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Proof step 2 from proof path-new-irrelevant-strong (previous step)

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

User applied rule structural induction

And obtained the following new goals

Proof step 3:
∀ r, H, r2, ce. r ∈ H → (path(r ' + x0, H[r2, ce]) ↔ path(r ' + x0, H)) ∨ r2 ∈ x0 ∨ r2 ∈ H, r ∈ H ⊦ 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 [], +;;%app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%in-empty: ⊦ ¬ r ∈ []; from specification list[reflist]%In-insert: ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; from specification store[linheap]