Typage et déduction dans le calcul de

Transcription

1 Typage et déduction dans le calcul de réécriture Benjamin Wack Encadrants : C. Kirchner, L. Liquori

2 Deduction and computation λ-calculus [Church 40] is a simple and powerful computational model Explicit notions of function, application, binding Turing equivalent Type systems and deduction in the rewriting calculus Introduction - 2

3 Deduction and computation λ-calculus [Church 40] is a simple and powerful computational model Explicit notions of function, application, binding Turing equivalent Simply typed λ-calculus [Church 40, Curry 34] Ensures strong normalization Isomorphism with natural deduction for intuitionistic logic [Curry, Howard, de Bruijn] Type systems and deduction in the rewriting calculus Introduction - 2

4 Deduction and computation λ-calculus [Church 40] is a simple and powerful computational model Explicit notions of function, application, binding Turing equivalent Simply typed λ-calculus [Church 40, Curry 34] Ensures strong normalization Isomorphism with natural deduction for intuitionistic logic [Curry, Howard, de Bruijn] Various extensions [de Bruijn 70, Girard 72, Coquand 85, Berardi 88, Paulin 90] To broaden the expressiveness of the logic To ease the definition of elaborated functions Type systems and deduction in the rewriting calculus Introduction - 2

5 More computational power? Explicit introduction of rewriting in the system [Breazu-Tannen, Jouannaud, Okada et al.] Term rewriting Higher-order rewriting Type systems and deduction in the rewriting calculus Introduction - 3

6 More computational power? Explicit introduction of rewriting in the system [Breazu-Tannen, Jouannaud, Okada et al.] Term rewriting Higher-order rewriting Removal of computational arguments from formal proofs Poincaré principle [Barendregt & Barendsen] Deduction modulo [Dowek, Hardin, Kirchner, Werner] Type systems and deduction in the rewriting calculus Introduction - 3

7 More computational power? Explicit introduction of rewriting in the system [Breazu-Tannen, Jouannaud, Okada et al.] Term rewriting Higher-order rewriting Removal of computational arguments from formal proofs Poincaré principle [Barendregt & Barendsen] Deduction modulo [Dowek, Hardin, Kirchner, Werner] The rewriting calculus [Cirstea, Kirchner, Liquori et al.] Designed as a semantics for rule-based languages Embeds the λ-calculus and various aspects of rewriting Type systems and deduction in the rewriting calculus Introduction - 3

8 Contents 1. Untyped rewriting calculus 2. Type systems for programming Properties and type inference Typed encoding of term rewriting systems 3. Pure Pattern Type Systems Strong normalization in ρ and ρp 4. Using the ρ-calculus for deduction P 2 T S-proof terms for deduction modulo Generalized Natural Deduction

9 The Untyped Syntax P T Patterns T ::= X K λp.t T T T T Terms 1. λp.a denotes an abstraction with pattern P and body A... the free variables of P are bound in A 2. The terms can also be structures built using the symbol 3. We work modulo α-conversion and Barendregt s hygiene-convention Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 5

10 Some ρ-terms (λx.x x) (λx.x x) the λ-term (ωω) (λ(f x y).(g y x)) (f a b) the application of a rewrite rule (λa.b λa.c) a the parallel application of two rules Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 6

11 Some ρ-terms (λx.x x) (λx.x x) the λ-term (ωω) (λ(f x y).(g y x)) (f a b) the application of a rewrite rule (λa.b λa.c) a the parallel application of two rules Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 6

12 Some ρ-terms (λx.x x) (λx.x x) the λ-term (ωω) (λ(f x y).(g y x)) (f a b) the application of a rewrite rule (λa.b λa.c) a the parallel application of two rules Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 6

13 The Small-step Reduction Semantics (λp.a) B ρ Aθ if P θ B (A B) C δ A C B C Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 7

14 Some ρ-reductions (λx.x x) (λx.x x) (λ(f x y).g y x) (f a b) (λa.b λa.c) a Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 8

15 Some ρ-reductions (λx.x x) (λx.x x) ρ {ω ω} ρδ... (λ(f x y).g y x) (f a b) (λa.b λa.c) a Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 8

16 Some ρ-reductions (λx.x x) (λx.x x) ρ {ω ω} ρδ... (λ(f x y).g y x) (f a b) ρ g b a (λa.b λa.c) a Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 8

17 Some ρ-reductions (λx.x x) (λx.x x) ρ {ω ω} ρδ... (λ(f x y).g y x) (f a b) ρ g b a (λa.b λa.c) a δ (λa.b) a (λa.c) a ρ b c Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 8

18 About preredexes ( λ(f x).(λa.b) x ) (f a) Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 9

19 About preredexes ( ) λ(f x).(λa.b) x (f a) a preredex (not reducible) Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 9

20 About preredexes ( ) λ(f x).(λa.b) x (f a) a preredex (not reducible) ρ (λa.b) a b Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 9

21 Ensuring confluence Strategies Call by value... Suitable for operational semantics but not adapted for logics Restrictions on patterns [van Oostrom 90] Algebraic and linear More restrictive but stable by reduction Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 10

22 About the expressiveness of the ρ-calculus The λ-calculus is fully embedded in the ρ-calculus [Cirstea & Kirchner 98] β-reductions are faithfully mimicked a λ-term ρ-reduces to λ-terms only Various aspects of rewriting can be represented [Cirstea & Kirchner 98] Rewriting paths Rewriting systems Rewriting strategies Various object calculi can be encoded [Cirstea, Kirchner & Liquori 01] Type systems and deduction in the rewriting calculus The untyped rewriting calculus - 11

23 Contents 1. Untyped rewriting calculus 2. Type systems for programming Properties and type inference Typed encoding of term rewriting systems 3. Pure Pattern Type Systems Strong normalization in ρ and ρp 4. Using the ρ-calculus for deduction P 2 T S-proof terms for deduction modulo Generalized Natural Deduction

24 A Simple Type System ρ 1 x : σ Γ Γ Σ x : σ (V ar) f:σ Σ Γ Σ f : σ (Const) Type systems and deduction in the rewriting calculus Types for programming - 13

25 A Simple Type System ρ 1 x : σ Γ Γ Σ x : σ (V ar) f:σ Σ Γ Σ f : σ (Const) Γ Σ A : σ τ Γ Σ B : σ Γ Σ A B : τ (Appl) Type systems and deduction in the rewriting calculus Types for programming - 13

26 A Simple Type System ρ 1 x : σ Γ Γ Σ x : σ (V ar) f:σ Σ Γ Σ f : σ (Const) Γ Σ A : σ τ Γ Σ B : σ Γ Σ A B : τ Γ, Σ P : σ Γ, Σ A : τ Γ Σ λ(p : ).A : σ τ (Appl) (Abs) Dom( ) = FV(P ) Type systems and deduction in the rewriting calculus Types for programming - 13

27 A Simple Type System ρ 1 x : σ Γ Γ Σ x : σ (V ar) f:σ Σ Γ Σ f : σ (Const) Γ Σ A : σ τ Γ Σ B : σ Γ Σ A B : τ Γ, Σ P : σ Γ, Σ A : τ Γ Σ λ(p : ).A : σ τ (Appl) (Abs) Dom( ) = FV(P ) Γ Σ A : σ Γ Σ B : σ Γ Σ A B : σ (Struct) Type systems and deduction in the rewriting calculus Types for programming - 13

28 Polymorphic extensions à la Church à la Curry Γ Σ A : σ α F V (Γ) Γ Σ λα.a : α.σ (Abs ) Γ Σ A : α.σ Γ Σ Aτ : σ[α := τ] (App ) Type systems and deduction in the rewriting calculus Types for programming - 14

29 Polymorphic extensions à la Church à la Curry Γ Σ A : σ α F V (Γ) Γ Σ λα.a : α.σ (Abs ) Γ Σ A : σ α F V (Γ) Γ Σ A : α.σ (Abs ) Γ Σ A : α.σ Γ Σ Aτ : σ[α := τ] (App ) Γ Σ A : α.σ Γ Σ A : σ[α := τ] (App ) Type systems and deduction in the rewriting calculus Types for programming - 14

30 Polymorphic extensions à la Church à la Curry Γ Σ A : σ α F V (Γ) Γ Σ λα.a : α.σ (Abs ) Γ Σ A : σ α F V (Γ) Γ Σ A : α.σ (Abs ) Γ Σ A : α.σ Γ Σ Aτ : σ[α := τ] (App ) Γ Σ A : α.σ Γ Σ A : σ[α := τ] (App ) (f:σ) Σ, σ α(σ 1... ι(β)) where β = BV(σ) Type systems and deduction in the rewriting calculus Types for programming - 14

31 Typing properties Well-typed matching If P θ A, then x P, Γ Σ x : σ Γ Σ xθ : σ Subject Reduction [Cirstea, Liquori & Wack 03] If Γ Σ A : σ and A ρδ B, then Γ Σ B : σ Uniqueness [Cirstea, Liquori & Wack 03] In systems à la Church, if Γ Σ A : σ and Γ Σ A : τ, then τ = α σ Decidability [Liquori & Wack 04] (typechecking) Γ In systems à la Church, Σ T : σ? (type reconstruction) Γ Σ T :? In systems à la Curry, both are undecidable } are decidable Type systems and deduction in the rewriting calculus Types for programming - 15

32 Type inference In systems à la Church, type inference is fully guided by syntax The type system à la Curry has to be restricted The only legal types are type-schemes α.τ where τ is a simple type Polymorphism is restricted to a new construction [P A]B (similar to let...in) Inference works in the style of the Damas-Milner algorithm Type systems and deduction in the rewriting calculus Types for programming - 16

33 Normalization failure ω = λ x.x x ω ω (λ x. x x) ω ρ ω ω ρ... Type systems and deduction in the rewriting calculus Types for programming - 17

34 Normalization failure Γ = x : α α, ω = λ x.x x ω ω (λ x. x x) ω ρ ω ω ρ... Type systems and deduction in the rewriting calculus Types for programming - 17

35 Normalization failure f : (α α) α and Γ = x : α α, ω = λ(f x).x (f x) ω (f ω) (λ(f x). x (f x))) (f ω) ρ ω (f ω) ρ... Type systems and deduction in the rewriting calculus Types for programming - 17

36 Normalization failure (cont d) f : (α α) α and Γ = x : α α, ω = λf x.x (f x) Γ Σ f : (α α) α Γ Σ x : α α Γ Σ x : α α Γ Σ f x : α Γ Σ x (f x) : α Σ ω (f ω) : α Type systems and deduction in the rewriting calculus Types for programming - 18

37 Encoding rewriting systems in the ρ-calculus Addition over Peano integers: plus = λrec z. ( Σ = {0, S, rec, add} λ(add 0 y). y λ(add(s x) y). S ( (z (rec z)) (add x y) ) ) Type systems and deduction in the rewriting calculus Encoding rewriting systems - 19

38 Encoding rewriting systems in the ρ-calculus Addition over Peano integers: plus = λrec z. ( Σ = {0, S, rec, add} λ(add 0 y). y λ(add(s x) y). S ( (z (rec z)) (add x y) ) ) (plus (rec plus)) (add N M) ρδ (λ0.m) N (λ0. M +1) Ñ 1 (λ0. M +N) 0 (λ(s x)....) 0 Type systems and deduction in the rewriting calculus Encoding rewriting systems - 19

39 Encoding rewriting systems in the ρ-calculus Addition over Peano integers: plus = λrec z. ( Σ = {0, S, rec, add} λ(add 0 y). y λ(add(s x) y). S ( (z (rec z)) (add x y) ) ) (plus (rec plus)) (add N M) ρδ (λ0.m) N (λ0. M +1) Ñ 1 (λ0. M +N) 0 (λ(s x)....) 0? M + N Type systems and deduction in the rewriting calculus Encoding rewriting systems - 19

40 Detecting matching failures: the symbol stk The relation P A detects (some) definitive matching failures Type systems and deduction in the rewriting calculus Encoding rewriting systems - 20

41 Detecting matching failures: the symbol stk The relation P A detects (some) definitive matching failures The relation stk treats matching failures uniformly: (λp :.A) B stk stk if P B stk A stk A A stk stk A stk A stk stk Type systems and deduction in the rewriting calculus Encoding rewriting systems - 20

42 Detecting matching failures: the symbol stk The relation P A detects (some) definitive matching failures The relation stk treats matching failures uniformly: (λp :.A) B stk stk if P B stk A stk A A stk stk A stk A stk stk Theorem [Cirstea, Liquori & Wack 03] The reduction stk ρδ is confluent Type systems and deduction in the rewriting calculus Encoding rewriting systems - 20

43 Systematic encoding There exists a ρ-term first (using stk) such that (first A 1 A 2... A n ) B stk ρδ A i+1 B if A i+1 B j i, A j B ρδ stk ρδ stk stk stk Type systems and deduction in the rewriting calculus Encoding rewriting systems - 21

44 Systematic encoding There exists a ρ-term first (using stk) such that (first A 1 A 2... A n ) B stk ρδ A i+1 B if A i+1 B j i, A j B ρδ stk ρδ stk stk stk The Term Rewrite System R = {l i r i } with signature {a j } is encoded by: λl 1. z (rec z) r 1 R = λ(rec z). first λ(a 1 x). z (rec z) a 1 (z (rec z) x) Type systems and deduction in the rewriting calculus Encoding rewriting systems - 21

45 Properties of the encoding Theorem [Cirstea, Liquori & Wack 03] This encoding is sound for left-linear TRS complete for convergent TRS typable if the TRS is well-typed Remark [Cirstea, Kirchner, Liquori & Wack 03] Various strategies can be encoded Type systems and deduction in the rewriting calculus Encoding rewriting systems - 22

46 Other cases of non termination under typing In CaML, ω can be written type t = F of (t -> t);; let omega x = match x with (F y) -> y (F y);; In CIC, type constructors must fulfill a positiveness condition [Mendler 87] Type systems and deduction in the rewriting calculus The source of non termination - 23

47 Logical inconsistency In this type system, the Curry-Howard isomorphism is not valid: Γ, Σ P : σ Γ, Σ A : τ Γ Σ λp :. A : σ τ (Abs) Γ, Σ σ Γ, Σ τ Γ Σ σ τ ( I) Type systems and deduction in the rewriting calculus The source of non termination - 24

48 Logical inconsistency In this type system, the Curry-Howard isomorphism is not valid: Γ, Σ P : σ Γ, Σ A : τ Γ Σ λp :. A : σ τ (Abs) Γ, Σ σ Γ, Σ τ Γ Σ σ τ ( I) How to fix it? Γ, X i : σ i Σ A : τ Γ Σ λp.a : ( σ i ) τ (Abs), FV(P ) = {X i} But how to type applications? Type systems and deduction in the rewriting calculus The source of non termination - 24

49 Contents 1. Untyped rewriting calculus 2. Type systems for programming Properties and type inference Typed encoding of term rewriting systems 3. Pure Pattern Type Systems Strong normalization in ρ and ρp 4. Using the ρ-calculus for deduction P 2 T S-proof terms for deduction modulo Generalized Natural Deduction

50 Dependent type discipline in P 2 T S Γ, Σ B : C Γ Σ ΠP :.C : s Γ Σ λp :.B : ΠP :.C (Abs) Γ Σ A : ΠP :.C Γ Σ [P B]C : s Γ Σ A B : [P B]C (Appl) Γ, Σ P : A Γ Σ B : A Γ, Σ A : s 1 Γ, Σ C : s 2 Γ Σ [P B]C : s 2 (Match) Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 26

51 Dependent type discipline in P 2 T S Γ, Σ B : C Γ Σ ΠP :.C : s Γ Σ λp :.B : ΠP :.C (Abs) Γ Σ A : ΠP :.C Γ Σ [P B]C : s Γ Σ A B : [P B]C (Appl) Γ, Σ P : A Γ Σ B : A Γ, Σ A : s 1 Γ, Σ C : s 2 Γ Σ [P B]C : s 2 (Match) With = {x:ι, l:list} we have Σ λ(cons x l):. x : Π(cons x l):. ι Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 26

52 The ρ-cube ρ2 ρω ρcc ρp 2 (, ) ρω (, ) ρ (, ) ρp ρp ω Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 27

53 Typing properties [Barthe, Cirstea, Kirchner & Liquori 03] Subject reduction: Γ Σ A : C A ρδ B Γ Σ B : C Correctness: Γ Σ A : B Γ Σ B : s B s Consistency: A Nf(ρδ) Σ A : ( = x:.x) Uniqueness: Γ Σ A : B Γ Σ A : B B = ρδ B Conservativity: Γ P T S A : B Γ P 2 T S A : B Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 28

54 Typing is more restrictive Here, with {x : Πz:α.α}: And: Σ ω = λ(f x):.x (f x) : Π(f x):.α Σ f : Π(y : Πz:α.α).α But to type f ω the pattern y and the argument ω must have a common type σ Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 29

55 Strong normalization : sketch of the proof Theorem [Wack 04]: In ρ and ρp, if Γ Σ A : C then A and C are SN Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 30

56 Strong normalization : sketch of the proof Theorem [Wack 04]: In ρ and ρp, if Γ Σ A : C then A and C are SN 1. Find a translation : P 2 T S λω correct w.r.t. reductions If A ρσδ B, then A β B in at least one step Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 30

57 Strong normalization : sketch of the proof Theorem [Wack 04]: In ρ and ρp, if Γ Σ A : C then A and C are SN 1. Find a translation : P 2 T S λω correct w.r.t. reductions If A ρσδ B, then A β B in at least one step 2. Typability of the translated terms Σ, Γ Σ A : C τ, Γ λω A : τ Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 30

58 Strong normalization : sketch of the proof Theorem [Wack 04]: In ρ and ρp, if Γ Σ A : C then A and C are SN 1. Find a translation : P 2 T S λω correct w.r.t. reductions If A ρσδ B, then A β B in at least one step 2. Typability of the translated terms Σ, Γ Σ A : C τ, Γ λω A : τ 3. Usual techniques can be adapted to reduce SN in ρp to SN in ρ Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 30

59 Correctness of reductions (λ(f x).x) (f a) = ( λu.(u(λx.x)) ) ( (λx 1.λz.(zx 1 ))(λv.v) ) β λv.v = a Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 31

60 Correctness of reductions (λ(f x).x) (f a) = ( λu.(u(λx.x)) ) ( (λx 1.λz.(zx 1 ))(λv.v) ) β λv.v = a The ρ-term ( λy.(λ(f x).x) y ) (f a) features a preredex Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 31

61 Correctness of reductions (λ(f x).x) (f a) = ( λu.(u(λx.x)) ) ( (λx 1.λz.(zx 1 ))(λv.v) ) β λv.v = a The ρ-term ( λy.(λ(f x).x) y ) (f a) features a preredex Thus, the reductions of the λ-term ( λy.(λ(f x).x) y ) (f a) must mimick first an external ρ-reduction Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 31

62 Correctness of reductions (λ(f x).x) (f a) = ( λu.(u(λx.x)) ) ( (λx 1.λz.(zx 1 ))(λv.v) ) β λv.v = a The ρ-term ( λy.(λ(f x).x) y ) (f a) features a preredex Thus, the reductions of the λ-term ( λy.(λ(f x).x) y ) (f a) must mimick first an external ρ-reduction Remark: a term produced by the translation may have additional reductions Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 31

63 The type of a translated pattern A naive translation gives λω f B : (σ β) β λω λ(f x).a : ((σ τ) γ) γ where τ is the type of A Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 32

64 The type of a translated pattern A naive translation gives λω f B : (σ β) β λω λ(f x).a : ((σ τ) γ) γ where τ is the type of A (σ τ) γ = (σ β) β thus τ = β = γ Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 32

65 The type of a translated pattern A naive translation gives λω f B : (σ β) β λω λ(f x).a : ((σ τ) γ) γ where τ is the type of A (σ τ) γ = (σ β) β thus τ = β = γ The actual translation features terms depending on types f B : β.(σ β β) λ(f x).a : β.(σ β β) τ Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 32

66 The type of a translated variable Naive translation x : Πy:ι.ι Σ x : Πy:ι. ι Σ λy:ι. y : Πy:ι. ι Σ λy:ι. a : Πy:ι. ι Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 33

67 The type of a translated variable Naive translation x : Πy:ι.ι Σ x : Πy:ι. ι Σ λy:ι. y : Πy:ι. ι Σ λy:ι. a : Πy:ι. ι Γ λω λy:β y.y : β y β y Γ λω λy:β y. a : β y α.(α α) Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 33

68 The type of a translated variable Naive translation x : Πy:ι.ι Σ x : Πy:ι. ι Σ λy:ι. y : Πy:ι. ι Σ λy:ι. a : Πy:ι. ι Γ λω λy:β y.y : β y β y Γ λω λy:β y. a : β y α.(α α) Use of types depending on types β x :, β y : λω x : β y β x β y Type systems and deduction in the rewriting calculus Pure Pattern Type Systems - 33

69 Contents 1. Untyped rewriting calculus 2. Type systems for programming Properties and type inference Typed encoding of term rewriting systems 3. Pure Pattern Type Systems Strong normalization in ρ and ρp 4. Using the ρ-calculus for deduction P 2 T S-proof terms for deduction modulo Generalized Natural Deduction

70 A linear representation of NDM proofs A proof in Natural Deduction Modulo: the congruence states that e is the neutral element of a group: e x = x y.(y e = y) = y.(y e = y) (Ax) y.(y e = y) = e e ( E) = e y.(y e = y) = e ( =) with e e = e = e = y.(y e = y) e ( I) = e Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 35

71 A linear representation of NDM proofs A proof in Natural Deduction Modulo: the congruence states that e is the neutral element of a group: e x = x y.(y e = y) = y.(y e = y) (Ax) y.(y e = y) = e e ( E) = e y.(y e = y) = e ( =) with e e = e = e = y.(y e = y) e ( I) = e λ-calculus is sufficient to write witnesses [Dowek & Werner 03] λα.(α e) the witness is short and focuses on reasoning but proof reconstruction can be tedious Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 35

72 A more explicit representation Using P 2 T S, conversions can be accounted for by dedicated constructs in the style of Leibniz s equality : Σ Rew φ t (λl.r) π : φ((λl.r) t) Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 36

73 A more explicit representation Using P 2 T S, conversions can be accounted for by dedicated constructs in the style of Leibniz s equality : Σ Rew φ t (λl.r) π : φ((λl.r) t) The new proof term for our example is ( λα. Rew ( λy.(y=e) ) (e e ) ( λ(e x).x ) ) (α e) Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 36

74 A more explicit representation Using P 2 T S, conversions can be accounted for by dedicated constructs in the style of Leibniz s equality : Σ Rew φ t (λl.r) π : φ((λl.r) t) The new proof term for our example is ( λα. Rew ( λy.(y=e) ) (e e ) ( λ(e x).x ) ) (α e) Proposition: For conversion on propositions, application of rewrite rules at top-level is sufficient Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 36

75 A Curry-Howard-de Bruijn correspondence Theorem [Wack 05]: Full proof representation Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 37

76 A Curry-Howard-de Bruijn correspondence Theorem [Wack 05]: Full proof representation Incomplete proof reduction Every redex represents a cut But some cuts are obfuscated by conversion rules ( I) ( =) ( E) p = p = p p = q p = p. = q? Conjecture : additional fold-unfold reduction rules allow to reduce every cut Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 37

77 Main benefits Proof checking reduces to type checking and matching Construction of the conversion steps can be delegated to an efficient rewriting-based software A λ-proof term can always be extracted from a ρ-proof term The set of used rewrite rules can also be extracted Type systems and deduction in the rewriting calculus Proof terms for deduction modulo - 38

78 A simple proof in Natural Deduction... The theory T contains at least { X Y x(x X x Y ) x(x ) Type systems and deduction in the rewriting calculus A generalization of natural deduction - 39

79 A simple proof in Natural Deduction... The theory T contains at least { X Y x(x X x Y ) x(x ) T A Type systems and deduction in the rewriting calculus A generalization of natural deduction - 39

80 A simple proof in Natural Deduction... The theory T contains at least { X Y x(x X x Y ) x(x ) (Ax) ( E) (Ax) T, x x(x ) ( E) ( E) T, x x ( I) ( I) ( E) T, x T, x x A T x x A T x(x x A) T... (Ax) T, x x T x(x x A) A T x(x x A) T A Type systems and deduction in the rewriting calculus A generalization of natural deduction - 39

81 ... shorter in deduction modulo In NDM the context is empty and R = { X Y x(x X x Y ) x Type systems and deduction in the rewriting calculus A generalization of natural deduction - 40

82 ... shorter in deduction modulo In NDM the context is empty and R = { X Y x(x X x Y ) x ( E) ( I) ( I) (Ax) x = x = x A = x x A = A x = A =... Type systems and deduction in the rewriting calculus A generalization of natural deduction - 40

83 ... shorter in deduction modulo In NDM the context is empty and R = { X Y x(x X x Y ) x ( E) ( I) ( I) (Ax) x = x = x A = x x A = A x = A =... The proof is shorter but not very informative Type systems and deduction in the rewriting calculus A generalization of natural deduction - 40

84 A generalization of Natural Deduction We consider some new rules about predicate symbols: ( I) Γ, x X x Y Γ X Y x / FV(Γ) ( E) Γ x Γ φ Type systems and deduction in the rewriting calculus A generalization of natural deduction - 41

85 A generalization of Natural Deduction We consider some new rules about predicate symbols: ( I) Γ, x X x Y Γ X Y x / FV(Γ) ( E) Γ x Γ φ A Type systems and deduction in the rewriting calculus A generalization of natural deduction - 41

86 A generalization of Natural Deduction We consider some new rules about predicate symbols: ( I) Γ, x X x Y Γ X Y x / FV(Γ) ( E) Γ x Γ φ ( I) x x A A Type systems and deduction in the rewriting calculus A generalization of natural deduction - 41

87 A generalization of Natural Deduction We consider some new rules about predicate symbols: ( I) Γ, x X x Y Γ X Y x / FV(Γ) ( E) Γ x Γ φ (Ax) x x ( E) ( I) x x A A Type systems and deduction in the rewriting calculus A generalization of natural deduction - 41

88 A generalization of Natural Deduction We consider some new rules about predicate symbols: ( I) Γ, x X x Y Γ X Y x / FV(Γ) ( E) Γ x Γ φ (Ax) x x ( E) ( I) x x A A The proof is even shorter than in NDM and bears some resemblance with an old-school mathematic style Type systems and deduction in the rewriting calculus A generalization of natural deduction - 41

89 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

90 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Example: X Y x.(x X x Y ) gives Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

91 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Example: X Y x.(x X x Y ) gives Γ, x X x Y Γ x X x Y Γ x.(x X x Y ) Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

92 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Example: X Y x.(x X x Y ) gives Γ, x X x Y Γ x X x Y Γ x.(x X x Y ) ( I) Γ, x X x Y Γ X Y Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

93 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Example: X Y x.(x X x Y ) gives Γ, x X x Y Γ x X x Y Γ x.(x X x Y ) Γ x.(x X x Y ) Γ t X t Y Γ t Y Γ t X ( I) Γ, x X x Y Γ X Y Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

94 Systematic generation of the new inference rules For each defined predicate P (i.e. there is a rewrite rule P φ): decompose φ along the connectives and and gather all the assumptions and side conditions to build a new rule Example: X Y x.(x X x Y ) gives Γ, x X x Y Γ x X x Y Γ x.(x X x Y ) Γ x.(x X x Y ) Γ t X t Y Γ t Y Γ t X ( I) Γ, x X x Y Γ X Y ( E) Γ X Y Γ t X Γ t Y Type systems and deduction in the rewriting calculus A generalization of natural deduction - 42

95 Conservativity w.r.t first-order logic Theorem: Every defined predicate is provably equivalent to its definition Thus, a GND system is correct and complete if and only if the corresponding NDM system is correct and complete Type systems and deduction in the rewriting calculus A generalization of natural deduction - 43

96 Cut elimination A new notion of cut appears for each defined predicate:. D 1 ( E). D 2 Γ t X ( I) Γ, x X x Y Γ X Y Γ t Y (x / FV(Γ)) Type systems and deduction in the rewriting calculus A generalization of natural deduction - 44

97 Cut elimination A new notion of cut appears for each defined predicate:. D 1 ( E). D 2 Γ t X ( I) Γ, x X x Y Γ X Y Γ t Y (x / FV(Γ)) reduces to. D D 2. D 1 Γ t Y Type systems and deduction in the rewriting calculus A generalization of natural deduction - 44

98 Cut elimination A new notion of cut appears for each defined predicate:. D 1 ( E). D 2 Γ t X ( I) Γ, x X x Y Γ X Y Γ t Y (x / FV(Γ)) reduces to. D D 2. D 1 Γ t Y Theorem: Cut elimination holds whenever it holds in the corresponding NDM system Type systems and deduction in the rewriting calculus A generalization of natural deduction - 44

99 Proof terms Definition of proof terms for Generalized Natural Deduction Add ad-hoc constructions in the language Use the λ-abstraction and store multiple assumptions and witnesses in patterns Type systems and deduction in the rewriting calculus A generalization of natural deduction - 45

100 Proof terms Definition of proof terms for Generalized Natural Deduction Add ad-hoc constructions in the language Use the λ-abstraction and store multiple assumptions and witnesses in patterns ( I) Γ, α : x X π : x Y Γ λ( x α).π : X Y ( E) Γ π : X Y Γ π : t X Γ π ( t π ) : t Y Type systems and deduction in the rewriting calculus A generalization of natural deduction - 45

101 Proof terms Definition of proof terms for Generalized Natural Deduction Add ad-hoc constructions in the language Use the λ-abstraction and store multiple assumptions and witnesses in patterns ( I) Γ, α : x X π : x Y Γ λ( x α).π : X Y ( E) Γ π : X Y Γ π : t X Γ π ( t π ) : t Y The reduction ( λ( x α).π ) ( t π ) π[x := t, α := π ] models cut elimination Type systems and deduction in the rewriting calculus A generalization of natural deduction - 45

102 Proof terms Definition of proof terms for Generalized Natural Deduction Add ad-hoc constructions in the language Use the λ-abstraction and store multiple assumptions and witnesses in patterns ( I) Γ, α : x X π : x Y Γ λ( x α).π : X Y ( E) Γ π : X Y Γ π : t X Γ π ( t π ) : t Y The reduction ( λ( x α).π ) ( t π ) π[x := t, α := π ] models cut elimination A collection of new type systems for the ρ-calculus, to be studied Type systems and deduction in the rewriting calculus A generalization of natural deduction - 45

103 Contributions Types for programming Properties and applications of these systems Type inference P 2 T S Detailed study of the usual properties Strong normalization in ρ and ρp Rewriting calculus and deduction Rich proof terms for deduction modulo A new way of embedding domain-specific information in the logic Type systems and deduction in the rewriting calculus Conclusions - 46

104 Perspectives Types Strong normalization in the remaining of the ρ-cube Conjunction types for structures Generalized Natural Deduction seen as a collection of type systems Type systems and deduction in the rewriting calculus Conclusions - 47

105 Perspectives Types Strong normalization in the remaining of the ρ-cube Conjunction types for structures Generalized Natural Deduction seen as a collection of type systems Generalized Natural Deduction Further decomposition of the propositions in the generation of new rules Tests on broader classes of rewrite rules Type systems and deduction in the rewriting calculus Conclusions - 47

106 Perspectives Types Strong normalization in the remaining of the ρ-cube Conjunction types for structures Generalized Natural Deduction seen as a collection of type systems Generalized Natural Deduction Further decomposition of the propositions in the generation of new rules Tests on broader classes of rewrite rules Implementation of proof assistants based on Natural Deduction Modulo, using ρ-proof terms based on Generalized Natural Deduction Type systems and deduction in the rewriting calculus Conclusions - 47

107 Thanks for your attention

108 Deduction modulo Let R be a rewriting system which rewrites: terms to terms (e.g. 0 + x x) atomic propositions to propositions (e.g. x y = 0 x = 0 y = 0) Type systems and deduction in the rewriting calculus Additional material - 49

109 Deduction modulo Let R be a rewriting system which rewrites: terms to terms (e.g. 0 + x x) atomic propositions to propositions (e.g. x y = 0 x = 0 y = 0) Let = be the congruence closure of R Type systems and deduction in the rewriting calculus Additional material - 49

110 Deduction modulo Let R be a rewriting system which rewrites: terms to terms (e.g. 0 + x x) atomic propositions to propositions (e.g. x y = 0 x = 0 y = 0) Let = be the congruence closure of R Every deduction rule is considered modulo = : Γ = ϑ Γ = φ ( E) Γ = ψ ϑ = φ ψ Type systems and deduction in the rewriting calculus Additional material - 49

111 Deduction modulo Let R be a rewriting system which rewrites: terms to terms (e.g. 0 + x x) atomic propositions to propositions (e.g. x y = 0 x = 0 y = 0) Let = be the congruence closure of R Every deduction rule is considered modulo = : Γ = ϑ Γ = φ ( E) Γ = ψ ϑ = φ ψ A large part of the theory can (or should) be represented in R Type systems and deduction in the rewriting calculus Additional material - 49

112 (Non-)Confluence of the ρ-calculus Active variables are troublesome (λx y.y) ((λa.a b) a) ρ out ρ in a (λx y.y) (a b) b ρ This kind of pattern (as well as abstractions) should be treated with higher-order matching Type systems and deduction in the rewriting calculus Additional material - 50

113 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A Type systems and deduction in the rewriting calculus Additional material - 51

114 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A C A Type systems and deduction in the rewriting calculus Additional material - 51

115 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A C A (λ(d z z).e) (d A (C A)) Type systems and deduction in the rewriting calculus Additional material - 51

116 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A C A (λ(d z z).e) (d A (C A)) (λ(d z z).e) (d (C A) (C A)) Type systems and deduction in the rewriting calculus Additional material - 51

117 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A C A (λ(d z z).e) (d A (C A)) (λ(d z z).e) (d (C A) (C A)) e Type systems and deduction in the rewriting calculus Additional material - 51

118 (Non-)Confluence of the ρ-calculus part II Non-linear patterns do not mix well with non-termination [Klop 80] C such that C ρδ A such that A ρδ λy.(λ(d x x).e) (d y (C y)) C A A C A (λ(d z z).e) (d A (C A)) C e (λ(d z z).e) (d (C A) (C A)) e Type systems and deduction in the rewriting calculus Additional material - 51

119 Expressiveness 1. Embedding the λ into the ρ. ϕ : λ ρ (a) ϕ(x) = x (b) ϕ(λx.m) = λx.ϕ(m) (c) ϕ(m N) = ϕ(m) ϕ(n) Theorem: If M β N, then ϕ(m) ρ ϕ(n) Type systems and deduction in the rewriting calculus Additional material - 52

120 Expressiveness 1. Embedding the λ into the ρ. ϕ : λ ρ (a) ϕ(x) = x (b) ϕ(λx.m) = λx.ϕ(m) (c) ϕ(m N) = ϕ(m) ϕ(n) Theorem: If M β N, then ϕ(m) ρ ϕ(n) 2. Encoding Rewriting (a) A rewrite system R can be represented as a structure containing all the rules (b) Reduction paths can be encoded If t 1 R t 2, then A such that A t 1 ρδ t 2 Type systems and deduction in the rewriting calculus Additional material - 52

121 Normalization failure f : (α α) α and Γ = x : α α, ω = λf x.x (f x) Γ Σ f : (α α) α Γ Σ f x : α Γ Σ x : α α Γ Σ x : α α Σ ω λf x.x (f x) : α α Γ Σ x (f x) : α. Γ Σ f x : α Σ ω (f ω) : α Type systems and deduction in the rewriting calculus Additional material - 53

122 The relation and first f P λq.b f P g B if f g i, P i B i P (λq.a) B if Q B P A first(a 1, A 2,..., A n ) = X ((stk A n X I) (... (stk A 2 X I) (A 1 X))) first(a 1, A 2,..., A n ) B ρσδ first(a 2,..., A n ) B if Type systems and deduction in the rewriting calculus Additional material - 54

123 Encoding of TRSs R = λrec z. first λrec z. first λl 1. z (rec z) r 1,, λa 1 x. z (rec z) a 1 (z (rec z) x), λl 1. z (rec z) r 1,, λy.y Type systems and deduction in the rewriting calculus Additional material - 55

124 Positiveness In CIC, the constructor F : (x 1 : A 1 )... (x n : A n ).R is accepted only if R is positive in each A i : 1. R is positive in T if R does not occur in T 2. R is positive in (R t) if R does not occur in t 3. R is positive in (x : A)C if R does not occur in A and R is positive in C Type systems and deduction in the rewriting calculus Additional material - 56

125 Encoding the P 2 T S into λ-calculus x = x f = λx 1... λx αf. (λz.(z x 1... x αf )) f B 1... B αf = λz.(z B 1... B αf ) Type systems and deduction in the rewriting calculus Additional material - 57

126 Encoding the P 2 T S into λ-calculus x = x f = λx 1... λx αf. (λz.(z x 1... x αf )) f B 1... B αf = λz.(z B 1... B αf ) λ(f P 1... P p ).A = λu.(u x... x λp 1... λp p.λx p+1... λx α f.a ) Type systems and deduction in the rewriting calculus Additional material - 57

127 Encoding the P 2 T S into λ-calculus x = x f = λx 1... λx αf. (λz.(z x 1... x αf )) f B 1... B αf = λz.(z B 1... B αf ) λ(f P 1... P p ).A = λu.(u x... x λp 1... λp p.λx p+1... λx α f.a ) λx.a = A B = λx. A A B Type systems and deduction in the rewriting calculus Additional material - 57

128 Encoding the P 2 T S into λ-calculus x = x f = λx 1... λx αf. (λz.(z x 1... x αf )) f B 1... B αf = λz.(z B 1... B αf ) λ(f P 1... P p ).A = λu.(u x... x λp 1... λp p.λx p+1... λx α f.a ) λx.a = λx. A A B = A B A B = λx 1... λx α. ( (λz.( A x1... x α ) ) ( B x 1... x α )) Type systems and deduction in the rewriting calculus Additional material - 57

129 An example of translated term λy.(λ(f x).x) y {}}{ ( λ(f x).x {}}{ ) (λy. (λu.(u(λx.x))) y ) ( f {}}{ (λx 1.λz.(zx 1 )) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

130 An example of translated term β λy.(λ(f x).x) y {}}{ λ(f x).x {}}{ ( ) (λy. (λu.(u(λx.x))) y ) ( ) ( λy.(y(λx.x)) (λx 1.λz.(zx 1 ))(λv.v) ( f {}}{ (λx 1.λz.(zx 1 )) ) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

131 An example of translated term β λy.(λ(f x).x) y {}}{ λ(f x).x {}}{ ( ) (λy. (λu.(u(λx.x))) y ) ( ) ( λy.(y(λx.x)) (λx 1.λz.(zx 1 ))(λv.v) β ( λy.(y(λx.x)) )( λz.(z(λv.v)) ) ( f {}}{ (λx 1.λz.(zx 1 )) ) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

132 An example of translated term β λy.(λ(f x).x) y {}}{ λ(f x).x {}}{ ( ) (λy. (λu.(u(λx.x))) y ) ( ) ( λy.(y(λx.x)) (λx 1.λz.(zx 1 ))(λv.v) ( )( ) β λy.(y(λx.x)) λz.(z(λv.v)) ( ) β λz.(z(λv.v)) (λx.x) ( f {}}{ (λx 1.λz.(zx 1 )) ) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

133 An example of translated term β λy.(λ(f x).x) y {}}{ λ(f x).x {}}{ ( ) (λy. (λu.(u(λx.x))) y ) ( ) ( λy.(y(λx.x)) (λx 1.λz.(zx 1 ))(λv.v) ( )( ) β λy.(y(λx.x)) λz.(z(λv.v)) ( ) β λz.(z(λv.v)) (λx.x) β (λx.x)(λv.v) ( f {}}{ (λx 1.λz.(zx 1 )) ) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

134 An example of translated term β λy.(λ(f x).x) y {}}{ λ(f x).x {}}{ ( ) (λy. (λu.(u(λx.x))) y ) ( ) ( λy.(y(λx.x)) (λx 1.λz.(zx 1 ))(λv.v) ( )( ) β λy.(y(λx.x)) λz.(z(λv.v)) ( ) β λz.(z(λv.v)) (λx.x) β β = a (λx.x)(λv.v) (λv.v) ( f {}}{ (λx 1.λz.(zx 1 )) ) a {}}{ ) (λv.v) Type systems and deduction in the rewriting calculus Additional material - 58

135 The type of a translated constant Supposing Σ f : Πx:ι.ι λω f = λx 1.λz.(z x 1 ) : λω f B : σ (σ β) β (σ β) β Type systems and deduction in the rewriting calculus Additional material - 59

136 Enhanced translation σ1,..., σ α = Π(β : ). ( (σ1... σ α β) β ) f = λf x.a = λx 1.λ(β : ) (λz.(z x 1 )) : σ σ λu. ( u τ λx. A ) : ( σ) τ where Γ λω A : τ x : = Π(β : ).β Type systems and deduction in the rewriting calculus Additional material - 60

137 Use of types depending on types Σ x : Πy:ι. ι β x :, β y : λω x : β y β x β y λy.y β x := λβ :.β λy.a β x := λβ :. f β x := λβ :. β Type systems and deduction in the rewriting calculus Additional material - 61

138 Disjunctive connectors When dealing with and, some part of the definition can not be decomposed properly Type systems and deduction in the rewriting calculus Additional material - 62

139 Disjunctive connectors When dealing with and, some part of the definition can not be decomposed properly With P (Q R) S the new rules are: (P I l ) Γ Q Γ R Γ P (P I r ) Γ S Γ P (P E) Γ P Γ, Q R U Γ, S U Γ U Type systems and deduction in the rewriting calculus Additional material - 62

140 Disjunctive connectors When dealing with and, some part of the definition can not be decomposed properly With P (Q R) S the new rules are: (P I l ) Γ Q Γ R Γ P (P I r ) Γ S Γ P (P E) Γ P Γ, Q R U Γ, S U Γ U The discrepancy between (P I l ) and the second assumption of (P E) may ruin cut elimination, and suggests further decomposition: (P E) Γ P Γ, Q, R U Γ, S U Γ U Type systems and deduction in the rewriting calculus Additional material - 62

141 Conservativity (K E) (P I) (Ax). def H 1... def H n def P. (P E) (K I) P, Γ P... P, Γ γ. P def Type systems and deduction in the rewriting calculus Additional material - 63

142 About unsound rules It is well-known that the rewrite rule R R gives an unsound deduction modulo Its associated introduction and elimination rules are (R I) Γ, R Γ R (R E) Γ R Γ R Γ Type systems and deduction in the rewriting calculus Additional material - 64

143 About unsound rules It is well-known that the rewrite rule R R gives an unsound deduction modulo Its associated introduction and elimination rules are (R I) Γ, R Γ R (R E) Γ R Γ R Γ and the (shortest) proof of has the proof term ( λr(α).α R(α) ) R ( λr(α).α R(α) ) Type systems and deduction in the rewriting calculus Additional material - 64

144 Curiosities Proof terms with patterns for the usual connectives Γ π : φ Γ ( I) π : ψ Γ (π, π ) : φ ψ ( E l ) Γ π : φ ψ Γ (λ (x, y).x)π : φ Type systems and deduction in the rewriting calculus Additional material - 65

145 Curiosities Proof terms with patterns for the usual connectives Γ π : φ Γ ( I) π : ψ Γ (π, π ) : φ ψ ( E l ) Γ π : φ ψ Γ (λ (x, y).x)π : φ The NDM formalization of higher-order logic gives the rules for higher-order quantifiers Predicates defined by induction give some natural rules (N E) Γ n N Γ 0 P Γ, m P S(m) P Γ n P Type systems and deduction in the rewriting calculus Additional material - 65