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

ce.nxt = H[H[r1].nxt].nxt, path(r ' + x, H), H[r1].nxt ∈ H, r1 ∈ H, (r ' + x).last ∈ H ⊦ r1 = null, H[r1].nxt = null, (r ' + x).last = H[r1].nxt, reachable(r, (r ' + x).last, H[r1, ce])

User applied rule structural induction

And obtained the following new goals

Proof step 6:
∀ ce, H, r1, r. ce.nxt = H[H[r1].nxt].nxt ∧ path(r ' + x0, H) ∧ H[r1].nxt ∈ H ∧ r1 ∈ H ∧ (r ' + x0).last ∈ H → r1 = null ∨ H[r1].nxt = null ∨ (r ' + x0).last = H[r1].nxt ∨ reachable(r, (r ' + x0).last, H[r1, ce]), ce.nxt = H[H[r1].nxt].nxt, path(r ' + (r0 + x0), H), H[r1].nxt ∈ H, r1 ∈ H, (r ' + (r0 + x0)).last ∈ H ⊦ r1 = null, H[r1].nxt = null, (r ' + (r0 + x0)).last = H[r1].nxt, reachable(r, (r ' + (r0 + x0)).last, H[r1, ce])

Rule argument

indvar: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ ¬ ¬(r ∈ H ∧ r ≠ null);%self-reachable: ⊦ reachable(r, r, H) ↔ ¬ ¬(r ≠ null ∧ r ∈ H);%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]%In-insert: ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; from specification store[linheap]