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Proof step 24 from proof BFS-correct-if-reachable (previous step)

⟪ADD(succ(v); S1, S)⟫ (INV(∅, S, S1, n0, source) ∧ INVR(∅, S, S1, n0, n, source, dest)), ∀ S0, n, n0, S, S1, source, dest, done0. 0 = # S0 ∧ 0 = n - n0 ∧ INVR(S0, S, S1, n0, n, source, dest) ∧ INV(S0, S, S1, n0, source) ∧ (done0 → dest ∈ S0) ∧ (S0 = ∅ → S = ∅) → ⟪while S0 ≠ ∅ ∧ ¬ done0 do { choose v with v ∈ S0 in if v = dest then done0 := true else { S0 := S0 -- v ; ADD(succ(v); S1, S) } ; if S0 = ∅ then S0 := S, S := ∅, n0 := n0 + 1 }⟫ ⟪norfail0 := (done0 ⊃ some(n0); fail)⟫ norfail0 = some(n), 〈ADD(succ(v); S1, S)〉 (S1 = s ∧ S = s0), ¬ done, n1 = n0 + 1, s1 = ∅, 0 = n - n0, v ≠ dest, INVR(∅, s0, s, n0, n, source, dest), INV(∅, s0, s, n0, source), INV(∅ ++ v, S, S1, n0, source), INVR(∅ ++ v, S, S1, n0, n, source, dest) ⊦ ⟪while s0 ≠ ∅ ∧ ¬ done do { choose v with v ∈ s0 in if v = dest then done := true else { s0 := s0 -- v ; ADD(succ(v); s, s1) } ; if s0 = ∅ then s0 := s1, s1 := ∅, n1 := n1 + 1 }⟫ ⟪norfail0 := (done ⊃ some(n1); fail)⟫ norfail0 = some(n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ INVR(C, N, V, n, n0, source, dest) ↔ ¬ ¬ ( reach(source, dest, n0) ∧ ¬ reach<(source, dest, n0) ∧ n ≤ n0 ∧ ((∃ a. a ∈ C ∧ reachS(a, {dest} , n0 - n)) ∨ n < n0 ∧ (∃ b. b ∈ N ∧ reachS(b, {dest} , n0 - n - 1))))

Rule argument

[x0lemmaarg: spec-name: INV instance-name: lemma-name: INVR-def antecedent: succedent: INVR(C, N, V, n, n0, source, dest) ↔ ¬ ¬ ( reach(source, dest, n0) ∧ ¬ reach<(source, dest, n0) ∧ n ≤ n0 ∧ ( (∃ a. a ∈ C ∧ reachS(a, {dest} , n0 - n)) ∨ n < n0 ∧ (∃ b. b ∈ N ∧ reachS(b, {dest} , n0 - n - 1)))) substlist: [dest, source, n0, n, V, N, C] → [dest, source, n, n0, S1, S, ∅ ++ v] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 11)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m ≤ n → (n - m = 0 ↔ n = m)], [CSIMPSEQ : ⊦ a ∈ ∅ ++ b ↔ a = b], [CSIMPSEQ : ⊦ {a} = ∅ ++ a], [CSIMPSEQ : ⊦ reachS(a, ∅ ++ b, n) ↔ ¬ ¬ ( reach(a, b, n) ∧ ¬ reach<(a, b, n))], [CSIMPSEQ : ⊦ m ≤ m], [CSIMPSEQ : ⊦ m - m = 0], [CSIMPSEQ : ⊦ reach(a, b, 0) ↔ a = b], [CSIMPSEQ : ⊦ ¬ n < n])], [EXTRATERMS : (dest, source, n, n0, S1, S, ∅ ++ v)]]