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Proof step 27 from proof rem-stays-reachable-aux (previous step)

ce.nxt = H[H[r0].nxt].nxt, (r ' + x0).last = H[r0].nxt, path(r ' + x0, H[r0, ce]), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, r ∈ x0, dups(x0), r ∈ x1, r0 ∈ x1, dups(x1), ∃ 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 28:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 29:
x0 ≠ [], ce.nxt = H[H[r0].nxt].nxt, x0.last = H[r0].nxt, path(r ' + x0, H[r0, ce]), path(r ' + x1 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H, H[r0].nxt ∈ H[r0, ce] ⊦ H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r = H[r0].nxt, r ∈ x0, dups(x0), r ∈ x1, r0 ∈ x1, dups(x1), ∃ x0. path(H[r0].nxt ' + x0 + H[r0].nxt ', H) ∧ ¬ r0 ∈ x0, ∃ x. path(r ' + x + H[r0].nxt ', H) ∧ ¬ r0 ∈ x

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x0, r '] flags: rewrite

Additional information

Used simplifier rules: app: ⊦ x0 + [] = x0; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%l: ⊦ r '.last = r; from specification list-last[reflist] Used terms: x0, r '