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Proof step 111 from proof BFS-correct-if-reachable (previous step)

∀ n, n0, S0, S, S1, source, dest, done0. m0 = 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), ∀ S2, m, n1, n2, S3, S4, source0, dest0, done. m1 = # 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), m0 + 1 = n - n0, m1 + 1 = # S0, S0 ≠ ∅, v ≠ dest, INVR(S0, S, S1, n0, n, source, dest), INV(S0, S, S1, n0, source), v ∈ S0 ⊦ ⟪ADD(succ(v); S1, S)⟫ ⟪S0 := S0 -- v ; if S0 = ∅ then S0 := S, S := ∅, n0 := n0 + 1⟫ ⟪done0 := false ; 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)

User applied rule insert lemma

And obtained the following new goals

Proof step 84:
INV(C, N, V, n, source), v ∈ C, INVR(C, N, V, n, n0, source, dest), v ≠ dest ⊦ ⟪ADD(succ(v); V, N)⟫ (INV(C -- v, N, V, n, source) ∧ INVR(C -- v, N, V, n, n0, source, dest))
Proof step 110:
⟪ADD(succ(v); S1, S)⟫ (INV(S0 -- v, S, S1, n0, source) ∧ INVR(S0 -- v, S, S1, n0, n, source, dest)), ∀ n, n0, S0, S, S1, source, dest, done0. m0 = 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), ∀ S2, m, n1, n2, S3, S4, source0, dest0, done. m1 = # 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), m0 + 1 = n - n0, m1 + 1 = # S0, S0 ≠ ∅, v ≠ dest, INVR(S0, S, S1, n0, n, source, dest), INV(S0, S, S1, n0, source), v ∈ S0 ⊦ ⟪ADD(succ(v); S1, S)⟫ ⟪S0 := S0 -- v ; if S0 = ∅ then S0 := S, S := ∅, n0 := n0 + 1⟫ ⟪done0 := false ; 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)

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: ADD-preserves-INV-INVR antecedent: INV(C, N, V, n, source), v ∈ C, INVR(C, N, V, n, n0, source, dest), v ≠ dest succedent: ⟪ADD(succ(v); V, N)⟫ ( INV(C -- v, N, V, n, source) ∧ INVR(C -- v, N, V, n, n0, source, dest)) substlist: [C, N, V, n, source, v, n0, dest] → [S0, S, S1, n0, source, v, n, dest] 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, n, dest)]]