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Proof step 12 from proof reach-reachnext (previous step)

H[(r ' + x).last].nxt ∈ H, path(r ' + x, H), (r ' + x).last ∈ H ⊦ H[(r ' + x).last].nxt = null, path(r ' + x + H[(r ' + x).last].nxt ', H)

User applied rule structural induction

And obtained the following new goals

Proof step 13:
∀ H, r. H[(r ' + x0).last].nxt ∈ H ∧ path(r ' + x0, H) ∧ (r ' + x0).last ∈ H → H[(r ' + x0).last].nxt = null ∨ path(r ' + x0 + H[(r ' + x0).last].nxt ', H), H[(r ' + (r0 + x0)).last].nxt ∈ H, path(r ' + (r0 + x0), H), (r ' + (r0 + x0)).last ∈ H ⊦ H[(r ' + (r0 + x0)).last].nxt = null, path(r ' + (r0 + x0) + H[(r ' + (r0 + x0)).last].nxt ', H)

Rule argument

indvar: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%path-two: ⊦ path(r ' + r0 ', H) ↔ ¬ ¬(r ∈ H ∧ r0 ∈ H ∧ r ≠ null ∧ r0 ≠ null ∧ H[r].nxt = r0);%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]