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Proof step 7 from proof path-new-irrelevant-strong (previous step)

r ∈ H, H[r].nxt ∈ H → (path(H[r].nxt ' + x0, H[r2, ce]) ↔ path(H[r].nxt ' + x0, H)), ∀ r, H, r2, ce. r ∈ H → (path(r ' + x0, H[r2, ce]) ↔ path(r ' + x0, H)) ∨ r2 ∈ H ∨ r2 ∈ x0 ⊦ r ∈ H[r2, ce] ∧ path(H[r].nxt ' + x0, H[r2, ce]) ↔ path(H[r].nxt ' + x0, H), r = null, r2 = H[r].nxt, r2 ∈ x0, r2 ∈ H

User applied rule case distinction

And obtained the following new goals

Proof step 8:
path(H[r].nxt ' + x0, H[r2, ce]), r ∈ H, r ∈ H[r2, ce], H[r].nxt ∈ H → (path(H[r].nxt ' + x0, H[r2, ce]) ↔ path(H[r].nxt ' + x0, H)), ∀ r, H, r2, ce. r ∈ H → (path(r ' + x0, H[r2, ce]) ↔ path(r ' + x0, H)) ∨ r2 ∈ H ∨ r2 ∈ x0 ⊦ r = null, r2 = H[r].nxt, path(H[r].nxt ' + x0, H), r2 ∈ H, r2 ∈ x0

Rule argument

pos: right 1

Additional information

Used simplifier rules: path-first-in-H: ⊦ path(r ' + x, H) → (r ∈ H ↔ true);%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r ∈ H[r0, ce] ↔ r = r0 ∨ r ∈ H; (ws)