Greg Restall Nodes and Subnodes Terms for Classical Sequents 17 of 66 ▶ A variable x of type A and a cut point • of type A are both A nodes . ▶ If n is an A ∧ B node, then L n is an A node and R n is a B node. ▶ If n is an A ∨ B node, then F n is an A node and S n is a B node. ▶ If n is an A ⊃ B node, then A n is an A node and C n is a B node. ▶ If n is a ¬ A node, then N n is an A node. ▶ For each complex node L n , R n , F n , S n , A n , C n and N n , n is its immediate subnode, and the subnodes of n are also subnodes of the original node.
Linkings, Inputs and Outputs If A Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N If A is in output position . is in input position , or N – L , R , F , S and C each preserve position . is also in output position . are in output position , or C If L , R , F , S is also in input position . are in input position , or C If L , R , F , S Positions generalise to subnodes as follows: is in output position . is in input position , and , In 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type.
Linkings, Inputs and Outputs or N Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N If A is in output position . is in input position , If A – L , R , F , S and C each preserve position . is also in output position . are in output position , or C If L , R , F , S is also in input position . are in input position , or C If L , R , F , S Positions generalise to subnodes as follows: 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position .
Linkings, Inputs and Outputs or N Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N If A is in output position . is in input position , If A – L , R , F , S and C each preserve position . is also in output position . are in output position , or C If L , R , F , S is also in input position . are in input position , or C If L , R , F , S 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows:
Linkings, Inputs and Outputs is in output position . Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N If A is in input position , or N If A – L , R , F , S and C each preserve position . is also in output position . are in output position , or C If L , R , F , S 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position .
Linkings, Inputs and Outputs If A Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N is in output position . is in input position , or N If A – L , R , F , S and C each preserve position . 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position .
Linkings, Inputs and Outputs If A Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . is in output position , or N is in output position . is in input position , or N If A – L , R , F , S and C each preserve position . 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position .
Linkings, Inputs and Outputs is in output position , Terms for Classical Sequents Greg Restall position of that linking. The inputs ( outputs ) of a linking are the variables in input ( output ) – A and N reverse position . is in input position . or N If A – L , R , F , S and C each preserve position . 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position . ▶ If A n or N n is in input position , n is in output position .
– L , R , F , S and C each preserve position . Linkings, Inputs and Outputs – A and N reverse position . The inputs ( outputs ) of a linking are the variables in input ( output ) position of that linking. Greg Restall Terms for Classical Sequents 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position . ▶ If A n or N n is in input position , n is in output position . ▶ If A n or N n is in output position , n is in input position .
– L , R , F , S and C each preserve position . Linkings, Inputs and Outputs – A and N reverse position . The inputs ( outputs ) of a linking are the variables in input ( output ) position of that linking. Greg Restall Terms for Classical Sequents 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position . ▶ If A n or N n is in input position , n is in output position . ▶ If A n or N n is in output position , n is in input position .
– L , R , F , S and C each preserve position . Linkings, Inputs and Outputs – A and N reverse position . position of that linking. Greg Restall Terms for Classical Sequents 18 of 66 ▶ A linking is a pair n ⌢ m of nodes of the same type. ▶ In n ⌢ m , n is in input position , and m is in output position . ▶ Positions generalise to subnodes as follows: ▶ If L n , R n , F n , S n or C n are in input position , n is also in input position . ▶ If L n , R n , F n , S n or C n are in output position , n is also in output position . ▶ If A n or N n is in input position , n is in output position . ▶ If A n or N n is in output position , n is in input position . ▶ The inputs ( outputs ) of a linking are the variables in input ( output )
Example Linkings AAA C CA C Greg Restall Terms for Classical Sequents 19 of 66 x of type (( p ⊃ q ) ⊃ p ) ⊃ p
Example Linkings CA C Greg Restall Terms for Classical Sequents 19 of 66 x of type (( p ⊃ q ) ⊃ p ) ⊃ p AAA x ⌢ C x
Example Linkings Greg Restall Terms for Classical Sequents 19 of 66 x of type (( p ⊃ q ) ⊃ p ) ⊃ p AAA x ⌢ C x CA x ⌢ C x
Preterms The inputs of a preterm are the inputs of its linkings. Its outputs are the outputs of its linkings. Greg Restall Terms for Classical Sequents 20 of 66 ▶ A preterm is a finite set of linkings.
Preterms Its outputs are the outputs of its linkings. Greg Restall Terms for Classical Sequents 20 of 66 ▶ A preterm is a finite set of linkings. ▶ The inputs of a preterm are the inputs of its linkings.
Preterms Greg Restall Terms for Classical Sequents 20 of 66 ▶ A preterm is a finite set of linkings. ▶ The inputs of a preterm are the inputs of its linkings. ▶ Its outputs are the outputs of its linkings.
derivations
Annotating Derivations: Identity Greg Restall Terms for Classical Sequents 22 of 66 x ⌢ y Σ, x : A � y : A, ∆
R Annotating Derivations: Conjunction F S Greg Restall Terms for Classical Sequents 23 of 66 π ( x, y ) Σ, x : A, y : B � ∆ ∧ L π ( F z, S z ) Σ, z : A ∧ B � ∆
Annotating Derivations: Conjunction Greg Restall Terms for Classical Sequents 23 of 66 π ( x, y ) π [ x ] π ′ [ y ] Σ ′ � y : B, ∆ ′ Σ, x : A, y : B � ∆ Σ � x : A, ∆ ∧ L ∧ R π ′ [ S z ] π ( F z, S z ) π [ F z ] Σ, Σ ′ � z : A ∧ B, ∆, ∆ ′ Σ, z : A ∧ B � ∆
Excursus on Weakening and Variables In a premise Terms for Classical Sequents Greg Restall There might be none . inputs to the proof term. and of the display all and the indicated F L can be S F L 24 of 66 [ x : p ] ⊃ I λy x : q ⊃ p ⊃ I λxλy x : p ⊃ ( q ⊃ p )
Excursus on Weakening and Variables In a premise Terms for Classical Sequents Greg Restall There might be none . inputs to the proof term. and of the display all and the indicated 24 of 66 can be [ x : p ] ⊃ I λy x : q ⊃ p ⊃ I λxλy x : p ⊃ ( q ⊃ p ) π ( x, y ) π ( x ) Σ, x : A, y : B � ∆ Σ, x : A � ∆ ∧ L ∧ L π ( F z, S z ) π ( F z ) Σ, z : A ∧ B � ∆ Σ, z : A ∧ B � ∆
Excursus on Weakening and Variables There might be none . Terms for Classical Sequents Greg Restall can be 24 of 66 [ x : p ] ⊃ I λy x : q ⊃ p ⊃ I λxλy x : p ⊃ ( q ⊃ p ) π ( x, y ) π ( x ) Σ, x : A, y : B � ∆ Σ, x : A � ∆ ∧ L ∧ L π ( F z, S z ) π ( F z ) Σ, z : A ∧ B � ∆ Σ, z : A ∧ B � ∆ In a premise π ( x, y ) the indicated x and y display all of the x and y inputs to the proof term.
Annotating Derivations: Negation Greg Restall Terms for Classical Sequents 25 of 66 π [ x ] π ( x ) Σ � x : A, ∆ Σ, x : A � ∆ ¬ L ¬ R π [ N z ] π ( N z ) Σ, z : ¬ A � ∆ Σ � z : ¬ A, ∆
Annotating Derivations: Disjunction Greg Restall Terms for Classical Sequents 26 of 66 π ( x ) π ′ ( y ) π [ x, y ] Σ, x : A � ∆ Σ ′ , y : B � ∆ ′ Σ � x : A, y : B, ∆ ∨ L ∨ R π ( L z ) π ′ ( R z ) π [ L z, R z ] Σ, Σ ′ , z : A ∨ B � ∆, ∆ ′ Σ � z : A ∨ B, ∆
Annotating Derivations: Conditional Greg Restall Terms for Classical Sequents 27 of 66 π [ x ] π ′ ( y ) π ( x )[ y ] Σ � x : A, ∆ Σ ′ , y : B � ∆ ′ Σ, x : A � y : B, ∆ ⊃ L ⊃ R π [ A z ] π ′ ( L z ) π ( A z )[ C z ] Σ, Σ ′ , z : A ⊃ B � ∆, ∆ ′ Σ � z : A ⊃ B, ∆
Example Annotation Greg Restall Terms for Classical Sequents 28 of 66 y ⌢ y z ⌢ z y : q � y : q z : r � z : r ∨ L x ⌢ x L w ⌢ y R w ⌢ z x : p � x : p w : q ∨ r � y : q, z : r ∧ R x ⌢ F v L w ⌢ S v R w ⌢ z x : p, w : q ∨ r � v : p ∧ q, z : r ∧ L F u ⌢ F v LS u ⌢ S v RS u ⌢ z u : p ∧ ( q ∨ r ) � v : p ∧ q, z : r ∨ R F u ⌢ FL t LS u ⌢ SL t RS u ⌢ R t u : p ∧ ( q ∨ r ) � t : ( p ∧ q ) ∨ r
Annotating Derivations: Cut Cut Greg Restall Terms for Classical Sequents 29 of 66 π [ x ] π ′ ( y ) Σ � x : A, ∆ Σ ′ , y : A � ∆ ′ π [ • ] π ′ ( • ) Σ, Σ ′ � ∆, ∆ ′
Example Annotation, with Cut Cut Terms for Classical Sequents Greg Restall 30 of 66 x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∨ L ∧ R L y ⌢ x R y ⌢ x x ⌢ F z x ⌢ S z y : p ∨ p � x : p x : p � z : p ∧ p L y ⌢ • R y ⌢ • • ⌢ F z • ⌢ S z y : p ∨ p � z : p ∧ p
31 of 66 R Terms for Classical Sequents Greg Restall R S L S R I E R S L R When is π 1 the same proof as π 2 (revisited)? z ⌢ z z ⌢ z z : p � z : p z : p � z : p p ∧ q ∨ R ∨ R ∧ E z ⌢ L y F x ⌢ z p z : p � y : p ∨ q x : p ∧ q � z : p ∨ I ∧ L ∧ L p ∨ q F x ⌢ L y F x ⌢ L y x : p ∧ q � y : p ∨ q x : p ∧ q � y : p ∨ q
31 of 66 Greg Restall Terms for Classical Sequents When is π 1 the same proof as π 2 (revisited)? z ⌢ z z ⌢ z z : p � z : p z : p � z : p p ∧ q ∨ R ∨ R ∧ E z ⌢ L y F x ⌢ z p z : p � y : p ∨ q x : p ∧ q � z : p ∨ I ∧ L ∧ L p ∨ q F x ⌢ L y F x ⌢ L y x : p ∧ q � y : p ∨ q x : p ∧ q � y : p ∨ q w ⌢ w w ⌢ w w : q � w : q w : q � w : q p ∧ q ∨ R ∨ R ∧ E w ⌢ R y S x ⌢ w q w : q � y : p ∨ q x : p ∧ q � w : q ∨ I ∧ L ∧ L p ∨ q S x ⌢ R y S x ⌢ R y x : p ∧ q � y : p ∨ q x : p ∧ q � y : p ∨ q
32 of 66 Greg Restall Terms for Classical Sequents When is π 1 the same proof as π 2 (revisited)? [ p ] 1 [ q ] 1 ∨ I ∨ I p ∨ q q ∨ p q ∨ p ∨ E 1 q ∨ p ∨ I ( q ∨ p ) ∨ r x ⌢ x y ⌢ y x : p � x : p y : q � y : q ∨ R ∨ R x ⌢ R z y ⌢ L z x : p � z : q ∨ p y : q � z : q ∨ p ∨ L L w ⌢ R z R w ⌢ L z w : p ∨ q � z : q ∨ p ∨ R L w ⌢ RL u R w ⌢ LL u w : p ∨ q � u : ( q ∨ p ) ∨ r
33 of 66 Greg Restall Terms for Classical Sequents When is π 1 the same proof as π 2 (revisited)? [ p ] 1 [ q ] 1 ∨ I ∨ I q ∨ p q ∨ p ∨ I ∨ I ( q ∨ p ) ∨ r ( q ∨ p ) ∨ r p ∨ q ∨ E 1 ( q ∨ p ) ∨ r x ⌢ x y ⌢ y x : p � x : p y : q � y : q ∨ R ∨ R x ⌢ R z x ⌢ LL u x : p � z : q ∨ p y : q � z : q ∨ p ∨ R ∨ R x ⌢ RL u y ⌢ LL u x : p � u : ( q ∨ p ) ∨ r y : q � u : ( q ∨ p ) ∨ r ∨ L L w ⌢ RL u R w ⌢ LL u w : p ∨ q � u : ( q ∨ p ) ∨ r
Sequentialisable Preterms A preterm is sequentialisable iff it is the conclusion of some derivation. Greg Restall Terms for Classical Sequents 34 of 66
terms
Nonsequentialisable Preterms This is connected, but it is not connected enough . Greg Restall Terms for Classical Sequents 36 of 66 L x ⌢ F y R x ⌢ S y x : p ∨ q � y : p ∧ q
Switching Example L Terms for Classical Sequents Greg Restall S R F L S R F S R F L S R F L 37 of 66 L x ⌢ F y R x ⌢ S y x : p ∨ q � y : p ∧ q
Switching Example Greg Restall Terms for Classical Sequents 37 of 66 L x ⌢ F y R x ⌢ S y x : p ∨ q � y : p ∧ q L x ⌢ F y ✚ ✚ ✚ R x ⌢ ✚ S y x : p ∨ − � y : p ∧ − F y ✚ ✚ L x ⌢ � � R x ⌢ S y x : p ∨ − � y : − ∧ q ✚ ✚ R x ⌢ ✚ L x ⌢ F y S y x : − ∨ q � y : p ∧ − ✚ L x ⌢ � � F y R x ⌢ S y x : − ∨ q � y : − ∧ q
Switchings and output position ), one item of the pair to keep, and the other to delete . A linking in a switching of a preterm survives if and only if neither side of the link involves a deletion. A preterm is spanned if every switching has at least one surviving linking. Greg Restall Terms for Classical Sequents 38 of 66 ▶ The switchings of a preterm π are found by selecting for each pair of subterms L n and R n in input position; F n and S n in output position , A n in output position and C n in input position ; or the cut point • (in both input
Switchings and output position ), one item of the pair to keep, and the other to delete . side of the link involves a deletion. A preterm is spanned if every switching has at least one surviving linking. Greg Restall Terms for Classical Sequents 38 of 66 ▶ The switchings of a preterm π are found by selecting for each pair of subterms L n and R n in input position; F n and S n in output position , A n in output position and C n in input position ; or the cut point • (in both input ▶ A linking in a switching of a preterm π survives if and only if neither
Switchings and output position ), one item of the pair to keep, and the other to delete . side of the link involves a deletion. Greg Restall Terms for Classical Sequents 38 of 66 ▶ The switchings of a preterm π are found by selecting for each pair of subterms L n and R n in input position; F n and S n in output position , A n in output position and C n in input position ; or the cut point • (in both input ▶ A linking in a switching of a preterm π survives if and only if neither ▶ A preterm is spanned if every switching has at least one surviving linking.
Example F Terms for Classical Sequents Greg Restall R RS SL LS FL F R RS SL LS FL R RS SL LS FL F R RS SL LS FL F This has two pairs for switching: 39 of 66 F u ⌢ FL t LS u ⌢ SL t RS u ⌢ R t LS u / RS u in input position . FL t / SL t in output position .
Example Greg Restall Terms for Classical Sequents 39 of 66 This has two pairs for switching: F u ⌢ FL t LS u ⌢ SL t RS u ⌢ R t LS u / RS u in input position . FL t / SL t in output position . FL t ✟✟ ✟ ✟ F u ⌢ ✟ LS u ⌢ SL t RS u ⌢ R t F u ⌢ FL t ✟✟ ✟ ✟ LS u ⌢ ✟ SL t RS u ⌢ R t ✘ ✟ FL t LS u ⌢ SL t ✘✘ F u ⌢ ✟ RS u ⌢ R t ✘ ✟ SL t ✘✘ F u ⌢ FL t LS u ⌢ ✟ RS u ⌢ R t
Terms Greg Restall Terms for Classical Sequents 40 of 66 A preterm π is a term when it is spanned .
Sequentialisable Preterms are Terms Greg Restall Terms for Classical Sequents 41 of 66 By induction on the derivation sequentialising π .
Sequentialisable Preterms are Terms: Identity Greg Restall Terms for Classical Sequents 42 of 66 x ⌢ y Σ, x : A � y : A, ∆
Sequentialisable Preterms are Terms: Conjunction Greg Restall Terms for Classical Sequents 43 of 66 π ( x, y ) π [ x ] π ′ [ y ] Σ ′ � y : B, ∆ ′ Σ, x : A, y : B � ∆ Σ � x : A, ∆ ∧ L ∧ R π ′ [ S z ] π ( F z, S z ) π [ F z ] Σ, Σ ′ � z : A ∧ B, ∆, ∆ ′ Σ, z : A ∧ B � ∆
Sequentialisable Preterms are Terms: Negation Greg Restall Terms for Classical Sequents 44 of 66 π [ x ] π ( x ) Σ � x : A, ∆ Σ, x : A � ∆ ¬ L ¬ R π [ N z ] π ( N z ) Σ, z : ¬ A � ∆ Σ � z : ¬ A, ∆
Sequentialisable Preterms are Terms: Disjunction Greg Restall Terms for Classical Sequents 45 of 66 π ( x ) π ′ ( y ) π [ x, y ] Σ, x : A � ∆ Σ ′ , y : B � ∆ ′ Σ � x : A, y : B, ∆ ∨ L ∨ R π ′ ( R z ) π ( L z ) π [ L z, R z ] Σ, Σ ′ , z : A ∨ B � ∆, ∆ ′ Σ � z : A ∨ B, ∆
Sequentialisable Preterms are Terms: Conditional Greg Restall Terms for Classical Sequents 46 of 66 π [ x ] π ′ ( y ) π ( x )[ y ] Σ � x : A, ∆ Σ ′ , y : B � ∆ ′ Σ, x : A � y : B, ∆ ⊃ L ⊃ R π ′ ( C z ) π [ A z ] π ( A z )[ C z ] Σ, Σ ′ , z : A ⊃ B � ∆, ∆ ′ Σ � z : A ⊃ B, ∆
Sequentialisable Preterms are Terms: Cut Cut Greg Restall Terms for Classical Sequents 47 of 66 π ′ ( y ) π [ x ] Σ � x : A, ∆ Σ ′ , y : A � ∆ ′ π [ • ] π ′ ( • ) Σ, Σ ′ � ∆, ∆ ′
Terms are Sequentialisable Except … Greg Restall Terms for Classical Sequents 48 of 66 By induction on the number of pairs for switching in π . x ⌢ y u ⌢ v
eliminating cuts
Conjunction Cut Reduction Cut Terms for Classical Sequents Greg Restall Cut Cut reduces to 50 of 66 π [ x ] π ′ [ y ] π ′′ ( u, v ) Σ ′ � y : B, ∆ Σ ′′ , u : A, v : B � ∆ ′′ Σ � x : A, ∆ ∧ R ∧ L π ′ [ S z ] π ′′ ( F w, S w ) π [ F z ] Σ, Σ ′ � z : A ∧ B, ∆, ∆ Σ ′′ , w : A ∧ B � ∆ ′′ π ′ [ S • ] π ′′ ( F • , S • ) π [ F • ] Σ, Σ ′ , Σ ′′ � ∆, ∆ ′ , ∆ ′′
Conjunction Cut Reduction Cut Terms for Classical Sequents Greg Restall Cut Cut reduces to 50 of 66 π [ x ] π ′ [ y ] π ′′ ( u, v ) Σ ′ � y : B, ∆ Σ ′′ , u : A, v : B � ∆ ′′ Σ � x : A, ∆ ∧ R ∧ L π ′ [ S z ] π ′′ ( F w, S w ) π [ F z ] Σ, Σ ′ � z : A ∧ B, ∆, ∆ Σ ′′ , w : A ∧ B � ∆ ′′ π ′ [ S • ] π ′′ ( F • , S • ) π [ F • ] Σ, Σ ′ , Σ ′′ � ∆, ∆ ′ , ∆ ′′ π ′ [ y ] π ′′ ( u, v ) Σ ′ � y : B, ∆ Σ ′′ , u : A, v : B � ∆ ′′ π ′ [ ⋆ ] π ′′ ( u, ⋆ ) π [ x ] Σ � x : A, ∆ Σ ′ , Σ ′′ , u : A � ∆ ′ , ∆ ′′ π [ ∗ ] π ′ [ ⋆ ] π ′′ ( ∗ , ⋆ ) Σ, Σ ′ , Σ ′′ � ∆, ∆ ′ , ∆ ′′
Identity Cut Reduction Cut reduces to Greg Restall Terms for Classical Sequents 51 of 66 π [ x ] y ⌢ z Σ � x : A, ∆ Σ ′ , y : A � z : A, ∆ ′ π [ • ] • ⌢ z Σ, Σ ′ � z : A, ∆, ∆ ′
Identity Cut Reduction Cut reduces to Greg Restall Terms for Classical Sequents 51 of 66 π [ x ] y ⌢ z Σ � x : A, ∆ Σ ′ , y : A � z : A, ∆ ′ π [ • ] • ⌢ z Σ, Σ ′ � z : A, ∆, ∆ ′ π [ z ] Σ, Σ ′ � z : A, ∆, ∆ ′
Difficult Cases: Contraction R L L R R L F F L L S R S Greg Restall Terms for Classical Sequents R L 52 of 66 Cut x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∨ L ∧ R L y ⌢ x R y ⌢ x x ⌢ F z x ⌢ S z y : p ∨ p � x : p x : p � z : p ∧ p L y ⌢ • R y ⌢ • • ⌢ F z • ⌢ S z y : p ∨ p � z : p ∧ p
Difficult Cases: Contraction Cut Terms for Classical Sequents Greg Restall 52 of 66 x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∨ L ∧ R L y ⌢ x R y ⌢ x x ⌢ F z x ⌢ S z y : p ∨ p � x : p x : p � z : p ∧ p L y ⌢ • R y ⌢ • • ⌢ F z • ⌢ S z y : p ∨ p � z : p ∧ p x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∨ L ∨ L L y ⌢ x R y ⌢ x L y ⌢ x R y ⌢ x y : p ∨ p � x : p y : p ∨ p � x : p ∧ R L y ⌢ F z R y ⌢ F z L y ⌢ S z R y ⌢ S z y : p ∨ p � z : p ∧ p
Difficult Cases: Contraction Cut Terms for Classical Sequents Greg Restall 53 of 66 x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∨ L ∧ R L y ⌢ x R y ⌢ x x ⌢ F z x ⌢ S z y : p ∨ p � x : p x : p � z : p ∧ p L y ⌢ • R y ⌢ • • ⌢ F z • ⌢ S z y : p ∨ p � z : p ∧ p x ⌢ x x ⌢ x x ⌢ x x ⌢ x x : p � x : p x : p � x : p x : p � x : p x : p � x : p ∧ R ∧ R x ⌢ F z x ⌢ S z x ⌢ F z x ⌢ S z x : p � z : p ∧ p x : p � z : p ∧ p ∨ L L y ⌢ F z R y ⌢ F z L y ⌢ S z R y ⌢ S z y : p ∨ p � z : p ∧ p
Difficult Cases: Weakening Cut Mix Greg Restall Terms for Classical Sequents 54 of 66 π π ′ Σ � ∆ Σ � ∆ π π ′ Σ � x : A, ∆ Σ, y : A � ∆ π π ′ Σ � ∆
Difficult Cases: Weakening Cut Terms for Classical Sequents Greg Restall Mix 54 of 66 π π ′ Σ � ∆ Σ � ∆ π π ′ Σ � x : A, ∆ Σ, y : A � ∆ π π ′ Σ � ∆ π ′ π Σ � ∆ Σ � ∆ π π ′ Σ � ∆
Back to Sequentialisation Mix Greg Restall Terms for Classical Sequents 55 of 66 x ⌢ y u ⌢ v
Back to Sequentialisation Mix Greg Restall Terms for Classical Sequents 55 of 66 x ⌢ y u ⌢ v x ⌢ y u ⌢ v x : A � y : A u : B � v : B x ⌢ y u ⌢ v x : A, u : B � y : B, v : A
Sequentialisation: Terms with No Switchings L Terms for Classical Sequents Greg Restall R S L NR F R and mixes . R and R , L , L , It has a derivation using the linear rules 56 of 66 The term contains no L n , R n , C n and • in input position or F n , S n , A n and • in output position.
Sequentialisation: Terms with No Switchings It has a derivation using the linear rules F L NR L S R Greg Restall Terms for Classical Sequents 56 of 66 The term contains no L n , R n , C n and • in input position or F n , S n , A n and • in output position. ∧ L , ¬ L , ¬ R , ∨ R and ⊃ R and mixes .
Sequentialisation: Terms with No Switchings It has a derivation using the linear rules Greg Restall Terms for Classical Sequents 56 of 66 The term contains no L n , R n , C n and • in input position or F n , S n , A n and • in output position. ∧ L , ¬ L , ¬ R , ∨ R and ⊃ R and mixes . F y ⌢ L z NR z ⌢ L z S y ⌢ R z y : p ∧ ¬ p � z : p ∨ ¬ p
Terms with No Switchings: Example Mix Terms for Classical Sequents Greg Restall Mix 57 of 66 u ⌢ u v ⌢ v u : p � u : p v : ¬ p � v : ¬ p ¬ R ∨ R x ⌢ x N v ⌢ u v ⌢ R z x : p � x : p � u : p, v : ¬ p v : ¬ p � z : p ∨ ¬ p ∨ R ∨ R ∧ L x ⌢ L z NR z ⌢ L z S y ⌢ R z x : p � z : p ∨ ¬ p � z : p ∨ ¬ p y : p ∧ ¬ p � z : p ∨ ¬ p ∧ L F y ⌢ L z NR z ⌢ L z S y ⌢ R z y : p ∧ ¬ p � z : p ∨ ¬ p y : p ∧ ¬ p � z : p ∨ ¬ p F y ⌢ L z NR z ⌢ L z S y ⌢ R z y : p ∧ ¬ p � z : p ∨ ¬ p
Terms with Switchings By induction on the number of switched pairs. Take a switched pair at the adjacent to variables or cut points (peel away unswitched steps if there aren’t any). Greg Restall Terms for Classical Sequents 58 of 66 π [ x ](−) π [−]( y ) Σ ′ , y : B � ∆ ′ Σ � x : A, ∆ ⊃ L π [ A z ]( L z ) Σ, Σ ′ , z : A ⊃ B � ∆, ∆ ′
Back to Eliminating Cuts: Cuts can be Complicated Cut Terms for Classical Sequents Greg Restall 59 of 66 π ′ [ x, v ] π ′′ ( y, z, x ) π [ x, u ] � x : A ∧ B, u : A � x : A ∧ B, v : B y : A, z : B, x : A ∧ B � ∧ R ∧ L π ′ [ x, S x ] π ′′ ( F x, S x, x ) π [ x, F x ] � x : A ∧ B x : A ∧ B � π [ • , F • ] π ′ [ • , S • ] π ′′ ( F • , S • , • ) Σ � ∆
Cut Reductions SN N S F S F FN S F S N S F F S Greg Restall Terms for Classical Sequents F S 60 of 66 F conjunction : for each F / S , add new cut points and . For any add for each link with as input. For any add for each link with as output. S F F S Given a term π ( • )[ • ] and a cut-point • , the • -reduction of π is found by: ▶ atomic : replace each pair n ⌢ • and • ⌢ m by n ⌢ m .
Recommend
More recommend
