[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 10 from proof validrefs-junkrefs (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
⊦ validrefs(str, [], ty, st)
Proof step 9:
st[r -- mkfs(class, ty1, fieldvar) '] = refval(r0), junkrefs(str, r0 ' + refs1, ty1, st), st[r -- _type] .type ≤ ty, is_obj(str, r, st), mkfs(class, ty1, fieldvar) ' ∈ instfields(st[r -- _type] .type .class), ¬ is_valid_ref(str, r0, st) ∨ ¬ st[r0 -- _type] .type ≤ ty1, ∀ str0, ty0, st0. junkrefs(str0, r0 ' + refs1, ty0, st0) → (∃ refs, r, refs0. r0 ' + refs1 = refs + r ' + refs0 ∧ validrefs(str0, refs, ty0, st0) ∧ r ≠ jvmref ∧ ( ¬ is_valid_ref(str0, r, st0) ∨ refs = [] ∧ ¬ st0[r -- _type] .type ≤ ty0 ∨ refs ≠ [] ∧ ( is_obj(refs .last, st0) ∧ (∃ fs. fs ' ∈ instfields(st0[refs .last -- _type] .type .class) ∧ is_referencevalue(st0[refs .last -- fs ']) ∧ r = 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 ']) ∧ r = st0[refs .last -- i '] .refval ∧ ¬ st0[st0[refs .last -- i '] .refval -- _type] .type ≤ st0[refs .last -- _type] .type .type)))) ⊦ r = jvmref, r0 = jvmref, validrefs(str, r ', ty, st) ∧ ( ¬ is_valid_ref(str, r0, st) ∨ is_obj(r, st) ∧ (∃ fs. fs ' ∈ instfields(st[r -- _type] .type .class) ∧ is_referencevalue(st[r -- fs ']) ∧ r0 = st[r -- fs '] .refval ∧ ¬ st[st[r -- fs '] .refval -- _type] .type ≤ fs .type) ∨ is_array(r, st) ∧ (∃ i. ¬ i < 0 ∧ ¬ st[r -- _length] .intval ≤ i ∧ is_referencevalue(st[r -- i ']) ∧ r0 = st[r -- i '] .refval ∧ ¬ st[st[r -- i '] .refval -- _type] .type ≤ st[r -- _type] .type .type))

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ' .last = r], [CSIMPSEQ : ⊦ refs + [] = refs], [CSIMPSEQ : ⊦ ¬ r ∈ []], [CSIMPSEQ : ⊦ r ' + refs = r0 ' + refs0 ↔ r = r0 ∧ refs = refs0], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ [] + refs = refs])]]