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Proof step 88 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 89:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 90:
x0 ≠ [], H[x0.last].nxt = H[r2].nxt, (x0.last ' + x1).last = x0.last, x1 + r2 ' = H[r2].nxt ' + r0 ' + y, path(x0.last ' + x1, H), x0.last ∈ H, path(x0.last ' + x1 + r2 ', H), H[r2].nxt ∈ H, r2 ∈ H, path(r2 ' + x0, H), r2 = H[r2].nxt ∨ r2 = r0 ∨ r2 ∈ y, H[r2].nxt ∈ x1 ∨ H[r2].nxt = r2 ⊦ x0.last = null, x0.last = r2, H[r2].nxt = null, dups(x1), r2 ∈ x0, dups(x0), x0.last ∈ x1, r2 ∈ x1

Rule argument

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