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Proof step 5 from proof reachable-trans (previous step)

path(r ' + x, H), reachable((r ' + x).last, r1, H), (r ' + x).last ∈ H, r1 ∈ H ⊦ reachable(r, r1, H)

User applied rule structural induction

And obtained the following new goals

Proof step 6:
∀ r, H, r1. path(r ' + x0, H) ∧ reachable((r ' + x0).last, r1, H) ∧ (r ' + x0).last ∈ H ∧ r1 ∈ H → reachable(r, r1, H), path(r ' + (r0 + x0), H), reachable((r ' + (r0 + x0)).last, r1, H), (r ' + (r0 + x0)).last ∈ H, r1 ∈ H ⊦ reachable(r, r1, H)

Rule argument

indvar: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%reflist freely generated by [], +;;%app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist]