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Proof step 39 from proof validrefs-nodups (previous step)

refs .last = jvmref, validrefs(str, r ' + refs1 + r0 ' + refs0 + r0 ' + refs, ty, st), validrefs(str, (r ' + refs1) + r0 ' + refs, 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)) ⊦ refs = [], r = r0, r ∈ refs, r ∈ refs0, r ∈ refs1, ∃ refs2. validrefs(str, refs2, ty, st) ∧ refs2 ≠ [] ∧ refs2 .last = jvmref ∧ r = refs2 .first ∧ ¬ dups(refs2) ∧ refs2 ⊆ refs2 .first ' + refs1 + r0 ' + refs0 + r0 ' + refs

User applied rule apply induction

And obtained the following new goals

Proof step 38:
∃ refs0. validrefs(str, refs0, ty, st) ∧ refs0 ≠ [] ∧ refs0 .last = jvmref ∧ ((r ' + refs1) + r0 ' + refs) .first = refs0 .first ∧ ¬ dups(refs0) ∧ refs0 ⊆ (r ' + refs1) + r0 ' + refs, refs .last = jvmref, validrefs(str, r ' + refs1 + r0 ' + refs0 + r0 ' + refs, ty, st), validrefs(str, (r ' + refs1) + r0 ' + refs, 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)) ⊦ refs = [], r = r0, r ∈ refs, r ∈ refs0, r ∈ refs1, ∃ refs2. validrefs(str, refs2, ty, st) ∧ refs2 ≠ [] ∧ refs2 .last = jvmref ∧ r = refs2 .first ∧ ¬ dups(refs2) ∧ refs2 ⊆ refs2 .first ' + refs1 + r0 ' + refs0 + r0 ' + refs

Rule argument

[substlist-arg: [st, ty, str, refs3] → [st, ty, str, (r ' + refs1) + r0 ' + refs] ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ refs ≠ [] → (refs0 + refs) .last = refs .last], [CSIMPSEQ : ⊦ r ' ≠ []], [CSIMPSEQ : ⊦ ¬ refs + refs0 ≠ [] ↔ refs = [] ∧ refs0 = []], [CSIMPSEQ : ⊦ # (refs + refs0) = # refs + # refs0], [CSIMPSEQ : ⊦ # r ' = 0 +1], [CSIMPSEQ : ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPSEQ : ⊦ r + refs = r ' + refs], [CSIMPSEQ : ⊦ m + n < m + n0 ↔ n < n0], [CSIMPSEQ : n < n0 ⊦ ((* n)) < m0 + ((* n0))], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m])], [EXTRATERMS : (st, ty, str, (r ' + refs1) + r0 ' + refs)]]