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Proof step 69 from proof BFS-correct-if-reachable (previous step)

m0 + 1 = n - n0, m1 + 1 = # S0, reach(source, dest, n) ∧ ¬ reach<(source, dest, n) ∧ n0 ≤ n ∧ ((∃ a. a ∈ S0 ∧ reachS(a, {dest} , n - n0)) ∨ n0 < n ∧ (∃ b. b ∈ S ∧ reachS(b, {dest} , n - n0 - 1))), reachS(source, S, n0 + 1), dest ∈ S0, reach(source, dest, n0), ∀ b. b ∈ S1 ↔ reach<(source, b, n0 + 1) ∨ (∃ a. b ∈ succ(a) ∧ reach(source, a, n0) ∧ ¬ reach<(source, a, n0) ∧ ¬ a ∈ S0), ∀ b. reach(source, b, n0 + 1) ∧ ¬ reach<(source, b, n0 + 1) → b ∈ S ∨ (∃ a. a ∈ S0 ∧ b ∈ succ(a) ∧ ¬ b ∈ S1) ⊦ n0 = n, S0 = ∅, reach<(source, dest, n0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 68:
m1 + 1 = # S0, m0 + 1 = n - n0, n0 < n, reach(source, dest, n), reach(source, dest, n0), dest ∈ S0, reachS(source, S, n0 + 1), a ∈ S0 ∧ reach(a, dest, n - n0) ∧ ¬ reach<(a, dest, n - n0) ∨ n0 < n ∧ b ∈ S ∧ reach(b, dest, n - n0 - 1) ∧ ¬ reach<(b, dest, n - n0 - 1), ∀ b. reach(source, b, n0 + 1) ∧ ¬ reach<(source, b, n0 + 1) → b ∈ S ∨ (∃ a. a ∈ S0 ∧ b ∈ succ(a) ∧ ¬ b ∈ S1), ∀ b. b ∈ S1 ↔ reach<(source, b, n0 + 1) ∨ (∃ a. b ∈ succ(a) ∧ reach(source, a, n0) ∧ ¬ reach<(source, a, n0) ∧ ¬ a ∈ S0) ⊦ S0 = ∅, n0 = n, reach<(source, dest, n), reach<(source, dest, n0)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m < n → n ≠ 0], [CSIMPFORWARD : ⊦ m < n → n ≠ 0], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ reachS(a, ∅ ++ b, n) ↔ ¬ ¬ ( reach(a, b, n) ∧ ¬ reach<(a, b, n))], [CSIMPSEQ : ⊦ {a} = ∅ ++ a], [CSIMPSEQ : ⊦ m ≠ n → (m ≤ n ↔ m < n)])]]