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Proof step 6 from proof add-preserves-reachable-rev (previous step)

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

User applied rule structural induction

And obtained the following new goals

Proof step 7:
H[r1].nxt = H[ce.nxt].nxt, path(r ' + (r0 + x0), H[r1, ce]), reachable(r, r1, H), (r ' + (r0 + x0)).last ∈ H[r1, ce], r1 ∈ H, ce.nxt ∈ H, ∀ H, r1, ce, r. H[r1].nxt = H[ce.nxt].nxt ∧ path(r ' + x0, H[r1, ce]) ∧ reachable(r, r1, H) ∧ ce.nxt ∈ H ∧ r1 ∈ H ∧ (r ' + x0).last ∈ H[r1, ce] → (r ' + x0).last = ce.nxt ∨ ce.nxt = null ∨ reachable(r, (r ' + x0).last, H) ∨ reachable(r, ce.nxt, H) ⊦ (r ' + (r0 + x0)).last = ce.nxt, ce.nxt = null, reachable(r, ce.nxt, H), reachable(r, (r ' + (r0 + x0)).last, H)

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 [], +;;% ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s)% ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; (ws)