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Proof step 44 from proof reachable-addnew-rev (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0)
Proof step 46:
path(r ' + x1 + H[(r ' + x1).last].nxt ' + x0, H[H[(r ' + x1).last].nxt, ce]), r ∈ H, (H[(r ' + x1).last].nxt ' + x0).last ∈ H, path(r ' + x1, H) ⊦ ∃ x. path(r ' + x, H) ∧ (r ' + x).last = (r ' + x1).last, H[(r ' + x1).last].nxt = null, (r ' + x1).last = H[(r ' + x1).last].nxt, H[(r ' + x1).last].nxt ∈ H, r ∈ x1, r ∈ x0, H[(r ' + x1).last].nxt ∈ x1, H[(r ' + x1).last].nxt ∈ x0, dups(x1 + x0), reachable(r, (H[(r ' + x1).last].nxt ' + x0).last, H), dups(x1), dups(x0)

Rule argument

spec: inst: name: reachable-def seq: ⊦ reachable(r, r0, H) ↔ (∃ x. path(r ' + x, H) ∧ (r ' + x).last = r0) subst: [H, r0, r] → [H, (r ' + x1).last, r] flags: rewrite, current

Additional information

Used terms: H, (r ' + x1).last, r