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Proof step 68 from proof rem-stays-reachable-aux2 (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 69:
path(r ' + x, H), path(r ' + y, H) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0)
Proof step 70:
x1 = x2 ∨ (∃ r0, y0. x1 = x2 + r0 ' + y0) ∨ (∃ r0, x0. x2 = x1 + r0 ' + x0), x2 = (x + H[x2.last].nxt ') + r0 ' + y, (x2.last ' + x0).last = (r ' + x).last, (r ' + x1).last = (r ' + x).last, path(r ' + x1, H), path(r ' + x + H[x2.last].nxt ', H), path(r ' + x2, H), H[x2.last].nxt ∈ H, x2.last ∈ H, path(x2.last ' + x0, H), (r ' + x).last ∈ H ⊦ H[x2.last].nxt = null, x2.last = r, x2 = [], x2.last ∈ x, x2.last ∈ x0, dups(x0), r ∈ x1, dups(x1), r ∈ x2, dups(x2)

Rule argument

spec: inst: name: linearity-path seq: path(r ' + x, H), path(r ' + y, H) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0) subst: [r, x, H, y] → [r, x1, H, x2] flags: current, []

Additional information

Used terms: r, x1, H, x2