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Proof step 10 from proof reachable-nodups-def (previous step)

path(H[r].nxt ' + x0, H), r ∈ H, ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

User applied rule all left

And obtained the following new goals

Proof step 11:
path(H[r].nxt ' + x0, H), r ∈ H, path(H[r].nxt ' + x0, H) → (∃ x1. path(H[r].nxt ' + x1, H) ∧ (H[r].nxt ' + x1).last = (H[r].nxt ' + x0).last ∧ ¬ H[r].nxt ∈ x1 ∧ ¬ dups(x1)), ∀ r, H. path(r ' + x0, H) → (∃ x1. path(r ' + x1, H) ∧ (r ' + x1).last = (r ' + x0).last ∧ ¬ r ∈ x1 ∧ ¬ dups(x1)) ⊦ r = null, ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (H[r].nxt ' + x0).last ∧ ¬ r ∈ x ∧ ¬ dups(x)

Rule argument

pos: left 3 terms: H[r].nxt, H abort: F discard: F computed: T

Additional information

Used terms: H[r].nxt, H