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Proof step 12 from proof validrefs-junkrefs (previous step)

st[refs .last -- mkfs(class0, ty2, fieldvar0) '] = refval(r2), st[r -- mkfs(class, ty1, fieldvar) '] = refval(r1), r1 ' + refs1 = refs + r2 ' + refs0, is_valid_ref(str, r2, st), r1 ∈ refs, junkrefs(str, r1 ' + refs1, ty1, st), st[r -- _type] .type ≤ ty, is_obj(str, r, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), validrefs(str, refs, ty1, st), validrefs(str, r ' + refs, ty, st), is_referencevalue(st[refs .last -- mkfs(class0, ty2, fieldvar0) ']), mkfs(class0, ty2, fieldvar0) ' ∈ instfields(st[refs .last -- _type] .type .class), is_obj(refs .last, st), ∀ str0, ty0, st0. junkrefs(str0, r1 ' + refs1, ty0, st0) → (∃ refs, r0, refs0. r1 ' + refs1 = refs + r0 ' + refs0 ∧ validrefs(str0, refs, ty0, st0) ∧ r0 ≠ jvmref ∧ ( ¬ is_valid_ref(str0, r0, st0) ∨ refs = [] ∧ ¬ st0[r0 -- _type] .type ≤ ty0 ∨ refs ≠ [] ∧ ( is_obj(refs .last, st0) ∧ (∃ fs. fs ' ∈ instfields(st0[refs .last -- _type] .type .class) ∧ is_referencevalue(st0[refs .last -- fs ']) ∧ r0 = st0[refs .last -- fs '] .refval ∧ ¬ st0[st0[refs .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs .last, st0) ∧ (∃ i. 0 ≤ i ∧ i < st0[refs .last -- _length] .intval ∧ is_referencevalue(st0[refs .last -- i ']) ∧ r0 = st0[refs .last -- i '] .refval ∧ ¬ st0[st0[refs .last -- i '] .refval -- _type] .type ≤ st0[refs .last -- _type] .type .type)))) ⊦ r = jvmref, r2 = jvmref, refs = [], st[r2 -- _type] .type ≤ ty2, ∃ fs. fs ' ∈ instfields(st[refs .last -- _type] .type .class) ∧ is_referencevalue(st[refs .last -- fs ']) ∧ r2 = st[refs .last -- fs '] .refval ∧ ¬ st[st[refs .last -- fs '] .refval -- _type] .type ≤ fs .type

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
r1 ' + refs1 = refs + r2 ' + refs0, st[r -- mkfs(class, ty1, fieldvar) '] = refval(r1), st[refs .last -- mkfs(class0, ty2, fieldvar0) '] = refval(r2), is_obj(refs .last, st), mkfs(class0, ty2, fieldvar0) ' ∈ instfields(st[refs .last -- _type] .type .class), validrefs(str, r ' + refs, ty, st), validrefs(str, refs, ty1, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), st[r -- _type] .type ≤ ty, junkrefs(str, r1 ' + refs1, ty1, st), r1 ∈ refs, is_valid_ref(str, r2, st), ∀ str0, ty0, st0. junkrefs(str0, r1 ' + refs1, ty0, st0) → (∃ refs, r0, refs0. r1 ' + refs1 = refs + r0 ' + refs0 ∧ validrefs(str0, refs, ty0, st0) ∧ r0 ≠ jvmref ∧ ( ¬ is_valid_ref(str0, r0, st0) ∨ refs = [] ∧ ¬ st0[r0 -- _type] .type ≤ ty0 ∨ refs ≠ [] ∧ ( is_obj(refs .last, st0) ∧ (∃ fs. fs ' ∈ instfields(st0[refs .last -- _type] .type .class) ∧ is_referencevalue(st0[refs .last -- fs ']) ∧ r0 = st0[refs .last -- fs '] .refval ∧ ¬ st0[st0[refs .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs .last, st0) ∧ (∃ i. 0 ≤ i ∧ i < st0[refs .last -- _length] .intval ∧ is_referencevalue(st0[refs .last -- i ']) ∧ r0 = st0[refs .last -- i '] .refval ∧ ¬ st0[st0[refs .last -- i '] .refval -- _type] .type ≤ st0[refs .last -- _type] .type .type)))) ⊦ refs = [], r2 = jvmref, r = jvmref, st[r2 -- _type] .type ≤ ty2, ∃ fs. fs ' ∈ instfields(st[refs .last -- _type] .type .class) ∧ is_referencevalue(st[refs .last -- fs ']) ∧ r2 = st[refs .last -- fs '] .refval ∧ ¬ st[st[refs .last -- fs '] .refval -- _type] .type ≤ fs .type

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ is_referencevalue(refval(r))], [CSIMPSEQ : ⊦ r ∈ refs → r ∈ refs + refs0], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPFORWARD : ⊦ r ' + refs = refs0 → r ∈ refs0])]]