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Proof step 91 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 92:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 93:
x1 + r2 ' = H[r2].nxt ' + r0 ' + y, (x0.last ' + x1).last = x0.last, H[x0.last].nxt = H[r2].nxt, path(x0.last ' + x1 + r2 ', H), path(x0.last ' + x1, H), path(r2 ' + x0, H), H[r2].nxt ∈ H, r2 ∈ H, x0.last ∈ H, H[r2].nxt ∈ x1 ∨ H[r2].nxt = r2, r2 = H[r2].nxt ∨ r2 = r0 ∨ r2 ∈ y ⊦ x0.last = r2, x0.last = null, H[r2].nxt = null, x0 = [], dups(x0), dups(x1), r2 ∈ x1, r2 ∈ x0, x0.last ∈ x1

Rule argument

spec: list-last, inst: reflist name: lapp seq: ⊦ x0 ≠ [] → (x1 + x0).last = x0.last subst: [x0, x1] → [x0, r2 '] flags: <all paths>

Additional information

Used simplifier rules: ⊦ x0 + [] = x0; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]% ⊦ r '.last = r; (s) Used terms: x0, r2 '