refs0 = [], r = jvmref, refs0 = [], r ∈ acc, dups(acc), validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]), is_array(str, r, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]) ∧ (st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)][r -- _type]) .type ≤ ty, ¬ st[r -- i '] .refval ∈ acc ∧ ¬ st[r -- i '] .refval ∈ refs0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 44:
st[r -- i '] = refval(r0), acc + refs0 = r0 ' + refs1, is_array(str, r, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_obj(str, (r ' + acc) .last, st), is_reftype(ty1), st[r -- _type] .type ≤ ty, i < st[r -- _length] .intval, is_referencevalue(st[r -- i ']), validrefs(str, acc + refs0, st[r -- _type] .type .type, st), 0 ≤ i, 0 < st[r -- _length] .intval, ∀ 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)]) ⊦ acc + refs0 = [], r = jvmref, refs0 = [], r ∈ acc, dups(acc), validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]), is_array(str, r, st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)]) ∧ (st[(r ' + acc) .last -- mkfs(class, ty1, fieldvar) ', refval(jvmref)][r -- _type]) .type ≤ ty, ¬ r0 ∈ acc ∧ ¬ r0 ∈ refs0

Rule argument

[speclemmaarg: ,is_referencevalue(val) → (r = val .refval ↔ val = refval(r)) ,[val, r] → [st[r -- i '], r0] ,T ,javavalue , ,elim-13 ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : ⊦ is_referencevalue(val) → ( r = val .refval ↔ val = refval(r))])], [EXTRATERMS : (st[r -- i '], r0)]]