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Proof step 101 from proof reachable-addnew-rev (previous step)

reachable(r, r0, H[H[r0].nxt, ce]), reachable(r, r0, H), r ∈ H, r0 ∈ H, r0 ∈ H[H[r0].nxt, ce] ⊦ H[r0].nxt = null, reachable(ce.nxt, H[r0].nxt, H[H[r0].nxt, ce]), reachable(r, H[r0].nxt, H[H[r0].nxt, ce]), H[r0].nxt ∈ H

User applied rule insert lemma

And obtained the following new goals

Proof step 102:
⊦ reachable(r, r1, H) → (reachable(r, H[r1].nxt, H) ↔ ¬(H[r1].nxt = null ∨ ¬ H[r1].nxt ∈ H))

Rule argument

spec: <toplevel> name: reach-reachnext seq: ⊦ reachable(r, r1, H) → ( reachable(r, H[r1].nxt, H) ↔ ¬(H[r1].nxt = null ∨ ¬ H[r1].nxt ∈ H)) subst: [r, r1, H] → [r, r0, H[H[r0].nxt, ce]]

Additional information

Used simplifier rules: reachable-addnew: ⊦ reachable(r, r1, H) ∧ ¬ r2 ∈ H → (reachable(r, r1, H[r2, ce]) ↔ true);% ⊦ r ∈ H ∧ ¬ r0 ∈ H → H[r0, ce][r] = H[r]; (s)% ⊦ r ∈ H[r, ce]; (s) Used terms: r, r0, H[H[r0].nxt, ce]