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Proof step 20 from proof BFS-returns-fail-implies-not-reachable (previous step)

S0 ≠ ∅, # S0 ≠ 0, v ≠ dest, INV(S0, S, S1, n0, source), v ∈ S0, ∀ n. ¬ reach(source, dest, n) ⊦ [ADD(succ(v); S1, S)] [S0 := S0 -- v ; if S0 = ∅ then S0 := S, S := ∅, n0 := n0 + 1] ((∀ n. ¬ reach(source, dest, n)) ∧ INV(S0, S, S1, n0, source))

User applied rule insert lemma

And obtained the following new goals

Proof step 9:
INV(C, N, V, n, source), v ∈ C ⊦ ⟪ADD(succ(v); V, N)⟫ INV(C -- v, N, V, n, source)
Proof step 19:
⟪ADD(succ(v); S1, S)⟫ INV(S0 -- v, S, S1, n0, source), S0 ≠ ∅, # S0 ≠ 0, v ≠ dest, INV(S0, S, S1, n0, source), v ∈ S0, ∀ n. ¬ reach(source, dest, n) ⊦ [ADD(succ(v); S1, S)] [S0 := S0 -- v ; if S0 = ∅ then S0 := S, S := ∅, n0 := n0 + 1] ((∀ n. ¬ reach(source, dest, n)) ∧ INV(S0, S, S1, n0, source))

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: ADD-preserves-INV antecedent: INV(C, N, V, n, source), v ∈ C succedent: ⟪ADD(succ(v); V, N)⟫ INV(C -- v, N, V, n, source) substlist: [C, N, V, n, source, v] → [S0, S, S1, n0, source, v] add preconditions?: F is rewrite lemma?: F current spec?: T rotate lemma?: F replace all?: F paths: () dummy: () ]

Additional information

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