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Proof step 9 from proof INV-nextlevel (previous step)

b ∈ succ(c), reach(source, c, n), reachS(source, N, n + 1), reach<(source, c, n) ⊦ succ(c) = ∅, # succ(c) = 0, b ∈ V, ∃ a. b ∈ succ(a) ∧ reach(source, a, n) ∧ ¬ reach<(source, a, n), ∃ b1. ¬ ((∃ c. reach(source, c, n) ∧ b1 ∈ succ(c)) ∧ ¬ reach<(source, b1, n + 1) → b1 ∈ N), ∃ b1. ¬ (b1 ∈ V ↔ reach<(source, b1, n + 1) ∨ (∃ a. b1 ∈ succ(a) ∧ reach(source, a, n) ∧ ¬ reach<(source, a, n))), ∃ m0. ¬ n < m0 ∧ reach(source, b, m0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ reach<(a, b, n) ↔ (∃ m. m < n ∧ reach(a, b, m))
Proof step 8:
b ∈ succ(c), reach(source, c, n), reachS(source, N, n + 1), ∃ m. m < n ∧ reach(source, c, m) ⊦ succ(c) = ∅, # succ(c) = 0, b ∈ V, ∃ a. b ∈ succ(a) ∧ reach(source, a, n) ∧ ¬ reach<(source, a, n), ∃ b1. ¬ ((∃ c. reach(source, c, n) ∧ b1 ∈ succ(c)) ∧ ¬ reach<(source, b1, n + 1) → b1 ∈ N), ∃ b1. ¬ (b1 ∈ V ↔ reach<(source, b1, n + 1) ∨ (∃ a. b1 ∈ succ(a) ∧ reach(source, a, n) ∧ ¬ reach<(source, a, n))), ∃ m0. ¬ n < m0 ∧ reach(source, b, m0)

Rule argument

[x0lemmaarg: spec-name: graph instance-name: lemma-name: reachls-def antecedent: succedent: reach<(a, b, n) ↔ (∃ m. m < n ∧ reach(a, b, m)) substlist: [n, b, a] → [n, c, source] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 3)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (n, c, source)]]