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Proof step 29 from proof pathexplus-pathex (previous step)

is_array(str, (r ' + refs) .last, st), validrefs(str, (r ' + refs) .last ' + r0 ', st[(r ' + refs) .last -- _type] .type, st), is_valid_ref(str, r, st), validrefs(str, r ' + refs + r0 ', mkclasstype(`java.lang.Object' .class), st), validrefs(str, r ' + refs, mkclasstype(`java.lang.Object' .class), st), is_arraytype(st[(r ' + refs) .last -- _type] .type) ⊦ (r ' + refs) .last = jvmref, r = jvmref, r = r0, ∃ i. ¬ i < 0 ∧ ¬ st[(r ' + refs) .last -- _length] .intval ≤ i ∧ is_referencevalue(st[(r ' + refs) .last -- i ']) ∧ st[(r ' + refs) .last -- i '] .refval = r0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
⊦ 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 28:
is_array(str, (r ' + refs) .last, st) ∧ st[(r ' + refs) .last -- _type] .type ≤ st[(r ' + refs) .last -- _type] .type, is_array(str, (r ' + refs) .last, st), r0 ' = [] ∨ (∃ i. 0 ≤ i ∧ i < st[(r ' + refs) .last -- _length] .intval ∧ is_referencevalue(st[(r ' + refs) .last -- i ']) ∧ r0 ' .first = st[(r ' + refs) .last -- i '] .refval ∧ validrefs(str, r0 ', st[(r ' + refs) .last -- _type] .type .type, st)), is_valid_ref(str, r, st), validrefs(str, r ' + refs + r0 ', mkclasstype(`java.lang.Object' .class), st), validrefs(str, r ' + refs, mkclasstype(`java.lang.Object' .class), st), is_arraytype(st[(r ' + refs) .last -- _type] .type) ⊦ (r ' + refs) .last = jvmref, r = jvmref, r = r0, ∃ i. ¬ i < 0 ∧ ¬ st[(r ' + refs) .last -- _length] .intval ≤ i ∧ is_referencevalue(st[(r ' + refs) .last -- i ']) ∧ st[(r ' + refs) .last -- i '] .refval = r0

Rule argument

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

Additional information

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