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Proof step 31 from proof validrefs-nodups (previous step)

refs .last = jvmref, refs4 .last = jvmref, validrefs(str, (r ' + refs1) + r0 ' + refs, ty, st), validrefs(str, r ' + refs1 + r0 ' + refs0 + r0 ' + refs, ty, st), refs4 ⊆ (r ' + refs1) + r0 ' + refs, validrefs(str, r ' + refs4, ty, st), ∀ st, ty, str, refs3. # refs3 < # (r + (refs1 + (r0 + (refs0 + (r0 + refs))))) → ( 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)) ⊦ refs4 = [], r = r0, refs = [], r ∈ refs1, r ∈ refs0, r ∈ refs, dups(refs4), r ∈ refs4, r ' + refs4 ⊆ r ' + refs1 + r0 ' + refs0 + r0 ' + refs

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

[speclemmaarg: ,refs ⊆ refs0 ↔ (∀ r. r ∈ refs → r ∈ refs0) ,[refs0, refs] → [r ' + refs1 + r0 ' + refs0 + r0 ' + refs, r ' + refs4] ,F ,list-set ,references ,subset ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ r ∈ refs + refs0 ↔ ¬ ¬ (r ∈ refs ∨ r ∈ refs0)], [CSIMPSEQ : ⊦ r ∈ r0 ' ↔ r = r0], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ ¬ r ∈ r0 ' + refs ↔ r ≠ r0 ∧ ¬ r ∈ refs], [CSIMPSEQ : ⊦ r ≠ r0 → (r ∈ r0 ' + refs ↔ r ∈ refs)], [CSIMPFORWARD : ⊦ refs ⊆ refs0 ∧ r ∈ refs → r ∈ refs0])], [EXTRATERMS : (r ' + refs1 + r0 ' + refs0 + r0 ' + refs, r ' + refs4)]]