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Proof step 92 from proof BFS-correct-if-reachable (previous step)

m0 = n - n0 + 1 ∧ INVR(s0, s1, s, n0 + 1, n, source, dest) ∧ INV(s0, s1, s, n0 + 1, source) ∧ (done → dest ∈ s0) ∧ (s0 = ∅ → s1 = ∅) → ⟪done0 := done, S1 := s, S := s1, S0 := s0, n0 := n0 + 1 ; 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)⟫ (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), 〈ADD(succ(v); S1, S)〉 (S1 = s ∧ S = s0), ¬ done, n1 = n0 + 1, s1 = ∅, m0 + 1 = 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 91:
⟪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), 〈ADD(succ(v); S1, S)〉 (S1 = s ∧ S = s0), m0 = n - n0 + 1 → ⟪done0 := false, S1 := s, S := ∅, S0 := s0, n0 := n0 + 1 ; 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), 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), 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)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ INV(∅, N, V, n, source) → ( ¬ INV(N, ∅, V, n + 1, source) ↔ false)], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ INVR (∅, N, V, n, n0, source, dest) ∧ INV(∅, N, V, n, source) → ( ¬ INVR (N, ∅, V, n + 1, n0, source, dest) ↔ false)])]]