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Proof step 41 from proof reachable-addnew-rev (previous step)

H[r2, ce][(r ' + x1).last].nxt = r2, path(r ' + x1 + r2 ' + x0, H[r2, ce]), path(r ' + x1, H), (r2 ' + x0).last ∈ H, r ∈ H ⊦ r2 = null, dups(x0), dups(x1), dups(x1 + x0), reachable(r, (r2 ' + x0).last, H), r2 ∈ x0, r2 ∈ x1, r ∈ x0, r ∈ x1, r2 ∈ H, reachable(r, (r ' + x1).last, H) ∧ H[(r ' + x1).last].nxt = r2 ∧ reachable(ce.nxt, (H[(r ' + x1).last].nxt ' + x0).last, H)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ r ≠ r0 → H[r0, ce][r] = H[r]
Proof step 43:
H[(r ' + x1).last].nxt = r2, path(r ' + x1, H), path(r ' + x1 + r2 ' + x0, H[r2, ce]), r ∈ H, (r2 ' + x0).last ∈ H ⊦ (r ' + x1).last = r2, r2 = null, dups(x1 + x0), dups(x1), dups(x0), reachable(r, (r2 ' + x0).last, H), r2 ∈ H, r ∈ x1, r ∈ x0, r2 ∈ x1, r2 ∈ x0, reachable(r, (r ' + x1).last, H) ∧ H[(r ' + x1).last].nxt = r2 ∧ reachable(ce.nxt, (H[(r ' + x1).last].nxt ' + x0).last, H)

Rule argument

spec: store, inst: linheap name: At-other seq: ⊦ r ≠ r0 → H[r0, ce][r] = H[r] subst: [r, ce, r0, H] → [(r ' + x1).last, ce, r2, H] flags: <all paths>

Additional information

Used simplifier rules: path-last: ⊦ path(r ' + x, H) → ((r ' + x).last ∈ H ↔ true);% ⊦ H[r, ce][r] = ce; (s)%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: (r ' + x1).last, ce, r2, H