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Proof step 98 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 99:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 100:
x0 ≠ [], H[r2].nxt ' + r0 ' + y = x + r2 ', (x0.last ' + x).last = x0.last, x0.last ∈ H, path(r ' + x2 + x0.last ' + x, H), path(r ' + (x2 + x0.last ') + H[r2].nxt ', H), path(r ' + (x2 + x0.last ' + x) + r2 ', H), H[r2].nxt ∈ H, r2 ∈ H, path(r2 ' + x0, H), H[r2].nxt ∈ x ∨ H[r2].nxt = r2, r2 = H[r2].nxt ∨ r2 = r0 ∨ r2 ∈ y ⊦ r2 = x0.last, r = r2, H[r2].nxt = null, r = x0.last, r2 ∈ x2, r2 ∈ x0, dups(x0), r ∈ x2, r ∈ x, r2 ∈ x, x0.last ∈ x2, x0.last ∈ x, dups(x2 + x), dups(x2), dups(x)

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 '