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Proof step 55 from proof reach-n-path-n (previous step)

1 + 1 + # x2 = 3 + n, ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n), 1 = 1 + # x2 ∧ r ∈ H ∧ H[r].nxt ∈ H ∧ r ≠ null ∧ H[r].nxt ≠ null ∧ H[r].nxt = r4 ∨ 1 ≤ 1 + # x2 ∧ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r4 ∧ path(H[r].nxt ' + x2, H) ⊦ ∃ r0. reachable-1(H, r, r0) ∧ (∃ r1. reachable-1(H, r0, r1) ∧ reachable-n(H, n, r1, (r4 ' + x2).last))

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 56:
x2 = [], 1 + 1 + # x2 = 3 + n, ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n), 1 = 1 + # x2 ∧ r ∈ H ∧ H[r].nxt ∈ H ∧ r ≠ null ∧ H[r].nxt ≠ null ∧ H[r].nxt = r4 ∨ 1 ≤ 1 + # x2 ∧ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r4 ∧ path(H[r].nxt ' + x2, H) ⊦ ∃ r0. reachable-1(H, r, r0) ∧ (∃ r1. reachable-1(H, r0, r1) ∧ reachable-n(H, n, r1, (r4 ' + x2).last))
Proof step 57:
1 + 1 + # x2 = 3 + n, ∀ H0, r1, r2. (∃ r3. reachable-1(H0, r1, r3) ∧ reachable-n(H0, n, r3, r2)) ↔ (∃ x. path(x, H0) ∧ hd x = r1 ∧ x.last = r2 ∧ # x = 2 + n), 1 = 1 + # x2 ∧ r ∈ H ∧ H[r].nxt ∈ H ∧ r ≠ null ∧ H[r].nxt ≠ null ∧ H[r].nxt = r4 ∨ 1 ≤ 1 + # x2 ∧ r ∈ H ∧ r ≠ null ∧ H[r].nxt = r4 ∧ path(H[r].nxt ' + x2, H) ⊦ x2 = [], ∃ r0. reachable-1(H, r, r0) ∧ (∃ r1. reachable-1(H, r0, r1) ∧ reachable-n(H, n, r1, (r4 ' + x2).last))

Rule argument

fma: x2 = []

Additional information

Used formulas: x2 = []