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

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Proof step 86 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), ¬ 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)

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

[x0lemmaarg: spec-name: nat instance-name: lemma-name: elim-sub antecedent: n ≤ m succedent: n0 = m - n ↔ m = n0 + n substlist: [m, n, n0] → [n, n0 + 1, n2] add preconditions?: F is rewrite lemma?: F current spec?: F rotate lemma?: T replace all?: T paths: () dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ m + n = m + n0 ↔ n = n0], [CSIMPSEQ : ⊦ m + n - m = n], [CSIMPSEQ : ⊦ m + 1 ≤ n ↔ m < n], [CSIMPSEQ : ⊦ ¬ m < n ↔ n ≤ m], [CSIMPSEQ : ⊦ m ≤ n → (n ≤ m ↔ m = n)], [CSIMPSEQ : ⊦ (m + n0) + n1 = m + n0 + n1], [CSIMPSEQ : ⊦ m + n = n + m], [CSIMPSEQ : ⊦ m - m = 0], [CSIMPSEQ : ⊦ m + 1 ≠ 0], [CSIMPELIM : n ≤ m ⊦ n0 = m - n ↔ m = n0 + n])], [EXTRATERMS : (n, n0 + 1, n2)]]