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Proof step 29 from proof rem-stays-reachable-aux1 (previous step)

path(H[H[r0].nxt].nxt ' + x0 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H ⊦ reachable(H[H[r0].nxt].nxt, H[r0].nxt, H), r0 = H[H[r0].nxt].nxt, H[r0].nxt = null, H[r0].nxt = H[H[r0].nxt].nxt, r0 ∈ x0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 31:
path(H[H[r0].nxt].nxt ' + x0 + H[r0].nxt ', H), H[r0].nxt ∈ H, r0 ∈ H ⊦ ∃ x. path(H[H[r0].nxt].nxt ' + x, H) ∧ (H[H[r0].nxt].nxt ' + x).last = H[r0].nxt, r0 = H[H[r0].nxt].nxt, H[r0].nxt = null, H[r0].nxt = H[H[r0].nxt].nxt, r0 ∈ x0

Rule argument

spec: inst: name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H, H[r0].nxt, H[H[r0].nxt].nxt] flags: rewrite, current

Additional information

Used terms: H, H[r0].nxt, H[H[r0].nxt].nxt