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Proof step 3 from proof path-new-irrelevant-strong (previous step)

r ∈ H ⊦ path(r ' + x, H[r2, ce]) ↔ path(r ' + x, H), r2 ∈ H, r2 ∈ x

User applied rule structural induction

And obtained the following new goals

Proof step 4:
r ∈ H, ∀ r, H, r2, ce. r ∈ H → (path(r ' + x0, H[r2, ce]) ↔ path(r ' + x0, H)) ∨ r2 ∈ H ∨ r2 ∈ x0 ⊦ path(r ' + (r0 + x0), H[r2, ce]) ↔ path(r ' + (r0 + x0), H), r2 ∈ r0 + x0, r2 ∈ 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 ∈ []; (s)% ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; (ws)