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Proof step 1 from proof reachable-n-rec-rev (conclusion)

⊦ reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-n(H, n, r, r1) ∧ reachable-1(H, r1, r0))

User applied rule insert lemma

And obtained the following new goals

Proof step 2:
reachable-n(H, n, r, r0), m < n ⊦ ∃ r1. reachable-n(H, m, r, r1) ∧ reachable-n(H, n - m, r1, r0)
Proof step 3:
⊦ reachable-n(H, n + 1, r, r0) ∧ n < n + 1, reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-n(H, n, r, r1) ∧ reachable-1(H, r1, r0))
Proof step 8:
∃ r1. reachable-n(H, n, r, r1) ∧ reachable-n(H, n + 1 - n, r1, r0) ⊦ reachable-n(H, n + 1, r, r0) ↔ (∃ r1. reachable-n(H, n, r, r1) ∧ reachable-1(H, r1, r0))

Rule argument

spec: inst: name: reach-n-split seq: reachable-n(H, n, r, r0), m < n ⊦ ∃ r1. reachable-n(H, m, r, r1) ∧ reachable-n(H, n - m, r1, r0) subst: [H, n, r, r0, m] → [H, n + 1, r, r0, n] flags: current, []

Additional information

Used terms: H, n + 1, r, r0, n