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Proof step 41 from proof rem-stays-reachable-aux1 (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 42:
path(r ' + x, H), path(r ' + y, H) ⊦ x = y ∨ (∃ r0, y0. x = y + r0 ' + y0) ∨ (∃ r0, x0. y = x + r0 ' + x0)
Proof step 43:
x0 + H[r].nxt ' = x1 + r ' ∨ (∃ r0, y0. x0 + H[r].nxt ' = (x1 + r ') + r0 ' + y0) ∨ (∃ r0, x2. x1 + r ' = (x0 + H[r].nxt ') + r0 ' + x2), path(H[H[r].nxt].nxt ' + x1 + r ', H), path(H[H[r].nxt].nxt ' + x0 + H[r].nxt ', H), H[r].nxt ∈ H, r ∈ H ⊦ r = H[H[r].nxt].nxt, H[r].nxt = null, H[r].nxt = H[H[r].nxt].nxt, r ∈ x0, H[H[r].nxt].nxt ∈ x1, r ∈ x1, dups(x1)

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[r].nxt].nxt, x0 + H[r].nxt ', H, x1 + r '] flags: current, []

Additional information

Used terms: H[H[r].nxt].nxt, x0 + H[r].nxt ', H, x1 + r '