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Proof step 128 from proof BFS-correct-if-reachable (previous step)

m1 + 1 = # S0, reach(source, dest, (m0 + 1) + n0), INV(S0, S, S1, n0, source), 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 118:
⊦ INV(C, N, V, n, source) ↔ ¬ ¬ ( (∀ b. b ∈ V ↔ reach<(source, b, n + 1) ∨ (∃ a. b ∈ succ(a) ∧ reachS(source, {a} , n) ∧ ¬ a ∈ C)) ∧ reachS(source, C, n) ∧ reachS(source, N, n + 1) ∧ (∀ b. reachS(source, {b} , n + 1) → b ∈ N ∨ (∃ a. a ∈ C ∧ b ∈ succ(a) ∧ ¬ b ∈ V)))
Proof step 127:
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)

Rule argument

[x0lemmaarg: spec-name: INV instance-name: lemma-name: INV-def antecedent: succedent: INV(C, N, V, n, source) ↔ ¬ ¬ ( (∀ b. b ∈ V ↔ reach<(source, b, n + 1) ∨ (∃ a. b ∈ succ(a) ∧ reachS(source, {a} , n) ∧ ¬ a ∈ C)) ∧ reachS(source, C, n) ∧ reachS(source, N, n + 1) ∧ (∀ b. reachS(source, {b} , n + 1) → b ∈ N ∨ (∃ a. a ∈ C ∧ b ∈ succ(a) ∧ ¬ b ∈ V))) substlist: [source, n, V, N, C] → [source, n0, S1, S, S0] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 2)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (source, n0, S1, S, S0)]]