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Proof step 127 from proof BFS-correct-if-reachable (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 126:
m1 + 1 = # S0, reach(source, dest, (m0 + 1) + n0), reachS(source, S0, n0), reachS(source, S, n0 + 1), dest ∈ S0, a ∈ S0 ∧ reach(a, dest, m0 + 1) ∧ ¬ reach<(a, dest, m0 + 1) ∨ b ∈ S ∧ reach(b, dest, m0) ∧ ¬ reach<(b, dest, m0), ∀ b0. b0 ∈ S1 ↔ reach<(source, b0, n0 + 1) ∨ (∃ a0. b0 ∈ succ(a0) ∧ reach(source, a0, n0) ∧ ¬ reach<(source, a0, n0) ∧ ¬ a0 ∈ S0), ∀ b0. reach(source, b0, n0 + 1) ∧ ¬ reach<(source, b0, n0 + 1) → b0 ∈ S ∨ (∃ a. a ∈ S0 ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ S1) ⊦ S0 = ∅, reach<(source, dest, (m0 + 1) + n0)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ reachS(a, ∅ ++ b, n) ↔ ¬ ¬ ( reach(a, b, n) ∧ ¬ reach<(a, b, n))], [CSIMPSEQ : ⊦ {a} = ∅ ++ a])]]