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Proof step 78 from proof rem-stays-reachable-aux (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 79:
path(x, H) ∧ x.last = r ∧ path(r ' + y, H) ⊦ path(x + y, H)
Proof step 80:
ce.nxt = H[H[r0].nxt].nxt, path(r ' + x2 + r0 ', H[r0, ce]), path(H[r0].nxt ' + x1 + H[r0].nxt ', H[r0, ce]), path(r ' + x2 + r0 ', H), H[r0].nxt ∈ H, r0 ∈ H ⊦ (r ' + x2 + r0 ').last = r0 ∧ path(r0 ' + x1 + H[r0].nxt ', H[r0, ce]), H[r0].nxt = ce.nxt, H[r0].nxt = null, r = r0, r0 ∈ x1, r ∈ x2, r0 ∈ x2, dups(x2), path(r ' + x2 + r0 ' + x1 + H[r0].nxt ', H[r0, ce])

Rule argument

spec: inst: name: path-trans seq: path(x, H) ∧ x.last = r ∧ path(r ' + y, H) ⊦ path(x + y, H) subst: [x, H, r, y] → [r ' + x2 + r0 ', H[r0, ce], r0, x1 + H[r0].nxt '] flags: current, []

Additional information

Used simplifier rules: a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist] Used terms: r ' + x2 + r0 ', H[r0, ce], r0, x1 + H[r0].nxt '