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Proof step 49 from proof validrefs-set-null (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 48:
fs = mkfs(class, ty1, fieldvar), (acc + refs0) .first = st[r -- i '] .refval, 0 < st[r -- _length] .intval, 0 ≤ i, validrefs(str, acc + refs0, st[r -- _type] .type .type, st), is_referencevalue(st[r -- i ']), i < st[r -- _length] .intval, st[r -- _type] .type ≤ ty, is_reftype(ty1), is_obj(str, (r ' + acc) .last, st), fs ' ∈ instfields(st[(r ' + acc) .last -- _type] .type .class), is_array(str, r, st), ∀ 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 = [], r = jvmref, validrefs(str, (r ' + acc) + jvmref ', ty, st[(r ' + acc) .last -- fs ', refval(jvmref)]), dups(acc), r ∈ acc, is_array(str, r, st[(r ' + acc) .last -- fs ', refval(jvmref)]) ∧ (st[(r ' + acc) .last -- fs ', refval(jvmref)][r -- _type]) .type ≤ ty

Rule argument

[speclemmaarg: , class = fs .class ∧ ty = fs .type ∧ fieldvar = fs .field ↔ fs = mkfs(class, ty, fieldvar) ,[fs, class, ty, fieldvar] → [fs, class, ty1, fieldvar] ,T ,fieldspec , ,elim ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPELIM : ⊦ class = fs .class ∧ ty = fs .type ∧ fieldvar = fs .field ↔ fs = mkfs(class, ty, fieldvar)])], [EXTRATERMS : (fs, class, ty1, fieldvar)]]