Introduction Reducibility Candidates Stability by Union Application to λ ⇒× Conclusion & Future work On the Stability by Union of Reducibility Candidates Colin Riba INPL & LORIA http://loria.fr/~riba/ Journée Déduction Modulo 14 Janvier 2006 Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Stability by Union Application to λ ⇒× Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Motivations Stability by Union The Calculus λ ⇒× Application to λ ⇒× Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Motivations Stability by Union The Calculus λ ⇒× Application to λ ⇒× Conclusion & Future work Our starting point: Strong normalization of λ -calculus plus rewriting in presence of union types [Blanqui & Riba 06]. More generally, Simple characterization of reducibility candidates and saturated sets. Better understanding of reducibility. Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Motivations Stability by Union The Calculus λ ⇒× Application to λ ⇒× Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Motivations Stability by Union The Calculus λ ⇒× Application to λ ⇒× Conclusion & Future work Terms Terms: t , u ∈ Λ ::= x | t u | λ x . t | π i t | � t , u � . Reductions: ( λ x . t ) u �→ β t [ u / x ] π i � t 1 , t 2 � �→ β t i . Two kinds of values: λ x . t and � t , u � Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates Motivations Stability by Union The Calculus λ ⇒× Application to λ ⇒× Conclusion & Future work Types Types: T , U ∈ T | T ⇒ U | T × U ::= B Typing rules: ( A X ) Γ , x : T ⊢ x : T Γ , x : U ⊢ t : T ( ⇒ E ) Γ ⊢ t : U ⇒ T Γ ⊢ u : U ( ⇒ I ) Γ ⊢ λ x . t : U ⇒ T Γ ⊢ t u : T ( × I ) Γ ⊢ t 1 : T 1 Γ ⊢ t 2 : T 2 ( × E ) Γ ⊢ t : T 1 × T 2 ( i ∈ { 1 , 2 } ) Γ ⊢ � t 1 , t 2 � : T 1 × T 2 Γ ⊢ π i t : T i Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Interpret types T ∈ T as sets of SN terms � T � ⊆ SN . Prove the soundness of the interpretation: If Γ ⊢ t : T and σ ( x ) ∈ � A � for all ( x : A ) ∈ Γ , then σ ( t ) ∈ � T � . � T � must satisfy some closure conditions. Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Interpret types T ∈ T as sets of SN terms � T � ⊆ SN . Prove the soundness of the interpretation: If Γ ⊢ t : T and σ ( x ) ∈ � A � for all ( x : A ) ∈ Γ , then σ ( t ) ∈ � T � . � T � must satisfy some closure conditions. Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Arrow: A ⇒ B { t | ∀ u ( u ∈ A ⇒ tu ∈ B ) } = def Product: A × B { t | π 1 t ∈ A ∧ π 2 t ∈ B} = def Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Outline Introduction 1 Motivations The Calculus λ ⇒× Reducibility Candidates 2 General Idea Interpretation of Types Girard’s Reducibility Candidates Stability by Union 3 Main Point General Considerations Weak Observational Preorder Application to λ ⇒× 4 Application to λ ⇒× Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Neutrality Atomic elimination contexts: ǫ [ ] ::= [ ] t | π i [ ] Elimination contexts: E [ ] ::= [ ] | E [ ǫ [ ]] . t is neutral ( t ∈ N ) iff t is not a value. If t ∈ N , then E [ t ] ∈ N 1 If E [ t ] → v , then v = E ′ [ t ′ ] with ( E [ ] , t ) → ( E ′ [ ] , t ′ ) . 2 Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Neutrality Atomic elimination contexts: ǫ [ ] ::= [ ] t | π i [ ] Elimination contexts: E [ ] ::= [ ] | E [ ǫ [ ]] . t is neutral ( t ∈ N ) iff t is not a value. If t ∈ N , then E [ t ] ∈ N 1 If E [ t ] → v , then v = E ′ [ t ′ ] with ( E [ ] , t ) → ( E ′ [ ] , t ′ ) . 2 Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Definitions C ∈ CR iff C ⊆ SN and ( CR 0 ) if t ∈ C and t → u then u ∈ C , ( CR 1 ) if t ∈ N and ( ∀ u ( t → u ⇒ u ∈ C )) then t ∈ C . If X ⊆ SN , X is the smallest set such that X ⊆ X ∈ CR . Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Application Let X ∈ {⇒ , ×} If A , B ∈ CR , then A X B ∈ CR . A X B ⊆ SN . A X B stable by reduction. Let t ∈ N and ( t ) → ⊆ A X B . Since ǫ [ t ] ∈ N , apply ( CR 1 ) on A , B . By induction on ǫ [ ] ∈ SN . Let ( ǫ [ ] , t ) → v . If v = ǫ [ t ′ ] with t → t ′ , we conclude by assumption. Otherwise, v = ǫ ′ [ t ] , and we conclude by induction hypothesis. Colin Riba On the Stability by Union of Reducibility Candidates
Introduction Reducibility Candidates General Idea Stability by Union Interpretation of Types Application to λ ⇒× Girard’s Reducibility Candidates Conclusion & Future work Application Let X ∈ {⇒ , ×} If A , B ∈ CR , then A X B ∈ CR . A X B ⊆ SN . A X B stable by reduction. Let t ∈ N and ( t ) → ⊆ A X B . Since ǫ [ t ] ∈ N , apply ( CR 1 ) on A , B . By induction on ǫ [ ] ∈ SN . Let ( ǫ [ ] , t ) → v . If v = ǫ [ t ′ ] with t → t ′ , we conclude by assumption. Otherwise, v = ǫ ′ [ t ] , and we conclude by induction hypothesis. Colin Riba On the Stability by Union of Reducibility Candidates
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