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Proof step 23 from proof reach-reachnext (previous step)

path(r ' + x, H), H[(r ' + x).last].nxt ∈ 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 24:
path(r ' + (r0 + x0), H), (r ' + (r0 + x0)).last ∈ H, H[(r ' + (r0 + x0)).last].nxt ∈ H, ∀ r, H. path(r ' + x0, H) ∧ H[(r ' + x0).last].nxt ∈ 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 = 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 [], +;;% ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s)% ⊦ [] + x0 = x0; (s)