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Proof step 24 from proof INVR-step (previous step)

c1 ∈ succ(v), reach(c1, dest, m), dest ∈ succ(c0), reach(source, c0, m + n), reachS(source, C, n), reachS(source, N, n + 1), dest ∈ succ(c), reach(v, c, m), v ∈ C, c1 ∈ N, reach(c1, dest, m0), m0 < m, c1 ∈ N ∪ (succ(v) \ V), ∀ b. ¬ b ∈ N ∪ (succ(v) \ V) ∨ ¬ reach(b, dest, m) ∨ reach<(b, dest, m), ∀ a. a = v ∨ ¬ a ∈ C ∨ (∀ c1. ¬ reach(a, c1, m) ∨ ¬ dest ∈ succ(c1)) ∨ reach<(a, dest, m + 1) ⊦ # (N ∪ (succ(v) \ V)) = 0, N = ∅, # succ(v) = 0, succ(c0) = ∅, succ(c) = ∅, # C = 0, v = dest, C = ∅, # succ(c0) = 0, # succ(c) = 0, succ(v) = ∅, # N = 0, N ∪ (succ(v) \ V) = ∅, reach<(source, dest, (m + 1) + n), reach<(v, dest, m + 1), ∃ b0. ¬ (b0 ∈ V ↔ reach<(source, b0, n + 1) ∨ (∃ a0. b0 ∈ succ(a0) ∧ reach(source, a0, n) ∧ ¬ reach<(source, a0, n) ∧ ¬ a0 ∈ C)), ∃ b0. ¬ ((∃ c. reach(source, c, n) ∧ b0 ∈ succ(c)) ∧ ¬ reach<(source, b0, n + 1) → b0 ∈ N ∨ (∃ a. a ∈ C ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ V))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ reach<(a, b, n) ↔ (∃ m. m < n ∧ reach(a, b, m))
Proof step 23:
c1 ∈ succ(v), reach(c1, dest, m), dest ∈ succ(c0), reach(source, c0, m + n), reachS(source, C, n), reachS(source, N, n + 1), dest ∈ succ(c), reach(v, c, m), v ∈ C, c1 ∈ N, reach(c1, dest, m0), m0 < m, c1 ∈ N ∪ (succ(v) \ V), ∀ b. ¬ b ∈ N ∪ (succ(v) \ V) ∨ ¬ reach(b, dest, m) ∨ reach<(b, dest, m), ∀ a. a = v ∨ ¬ a ∈ C ∨ (∀ c1. ¬ reach(a, c1, m) ∨ ¬ dest ∈ succ(c1)) ∨ reach<(a, dest, m + 1) ⊦ # (N ∪ (succ(v) \ V)) = 0, N = ∅, # succ(v) = 0, succ(c0) = ∅, succ(c) = ∅, # C = 0, v = dest, C = ∅, # succ(c0) = 0, # succ(c) = 0, succ(v) = ∅, # N = 0, N ∪ (succ(v) \ V) = ∅, ∃ m0. m0 < (m + 1) + n ∧ reach(source, dest, m0), reach<(v, dest, m + 1), ∃ b0. ¬ (b0 ∈ V ↔ reach<(source, b0, n + 1) ∨ (∃ a0. b0 ∈ succ(a0) ∧ reach(source, a0, n) ∧ ¬ reach<(source, a0, n) ∧ ¬ a0 ∈ C)), ∃ b0. ¬ ((∃ c. reach(source, c, n) ∧ b0 ∈ succ(c)) ∧ ¬ reach<(source, b0, n + 1) → b0 ∈ N ∨ (∃ a. a ∈ C ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ V))

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] → [(m + 1) + n, dest, source] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 13)) dummy: () ]

Additional information

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