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Proof step 153 from proof rem-stays-reachable-aux2 (previous step)

(r ' + x1).last = (r0 ' + x0).last, path(r0 ' + x0, H), r0 ∈ H, H[r0].nxt ∈ H, path(r ' + x1 + r2 ' + ((x2 + (r0 ' + x0).last ') + H[r0].nxt ') + r0 ', H), path(r ' + x1 + r2 ' + (x2 + (r0 ' + x0).last ') + H[r0].nxt ', H), path(r ' + x1, H), (r0 ' + x0).last ∈ H ⊦ r2 = r0, r = r2, r0 = r, H[r0].nxt = null, r0 = H[r0].nxt, r2 = H[r0].nxt, r = H[r0].nxt, r0 = (r0 ' + x0).last, H[r0].nxt = (r0 ' + x0).last, r2 = (r0 ' + x0).last, r = (r0 ' + x0).last, r ∈ x1, dups(x0), r0 ∈ x0, r0 ∈ x2, H[r0].nxt ∈ x2, H[r0].nxt ∈ x1, (r0 ' + x0).last ∈ x1, (r0 ' + x0).last ∈ x2, dups(x1 + x2), r0 ∈ x1, r2 ∈ x2, r2 ∈ x1, r ∈ x2, dups(x1), dups(x2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 154:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last

Rule argument

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

Additional information

Used simplifier rules: lin-02: ⊦ x0.last = r ∧ x0 ≠ [] → r ∈ x0; from specification list-last[reflist]%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: x1, r '