Types in Proof Mining Ulrich Kohlenbach Department of Mathematics - - PowerPoint PPT Presentation

Types in Proof Mining

Ulrich Kohlenbach Department of Mathematics Technische Universit¨ at Darmstadt

TYPES 2013, Toulouse, April 22-26, 2013

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C:

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds,

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms,

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters,

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises.

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises. E.g. Let C ≡ ∀x ∈ I N∃y ∈ I N F(x, y)

Types in Proof Mining

Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs

Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises. E.g. Let C ≡ ∀x ∈ I N∃y ∈ I N F(x, y) Naive Attempt: try to extract an explicit computable function realizing (or bounding) ‘∃y’: ∀x ∈ I N F(x, f(x)).

Types in Proof Mining

Naive attempt fails

Let (an) be a nonincreasing sequence in [0, 1]. Then, clearly, (an) is convergent and so a Cauchy sequence which we write as: (1) ∀k ∈ I N∃n ∈ I N∀m ∈ I N∀i, j ∈ [n; n + m] (|ai − aj| ≤ 2−k), where [n; n + m] := {n, n + 1, . . . , n + m}.

Types in Proof Mining

Naive attempt fails

Let (an) be a nonincreasing sequence in [0, 1]. Then, clearly, (an) is convergent and so a Cauchy sequence which we write as: (1) ∀k ∈ I N∃n ∈ I N∀m ∈ I N∀i, j ∈ [n; n + m] (|ai − aj| ≤ 2−k), where [n; n + m] := {n, n + 1, . . . , n + m}. By E. Specker 1949 there exist computable such sequences (an) even in Q ∩ [0, 1] without computable bound on ‘∃n’ in (1).

Types in Proof Mining

Consider the (partial) Herbrand normal form of (1) : (2) ∀k ∈ I N∀g ∈ I NI

N∃n ∈ I

N∀i, j ∈ [n; n + g(n)] (|ai − aj| ≤ 2−k).

Types in Proof Mining

Consider the (partial) Herbrand normal form of (1) : (2) ∀k ∈ I N∀g ∈ I NI

N∃n ∈ I

N∀i, j ∈ [n; n + g(n)] (|ai − aj| ≤ 2−k). There is a simple (primitive recursive) bound Φ∗(g, k) on (2) : no-counterexample interpretation (Kreisel) or metastability (Tao):

Types in Proof Mining

Consider the (partial) Herbrand normal form of (1) : (2) ∀k ∈ I N∀g ∈ I NI

N∃n ∈ I

N∀i, j ∈ [n; n + g(n)] (|ai − aj| ≤ 2−k). There is a simple (primitive recursive) bound Φ∗(g, k) on (2) : no-counterexample interpretation (Kreisel) or metastability (Tao): Proposition Let (an) be any nonincreasing sequence in [0, 1] then ∀k ∈ I N∀g ∈ I NI

N∃n ≤ Φ∗(g, k)∀i, j ∈ [n; n+g(n)] (|ai−aj| ≤ 2−k),

where Φ∗(g, k) := ˜ g(2k−1)(0) with ˜ g(n) := n + g(n). Moreover, there exists an i < 2k such that n can be taken as ˜ g (i)(0).

Types in Proof Mining

Consider the (partial) Herbrand normal form of (1) : (2) ∀k ∈ I N∀g ∈ I NI

N∃n ∈ I

N∀i, j ∈ [n; n + g(n)] (|ai − aj| ≤ 2−k). There is a simple (primitive recursive) bound Φ∗(g, k) on (2) : no-counterexample interpretation (Kreisel) or metastability (Tao): Proposition Let (an) be any nonincreasing sequence in [0, 1] then ∀k ∈ I N∀g ∈ I NI

N∃n ≤ Φ∗(g, k)∀i, j ∈ [n; n+g(n)] (|ai−aj| ≤ 2−k),

where Φ∗(g, k) := ˜ g(2k−1)(0) with ˜ g(n) := n + g(n). Moreover, there exists an i < 2k such that n can be taken as ˜ g (i)(0). Generalized (variable length) Herbrand disjunction!

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An Example from Ergodic Theory

X Hilbert space, f : X → X linear and f (x) ≤ x for all x ∈ X. An(x) := 1 n + 1Sn(x), where Sn(x) :=

n

• i=0

fi(x) (n ≥ 0).

Types in Proof Mining

An Example from Ergodic Theory

X Hilbert space, f : X → X linear and f (x) ≤ x for all x ∈ X. An(x) := 1 n + 1Sn(x), where Sn(x) :=

n

• i=0

fi(x) (n ≥ 0). Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X, the sequence (An(x))n converges.

Types in Proof Mining

An Example from Ergodic Theory

X Hilbert space, f : X → X linear and f (x) ≤ x for all x ∈ X. An(x) := 1 n + 1Sn(x), where Sn(x) :=

n

• i=0

fi(x) (n ≥ 0). Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X, the sequence (An(x))n converges. Avigad/Gerhardy/Towsner (TAMS 2010): in general no computable rate of convergence.

Types in Proof Mining

An Example from Ergodic Theory

X Hilbert space, f : X → X linear and f (x) ≤ x for all x ∈ X. An(x) := 1 n + 1Sn(x), where Sn(x) :=

n

• i=0

fi(x) (n ≥ 0). Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X, the sequence (An(x))n converges. Avigad/Gerhardy/Towsner (TAMS 2010): in general no computable rate of convergence. Theorem (Garrett Birkhoff 1939) Mean Ergodic Theorem holds for uniformly convex Banach spaces.

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Based on a logical metatheorem to be discussed below: Theorem (K./Leu ¸ stean, Ergodic Theor. Dynam. Syst. 2009) X uniformly convex Banach space, η a modulus of uniform convexity and f : X → X as above, b > 0. Then for all x ∈ X with x ≤ b, all ε > 0, all g : I N → I N : ∃n ≤ Φ(ε, g, b, η) ∀i, j ∈ [n; n + g(n)]

• Ai(x) − Aj(x) < ε
• ,

Types in Proof Mining

Based on a logical metatheorem to be discussed below: Theorem (K./Leu ¸ stean, Ergodic Theor. Dynam. Syst. 2009) X uniformly convex Banach space, η a modulus of uniform convexity and f : X → X as above, b > 0. Then for all x ∈ X with x ≤ b, all ε > 0, all g : I N → I N : ∃n ≤ Φ(ε, g, b, η) ∀i, j ∈ [n; n + g(n)]

• Ai(x) − Aj(x) < ε
• ,

where Φ(ε, g, b, η) := M · ˜ hK(0), with M := 16b

ε

• , γ :=

ε 16η

ε

8b

• ,

K :=

• b

γ

• ,

h, ˜ h : I N → I N, h(n) := 2(Mn + g(Mn)), ˜ h(n) := maxi≤n h(i).

Types in Proof Mining

Based on a logical metatheorem to be discussed below: Theorem (K./Leu ¸ stean, Ergodic Theor. Dynam. Syst. 2009) X uniformly convex Banach space, η a modulus of uniform convexity and f : X → X as above, b > 0. Then for all x ∈ X with x ≤ b, all ε > 0, all g : I N → I N : ∃n ≤ Φ(ε, g, b, η) ∀i, j ∈ [n; n + g(n)]

• Ai(x) − Aj(x) < ε
• ,

where Φ(ε, g, b, η) := M · ˜ hK(0), with M := 16b

ε

• , γ :=

ε 16η

ε

8b

• ,

K :=

• b

γ

• ,

h, ˜ h : I N → I N, h(n) := 2(Mn + g(Mn)), ˜ h(n) := maxi≤n h(i). Special Hilbert case: treated prior by Avigad/Gerhardy/Towsner (again based on logical metatheorem).

Types in Proof Mining

Problems of the no-counterexample interpretation

For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. Cn := {0, 1, . . . , n}.

Types in Proof Mining

Problems of the no-counterexample interpretation

For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. Cn := {0, 1, . . . , n}. Direct example: Infinitary Pigeonhole Principle (IPP): ∀n ∈ I N∀f : I N → Cn∃i ≤ n∀k ∈ I N∃m ≥ k

• f(m) = i
• .

Types in Proof Mining

Problems of the no-counterexample interpretation

For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. Cn := {0, 1, . . . , n}. Direct example: Infinitary Pigeonhole Principle (IPP): ∀n ∈ I N∀f : I N → Cn∃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 → Cn∀F : Cn → I N∃i ≤ n∃m ≥ F(i)

• f(m) = i
• Types in Proof Mining

Problems of the no-counterexample interpretation

For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. Cn := {0, 1, . . . , n}. Direct example: Infinitary Pigeonhole Principle (IPP): ∀n ∈ I N∀f : I N → Cn∃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 → Cn∀F : Cn → 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’. Cn := {0, 1, . . . , n}. Direct example: Infinitary Pigeonhole Principle (IPP): ∀n ∈ I N∀f : I N → Cn∃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 → Cn∀F : Cn → 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’. Cn := {0, 1, . . . , n}. Direct example: Infinitary Pigeonhole Principle (IPP): ∀n ∈ I N∀f : I N → Cn∃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 → Cn∀F : Cn → 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 → AI,

Types in Proof Mining

A Modular Approach: Proof Interpretations

Interpret the formulas A in P : A → AI, Interpretation C I contains the additional information,

Types in Proof Mining

A Modular Approach: Proof Interpretations

Interpret the formulas A in P : A → AI, Interpretation C I contains the additional information, Construct by recursion on P a new proof PI of C I.

Types in Proof Mining

A Modular Approach: Proof Interpretations

Interpret the formulas A in P : A → AI, Interpretation C I contains the additional information, Construct by recursion on P a new proof PI of C I. In particular: solve modus ponens problem: AI , (A → B)I BI .

Types in Proof Mining

A Modular Approach: Proof Interpretations

Interpret the formulas A in P : A → AI, Interpretation C I contains the additional information, Construct by recursion on P a new proof PI of C I. In particular: solve modus ponens problem: AI , (A → B)I BI . Our approach is based on novel forms and extensions of:

• K. G¨
• del’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. HAω 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. HAω 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. HAω is the extension of HA to all finite types over I N. Types T: (i) I N ∈ T, ρ, τ ∈ T ⇒ (ρ → τ) ∈ T. HAω has λ-abstraction (λxρ.t[x]τ)(sρ) =τ t[s/x] and primitive recursion in all finite types (Hilbert 1926, G¨

• del 1958): for x ∈ I

N Rρ(0, y, z) =ρ y, Rρ(x + 1, y, z) =ρ z(Rρxyz, 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. HAω is the extension of HA to all finite types over I N. Types T: (i) I N ∈ T, ρ, τ ∈ T ⇒ (ρ → τ) ∈ T. HAω has λ-abstraction (λxρ.t[x]τ)(sρ) =τ t[s/x] and primitive recursion in all finite types (Hilbert 1926, G¨

• del 1958): for x ∈ I

N Rρ(0, y, z) =ρ y, Rρ(x + 1, y, z) =ρ z(Rρxyz, x), where =ρ is defined as pointwise (extensional) equality (with a weak extensionality rule; see later). PAω =HAω + (A ∨ ¬A).

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Towards proofs based on classical logic

Entrance door for classical logic: Markov’s principle Mω! Mω : ¬¬∃xρ Aqf (x) → ∃xρ Aqf (x), Aqf quantifier-free.

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Towards proofs based on classical logic

Entrance door for classical logic: Markov’s principle Mω! Mω : ¬¬∃xρ Aqf (x) → ∃xρ Aqf (x), Aqf 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 Mω! Mω : ¬¬∃xρ Aqf (x) → ∃xρ Aqf (x), Aqf 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!

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G¨

• del’s functional (‘Dialectica’) interpretation D

(G¨

• del 1941, 1958)

Solution: Don’t try to solve Mω but eliminate it from proofs!

Types in Proof Mining

G¨

• del’s functional (‘Dialectica’) interpretation D

(G¨

• del 1941, 1958)

Solution: Don’t try to solve Mω 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¨

• del’s functional (‘Dialectica’) interpretation D

(G¨

• del 1941, 1958)

Solution: Don’t try to solve Mω 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 Aqf(x, y) an explicit effective functional Φ realizing AG, i.e. ∀x Aqf(x, Φ(x)).

Types in Proof Mining

Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that

Types in Proof Mining

Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that AG ≡ ∀x∃y AG(x, y), where AG is quantifier-free,

Types in Proof Mining

Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that AG ≡ ∀x∃y AG(x, y), where AG is quantifier-free, For A ≡ ∀x∃y Aqf(x, y) one has AG ≡ A.

Types in Proof Mining

Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that AG ≡ ∀x∃y AG(x, y), where AG is quantifier-free, For A ≡ ∀x∃y Aqf(x, y) one has AG ≡ A. A ↔ AG by quantifier-free choice in all types QF-AC : ∀a∃b Fqf(a, b) → ∃B∀a Fqf(a, B(a)).

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Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that AG ≡ ∀x∃y AG(x, y), where AG is quantifier-free, For A ≡ ∀x∃y Aqf(x, y) one has AG ≡ A. A ↔ AG by quantifier-free choice in all types QF-AC : ∀a∃b Fqf(a, b) → ∃B∀a Fqf(a, B(a)). x, y are tuples of functionals of finite type.

Types in Proof Mining

Functional interpretation in five minutes

G¨

• del’s functional interpretation G is a map

G : Form(PAω) → Form(PAω), A → AG such that AG ≡ ∀x∃y AG(x, y), where AG is quantifier-free, For A ≡ ∀x∃y Aqf(x, y) one has AG ≡ A. A ↔ AG by quantifier-free choice in all types QF-AC : ∀a∃b Fqf(a, b) → ∃B∀a Fqf(a, B(a)). x, y are tuples of functionals of finite type. N := Krivine’s negative transl.

Streicher/K.

⇒ G = Shoenfield Variant!

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y).

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u))

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u)) (G3) (A ∨ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∨ BG(v, y)
• Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u)) (G3) (A ∨ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∨ BG(v, y)
• (G4) (∀z A)G ≡ ∀z, u∃x AG(z, u, x)

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u)) (G3) (A ∨ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∨ BG(v, y)
• (G4) (∀z A)G ≡ ∀z, u∃x AG(z, u, x)

(G5) (A→B)G ≡ ∀f, v∃u, y

• AG(u, f(u)) → BG(v, y)
• Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u)) (G3) (A ∨ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∨ BG(v, y)
• (G4) (∀z A)G ≡ ∀z, u∃x AG(z, u, x)

(G5) (A→B)G ≡ ∀f, v∃u, y

• AG(u, f(u)) → BG(v, y)
• (G6) (∃zA)G ≡ ∀U∃z, f AG(z, U(z, f), f(U(z, f)))

Types in Proof Mining

AG ≡ ∀u∃x AG(u, x), BG ≡ ∀v∃y BG(v, y). (G1) PG ≡ P ≡ PG for atomic P (G2) (¬A)G ≡ ∀f∃u ¬AG(u, f(u)) (G3) (A ∨ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∨ BG(v, y)
• (G4) (∀z A)G ≡ ∀z, u∃x AG(z, u, x)

(G5) (A→B)G ≡ ∀f, v∃u, y

• AG(u, f(u)) → BG(v, y)
• (G6) (∃zA)G ≡ ∀U∃z, f AG(z, U(z, f), f(U(z, f)))

(G7) (A ∧ B)G ≡ ∀u, v∃x, y

• AG(u, x) ∧ BG(v, y)
• .

Types in Proof Mining

Comments

The program extraction theorem scales down to weak systems such as RCA0 (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 RCA0 (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 Mω or Cω): Spector 1962.

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Connection to no-counterexample interpretation

Let A be a prenex (arithmetical) formula and AS, AG, An.c.i its Skolem, G and n.c.i. interpretations resp., then HAω ⊢ AS → AG → An.c.i, but the converse implications in general fail to hold even in PAω+QF-AC!

Types in Proof Mining

Connection to no-counterexample interpretation

Let A be a prenex (arithmetical) formula and AS, AG, An.c.i its Skolem, G and n.c.i. interpretations resp., then HAω ⊢ AS → AG → An.c.i, but the converse implications in general fail to hold even in PAω+QF-AC! AS too strong (for a computable solution): Specker!

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Connection to no-counterexample interpretation

Let A be a prenex (arithmetical) formula and AS, AG, An.c.i its Skolem, G and n.c.i. interpretations resp., then HAω ⊢ AS → AG → An.c.i, but the converse implications in general fail to hold even in PAω+QF-AC! AS too strong (for a computable solution): Specker! An.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 AS, AG, An.c.i its Skolem, G and n.c.i. interpretations resp., then HAω ⊢ AS → AG → An.c.i, but the converse implications in general fail to hold even in PAω+QF-AC! AS too strong (for a computable solution): Specker! An.c.i. too weak (see IPP above; modus ponens problem). AG just right: PAω+QF-AC ⊢ A ↔ AG.

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Majorizability

The functionals occurring in functional interpretation (such as the primitive recursive ones from PAω 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.

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Majorizability

The functionals occurring in functional interpretation (such as the primitive recursive ones from PAω 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 PAω 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 PAω 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¨

• nig’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. PAω+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. PAω+QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀x ∈ P∀y ∈ K∃m ∈ I NA∃(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. PAω+QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀x ∈ P∀y ∈ K∃m ∈ I NA∃(x, y, m)

• ne can extract a closed term Φ of BA

BA ⊢ ∀x ∈ P∀y ∈ K∃m ≤ Φ(fx)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. PAω+QF-AC, HBC Heine/Borel compactness (via WKL). Theorem (K., APAL 1993) From a proof BA + HBC ⊢ ∀x ∈ P∀y ∈ K∃m ∈ I NA∃(x, y, m)

• ne can extract a closed term Φ of BA

BA ⊢ ∀x ∈ P∀y ∈ K∃m ≤ Φ(fx)A∃(x, y, m). Important: Φ(fx) does not depend on y∈K but on representation fx 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. PAω,X is the extension of Peano Arithmetic to all types. Aω,X:=PAω,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. PAω,X is the extension of Peano Arithmetic to all types. Aω,X:=PAω,X+DC, where DC: axiom of dependent choice for all types Implies full comprehension for numbers (higher order arithmetic). Aω[X, d, . . .] results by adding constants dX, . . . 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

xX =X yX :≡ dX(x, y) =I

R 0I R,

x =ρ→τ y :≡ ∀vρ(s(v) =τ t(v)).

Types in Proof Mining

A novel form of majorization

y, x functionals of types ρ, ρ := ρ[I N/X] and aX of type X: xI

N a I N yI N :≡ x ≥ y

xI

N a X yX :≡ x ≥ d(y, a).

Types in Proof Mining

A novel form of majorization

y, x functionals of types ρ, ρ := ρ[I N/X] and aX of type X: xI

N a I N yI N :≡ x ≥ y

xI

N a X yX :≡ x ≥ d(y, a).

For complex types ρ → τ this is extended in a hereditary fashion.

Types in Proof Mining

A novel form of majorization

y, x functionals of types ρ, ρ := ρ[I N/X] and aX of type X: xI

N a I N yI N :≡ x ≥ y

xI

N a X yX :≡ x ≥ d(y, a).

For complex types ρ → τ this is extended in a hereditary fashion. Example: f∗ a

X→X f ≡ ∀n ∈ I

N, x ∈ X[n ≥ d(a, x) → f∗(n) ≥ d(a, f(x))].

Types in Proof Mining

A novel form of majorization

y, x functionals of types ρ, ρ := ρ[I N/X] and aX of type X: xI

N a I N yI N :≡ x ≥ y

xI

N a X yX :≡ x ≥ d(y, a).

For complex types ρ → τ this is extended in a hereditary fashion. Example: f∗ a

X→X f ≡ ∀n ∈ I

N, x ∈ X[n ≥ d(a, x) → f∗(n) ≥ d(a, f(x))]. f : X → X is nonexpansive (n.e.) if d(f(x), f(y)) ≤ d(x, y). Then λn.n + b a

X→X f , if d(a, f (a)) ≤ b.

Types in Proof Mining

A novel form of majorization

y, x functionals of types ρ, ρ := ρ[I N/X] and aX of type X: xI

N a I N yI N :≡ x ≥ y

xI

N a X yX :≡ x ≥ d(y, a).

For complex types ρ → τ this is extended in a hereditary fashion. Example: f∗ a

X→X f ≡ ∀n ∈ I

N, x ∈ X[n ≥ d(a, x) → f∗(n) ≥ d(a, f(x))]. f : X → X is nonexpansive (n.e.) if d(f(x), f(y)) ≤ d(x, y). Then λn.n + b a

X→X f , if d(a, f (a)) ≤ b.

Normed linear case: a := 0X.

Types in Proof Mining

Treatment of several metric structures X1, X2, . . . , Xn

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 X1, X2, . . . , Xn

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 Xi one selects a reference point aXi

i

to define the 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 X1, X2, . . . , Xn

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 Xi one selects a reference point aXi

i

to define the 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 Aω[X, d, W ] proves ∀x ∈ P∀y ∈ K∀zτ∃vI

NA∃(x, y, z, v),

then one can extract a computable Φ : I NI

N × I

N(I

N) → I

N s.t. the following holds in every nonempty hyperbolic space: for all representatives rx ∈ I NI

N of x ∈ P and all zτ and z∗ ∈ I

N(I

N) s.t. ∃a ∈ X(z∗ a τ z):

∀y ∈ K∃v ≤ Φ(rx, 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 Aω[X, d, W ] proves ∀x ∈ P∀y ∈ K∀zτ∃vI

NA∃(x, y, z, v),

then one can extract a computable Φ : I NI

N × I

N(I

N) → I

N s.t. the following holds in every nonempty hyperbolic space: for all representatives rx ∈ I NI

N of x ∈ P and all zτ and z∗ ∈ I

N(I

N) s.t. ∃a ∈ X(z∗ a τ z):

∀y ∈ K∃v ≤ Φ(rx, 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 dX 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 dX 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

Warning on extensionality

Full extensionality is in conflict with the metatheorem: ∀fX→X, xX, yX(x =X y → f(x) =X f(y)) yields with Markov’s principle ∀fX→X, xX, yX, kI

N∃nI N (dX(x, y) ≤ 2−n → dX(f(x), f(y)) < 2−k)

Types in Proof Mining

Warning on extensionality

Full extensionality is in conflict with the metatheorem: ∀fX→X, xX, yX(x =X y → f(x) =X f(y)) yields with Markov’s principle ∀fX→X, xX, yX, kI

N∃nI N (dX(x, y) ≤ 2−n → dX(f(x), f(y)) < 2−k)

and hence with metatheorem for bounded metric spaces X: All functions f : X → X are equicontinuous (and the modulus only depends on the bound b of the metric).

Types in Proof Mining

Baillon’s nonlinear ergodic theorem

Theorem (J.-B. Baillon 1975): X Hilbert space, C ⊂ X bounded closed and convex, U : C → C nonexpansive. Then for every u0 ∈ C, the sequence of Ces` aro means (un) un := 1 n + 1

n

• k=0

U(k)(u0) converges weakly to a fixed point of U.

Types in Proof Mining

Baillon’s nonlinear ergodic theorem

Theorem (J.-B. Baillon 1975): X Hilbert space, C ⊂ X bounded closed and convex, U : C → C nonexpansive. Then for every u0 ∈ C, the sequence of Ces` aro means (un) un := 1 n + 1

n

• k=0

U(k)(u0) converges weakly to a fixed point of U. All proofs of this celebrated theorem use weak sequential

• compactness. A particular simple proof is due to Br´

ezis and Browder (1976).

Types in Proof Mining

Baillon’s nonlinear ergodic theorem

Theorem (J.-B. Baillon 1975): X Hilbert space, C ⊂ X bounded closed and convex, U : C → C nonexpansive. Then for every u0 ∈ C, the sequence of Ces` aro means (un) un := 1 n + 1

n

• k=0

U(k)(u0) converges weakly to a fixed point of U. All proofs of this celebrated theorem use weak sequential

• compactness. A particular simple proof is due to Br´

ezis and Browder (1976). Strong convergence in general fails (counterexample by Baillon).

Types in Proof Mining

Baillon’s nonlinear ergodic theorem

Theorem (J.-B. Baillon 1975): X Hilbert space, C ⊂ X bounded closed and convex, U : C → C nonexpansive. Then for every u0 ∈ C, the sequence of Ces` aro means (un) un := 1 n + 1

n

• k=0

U(k)(u0) converges weakly to a fixed point of U. All proofs of this celebrated theorem use weak sequential

• compactness. A particular simple proof is due to Br´

ezis and Browder (1976). Strong convergence in general fails (counterexample by Baillon). In important special cases (see talk on Friday) strong convergence can be established.

Types in Proof Mining

A quantitative ‘metastable’ version of Baillon’s theorem

Theorem (K., Comm.Contemp.Math.2012) Logical analysis of the proof of Baillon’s theorem due to Br´ ezis-Browder yields a primitive recursive functional ϕ such that for a bar-recursive bound Ω∗ extracted from the weak compactness proof and ε > 0, g : I N → I N, we get ϕ(Ω∗, ε, b, g) as a bound on the metastable version of the weak Cauchy property of the Ces` aro means (un), i.e. ∀w ∈ B1(0) ∃n ≤ ϕ(Ω∗, ε, b, g) ∀i, j ∈ [n; n+g(n)]

• |ui−uj, w| < ε
• .

Types in Proof Mining

A quantitative ‘metastable’ version of Baillon’s theorem

Theorem (K., Comm.Contemp.Math.2012) Logical analysis of the proof of Baillon’s theorem due to Br´ ezis-Browder yields a primitive recursive functional ϕ such that for a bar-recursive bound Ω∗ extracted from the weak compactness proof and ε > 0, g : I N → I N, we get ϕ(Ω∗, ε, b, g) as a bound on the metastable version of the weak Cauchy property of the Ces` aro means (un), i.e. ∀w ∈ B1(0) ∃n ≤ ϕ(Ω∗, ε, b, g) ∀i, j ∈ [n; n+g(n)]

• |ui−uj, w| < ε
• .

Based on the detailed construction of Ω∗ and results of W.A. Howard it follows that Φ(ε, b, g) := ϕ(Ω∗, ε, b, g) is definable in G¨

• del’s T4 (note

that Φ has type level 2).

Types in Proof Mining

A strong nonlinear ergodic theorem

Already in 1976 Baillon proved strong convergence of Ces` aro means if ∀v ∈ C (−v ∈ C) and f is nonexpansive and odd, i.e. f (−v) = −f (v).

Types in Proof Mining

A strong nonlinear ergodic theorem

Already in 1976 Baillon proved strong convergence of Ces` aro means if ∀v ∈ C (−v ∈ C) and f is nonexpansive and odd, i.e. f (−v) = −f (v). This was generalized by Wittmann in 1990: Theorem [R. Wittmann 1990]: Let C ⊆ X be an arbitrary subset and f : C → C s.t. ∀u, v ∈ C (f(u) + f(v) ≤ u + v). Then the sequence of Ces` aro means (xn) converges strongly.

Types in Proof Mining

A strong nonlinear ergodic theorem

Already in 1976 Baillon proved strong convergence of Ces` aro means if ∀v ∈ C (−v ∈ C) and f is nonexpansive and odd, i.e. f (−v) = −f (v). This was generalized by Wittmann in 1990: Theorem [R. Wittmann 1990]: Let C ⊆ X be an arbitrary subset and f : C → C s.t. ∀u, v ∈ C (f(u) + f(v) ≤ u + v). Then the sequence of Ces` aro means (xn) converges strongly. Holds for C closed under v → −v and nonexpansive, odd f .

Types in Proof Mining

A strong nonlinear ergodic theorem

Already in 1976 Baillon proved strong convergence of Ces` aro means if ∀v ∈ C (−v ∈ C) and f is nonexpansive and odd, i.e. f (−v) = −f (v). This was generalized by Wittmann in 1990: Theorem [R. Wittmann 1990]: Let C ⊆ X be an arbitrary subset and f : C → C s.t. ∀u, v ∈ C (f(u) + f(v) ≤ u + v). Then the sequence of Ces` aro means (xn) converges strongly. Holds for C closed under v → −v and nonexpansive, odd f . The condition above does not even imply that f is continuous. But f has a trivial majorant f ∗ := Id (v := u). Hence: Metatheorem applicable!

Types in Proof Mining

Theorem (P. Safarik, J. Math. Anal. Appl. 2012) ∀k ∈ I N ∀g ∈ I NI

N ∃m ≤ Φ(k, b, gM) (xm − xm+g(m) ≤ 2−k),

Types in Proof Mining

Theorem (P. Safarik, J. Math. Anal. Appl. 2012) ∀k ∈ I N ∀g ∈ I NI

N ∃m ≤ Φ(k, b, gM) (xm − xm+g(m) ≤ 2−k),

where (for b ≥ x0) Φ(k, b, g) := (N(2k + 7, g) + P(2k + 7, g)) · b · 22k+8 + 1, P(k, g) := P0(k, F(k, g, N(k, g))), F(k, g, n)(p) := p + n + ˜ g((n + p) · b · 2k+1), L(k, g)(n) := n +P0(k, F(k, g, n)) + ˜ g((n +P0(k, F(k, g, n)))·b·2k+1), N(k, g) := (L(k, g))(b22k+2)(0), P0(k, f) := ˜ f(b22k)(0), ˜ f(n) := n + f(n), fM(n) := max

i≤n+1 f(i).

Types in Proof Mining

1 23

Springer Monographs in Mathematics SMM ulrich kohlenbach

kohlenbach

• u. kohlenbach

Applied Proof Theory: Proof Interpretations and their Use in Mathematics Ulrich Kohlenbach presents an applied form of proof theory that has led in recent years to new results in number theory, approxi- mation theory, nonlinear analysis, geodesic geometry and ergodic theory (among others). This applied approach is based on logical transformations (so-called proof interpretations) and concerns the extraction of effective data (such as bounds) from prima facie ineffective proofs as well as new qualitative results such as inde- pendence of solutions from certain parameters, generalizations

• f proofs by elimination of premises.

The book first develops the necessary logical machinery empha- sizing novel forms of Gödel‘s famous functional (‚Dialectica‘)

• interpretation. It then establishes general logical metatheorems

that connect these techniques with concrete mathematics. Finally, two extended case studies (one in approximation theory and one in fixed point theory) show in detail how this machinery can be applied to concrete proofs in different areas of mathematics.

1

Applied Proof Theory: Proof Interpretations and their Use in Mathematics Applied Proof Theory: Proof Interpretations and their Use in Mathematics 54205 WMXDesign GmbH Heidelberg – Bender 06.12.07

Dieser pdf-file gibt nur annähernd das endgültige Druckergebnis wieder ! issn 1439-7382

› springer.com

ISBN 978-3-540-77532-4

Applied Proof Theory: Proof Interpretations and their Use in Mathematics Applied Proof Theory: Proof Interpretations and their Use in Mathematics

Types in Proof Mining