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Proof step 36 from proof INVR-step (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 33:
⊦ reach(a, b, n) ∧ reach(b, c, m) → (¬ reach(a, c, m + n) ↔ false)
Proof step 35:
reach(source, c1, n + 1) ∧ reach(c1, dest, m0) → (¬ reach(source, dest, m0 + n + 1) ↔ false), reach(c1, dest, m0), m0 < m, v ∈ C, reach(v, c, m), dest ∈ succ(c), reachS(source, N, n + 1), reach(source, c0, m + n), dest ∈ succ(c0), reach(c1, dest, m), c1 ∈ succ(v), a ∈ C, c1 ∈ succ(a), reach(source, v, n), ∀ b. b ∈ C → reach(source, b, n) ∧ ¬ reach<(source, b, n), ∀ a. a = v ∨ ¬ a ∈ C ∨ (∀ c1. ¬ reach(a, c1, m) ∨ ¬ dest ∈ succ(c1)) ∨ reach<(a, dest, m + 1), ∀ b. ¬ b ∈ N ∪ (succ(v) \ V) ∨ ¬ reach(b, dest, m) ∨ reach<(b, dest, m) ⊦ # succ(a) = 0, succ(v) = ∅, # succ(c) = 0, # succ(c0) = 0, C = ∅, v = dest, # C = 0, succ(c) = ∅, succ(c0) = ∅, # succ(v) = 0, succ(a) = ∅, c1 ∈ N, reach<(v, dest, m + 1), c1 ∈ V, reach<(source, v, n), ∃ b0. ¬ ((∃ c. reach(source, c, n) ∧ b0 ∈ succ(c)) ∧ ¬ reach<(source, b0, n + 1) → b0 ∈ N ∨ (∃ a. a ∈ C ∧ b0 ∈ succ(a) ∧ ¬ b0 ∈ V)), ∃ b0. ¬ (b0 ∈ V ↔ reach<(source, b0, n + 1) ∨ (∃ a0. b0 ∈ succ(a0) ∧ reach(source, a0, n) ∧ ¬ reach<(source, a0, n) ∧ ¬ a0 ∈ C)), ∃ c. reach(source, c, m0 + n) ∧ dest ∈ succ(c)

Rule argument

[x0lemmaarg: spec-name: graph instance-name: lemma-name: reach-trans antecedent: succedent: reach(a, b, n) ∧ reach(b, c, m) → (¬ reach(a, c, m + n) ↔ false) substlist: [a, b, n, c, m] → [source, c1, n + 1, dest, m0] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (source, c1, n + 1, dest, m0)]]