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Proof step 2 from proof path-split-last (previous step)

⊦ path(r ' + x + r0 ', H) ↔ path(r ' + x, H) ∧ H[(r ' + x).last].nxt = r0 ∧ H[(r ' + x).last].nxt ∈ H ∧ H[(r ' + x).last].nxt ≠ null

User applied rule structural induction

And obtained the following new goals

Proof step 3:
∀ r, r0, H. path(r ' + x0 + r0 ', H) ↔ path(r ' + x0, H) ∧ H[(r ' + x0).last].nxt = r0 ∧ H[(r ' + x0).last].nxt ∈ H ∧ H[(r ' + x0).last].nxt ≠ null ⊦ path(r ' + (r1 + x0) + r0 ', H) ↔ path(r ' + (r1 + x0), H) ∧ H[(r ' + (r1 + x0)).last].nxt = r0 ∧ H[(r ' + (r1 + x0)).last].nxt ∈ H ∧ H[(r ' + (r1 + x0)).last].nxt ≠ null

Rule argument

indvar: x

Additional information

Used simplifier rules: path-two: ⊦ path(r ' + r0 ', H) ↔ ¬ ¬(r ∈ H ∧ r0 ∈ H ∧ r ≠ null ∧ r0 ≠ null ∧ H[r].nxt = r0);%path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%reflist freely generated by [], +;;%Nil: ⊦ [] + x0 = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%app: ⊦ x0 + [] = x0; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist]