[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 87 from proof BFS-correct-if-reachable (previous step)

〈ADD(succ(v); S1, S)〉 (S1 = s ∧ S = s0), ⟪ADD(succ(v); S1, S)⟫ (INV(∅, S, S1, n0, source) ∧ INVR(∅, S, S1, n0, n, source, dest)), ∀ S2, m, n1, n2, S3, S4, source0, dest0, done. 0 = # S2 ∧ (∀ n, n0, S0, S, S1, source, dest, done0. m = 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)) ∧ m + 1 = n1 - n2 ∧ INVR(S2, S3, S4, n2, n1, source0, dest0) ∧ INV(S2, S3, S4, n2, source0) ∧ (done → dest0 ∈ S2) ∧ (S2 = ∅ → S3 = ∅) → ⟪while S2 ≠ ∅ ∧ ¬ done do { choose v with v ∈ S2 in if v = dest0 then done := true else { S2 := S2 -- v ; ADD(succ(v); S4, S3) } ; if S2 = ∅ then S2 := S3, S3 := ∅, n2 := n2 + 1 }⟫ ⟪norfail0 := (done ⊃ some(n2); fail)⟫ norfail0 = some(n1), s1 = ∅, ¬ done, n1 = n0 + 1, m0 + 1 = n - n0, v ≠ dest, INVR(∅, s0, s, n0, n, source, dest), INV(∅, s0, s, n0, source), INV(∅ ++ v, S, S1, n0, source), reach(source, dest, n) ∧ ¬ reach<(source, dest, n) ∧ n0 ≤ n ∧ ((∃ a. a ∈ ∅ ++ v ∧ reachS(a, {dest} , n - n0)) ∨ n0 < n ∧ (∃ b. b ∈ S ∧ reachS(b, {dest} , n - n0 - 1))), m0 ≠ n - n0 + 1 ⊦ ⟪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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 86:
〈ADD(succ(v); S1, S)〉 (S1 = s ∧ S = s0), ⟪ADD(succ(v); S1, S)⟫ (INV(∅, S, S1, n0, source) ∧ INVR(∅, S, S1, n0, n, source, dest)), ∀ S2, m, n1, n2, S3, S4, source0, dest0, done. 0 = # S2 ∧ (∀ n, n0, S0, S, S1, source, dest, done0. m = 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)) ∧ m + 1 = n1 - n2 ∧ INVR(S2, S3, S4, n2, n1, source0, dest0) ∧ INV(S2, S3, S4, n2, source0) ∧ (done → dest0 ∈ S2) ∧ (S2 = ∅ → S3 = ∅) → ⟪while S2 ≠ ∅ ∧ ¬ done do { choose v with v ∈ S2 in if v = dest0 then done := true else { S2 := S2 -- v ; ADD(succ(v); S4, S3) } ; if S2 = ∅ then S2 := S3, S3 := ∅, n2 := n2 + 1 }⟫ ⟪norfail0 := (done ⊃ some(n2); fail)⟫ norfail0 = some(n1), ¬ done, s1 = ∅, n1 = n0 + 1, m0 + 1 = n - n0, v ≠ dest, m0 ≠ n - n0 + 1, INVR(∅, s0, s, n0, n, source, dest), INV(∅, s0, s, n0, source), INV(∅ ++ v, S, S1, n0, source), reach(source, dest, n), n0 ≤ n, ¬ reach<(source, dest, n), a = v ∧ reach(v, dest, n - n0) ∧ ¬ reach<(v, dest, n - n0) ∨ n0 < n ∧ b ∈ S ∧ reach(b, dest, n - n0 - 1) ∧ ¬ reach<(b, dest, n - n0 - 1) ⊦ ⟪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)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ reachS(a, ∅ ++ b, n) ↔ ¬ ¬ ( reach(a, b, n) ∧ ¬ reach<(a, b, n))], [CSIMPSEQ : ⊦ {a} = ∅ ++ a], [CSIMPSEQ : ⊦ a ∈ ∅ ++ b ↔ a = b])]]