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Proof step 183 from proof rem-stays-reachable-aux2 (previous step)

H[r].nxt = H[r1].nxt, path(H[H[r1].nxt].nxt ' + x + r ', H), r1 ∈ H, r ∈ H, H[r1].nxt ∈ H, path(H[H[r1].nxt].nxt ' + x3 + r1 ', H) ⊦ r = null, H[r1].nxt = r1, r1 = r, r1 = H[H[r1].nxt].nxt, H[r1].nxt = H[H[r1].nxt].nxt, r = H[H[r1].nxt].nxt, H[r1].nxt = null, r1 = null, r ∈ x, H[r1].nxt ∈ x, dups(x), H[H[r1].nxt].nxt ∈ x, r1 ∈ x, H[H[r1].nxt].nxt ∈ x3, dups(x3), r1 ∈ x3, H[r1].nxt ∈ x3, r ∈ x3

User applied rule insert lemma

And obtained the following new goals

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] → [H[H[r1].nxt].nxt, x3 + r1 ', H, x + r '] flags: current, []

Additional information

Used simplifier rules: neq-03: ⊦ ¬ r ∈ x ∧ r ≠ r0 → x + r0 ' ≠ x0 + r ' + x1; from specification list[reflist]%a: ⊦ (x0 + x1) + x2 = x0 + x1 + x2; from specification list[reflist]%app-06: ⊦ x0 + r ' = x1 + r0 ' ↔ x0 = x1 ∧ r = r0; from specification list[reflist] Used terms: H[H[r1].nxt].nxt, x3 + r1 ', H, x + r '