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Proof step 94 from proof hassharing-validrefs-dups (previous step)

validrefs(str, refs0, st[refs0 .first -- _type] .type, st), validrefs(str, r ' + refs0, st[r -- _type] .type, st), path∃(str, r, r, st), dups(refs0), ∀ st, str, refs, r0. # (r0 ' + refs) < # (r ' + refs0) → (validrefs(str, r0 ' + refs, st[r0 -- _type] .type, st) → hasSharing(r0, st) ∨ ¬ r0 ∈ refs ∧ ¬ dups(refs)) ⊦ refs0 = [], r ∈ refs0, hasSharing(r, st)

User applied rule apply induction

And obtained the following new goals

Proof step 93:
hasSharing(refs0 .first, st) ∨ ¬ refs0 .first ∈ refs0 .rest ∧ ¬ dups(refs0 .rest), validrefs(str, refs0, st[refs0 .first -- _type] .type, st), validrefs(str, r ' + refs0, st[r -- _type] .type, st), path∃(str, r, r, st), dups(refs0), ∀ st, str, refs, r0. # (r0 ' + refs) < # (r ' + refs0) → (validrefs(str, r0 ' + refs, st[r0 -- _type] .type, st) → hasSharing(r0, st) ∨ ¬ r0 ∈ refs ∧ ¬ dups(refs)) ⊦ refs0 = [], r ∈ refs0, hasSharing(r, st)

Rule argument

[substlist-arg: [st, str, refs, r0] → [st, str, refs0 .rest, refs0 .first] ]

Additional information

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