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Proof step 54 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ x0 ≠ [] → (x1 + x0).last = x0.last
Proof step 56:
x1 ≠ [], r0 = H[r0].nxt, (r0 ' + x0).last = x1.last, x1.last ∈ H, path(r0 ' + x0, H), r0 ∈ H, path(r ' + (x2 + x1.last ') + r0 ', H), path(r ' + x1, H) ⊦ r0 = r, r0 = null, r = x1.last, r0 = x1.last, x1.last ∈ x2, dups(x2), r ∈ x2, dups(x1), r ∈ x1, dups(x0), r0 ∈ x0, r0 ∈ x2

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: 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 '