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Proof step 15 from proof validrefs-nodups (previous step)

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

User applied rule apply induction

And obtained the following new goals

Proof step 14:
∃ refs3. validrefs(str, refs3, ty, st) ∧ refs3 ≠ [] ∧ refs3 .last = jvmref ∧ (r ' + refs) .first = refs3 .first ∧ ¬ dups(refs3) ∧ refs3 ⊆ r ' + refs, refs .last = jvmref, validrefs(str, r ' + refs0 + r ' + refs, ty, st), validrefs(str, r ' + refs, ty, st), ∀ st, ty, str, refs1. # refs1 < # (r + (refs0 + (r + refs))) → ( validrefs(str, refs1, ty, st) ∧ refs1 ≠ [] ∧ refs1 .last = jvmref → (∃ refs3. validrefs(str, refs3, ty, st) ∧ refs3 ≠ [] ∧ refs3 .last = jvmref ∧ refs1 .first = refs3 .first ∧ ¬ dups(refs3) ∧ refs3 ⊆ refs1)) ⊦ refs = [], ∃ refs1. validrefs(str, refs1, ty, st) ∧ refs1 ≠ [] ∧ refs1 .last = jvmref ∧ r = refs1 .first ∧ ¬ dups(refs1) ∧ refs1 ⊆ refs1 .first ' + refs0 + refs1 .first ' + refs

Rule argument

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

Additional information

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