Unions of Reducibility Families for -Calculus with Orthogonal - PowerPoint PPT Presentation

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Girard’s Reducibility Candidates ◮ Consider a set of contexts E [ ] ∈ E and a rewrite relation → R . ◮ A term t is Neutral if it interacts with no contexts E [ ] ∈ E : if then the reduction is either in E [ ] or in t . E [ t ] → R v 12 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Girard’s Reducibility Candidates ◮ Consider a set of contexts E [ ] ∈ E and a rewrite relation → R . ◮ A term t is Neutral if it interacts with no contexts E [ ] ∈ E : if then the reduction is either in E [ ] or in t . E [ t ] → R v ◮ C ⊆ SN is a Reducibility Candidate ( C ∈ C R ) iff C is stable by reduction (if t ∈ C and t → R u then u ∈ C ) and 12 / 32

� Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Girard’s Reducibility Candidates ◮ Consider a set of contexts E [ ] ∈ E and a rewrite relation → R . ◮ A term t is Neutral if it interacts with no contexts E [ ] ∈ E : if then the reduction is either in E [ ] or in t . E [ t ] → R v ◮ C ⊆ SN is a Reducibility Candidate ( C ∈ C R ) iff C is stable by reduction (if t ∈ C and t → R u then u ∈ C ) and C has the neutral term property: for all neutral term t , t ∈ C t ⇒ � � = � ���������� � � R R � � � � � � . . . u 1 ∈ C u n ∈ C 12 / 32

� Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Girard’s Reducibility Candidates ◮ Consider a set of contexts E [ ] ∈ E and a rewrite relation → R . ◮ A term t is Neutral if it interacts with no contexts E [ ] ∈ E : if then the reduction is either in E [ ] or in t . E [ t ] → R v ◮ C ⊆ SN is a Reducibility Candidate ( C ∈ C R ) iff C is stable by reduction (if t ∈ C and t → R u then u ∈ C ) and C has the neutral term property: for all neutral term t , t ∈ C t ⇒ � � = � ���������� � � R R � � � � � � . . . u 1 ∈ C u n ∈ C . . . E [ u 1 ] ∈ SN E [ u n ] ∈ SN E [ t ] ∈ SN ⇒ = 12 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } A ⊆ Λ ( Σ ) 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } A ⊆ Λ ( Σ ) ↓ ⊥ ⊥ ⊥ ⊆ E A ⊥ 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } A ⊆ Λ ( Σ ) ↓ ⊥ ⊥ ⊥ ⊆ E A ⊥ ↓ ⊥ ⊥ ⊥ ⊆ Λ ( Σ ) A ⊥ ⊥ ⊥ 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } ⊥ is a closure operator. ◮ ( _ ) ⊥ ⊥ ⊥ 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Gir87, Par97, DK00, Pit00, MV05] ◮ Choose a pole t ⊥ ⊥ E [ ] E [ t ] ∈ SN ⇐ ⇒ def ◮ Given A ⊆ Λ ( Σ ) and P ⊆ E , let A ⊥ ⊥ { E [ ] ∈ E ∀ t ∈ A. t ⊥ ⊥ E [ ] } = def | P ⊥ ⊥ = def { t ∈ Λ ( Σ ) | ∀ E [ ] ∈ P. t ⊥ ⊥ E [ ] } ⊥ is a closure operator. ◮ ( _ ) ⊥ ⊥ ⊥ Lemma ⊥ ∈ C A ⊥ ⊥ ⊥ ∅ � = A ⊆ SN R ⇒ = 13 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Outline Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion 14 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion ◮ Given a typed rewrite system R , ◮ find a reducibility family R ed which leads to an adequate type interpretation and such that � ∅ � = R ⊆ R ed R ∈ R ed ⇒ = 15 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Union Types T 1 , T 2 ∈ T ::= . . . | T 1 ⊔ T 2 ◮ We put � T 1 ⊔ T 2 � = def R ed ( � T 1 � ∪ � T 2 � ) This validates Γ ⊢ t : T i ( ⊔ I ) Γ ⊢ t : T 1 ⊔ T 2 16 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Union Types T 1 , T 2 ∈ T ::= . . . | T 1 ⊔ T 2 ◮ We put � T 1 ⊔ T 2 � = def R ed ( � T 1 � ∪ � T 2 � ) This validates Γ ⊢ t : T i ( ⊔ I ) Γ ⊢ t : T 1 ⊔ T 2 ◮ If R ed stable by union, we have � T 1 ⊔ T 2 � � T 1 � ∪ � T 2 � = This is sufficient to validate Γ, x : T 1 ⊢ c : C Γ ⊢ t : T 1 ⊔ T 2 Γ, x : T 2 ⊢ c : C ( ⊔ E ) Γ ⊢ c [ t/x ] : C 16 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 = def λx.x a δ t 2 = def λy.δ 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 = def λx.x a δ t 2 = def λy.δ t 1 : T 1 t 2 : T 2 x : T 1 ⊢ xx : C x : T 2 ⊢ xx : C t 1 + t 2 : T 1 ⊔ T 2 Because and t 1 t 1 ∈ SN t 2 t 2 ∈ SN 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 = def λx.x a δ t 2 = def λy.δ t 1 : T 1 t 2 : T 2 x : T 1 ⊢ xx : C x : T 2 ⊢ xx : C t 1 + t 2 : T 1 ⊔ T 2 ( ⊔ E ) ( t 1 + t 2 )( t 1 + t 2 ) : C While ∈ SN ( t 1 + t 2 )( t 1 + t 2 ) → t 1 t 2 → ( λy.δ ) a δ → δ δ / 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 = def λx.x a δ t 2 = def λy.δ t 1 : T 1 t 2 : T 2 x : T 1 ⊢ xx : C x : T 2 ⊢ xx : C t 1 + t 2 : T 1 ⊔ T 2 ( ⊔ E ) ( t 1 + t 2 )( t 1 + t 2 ) : C While ∈ SN ( t 1 + t 2 )( t 1 + t 2 ) → t 1 t 2 → ( λy.δ ) a δ → δ δ / Similar example with a confluent rewrite system. 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 : T 1 t 2 : T 2 x : T 1 ⊢ c : C x : T 2 ⊢ c : C t 1 + t 2 : T 1 ⊔ T 2 ( ⊔ E ) c [( t 1 + t 2 ) /x ] : C 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 : T 1 t 2 : T 2 x : T 1 ⊢ c : C x : T 2 ⊢ c : C t 1 + t 2 : T 1 ⊔ T 2 ( ⊔ E ) c [( t 1 + t 2 ) /x ] : C But     does not   and   imply       Safe Safe Safe 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Unsafe Interaction [Rib07b] t 1 + t 2 � → R t 1 t 1 + t 2 � → R t 2 t 1 : T 1 t 2 : T 2 x : T 1 ⊢ c : C x : T 2 ⊢ c : C t 1 + t 2 : T 1 ⊔ T 2 ( ⊔ E ) c [( t 1 + t 2 ) /x ] : C But     does not   and   imply       Safe Safe Safe Prevents from having � T 1 ⊔ T 2 � = � T 1 � ∪ � T 2 � . 17 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Sufficient Conditions for � T 1 ⊔ T 2 � = � T 1 � ∪ � T 2 � Let → R be a rewrite relation on Λ ( Σ ) and E be a set of elimination contexts. 18 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Sufficient Conditions for � T 1 ⊔ T 2 � = � T 1 � ∪ � T 2 � Let → R be a rewrite relation on Λ ( Σ ) and E be a set of elimination contexts.       and implies         Safe Safe Safe 18 / 32

� � Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Sufficient Conditions for � T 1 ⊔ T 2 � = � T 1 � ∪ � T 2 � Let → R be a rewrite relation on Λ ( Σ ) and E be a set of elimination contexts.       and implies         Safe Safe Safe OK if i.e. if t neutral has a "principal reduct" u : " � " t � � � � � � � � �������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � . . . � u n � ( u 1 . . . ) � u 18 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property ◮ Characterize the membership to a candidate 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with some contexts E [ ] ∈ E . 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with some contexts E [ ] ∈ E . ◮ Weak Observational Preorder Let u � C R t iff every value of u is a value of t . 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with some contexts E [ ] ∈ E . ◮ Weak Observational Preorder Let u � C R t iff every value of u is a value of t . Lemma If C is a reducibility candidate, then C is downward closed wrt. � C R 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Reducibility Candidates [Rib07a] ◮ Neutral Term Property ◮ Characterize the membership to a candidate ◮ A Value is an observable term, ie a term which interacts with some contexts E [ ] ∈ E . ◮ Weak Observational Preorder Let u � C R t iff every value of u is a value of t . Lemma If C is a reducibility candidate, then C is downward closed wrt. � C R ◮ But not every such C is a reducibility candidate. 19 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Stability by Union of Reducibility Candidates [Rib07a] Theorem The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. � C R . (iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that and t → R u t � C R u 20 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Stability by Union of Reducibility Candidates [Rib07a] Theorem The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. � C R . (iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that and t → R u t � C R u ◮ u is a strong principal reduct of t . 20 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Stability by Union of Reducibility Candidates [Rib07a] Theorem The following are equivalent: (i) C R is stable by union, (ii) C R is the set of all non-empty subsets C of SN that are downward closed wrt. � C R . (iii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that and t → R u t � C R u ◮ u is a strong principal reduct of t . ◮ This holds for the λ -calculus with products and sums (also [Tat07]). 20 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals ◮ Biorthogonals are reducibility candidates. 21 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals ◮ Biorthogonals are reducibility candidates. ◮ In general, biorthogonals are not stable by union. 21 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals ◮ Biorthogonals are reducibility candidates. ◮ In general, biorthogonals are not stable by union. ◮ Is the closure by union of biorthogonals a reducibility family ? 21 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Rib07b] ◮ Is the closure by union of biorthogonals a reducibility family ? 22 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Rib07b] ◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u � SN t iff for all if then E [ ] ∈ E , E [ u ] ∈ SN E [ t ] ∈ SN 22 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Rib07b] ◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u � SN t iff for all if then E [ ] ∈ E , E [ u ] ∈ SN E [ t ] ∈ SN Theorem The following are equivalent: (i) unions of biorthogonals are reducibility candidates, (ii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t → R u and u � SN t 22 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Biorthogonals [Rib07b] ◮ Is the closure by union of biorthogonals a reducibility family ? ◮ Let u � SN t iff for all if then E [ ] ∈ E , E [ u ] ∈ SN E [ t ] ∈ SN Theorem The following are equivalent: (i) unions of biorthogonals are reducibility candidates, (ii) for every t which is non-normal, strongly normalizing and neutral, there is a term u such that t → R u and u � SN t u is a principal reduct of t . 22 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Comparison [Rib07b] Lemma ∀ t, u ∈ SN . t � C R u u � SN t ⇒ = 23 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Comparison [Rib07b] Lemma ∀ t, u ∈ SN . t � C R u u � SN t ⇒ = Every strong principal reduct is a principal reduct. Lemma O ⊆ C C R stable by union ⇒ R = 23 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Comparison [Rib07b] Lemma ∀ t, u ∈ SN . t � C R u u � SN t ⇒ = Every strong principal reduct is a principal reduct. Lemma O ⊆ C C R stable by union ⇒ R = The converse is false, consider p � → R λx. c 1 p � → R λx. c 2 c i � → R d Indeed, but p � � C R λx. c i λx. c i � SN p 23 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Outline Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion 24 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Constructor Rewriting ◮ Constructors are symbols c of type Γ ⊢ t 1 : T 1 Γ ⊢ t n : T n . . . Γ ⊢ c ( t 1 , . . . , t n ) : B 25 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Constructor Rewriting ◮ Constructors are symbols c of type Γ ⊢ t 1 : T 1 Γ ⊢ t n : T n . . . Γ ⊢ c ( t 1 , . . . , t n ) : B ◮ Constructor Patterns p ::= x | c ( p 1 , . . . , p n ) 25 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Constructor Rewriting ◮ Constructors are symbols c of type Γ ⊢ t 1 : T 1 Γ ⊢ t n : T n . . . Γ ⊢ c ( t 1 , . . . , t n ) : B ◮ Constructor Patterns p ::= x | c ( p 1 , . . . , p n ) ◮ Rewrite Rules f ( p 1 , . . . , p n ) � → R r where p 1 , . . . , p n are constructor patterns. 25 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Values ◮ Destructors For each n -ary c with n ≥ 1 and each i ∈ { 1, . . . , n } , 26 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Values ◮ Destructors For each n -ary c with n ≥ 1 and each i ∈ { 1, . . . , n } , d c ,i ( c ( x 1 , . . . , x n )) � → D x i 26 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Values ◮ Destructors For each n -ary c with n ≥ 1 and each i ∈ { 1, . . . , n } , d c ,i ( c ( x 1 , . . . , x n )) � → D x i For each nullary c , d c c � → D ℧ 26 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Values ◮ Destructors For each n -ary c with n ≥ 1 and each i ∈ { 1, . . . , n } , d c ,i ( c ( x 1 , . . . , x n )) � → D x i For each nullary c , d c c � → D ℧ ◮ Elimination Contexts E [ ] ∈ E ⇒ C ::= [ ] | E [ ] t | d ( E [ ]) 26 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Values ◮ Destructors For each n -ary c with n ≥ 1 and each i ∈ { 1, . . . , n } , d c ,i ( c ( x 1 , . . . , x n )) � → D x i For each nullary c , d c c � → D ℧ ◮ Elimination Contexts E [ ] ∈ E ⇒ C ::= [ ] | E [ ] t | d ( E [ ]) ◮ Values λx.t c ( t 1 , . . . , t n ) 26 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. ◮ A subterm (position) u occurs in a Redex Argument of a term t if either 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. ◮ A subterm (position) u occurs in a Redex Argument of a term t if either t has a subterm ( λx.t 1 ) t 2 and u occurs in some t i or 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. ◮ A subterm (position) u occurs in a Redex Argument of a term t if either t has a subterm ( λx.t 1 ) t 2 and u occurs in some t i or t has a subterm f ( l 1 σ, . . . , l n σ ) and u occurs in some l i σ where f ( l 1 , . . . , l n ) r � → R 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. ◮ A subterm (position) u occurs in a Redex Argument of a term t if either t has a subterm ( λx.t 1 ) t 2 and u occurs in some t i or t has a subterm f ( l 1 σ, . . . , l n σ ) and u occurs in some l i σ where f ( l 1 , . . . , l n ) r � → R ◮ A redex (position) u is External in a term t if no residual of u occurs in a redex argument of a reduct of t . 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Redexes (CCERSs) [KOO01] Let R be a constructor rewrite system. ◮ A subterm (position) u occurs in a Redex Argument of a term t if either t has a subterm ( λx.t 1 ) t 2 and u occurs in some t i or t has a subterm f ( l 1 σ, . . . , l n σ ) and u occurs in some l i σ where f ( l 1 , . . . , l n ) r � → R ◮ A redex (position) u is External in a term t if no residual of u occurs in a redex argument of a reduct of t . ◮ If t → β R u by contracting an external redex of t , then u is an External Reduct of t . 27 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Reducts are Strong Principal [Rib08] Let R be an orthogonal constructor rewrite system. 28 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Reducts are Strong Principal [Rib08] Let R be an orthogonal constructor rewrite system. Lemma Let t be a neutral term and u be an external reduct of t . If t reduces to a value v then u reduces to v . 28 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Reducts are Strong Principal [Rib08] Let R be an orthogonal constructor rewrite system. Lemma Let t be a neutral term and u be an external reduct of t . If t reduces to a value v then u reduces to v . ◮ If t is neutral and u is an external reduct of t then t � C R u . 28 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Reducts are Strong Principal [Rib08] Let R be an orthogonal constructor rewrite system. Lemma Let t be a neutral term and u be an external reduct of t . If t reduces to a value v then u reduces to v . ◮ If t is neutral and u is an external reduct of t then t � C R u . ◮ Theorem [KOO01] If R is orthogonal then every reducible term has an external redex. 28 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion External Reducts are Strong Principal [Rib08] Let R be an orthogonal constructor rewrite system. Lemma Let t be a neutral term and u be an external reduct of t . If t reduces to a value v then u reduces to v . ◮ If t is neutral and u is an external reduct of t then t � C R u . ◮ Theorem [KOO01] If R is orthogonal then every reducible term has an external redex. Corollary If R is an orthogonal constructor rewrite system then C R is stable by union. 28 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Outline Introduction Reducibility Stability by Union Application to Orthogonal Constructor Rewriting Conclusion 29 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Conclusion We have studied different reducibility families, and compared them wrt. stability by union. 30 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Conclusion We have studied different reducibility families, and compared them wrt. stability by union. ◮ Some rewrite systems do not admit reducibility families stable by union. 30 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Conclusion We have studied different reducibility families, and compared them wrt. stability by union. ◮ Some rewrite systems do not admit reducibility families stable by union. ◮ Sufficient conditions to have a reducibility family stable by union. 30 / 32

Introduction Reducibility Stability by Union Orthogonal Constructor Rewriting Conclusion Conclusion We have studied different reducibility families, and compared them wrt. stability by union. ◮ Some rewrite systems do not admit reducibility families stable by union. ◮ Sufficient conditions to have a reducibility family stable by union. ◮ Investigation of the structure of Girard’s Candidates. 30 / 32