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Proof step 3 from proof path-modify-strong (previous step)

r0 ∈ H ⊦ path(x + r1 ', H[r0, ce]) ↔ path(x + r1 ', H), r0 ∈ x

User applied rule structural induction

And obtained the following new goals

Proof step 4:
r0 ∈ H, ∀ r0, H, r1, ce. r0 ∈ H → (path(x0 + r1 ', H[r0, ce]) ↔ path(x0 + r1 ', H)) ∨ r0 ∈ x0 ⊦ path((r + x0) + r1 ', H[r0, ce]) ↔ path((r + x0) + r1 ', H), r0 ∈ r + x0

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 ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; (ws)% ⊦ ¬ r ∈ []; (s)