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Local proof transformations for flexible interpolation and proof reduction N. Sharygina Formal Verification and Security Group University of Lugano June 21, 2011 Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 1 / 72

  1. Notation Interpolation • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • A ⇒ I , I ∧ B unsatisfiable • I defined over common symbols of A and B • I as over-approximation A conflicting with B Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

  2. Notation Interpolation • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • A ⇒ I , I ∧ B unsatisfiable • I defined over common symbols of A and B • I as over-approximation A conflicting with B • Example Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

  3. Notation Interpolation • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • A ⇒ I , I ∧ B unsatisfiable • I defined over common symbols of A and B • I as over-approximation A conflicting with B • Example • A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

  4. Notation Interpolation • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • A ⇒ I , I ∧ B unsatisfiable • I defined over common symbols of A and B • I as over-approximation A conflicting with B • Example • A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Interpolant q Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

  5. Notation Interpolation • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • A ⇒ I , I ∧ B unsatisfiable • I defined over common symbols of A and B • I as over-approximation A conflicting with B • Example • A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Interpolant q • A ⇒ q q ∧ B unsatisfiable Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 11 / 72

  6. Interpolation Background • Craig’s interpolant I for unsatisfiable conjunction of formulae A ∧ B [Craig57] • I as over-approximation A conflicting with B B A I Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 12 / 72

  7. Interpolation Background • Applications in symbolic model checking Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  8. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  9. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] • Predicate abstraction refinement based on spurious behaviors [Henzinger04] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  10. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] • Predicate abstraction refinement based on spurious behaviors [Henzinger04] • Property-based transition relation approximation [Jhala05] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  11. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] • Predicate abstraction refinement based on spurious behaviors [Henzinger04] • Property-based transition relation approximation [Jhala05] • Forementioned applications involve Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  12. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] • Predicate abstraction refinement based on spurious behaviors [Henzinger04] • Property-based transition relation approximation [Jhala05] • Forementioned applications involve • Problem encoding into logic (SAT, SMT) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  13. Interpolation Background • Applications in symbolic model checking • Bounded model checking: approximate cheaper reachability set computation [McMillan03] • Predicate abstraction refinement based on spurious behaviors [Henzinger04] • Property-based transition relation approximation [Jhala05] • Forementioned applications involve • Problem encoding into logic (SAT, SMT) • Problem solving by means of resolution based engines (SAT solvers, SMT solvers) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 13 / 72

  14. SAT and SMT Background • Satisfiability (SAT) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  15. SAT and SMT Background • Satisfiability (SAT) • Example A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  16. SAT and SMT Background • Satisfiability (SAT) • Example A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Satisfiability Modulo Theories (SMT): more expressivity than boolean logic Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  17. SAT and SMT Background • Satisfiability (SAT) • Example A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Satisfiability Modulo Theories (SMT): more expressivity than boolean logic • Timed automata, hybrid systems, . . . Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  18. SAT and SMT Background • Satisfiability (SAT) • Example A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Satisfiability Modulo Theories (SMT): more expressivity than boolean logic • Timed automata, hybrid systems, . . . • Arbitrary precision arithmetic, data structures . . . Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  19. SAT and SMT Background • Satisfiability (SAT) • Example A � ( p ∨ q ) ∧ ( p ∨ q ) B � ( q ∨ r ) ∧ ( q ∨ r ) • Satisfiability Modulo Theories (SMT): more expressivity than boolean logic • Timed automata, hybrid systems, . . . • Arbitrary precision arithmetic, data structures . . . • Example A � (5 x − y ≤ 1) ∧ ( y − 5 x ≤ − 1) B � ( y − 5 z ≤ 3) ∧ (5 z − y ≤ − 2) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 14 / 72

  20. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  21. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability • Propositional proof of unsatisfiability • Generated by logging steps at solving time Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  22. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability • Propositional proof of unsatisfiability • Generated by logging steps at solving time • DPLL SAT solver [Davis60,62] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  23. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability • Propositional proof of unsatisfiability • Generated by logging steps at solving time • DPLL SAT solver [Davis60,62] • Search space boolean assignments • Backtracking Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  24. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability • Propositional proof of unsatisfiability • Generated by logging steps at solving time • DPLL SAT solver [Davis60,62] • Search space boolean assignments • Backtracking • SMT solver Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  25. SAT and SMT Proofs and Solving Engines • A ∧ B unsatisfiable: certificate of unsatisfiability • Propositional proof of unsatisfiability • Generated by logging steps at solving time • DPLL SAT solver [Davis60,62] • Search space boolean assignments • Backtracking • SMT solver • DPLL SAT solver • Theory solver Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 15 / 72

  26. Interpolation Generation • Interpolant I for unsatisfiable conjunction of formulae A ∧ B Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

  27. Interpolation Generation • Interpolant I for unsatisfiable conjunction of formulae A ∧ B • State-of-the-art approach [Pudl´ ak97, McMillan04] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

  28. Interpolation Generation • Interpolant I for unsatisfiable conjunction of formulae A ∧ B • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability resolution proof of A ∧ B Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

  29. Interpolation Generation • Interpolant I for unsatisfiable conjunction of formulae A ∧ B • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability resolution proof of A ∧ B • Computation of I from proof structure in linear time Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 16 / 72

  30. Resolution System Background • Literal p p Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  31. Resolution System Background • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  32. Resolution System Background • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥ • Input formula ( p ∨ q ) ∧ ( r ∨ p ) . . . → { pq , rp } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  33. Resolution System Background • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥ • Input formula ( p ∨ q ) ∧ ( r ∨ p ) . . . → { pq , rp } pC pD • Resolution rule p CD Antecedents: pC pD Resolvent: CD Pivot: p Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  34. Resolution System Background • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥ • Input formula ( p ∨ q ) ∧ ( r ∨ p ) . . . → { pq , rp } pC pD • Resolution rule p CD Antecedents: pC pD Resolvent: CD Pivot: p • Resolution proof of unsatisfiability of a set of clauses S Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  35. Resolution System Background • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . Empty clause ⊥ • Input formula ( p ∨ q ) ∧ ( r ∨ p ) . . . → { pq , rp } pC pD • Resolution rule p CD Antecedents: pC pD Resolvent: CD Pivot: p • Resolution proof of unsatisfiability of a set of clauses S • Tree • Leaves as clauses of S • Intermediate nodes as resolvents • Root as unique empty clause Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 17 / 72

  36. Resolution Proofs SAT • A � { pq , pq } B � { qr , qr } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 18 / 72

  37. Resolution Proofs SAT • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability pq pq qr qr p r q q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 18 / 72

  38. Interpolant Generation SAT [Pudl´ ak97] • Computation of interpolant I for A ∧ B from proof structure Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

  39. Interpolant Generation SAT [Pudl´ ak97] • Computation of interpolant I for A ∧ B from proof structure • Partial interpolant for leaf Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

  40. Interpolant Generation SAT [Pudl´ ak97] • Computation of interpolant I for A ∧ B from proof structure • Partial interpolant for leaf • Partial interpolant for resolvent • Pivot • Partial interpolants for antecedents Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

  41. Interpolant Generation SAT [Pudl´ ak97] • Computation of interpolant I for A ∧ B from proof structure • Partial interpolant for leaf • Partial interpolant for resolvent • Pivot • Partial interpolants for antecedents • Partial interpolant for ⊥ is I Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 19 / 72

  42. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr qr pq pq p r q q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  43. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr qr pq {⊥} pq {⊥} p r q q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  44. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  45. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥ ∨ ⊥} q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  46. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥} q q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  47. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥} q {⊤ ∧ ⊤} q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  48. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥} q {⊤} q ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  49. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥} q {⊤} q ⊥ { ( ⊥ ∨ q ) ∧ ( ⊤ ∨ q ) } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  50. Interpolant Generation SAT [Pudl´ ak97] • A � { pq , pq } B � { qr , qr } • Proof of unsatisfiability qr {⊤} qr {⊤} pq {⊥} pq {⊥} p r q {⊥} q {⊤} q ⊥ { q } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 20 / 72

  51. Resolution Proofs SMT p q r s z }| { z }| { z }| { z }| { • A � { ( y − 5 x ≤ − 1) } B � { (5 x − y ≤ 1) , ( y − 5 z ≤ 3) , (5 z − y ≤ − 2) } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

  52. Resolution Proofs SMT p q r s z }| { z }| { z }| { z }| { • A � { ( y − 5 x ≤ − 1) } B � { (5 x − y ≤ 1) , ( y − 5 z ≤ 3) , (5 z − y ≤ − 2) } • Theory lemmata Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

  53. Resolution Proofs SMT p q r s z }| { z }| { z }| { z }| { • A � { ( y − 5 x ≤ − 1) } B � { (5 x − y ≤ 1) , ( y − 5 z ≤ 3) , (5 z − y ≤ − 2) } • Theory lemmata t u � �� � � �� � • LIA: ( x − z ≤ 0) ( x − z ≥ 1) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

  54. Resolution Proofs SMT p q r s z }| { z }| { z }| { z }| { • A � { ( y − 5 x ≤ − 1) } B � { (5 x − y ≤ 1) , ( y − 5 z ≤ 3) , (5 z − y ≤ − 2) } • Theory lemmata t u � �� � � �� � • LIA: ( x − z ≤ 0) ( x − z ≥ 1) p r u � �� � � �� � � �� � • LRA: (5 x − y � 1) ( y − 5 z � 3) ( x − z � 1) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

  55. Resolution Proofs SMT p q r s z }| { z }| { z }| { z }| { • A � { ( y − 5 x ≤ − 1) } B � { (5 x − y ≤ 1) , ( y − 5 z ≤ 3) , (5 z − y ≤ − 2) } • Theory lemmata t u � �� � � �� � • LIA: ( x − z ≤ 0) ( x − z ≥ 1) p r u � �� � � �� � � �� � • LRA: (5 x − y � 1) ( y − 5 z � 3) ( x − z � 1) q s t � �� � � �� � � �� � • LRA: ( y − 5 x � − 1) (5 z − y � − 2) ( x − z � 0) Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 21 / 72

  56. Resolution Proofs SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 22 / 72

  57. Resolution Proofs SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } • Proof of unsatisfiability p pru p ru r r u tu u t qst t qs q q s s s ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 22 / 72

  58. Interpolant Generation SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } • Proof of unsatisfiability p pru p ru r r u tu u t qst t qs q q s s s ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

  59. Interpolant Generation SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } • Proof of unsatisfiability p {⊥} pru p ru r r u tu u t qst t q {⊥} qs q s s s ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

  60. Interpolant Generation SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } • Proof of unsatisfiability p {⊥} pru p r {⊤} ru r u tu u t qst t q {⊥} qs q s s {⊤} s ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

  61. Interpolant Generation SMT • A � { p , q } B � { r , s } L � { tu , pru , qst } • Proof of unsatisfiability p {⊥} pru p r {⊤} ru r tu ? u u t qst t q {⊥} qs q s s {⊤} s ⊥ Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 23 / 72

  62. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  63. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability proof of A ∧ B • Computation of interpolant from proof structure in linear time Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  64. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability proof of A ∧ B • Computation of interpolant from proof structure in linear time • Restriction Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  65. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability proof of A ∧ B • Computation of interpolant from proof structure in linear time • Restriction • Need for proof not to contain AB-mixed predicates A-local B-local AB-common AB-mixed Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  66. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability proof of A ∧ B • Computation of interpolant from proof structure in linear time • Restriction • Need for proof not to contain AB-mixed predicates A-local B-local AB-common AB-mixed A � { (5 x − y ≤ 1) , . . . } B � { ( y − 5 z ≤ 3) , . . . } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  67. Interpolation Challenge • State-of-the-art approach [Pudl´ ak97, McMillan04] • Derivation of unsatisfiability proof of A ∧ B • Computation of interpolant from proof structure in linear time • Restriction • Need for proof not to contain AB-mixed predicates A-local B-local AB-common AB-mixed A � { (5 x − y ≤ 1) , . . . } B � { ( y − 5 z ≤ 3) , . . . } L � { ( x − z ≤ 0) , . . . } Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 24 / 72

  68. Interpolation Possible Solutions • Need for proof not to contain AB-mixed predicates Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

  69. Interpolation Possible Solutions • Need for proof not to contain AB-mixed predicates • Tune solvers to avoid generating AB-mixed predicates [Cimatti08,Beyer08] Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

  70. Interpolation Possible Solutions • Need for proof not to contain AB-mixed predicates • Tune solvers to avoid generating AB-mixed predicates [Cimatti08,Beyer08] • Transform proof to remove AB-mixed predicates Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 25 / 72

  71. Proof Transformation Motivation • Proof transformation approach Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

  72. Proof Transformation Motivation • Proof transformation approach • Motivation: more flexibility by decoupling SMT solving and interpolant generation Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

  73. Proof Transformation Motivation • Proof transformation approach • Motivation: more flexibility by decoupling SMT solving and interpolant generation • Motivation: standard SMT techniques can require addition of AB-mixed predicates Natasha Sharygina (USI) Flexible Proof Transformation June 21, 2011 26 / 72

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