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Proof step 188 from proof rem-stays-reachable-aux2 (previous step)

H[(r1 ' + x0).last].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), H[r1].nxt ∈ H, (r1 ' + x0).last ∈ H, path(r ' + x1 + (r1 ' + x0).last ', H), path(r ' + (x1 + (r1 ' + x0).last ') + H[r1].nxt ' + (H[H[r1].nxt].nxt ' + x3) + r1 ', H), r1 ∈ H, path(r1 ' + x0, H) ⊦ H[r1].nxt = null, r = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r1 = H[H[r1].nxt].nxt, r1 = r, r = H[r1].nxt, H[r1].nxt = r1, H[H[r1].nxt].nxt = (r1 ' + x0).last, r1 = (r1 ' + x0).last, r = (r1 ' + x0).last, H[r1].nxt = (r1 ' + x0).last, H[r1].nxt ∈ x3, r1 ∈ x3, (r1 ' + x0).last ∈ x1, (r1 ' + x0).last ∈ x3, dups(x1 + x3), H[H[r1].nxt].nxt ∈ x1, H[H[r1].nxt].nxt ∈ x3, r1 ∈ x1, r1 ∈ x0, dups(x0), r ∈ x1, r ∈ x3, H[r1].nxt ∈ x1, dups(x1), dups(x3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 189:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 190:
x0 ≠ [], H[x0.last].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H), H[r1].nxt ∈ H, x0.last ∈ H, path(r ' + x1 + x0.last ', H), path(r ' + (x1 + x0.last ') + H[r1].nxt ' + (H[H[r1].nxt].nxt ' + x3) + r1 ', H), r1 ∈ H, path(r1 ' + x0, H) ⊦ H[r1].nxt = null, r = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r1 = H[H[r1].nxt].nxt, r1 = r, r = H[r1].nxt, H[r1].nxt = r1, H[H[r1].nxt].nxt = x0.last, r1 = x0.last, r = x0.last, H[r1].nxt = x0.last, H[r1].nxt ∈ x3, r1 ∈ x3, x0.last ∈ x1, x0.last ∈ x3, dups(x1 + x3), H[H[r1].nxt].nxt ∈ x1, H[H[r1].nxt].nxt ∈ x3, r1 ∈ x1, r1 ∈ x0, dups(x0), r ∈ x1, r ∈ x3, H[r1].nxt ∈ x1, dups(x1), dups(x3)

Rule argument

spec: list-last inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x0, r1 '] 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, r1 '