[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 243 from proof exvalidrefs-maxlength (previous step)

st[r -- 0 '] = refval(r0), validrefs(str, r0 ' + refs1 + r1 ', st[r0 -- _type] .type, st -- r), st[r0 -- _type] .type ≤ st[r -- _type] .type .type, is_array(r, st), is_array(str, r, st), 0 < st[r -- _length] .intval, validrefs(str, r0 ' + refs1 + r1 ', st[r0 -- _type] .type, st), is_reftype(st[r -- _type] .type .type) ∨ st[r -- _type] .type .type = void_type, ∀ r, str, st0. # st0 .refs < # st .refs → ( true → cyclic(str, r, st0) ∨ junkref(str, r, st0) ∨ (∃ refs. validrefs(str, refs, st0[refs .first -- _type] .type, st0) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r ∧ ¬ has_reffield(refs .last, st0))) ⊦ r0 = r1, r = r1, r = r0, r0 = jvmref, r = jvmref, r0 ∈ refs1, dups(refs1), r1 ∈ refs1, has_reffield(r1, st), junkref(str, r0, st), cyclic(str, r, st), r ∈ refs1, validrefs(str, r ' + r0 ' + refs1 + r1 ', st[r -- _type] .type, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 239:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 242:
is_array(str, r, st) ∧ st[r -- _type] .type ≤ st[r -- _type] .type, st[r -- 0 '] = refval(r0), validrefs(str, r0 ' + refs1 + r1 ', st[r0 -- _type] .type, st -- r), st[r0 -- _type] .type ≤ st[r -- _type] .type .type, is_array(r, st), is_array(str, r, st), 0 < st[r -- _length] .intval, validrefs(str, r0 ' + refs1 + r1 ', st[r0 -- _type] .type, st), is_reftype(st[r -- _type] .type .type) ∨ st[r -- _type] .type .type = void_type, ∀ r, str, st0. # st0 .refs < # st .refs → ( true → cyclic(str, r, st0) ∨ junkref(str, r, st0) ∨ (∃ refs. validrefs(str, refs, st0[refs .first -- _type] .type, st0) ∧ refs ≠ [] ∧ ¬ dups(refs) ∧ refs .first = r ∧ ¬ has_reffield(refs .last, st0))) ⊦ r0 = r1, r = r1, r = r0, r0 = jvmref, r = jvmref, r0 ∈ refs1, dups(refs1), r1 ∈ refs1, has_reffield(r1, st), junkref(str, r0, st), cyclic(str, r, st), r ∈ refs1, r0 ' + refs1 + r1 ' = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ (r0 ' + refs1 + r1 ') .first = st[r -- i '] .refval ∧ validrefs(str, r0 ' + refs1 + r1 ', st[r -- _type] .type .type, st))

Rule argument

[crewrite-arg: validrefs-array , F , [st, ty, refs, r, str] → [st, st[r -- _type] .type, r0 ' + refs1 + r1 ', r, str] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ ty ≤ ty])], [EXTRATERMS : (st, st[r -- _type] .type, r0 ' + refs1 + r1 ', r, str)]]