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Proof step 101 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 102:
⊦ 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] → [x, (r2 ' + x0).last '] flags: <all paths>

Additional information

Used simplifier rules: ⊦ x0.last = r ∧ x0 ≠ [] → r ∈ x0; (s)% ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s)% ⊦ [] + x0 = x0; (s)% ⊦ r ' = r0 ' + x0 ↔ r = r0 ∧ x0 = []; (ws)% ⊦ r ' + x0 ≠ []; (s) Used terms: x, (r2 ' + x0).last '