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Proof step 25 from proof validrefs-nodups (previous step)

validrefs(str, r ' + refs1 + jvmref ' + refs0 + jvmref ', ty, st), ∀ st, ty, str, refs3. # refs3 < # (r + (refs1 + (jvmref + (refs0 + (jvmref + []))))) → ( validrefs(str, refs3, ty, st) ∧ refs3 ≠ [] ∧ refs3 .last = jvmref → (∃ refs1. validrefs(str, refs1, ty, st) ∧ refs1 ≠ [] ∧ refs1 .last = jvmref ∧ refs3 .first = refs1 .first ∧ ¬ dups(refs1) ∧ refs1 ⊆ refs3)) ⊦ r = jvmref, r ∈ refs1, r ∈ refs0, ∃ refs. validrefs(str, refs, ty, st) ∧ refs ≠ [] ∧ refs .last = jvmref ∧ r = refs .first ∧ ¬ dups(refs) ∧ refs ⊆ refs .first ' + refs1 + jvmref ' + refs0 + jvmref '

User applied rule insert lemma

And obtained the following new goals

Rule argument

[LEMMAARG : validrefs-contains-null, [str, refs, refs0, ty, st] → [str, r ' + refs1, refs0 + jvmref ', ty, st]]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ r ' ≠ []])], [EXTRATERMS : (str, r ' + refs1, refs0 + jvmref ', ty, st)]]