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Proof step 4 from proof reach-n-split (previous step)

∀ H, n0, r, r0. reachable-n(H, n0 + m0, r, r0) → n0 = 0 ∨ (∃ r1. reachable-n(H, m0, r, r1) ∧ reachable-n(H, n0, r1, r0)), reachable-n(H, n0 + m0 +1, r, r0) ⊦ n0 = 0, ∃ r1. reachable-n(H, m0 +1, r, r1) ∧ reachable-n(H, n0, r1, r0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
H[r].nxt ∈ H, r ∈ H, reachable-n(H, n0 + m0, H[r].nxt, r0), ∀ H, n0, r, r0. reachable-n(H, n0 + m0, r, r0) → n0 = 0 ∨ (∃ r1. reachable-n(H, m0, r, r1) ∧ reachable-n(H, n0, r1, r0)) ⊦ r = null, H[r].nxt = null, n0 = 0, ∃ r2. (∃ r1. reachable-1(H, r, r1) ∧ reachable-n(H, m0, r1, r2)) ∧ reachable-n(H, n0, r2, r0)

Rule argument

Additional information

Used simplifier rules: reachable-n-rec: ⊦ reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-1(H, r, r1) ∧ reachable-n(H, n, r1, r0));%reachable-1-def: ⊦ reachable-1(H, r, r0) ↔ r ≠ null ∧ r0 ≠ null ∧ r ∈ H ∧ r0 ∈ H ∧ H[r].nxt = r0;%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%succ: ⊦ m +1 = m + 1; from specification nat-basic2