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Proof step 145 from proof validref-refs-last-valid-subset (previous step)

(acc + r1 ') .first = st0[r0 -- mkfs(class, ty0, fieldvar) '] .refval, is_obj(str, r0, st0), validref(str, r, st[r -- _type] .type, st), st ⊆ st0, st0[r0 -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st0[r0 -- _type] .type .class), is_referencevalue(st0[r0 -- mkfs(class, ty0, fieldvar) ']), validrefs(str, acc + r1 ', ty0, st0), ∀ str, r, st, st0, r1, ty. validref(str, r, st[r -- _type] .type, st) ∧ st ⊆ st0 ∧ r ∈ acc ∧ validrefs(str, acc + r1 ', ty, st0) → validref(str, r1, st[r1 -- _type] .type, st) ⊦ r0 = jvmref, validref(str, r1, st[r1 -- _type] .type, st), r ≠ r0 ∧ ¬ r ∈ acc

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 144:
acc + r1 ' = r2 + refs ∧ acc + r1 ' ≠ [], r2 = st0[r0 -- mkfs(class, ty0, fieldvar) '] .refval, is_obj(str, r0, st0), validref(str, r, st[r -- _type] .type, st), st ⊆ st0, st0[r0 -- _type] .type ≤ ty, mkfs(class, ty0, fieldvar) ' ∈ instfields(st0[r0 -- _type] .type .class), is_referencevalue(st0[r0 -- mkfs(class, ty0, fieldvar) ']), validrefs(str, acc + r1 ', ty0, st0), ∀ str, r, st, st0, r1, ty. validref(str, r, st[r -- _type] .type, st) ∧ st ⊆ st0 ∧ r ∈ acc ∧ validrefs(str, acc + r1 ', ty, st0) → validref(str, r1, st[r1 -- _type] .type, st) ⊦ r0 = jvmref, validref(str, r1, st[r1 -- _type] .type, st), r ≠ r0 ∧ ¬ r ∈ acc

Rule argument

[speclemmaarg: refs ≠ [] ,r = refs .first ∧ refs0 = refs .rest ↔ refs = r + refs0 ,[refs, r, refs0] → [acc + r1 ', r2, refs] ,T ,list-data ,references-olista ,elim-list-c ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ r ' ≠ []], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPELIM : refs ≠ [] ⊦ r = refs .first ∧ refs0 = refs .rest ↔ refs = r + refs0])], [EXTRATERMS : (acc + r1 ', r2, refs)]]