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Proof step 44 from proof validrefs-junkrefs (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 43:
fs = mkfs(class, ty1, fieldvar), is_referencevalue(st[r -- fs ']), fs ' ∈ instfields(st[r -- _type] .type .class), is_obj(str, r, st), st[r -- _type] .type ≤ ty, junkrefs(str, st[r -- fs '] .refval ' + refs1, ty1, st), ∀ str0, ty0, st0. junkrefs(str0, st[r -- fs '] .refval ' + refs1, ty0, st0) → (∃ refs, r0, refs0. st[r -- fs '] .refval ' + 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, ∃ refs, r0, refs0. r ' + st[r -- fs '] .refval ' + refs1 = refs + r0 ' + refs0 ∧ validrefs(str, refs, ty, st) ∧ r0 ≠ jvmref ∧ ( ¬ is_valid_ref(str, r0, st) ∨ refs = [] ∧ ¬ st[r0 -- _type] .type ≤ ty ∨ refs ≠ [] ∧ ( is_obj(refs .last, st) ∧ (∃ fs. fs ' ∈ instfields(st[refs .last -- _type] .type .class) ∧ is_referencevalue(st[refs .last -- fs ']) ∧ r0 = st[refs .last -- fs '] .refval ∧ ¬ st[st[refs .last -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(refs .last, st) ∧ (∃ i. ¬ i < 0 ∧ ¬ st[refs .last -- _length] .intval ≤ i ∧ is_referencevalue(st[refs .last -- i ']) ∧ r0 = st[refs .last -- i '] .refval ∧ ¬ st[st[refs .last -- i '] .refval -- _type] .type ≤ st[refs .last -- _type] .type .type)))

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)]]