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Proof step 67 from proof BFS-correct-if-reachable (previous step)

n = n1 + n0 ∧ n0 ≤ n, m1 + 1 = # S0, m0 + 1 = n1, n0 < n, reach(source, dest, n), reach(source, dest, n0), dest ∈ S0, reachS(source, S, n0 + 1), a ∈ S0 ∧ reach(a, dest, n1) ∧ ¬ reach<(a, dest, n1) ∨ n0 < n ∧ b ∈ S ∧ reach(b, dest, n1 - 1) ∧ ¬ reach<(b, dest, n1 - 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 66:
m1 + 1 = # S0, reach(source, dest, (m0 + 1) + n0), reach(source, dest, n0), dest ∈ S0, reachS(source, S, 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), a ∈ S0 ∧ reach(a, dest, m0 + 1) ∧ ¬ reach<(a, dest, m0 + 1) ∨ b ∈ S ∧ reach(b, dest, m0) ∧ ¬ reach<(b, dest, m0) ⊦ S0 = ∅, reach<(source, dest, (m0 + 1) + n0), reach<(source, dest, n0)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ 0 + m = m], [CSIMPSEQ : ⊦ n ≤ m → m + n0 - n = (m - n) + n0], [CSIMPSEQ : ⊦ ¬ m + n < m], [CSIMPSEQ : ⊦ m < n + 1 ↔ ¬ n < m], [CSIMPSEQ : ⊦ m < n → m ≤ n], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + 1 ≠ 0], [CSIMPSEQ : ⊦ m = m + n ↔ n = 0])]]