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Proof step 35 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, path(r ' + x2 + H[r0].nxt ', H[r0, ce]), path(r ' + x1 + r0 ', H), r0 ∈ x2, H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = H[r0].nxt, dups(x1), dups(x2), r ∈ x1, r ∈ x2, r0 ∈ x1, H[r0].nxt ∈ x2, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 36:
⊦ r ∈ x0 ↔ (∃ x1, x2. x0 = x1 + r ' + x2 ∧ ¬ r ∈ x1)
Proof step 37:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x1 + r0 ', H), path(r ' + x2 + H[r0].nxt ', H[r0, ce]), H[r0].nxt ∈ H[r0, ce], r0 ∈ H, H[r0].nxt ∈ H, ∃ x1, x. x2 = x1 + r0 ' + x ∧ ¬ r0 ∈ x1 ⊦ H[r0].nxt = null, H[r0].nxt = ce.nxt, r = H[r0].nxt, dups(x2), dups(x1), H[r0].nxt ∈ x2, r0 ∈ x1, r ∈ x2, r ∈ x1, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x, ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0

Rule argument

spec: list, inst: reflist name: firstoc-rew seq: ⊦ r ∈ x0 ↔ (∃ x1, x2. x0 = x1 + r ' + x2 ∧ ¬ r ∈ x1) subst: [x0, r] → [x2, r0] flags: <all paths>

Additional information

Used terms: x2, r0