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Proof step 30 from proof pathex-put-pathex-array-refval (previous step)

st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- i0 '] = refval(r), validrefs(str, r ' + refs, ty, st), 0 ≤ i0, 0 ≤ i, 0 < st[r0 -- _length] .intval, validref(str, r2, ty, st), i < st[r0 -- _length] .intval, is_array(str, r0, st), is_array(str, r0, st[r0 -- i ', refval(r2)]), i0 < st[r0 -- _length] .intval, r2 = jvmref → is_reftype(ty) ⊦ r0 = r, r0 = (r ' + refs) .last, r0 = jvmref, i0 = i, r ∈ refs, dups(refs), r0 ∈ refs, path∃(str, r2, (r ' + refs) .last, st), validrefs(str, r0 ' + r ' + refs, mkarraytype(ty), st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ 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 29:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ mkarraytype(ty), st[r0 -- _type] .type = mkarraytype(ty), st[r0 -- i0 '] = refval(r), validrefs(str, r ' + refs, ty, st), 0 ≤ i0, 0 ≤ i, 0 < st[r0 -- _length] .intval, validref(str, r2, ty, st), i < st[r0 -- _length] .intval, is_array(str, r0, st), is_array(str, r0, st[r0 -- i ', refval(r2)]), i0 < st[r0 -- _length] .intval, r2 = jvmref → is_reftype(ty) ⊦ r0 = r, r0 = (r ' + refs) .last, r0 = jvmref, i0 = i, r ∈ refs, dups(refs), r0 ∈ refs, path∃(str, r2, (r ' + refs) .last, st), r ' + refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r0 -- _length] .intval ∧ is_referencevalue(st[r0 -- i ']) ∧ (r ' + refs) .first = st[r0 -- i '] .refval ∧ validrefs(str, r ' + refs, st[r0 -- _type] .type .type, st))

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: validrefs-array antecedent: succedent: 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))) substlist: [st, ty, refs, r, str] → [st, mkarraytype(ty), r ' + refs, r0, str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((1, 0, 8)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : typefuns, , subarray1, ⊦ mkarraytype(ty) ≤ mkarraytype(ty0) ↔ ty ≤ ty0], [CSIMPNAMEDSEQ : typefuns, , ref, ⊦ ty ≤ ty])], [EXTRATERMS : (st, mkarraytype(ty), r ' + refs, r0, str)]]