Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’,‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . Types in Proof Mining
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’,‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . M , I do not reflect true complexity of IPP! Types in Proof Mining
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’,‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . M , I do not reflect true complexity of IPP! Related problem: bad behavior w.r.t. modus ponens! Types in Proof Mining
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Types in Proof Mining
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Types in Proof Mining
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Construct by recursion on P a new proof P I of C I . Types in Proof Mining
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Construct by recursion on P a new proof P I of C I . In particular: solve modus ponens problem: A I (A → B) I , . B I Types in Proof Mining
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Construct by recursion on P a new proof P I of C I . In particular: solve modus ponens problem: A I (A → B) I , . B I Our approach is based on novel forms and extensions of: K. G¨ odel’s functional interpretation! Types in Proof Mining
Detour through intuitionistic systems and higher types Types in Proof Mining
Detour through intuitionistic systems and higher types HA (‘Heyting arithmetic’ is defined as Peano arithmetic but with intuitionistic (constructive) logic . Types in Proof Mining
Detour through intuitionistic systems and higher types HA (‘Heyting arithmetic’ is defined as Peano arithmetic but with intuitionistic (constructive) logic . HA ω is the extension of HA to all finite types over I N . Types in Proof Mining
Detour through intuitionistic systems and higher types HA (‘Heyting arithmetic’ is defined as Peano arithmetic but with intuitionistic (constructive) logic . HA ω is the extension of HA to all finite types over I N . Types T: (i) I N ∈ T, ρ, τ ∈ T ⇒ ( ρ → τ ) ∈ T. Types in Proof Mining
Detour through intuitionistic systems and higher types HA (‘Heyting arithmetic’ is defined as Peano arithmetic but with intuitionistic (constructive) logic . HA ω is the extension of HA to all finite types over I N . Types T: (i) I N ∈ T, ρ, τ ∈ T ⇒ ( ρ → τ ) ∈ T. HA ω has λ -abstraction ( λ x ρ . t [ x ] τ )( s ρ ) = τ t [ s / x ] and primitive recursion in all finite types (Hilbert 1926, G¨ odel 1958): for x ∈ I N R ρ (0 , y , z) = ρ y , R ρ (x + 1 , y , z) = ρ z(R ρ xyz , x) , where = ρ is defined as pointwise (extensional) equality (with a weak extensionality rule; see later). Types in Proof Mining
Detour through intuitionistic systems and higher types HA (‘Heyting arithmetic’ is defined as Peano arithmetic but with intuitionistic (constructive) logic . HA ω is the extension of HA to all finite types over I N . Types T: (i) I N ∈ T, ρ, τ ∈ T ⇒ ( ρ → τ ) ∈ T. HA ω has λ -abstraction ( λ x ρ . t [ x ] τ )( s ρ ) = τ t [ s / x ] and primitive recursion in all finite types (Hilbert 1926, G¨ odel 1958): for x ∈ I N R ρ (0 , y , z) = ρ y , R ρ (x + 1 , y , z) = ρ z(R ρ xyz , x) , where = ρ is defined as pointwise (extensional) equality (with a weak extensionality rule; see later). PA ω = HA ω + ( A ∨ ¬ A ) . Types in Proof Mining
Towards proofs based on classical logic Entrance door for classical logic: Markov’s principle M ω ! M ω : ¬¬∃ x ρ A qf ( x ) → ∃ x ρ A qf ( x ) , A qf quantifier-free. Types in Proof Mining
Towards proofs based on classical logic Entrance door for classical logic: Markov’s principle M ω ! M ω : ¬¬∃ x ρ A qf ( x ) → ∃ x ρ A qf ( x ) , A qf quantifier-free. For ρ = I N , this has a partial computable solution by unbounded search ( Kleene realizability ) (no complexity information), but no total computable solution, i.e. no modified realizability! Types in Proof Mining
Towards proofs based on classical logic Entrance door for classical logic: Markov’s principle M ω ! M ω : ¬¬∃ x ρ A qf ( x ) → ∃ x ρ A qf ( x ) , A qf quantifier-free. For ρ = I N , this has a partial computable solution by unbounded search ( Kleene realizability ) (no complexity information), but no total computable solution, i.e. no modified realizability! For ρ � = I N : not even unbounded search possible! Types in Proof Mining
G¨ odel’s functional (‘Dialectica’) interpretation D (G¨ odel 1941, 1958) Solution: Don’t try to solve M ω but eliminate it from proofs! Types in Proof Mining
G¨ odel’s functional (‘Dialectica’) interpretation D (G¨ odel 1941, 1958) Solution: Don’t try to solve M ω but eliminate it from proofs! Combined with negative translation G := D ◦ N one obtains Program Extraction Theorems for classical proofs! Types in Proof Mining
G¨ odel’s functional (‘Dialectica’) interpretation D (G¨ odel 1941, 1958) Solution: Don’t try to solve M ω but eliminate it from proofs! Combined with negative translation G := D ◦ N one obtains Program Extraction Theorems for classical proofs! G extracts from a given proof p p ⊢ ∀ x ∃ y A qf (x , y) an explicit effective functional Φ realizing A G , i.e. ∀ x A qf (x , Φ(x)) . Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that A G ≡ ∀ x ∃ y A G (x , y) , where A G is quantifier-free , Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that A G ≡ ∀ x ∃ y A G (x , y) , where A G is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A G ≡ A . Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that A G ≡ ∀ x ∃ y A G (x , y) , where A G is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A G ≡ A . A ↔ A G by quantifier-free choice in all types QF-AC : ∀ a ∃ b F qf (a , b) → ∃ B ∀ a F qf (a , B(a)) . Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that A G ≡ ∀ x ∃ y A G (x , y) , where A G is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A G ≡ A . A ↔ A G by quantifier-free choice in all types QF-AC : ∀ a ∃ b F qf (a , b) → ∃ B ∀ a F qf (a , B(a)) . x , y are tuples of functionals of finite type. Types in Proof Mining
Functional interpretation in five minutes G¨ odel’s functional interpretation G is a map G : Form( PA ω ) → Form( PA ω ) , A �→ A G such that A G ≡ ∀ x ∃ y A G (x , y) , where A G is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A G ≡ A . A ↔ A G by quantifier-free choice in all types QF-AC : ∀ a ∃ b F qf (a , b) → ∃ B ∀ a F qf (a , B(a)) . x , y are tuples of functionals of finite type. Streicher / K . N := Krivine’s negative transl. ⇒ G = Shoenfield Variant! Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) � � (G3) (A ∨ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∨ B G (v , y) Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) � � (G3) (A ∨ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∨ B G (v , y) (G4) ( ∀ z A) G ≡ ∀ z , u ∃ x A G (z , u , x) Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) � � (G3) (A ∨ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∨ B G (v , y) (G4) ( ∀ z A) G ≡ ∀ z , u ∃ x A G (z , u , x) � � (G5) (A → B) G ≡ ∀ f , v ∃ u , y A G (u , f(u)) → B G (v , y) Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) � � (G3) (A ∨ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∨ B G (v , y) (G4) ( ∀ z A) G ≡ ∀ z , u ∃ x A G (z , u , x) � � (G5) (A → B) G ≡ ∀ f , v ∃ u , y A G (u , f(u)) → B G (v , y) (G6) ( ∃ zA) G ≡ ∀ U ∃ z , f A G (z , U(z , f) , f(U(z , f))) Types in Proof Mining
A G ≡ ∀ u ∃ x A G (u , x) , B G ≡ ∀ v ∃ y B G (v , y) . (G1) P G ≡ P ≡ P G for atomic P (G2) ( ¬ A) G ≡ ∀ f ∃ u ¬ A G (u , f(u)) � � (G3) (A ∨ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∨ B G (v , y) (G4) ( ∀ z A) G ≡ ∀ z , u ∃ x A G (z , u , x) � � (G5) (A → B) G ≡ ∀ f , v ∃ u , y A G (u , f(u)) → B G (v , y) (G6) ( ∃ zA) G ≡ ∀ U ∃ z , f A G (z , U(z , f) , f(U(z , f))) � � (G7) (A ∧ B) G ≡ ∀ u , v ∃ x , y A G (u , x) ∧ B G (v , y) . Types in Proof Mining
Comments The program extraction theorem scales down to weak systems such as RCA 0 (where then Φ is ordinarily prim. rec., Parsons 1971) or of bounded arithmetic (where then Φ is basic feasible, Cook/Urquhart 1993). Types in Proof Mining
Comments The program extraction theorem scales down to weak systems such as RCA 0 (where then Φ is ordinarily prim. rec., Parsons 1971) or of bounded arithmetic (where then Φ is basic feasible, Cook/Urquhart 1993). It also scales up all the way to full countable and even dependent choice (including full 2nd order arithmetic), where then Φ is bar recursive (and holds in M ω or C ω ): Spector 1962. Types in Proof Mining
Connection to no-counterexample interpretation Let A be a prenex (arithmetical) formula and A S , A G , A n . c . i its Skolem, G and n . c . i . interpretations resp., then HA ω ⊢ A S → A G → A n . c . i , but the converse implications in general fail to hold even in PA ω +QF-AC! Types in Proof Mining
Connection to no-counterexample interpretation Let A be a prenex (arithmetical) formula and A S , A G , A n . c . i its Skolem, G and n . c . i . interpretations resp., then HA ω ⊢ A S → A G → A n . c . i , but the converse implications in general fail to hold even in PA ω +QF-AC! A S too strong (for a computable solution): Specker! Types in Proof Mining
Connection to no-counterexample interpretation Let A be a prenex (arithmetical) formula and A S , A G , A n . c . i its Skolem, G and n . c . i . interpretations resp., then HA ω ⊢ A S → A G → A n . c . i , but the converse implications in general fail to hold even in PA ω +QF-AC! A S too strong (for a computable solution): Specker! A n . c . i . too weak (see IPP above; modus ponens problem). Types in Proof Mining
Connection to no-counterexample interpretation Let A be a prenex (arithmetical) formula and A S , A G , A n . c . i its Skolem, G and n . c . i . interpretations resp., then HA ω ⊢ A S → A G → A n . c . i , but the converse implications in general fail to hold even in PA ω +QF-AC! A S too strong (for a computable solution): Specker! A n . c . i . too weak (see IPP above; modus ponens problem). A G just right : PA ω +QF-AC ⊢ A ↔ A G . Types in Proof Mining
Majorizability The functionals occurring in functional interpretation (such as the primitive recursive ones from PA ω but also the bar recursive ones) have a striking mathematical structure property: Definition (W.A. Howard 1973) � x ∗ � I N x : ≡ x ∗ ≥ x , x ∗ � ρ → τ x : ≡ ∀ y ∗ , y(y ∗ � ρ y → x ∗ (y ∗ ) � τ x(y)) . Read: ‘ x ∗ majorizes x ’ for x ∗ � x . Types in Proof Mining
Majorizability The functionals occurring in functional interpretation (such as the primitive recursive ones from PA ω but also the bar recursive ones) have a striking mathematical structure property: Definition (W.A. Howard 1973) � x ∗ � I N x : ≡ x ∗ ≥ x , x ∗ � ρ → τ x : ≡ ∀ y ∗ , y(y ∗ � ρ y → x ∗ (y ∗ ) � τ x(y)) . Read: ‘ x ∗ majorizes x ’ for x ∗ � x . Monotone functional interpretation MD (K.96) directly extracts majorants for functionals satisfying D . Types in Proof Mining
Majorizability The functionals occurring in functional interpretation (such as the primitive recursive ones from PA ω but also the bar recursive ones) have a striking mathematical structure property: Definition (W.A. Howard 1973) � x ∗ � I N x : ≡ x ∗ ≥ x , x ∗ � ρ → τ x : ≡ ∀ y ∗ , y(y ∗ � ρ y → x ∗ (y ∗ ) � τ x(y)) . Read: ‘ x ∗ majorizes x ’ for x ∗ � x . Monotone functional interpretation MD (K.96) directly extracts majorants for functionals satisfying D . Provides uniform bounds. Types in Proof Mining
Majorizability The functionals occurring in functional interpretation (such as the primitive recursive ones from PA ω but also the bar recursive ones) have a striking mathematical structure property: Definition (W.A. Howard 1973) � x ∗ � I N x : ≡ x ∗ ≥ x , x ∗ � ρ → τ x : ≡ ∀ y ∗ , y(y ∗ � ρ y → x ∗ (y ∗ ) � τ x(y)) . Read: ‘ x ∗ majorizes x ’ for x ∗ � x . Monotone functional interpretation MD (K.96) directly extracts majorants for functionals satisfying D . Provides uniform bounds. Applied to principles such as the binary K¨ onig’s Lemma WKL which do not have a computable D -interpretation. Types in Proof Mining
A logical metatheorem for concrete spaces P , K P Polish, K compact metric space, A ∃ existential, = X , = K -extensional. BA:= basic arithmetic, e.g. PA ω +QF-AC, HBC Heine/Borel compactness (via WKL). Types in Proof Mining
A logical metatheorem for concrete spaces P , K P Polish, K compact metric space, A ∃ existential, = X , = K -extensional. BA:= basic arithmetic, e.g. PA ω +QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀ x ∈ P ∀ y ∈ K ∃ m ∈ I N A ∃ (x , y , m) Types in Proof Mining
A logical metatheorem for concrete spaces P , K P Polish, K compact metric space, A ∃ existential, = X , = K -extensional. BA:= basic arithmetic, e.g. PA ω +QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀ x ∈ P ∀ y ∈ K ∃ m ∈ I N A ∃ (x , y , m) one can extract a closed term Φ of BA BA ⊢ ∀ x ∈ P ∀ y ∈ K ∃ m ≤ Φ(f x )A ∃ (x , y , m) . Types in Proof Mining
A logical metatheorem for concrete spaces P , K P Polish, K compact metric space, A ∃ existential, = X , = K -extensional. BA:= basic arithmetic, e.g. PA ω +QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀ x ∈ P ∀ y ∈ K ∃ m ∈ I N A ∃ (x , y , m) one can extract a closed term Φ of BA BA ⊢ ∀ x ∈ P ∀ y ∈ K ∃ m ≤ Φ(f x )A ∃ (x , y , m) . Important: Φ( f x ) does not depend on y ∈ K but on representation f x of x ! Types in Proof Mining
Abstract (nonseparable) structures For separable structures, the compactness (both total boundedness and completeness) is necessary for the independence from y ∈ K . Types in Proof Mining
Abstract (nonseparable) structures For separable structures, the compactness (both total boundedness and completeness) is necessary for the independence from y ∈ K . In the mean ergodic theorem, X was a totally general Hilbert space and the independence of x ∈ X (and f ) only depended on b ≥ � x � . Types in Proof Mining
Abstract (nonseparable) structures For separable structures, the compactness (both total boundedness and completeness) is necessary for the independence from y ∈ K . In the mean ergodic theorem, X was a totally general Hilbert space and the independence of x ∈ X (and f ) only depended on b ≥ � x � . Crucially used for this that the proof treats X as abstract structure that is not represented as separable space. Types in Proof Mining
Formal systems for analysis with abstract spaces X Types: (i) I N , X are types, (ii) with ρ, τ also ρ → τ is a type. Types in Proof Mining
Formal systems for analysis with abstract spaces X Types: (i) I N , X are types, (ii) with ρ, τ also ρ → τ is a type. PA ω, X is the extension of Peano Arithmetic to all types. A ω, X :=PA ω, X + DC , where DC: axiom of dependent choice for all types Implies full comprehension for numbers (higher order arithmetic). Types in Proof Mining
Formal systems for analysis with abstract spaces X Types: (i) I N , X are types, (ii) with ρ, τ also ρ → τ is a type. PA ω, X is the extension of Peano Arithmetic to all types. A ω, X :=PA ω, X + DC , where DC: axiom of dependent choice for all types Implies full comprehension for numbers (higher order arithmetic). A ω [ X , d , . . . ] results by adding constants d X , . . . with axioms expressing that ( X , d , . . . ) is a nonempty metric, hyperbolic . . . space. Types in Proof Mining
A warning concerning equality Extensionality rule ( only! ): s = ρ t r(s) = τ r(t) , where only x = I N y primitive equality predicate but for ρ → τ x X = X y X : ≡ d X (x , y) = I R 0 I R , x = ρ → τ y : ≡ ∀ v ρ (s(v) = τ t(v)) . Types in Proof Mining
A novel form of majorization N / X ] and a X of type X : y , x functionals of types ρ, � ρ := ρ [ I N � a N : ≡ x ≥ y x I N y I I N � a X y X : ≡ x ≥ d(y , a) . x I Types in Proof Mining
A novel form of majorization N / X ] and a X of type X : y , x functionals of types ρ, � ρ := ρ [ I N � a N : ≡ x ≥ y x I N y I I N � a X y X : ≡ x ≥ d(y , a) . x I For complex types ρ → τ this is extended in a hereditary fashion . Types in Proof Mining
A novel form of majorization N / X ] and a X of type X : y , x functionals of types ρ, � ρ := ρ [ I N � a N : ≡ x ≥ y x I N y I I N � a X y X : ≡ x ≥ d(y , a) . x I For complex types ρ → τ this is extended in a hereditary fashion . Example: f ∗ � a N , x ∈ X[n ≥ d(a , x) → f ∗ (n) ≥ d(a , f(x))] . X → X f ≡ ∀ n ∈ I Types in Proof Mining
A novel form of majorization N / X ] and a X of type X : y , x functionals of types ρ, � ρ := ρ [ I N � a N : ≡ x ≥ y x I N y I I N � a X y X : ≡ x ≥ d(y , a) . x I For complex types ρ → τ this is extended in a hereditary fashion . Example: f ∗ � a N , x ∈ X[n ≥ d(a , x) → f ∗ (n) ≥ d(a , f(x))] . X → X f ≡ ∀ n ∈ I f : X → X is nonexpansive (n.e.) if d(f(x) , f(y)) ≤ d(x , y) . Then λ n . n + b � a X → X f , if d ( a , f ( a )) ≤ b . Types in Proof Mining
A novel form of majorization N / X ] and a X of type X : y , x functionals of types ρ, � ρ := ρ [ I N � a N : ≡ x ≥ y x I N y I I N � a X y X : ≡ x ≥ d(y , a) . x I For complex types ρ → τ this is extended in a hereditary fashion . Example: f ∗ � a N , x ∈ X[n ≥ d(a , x) → f ∗ (n) ≥ d(a , f(x))] . X → X f ≡ ∀ n ∈ I f : X → X is nonexpansive (n.e.) if d(f(x) , f(y)) ≤ d(x , y) . Then λ n . n + b � a X → X f , if d ( a , f ( a )) ≤ b . Normed linear case: a := 0 X . Types in Proof Mining
Treatment of several metric structures X 1 , X 2 , . . . , X n Instead of one base type X one can also have several of which some are metric spaces, other normed spaces etc. together with the product spaces (where the majorants depend on the product metric chosen but not the majorizability). Types in Proof Mining
Treatment of several metric structures X 1 , X 2 , . . . , X n Instead of one base type X one can also have several of which some are metric spaces, other normed spaces etc. together with the product spaces (where the majorants depend on the product metric chosen but not the majorizability). For each base type X i one selects a reference point a X i to define the i majorization relation. Nonexpansive maps f : X → Y are ( a , b ) -majorized by Id ( a ∈ X , b ∈ Y ). Types in Proof Mining
Treatment of several metric structures X 1 , X 2 , . . . , X n Instead of one base type X one can also have several of which some are metric spaces, other normed spaces etc. together with the product spaces (where the majorants depend on the product metric chosen but not the majorizability). For each base type X i one selects a reference point a X i to define the i majorization relation. Nonexpansive maps f : X → Y are ( a , b ) -majorized by Id ( a ∈ X , b ∈ Y ). Convex subsets C ⊆ X can be added as new types which are related with X via isometric isomorphic embeddings. For uniformly convex Banach spaces X and closed convex C one can add (easy to majorize) metric projection operators characterized by universal axioms (Master Thesis D. G¨ unzel 2013). Types in Proof Mining
Suitable metric and normed structures Structures need to have an axiomatization with effective monotone functional interpretation . Types in Proof Mining
Suitable metric and normed structures Structures need to have an axiomatization with effective monotone functional interpretation . This covers metric spaces, hyperbolic metric spaces (Takahayshi, Kirk, Reich), CAT(0)-spaces (Bruhat-Tits), δ -hyperbolic spaces (Gromov), normed spaces, uniformly convex spaces, uniformly smooth, inner product spaces and the complete versions. Types in Proof Mining
Suitable metric and normed structures Structures need to have an axiomatization with effective monotone functional interpretation . This covers metric spaces, hyperbolic metric spaces (Takahayshi, Kirk, Reich), CAT(0)-spaces (Bruhat-Tits), δ -hyperbolic spaces (Gromov), normed spaces, uniformly convex spaces, uniformly smooth, inner product spaces and the complete versions. Unless the axioms can be written in purely universal form, the extracted bounds then depend on the appropriate moduli of uniform convexity, smoothness etc. (usually number-theoretic functions). Types in Proof Mining
Suitable metric and normed structures Structures need to have an axiomatization with effective monotone functional interpretation . This covers metric spaces, hyperbolic metric spaces (Takahayshi, Kirk, Reich), CAT(0)-spaces (Bruhat-Tits), δ -hyperbolic spaces (Gromov), normed spaces, uniformly convex spaces, uniformly smooth, inner product spaces and the complete versions. Unless the axioms can be written in purely universal form, the extracted bounds then depend on the appropriate moduli of uniform convexity, smoothness etc. (usually number-theoretic functions). Structures which are not sufficiently uniform get uniformized : Types in Proof Mining
Suitable metric and normed structures Structures need to have an axiomatization with effective monotone functional interpretation . This covers metric spaces, hyperbolic metric spaces (Takahayshi, Kirk, Reich), CAT(0)-spaces (Bruhat-Tits), δ -hyperbolic spaces (Gromov), normed spaces, uniformly convex spaces, uniformly smooth, inner product spaces and the complete versions. Unless the axioms can be written in purely universal form, the extracted bounds then depend on the appropriate moduli of uniform convexity, smoothness etc. (usually number-theoretic functions). Structures which are not sufficiently uniform get uniformized : strict convex → uniformly convex; uniformly Gˆ ateaux differentiable norm → uniformly smooth; separability → total boundedness (of bounded substructures) etc. Types in Proof Mining
Small types (over I N , X ): I N , I N → I N , X , I N → X , X → X . Theorem (K., Trans.AMS 2005, Gerhardy/K.,Trans.AMS 2008) Let P , K be Polish resp. compact metric spaces, A ∃ ∃ -formula, τ small. If A ω [ X , d , W ] proves ∀ x ∈ P ∀ y ∈ K ∀ z τ ∃ v I N A ∃ (x , y , z , v) , N × I N ) → I N ( I N I then one can extract a computable Φ : I N s.t. the following holds in every nonempty hyperbolic space: for all representatives N of x ∈ P and all z τ and z ∗ ∈ I N ) s.t. ∃ a ∈ X ( z ∗ � a N I N ( I r x ∈ I τ z ): ∀ y ∈ K ∃ v ≤ Φ(r x , z ∗ ) A ∃ (x , y , z , v) . Types in Proof Mining
Small types (over I N , X ): I N , I N → I N , X , I N → X , X → X . Theorem (K., Trans.AMS 2005, Gerhardy/K.,Trans.AMS 2008) Let P , K be Polish resp. compact metric spaces, A ∃ ∃ -formula, τ small. If A ω [ X , d , W ] proves ∀ x ∈ P ∀ y ∈ K ∀ z τ ∃ v I N A ∃ (x , y , z , v) , N × I N ) → I N ( I N I then one can extract a computable Φ : I N s.t. the following holds in every nonempty hyperbolic space: for all representatives N of x ∈ P and all z τ and z ∗ ∈ I N ) s.t. ∃ a ∈ X ( z ∗ � a N I N ( I r x ∈ I τ z ): ∀ y ∈ K ∃ v ≤ Φ(r x , z ∗ ) A ∃ (x , y , z , v) . Explains the unwinding of Birkhoff’s proof above. Types in Proof Mining
Elements of the proof Proceeds by monotone functional interpretation . Types in Proof Mining
Elements of the proof Proceeds by monotone functional interpretation . The result is interpreted in an extension of the Bezem-Howard type structure of strongly majorizable functionals to the new types and the new majorizability relation (needed to satisfy bar recursion stemmming from DC). Types in Proof Mining
Elements of the proof Proceeds by monotone functional interpretation . The result is interpreted in an extension of the Bezem-Howard type structure of strongly majorizable functionals to the new types and the new majorizability relation (needed to satisfy bar recursion stemmming from DC). For sufficiently restricted types, this yields the validity over the full set-theoretic type structure over I N and X . Types in Proof Mining
Elements of the proof Proceeds by monotone functional interpretation . The result is interpreted in an extension of the Bezem-Howard type structure of strongly majorizable functionals to the new types and the new majorizability relation (needed to satisfy bar recursion stemmming from DC). For sufficiently restricted types, this yields the validity over the full set-theoretic type structure over I N and X . The interpretation of d X or � · � X uses an ineffective discontinuous (but trivially majorizable ) operator ◦ to select a canonical name for the resulting real. Types in Proof Mining
Elements of the proof Proceeds by monotone functional interpretation . The result is interpreted in an extension of the Bezem-Howard type structure of strongly majorizable functionals to the new types and the new majorizability relation (needed to satisfy bar recursion stemmming from DC). For sufficiently restricted types, this yields the validity over the full set-theoretic type structure over I N and X . The interpretation of d X or � · � X uses an ineffective discontinuous (but trivially majorizable ) operator ◦ to select a canonical name for the resulting real. The constants related to X are all majorized by simple terms not involving X . Types in Proof Mining
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