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Proof step 50 from proof INV-nextlevel (previous step)

reachS(source, N, n + 1), reach(source, b0, 2 + n) ⊦ reach<(source, b0, 2 + n), ∃ a. a ∈ N ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ V, ∃ 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)))

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 30:
⊦ reach(a, b, n + 1) ↔ (∃ c. reach(a, c, n) ∧ b ∈ succ(c))
Proof step 49:
reach(source, b0, (n + 1) + 1) ↔ (∃ c. reach(source, c, n + 1) ∧ b0 ∈ succ(c)), reachS(source, N, n + 1), reach(source, b0, 2 + n) ⊦ reach<(source, b0, 2 + n), ∃ a. a ∈ N ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ V, ∃ 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)))

Rule argument

[x0lemmaarg: spec-name: graph instance-name: lemma-name: reach-succ antecedent: succedent: reach(a, b, n + 1) ↔ (∃ c. reach(a, c, n) ∧ b ∈ succ(c)) substlist: [a, b, n] → [source, b0, n + 1] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (source, b0, n + 1)]]