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Proof step 11 from proof pathex-def-nodups (previous step)

list2set(refs0) ∩ list2set(refs) = ∅, validrefs(str, r ' + refs0 + r ' + refs, st[r -- _type] .type, st) ⊦ r = (r ' + refs) .last, r ∈ refs, r ∈ refs0, dups(refs), dups(refs0), ∃ refs1. refs1 ≠ [] ∧ refs1 .first = r ∧ refs1 .last = (refs1 .first ' + refs) .last ∧ ¬ dups(refs1) ∧ validrefs(str, refs1, st[refs1 .first -- _type] .type, st)

User applied rule insert lemma

And obtained the following new goals

Proof step 8:
⊦ ¬ validrefs(str, refs + r ' + refs1, ty, st) → ¬ validrefs(str, refs + r ' + refs0 + r ' + refs1, ty, st)
Proof step 10:
¬ validrefs(str, [] + r ' + refs, st[r -- _type] .type, st) → ¬ validrefs(str, [] + r ' + refs0 + r ' + refs, st[r -- _type] .type, st), list2set(refs0) ∩ list2set(refs) = ∅, validrefs(str, r ' + refs0 + r ' + refs, st[r -- _type] .type, st) ⊦ r = (r ' + refs) .last, r ∈ refs, r ∈ refs0, dups(refs), dups(refs0), ∃ refs1. refs1 ≠ [] ∧ refs1 .first = r ∧ refs1 .last = (refs1 .first ' + refs) .last ∧ ¬ dups(refs1) ∧ validrefs(str, refs1, st[refs1 .first -- _type] .type, st)

Rule argument

[LEMMAARG : validrefs-delcycle-01, [str, refs, r, refs1, ty, st, refs0] → [str, [], r, refs, st[r -- _type] .type, st, refs0]]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (str, [], r, refs, st[r -- _type] .type, st, refs0)]]