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Proof step 10 from proof reachable-modbehind (previous step)

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

User applied rule structural induction

And obtained the following new goals

Proof step 11:
path(r ' + (r0 + x0), H), reachable((r ' + (r0 + x0)).last, r1, H), r1 ∈ H, (r ' + (r0 + x0)).last ∈ H, ∀ r, H, r1, ce. path(r ' + x0, H) ∧ reachable((r ' + x0).last, r1, H) ∧ (r ' + x0).last ∈ H ∧ r1 ∈ H → path(r ' + x0, H[r1, ce]) ∨ reachable(r1, (r ' + x0).last, H) ⊦ path(r ' + (r0 + x0), H[r1, ce]), reachable(r1, (r ' + (r0 + x0)).last, H)

Rule argument

indvar: x

Additional information

Used simplifier rules: path-one: ⊦ path(r ', H) ↔ r ∈ H ∧ r ≠ null;%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)