Proof Mining for Nonexpansive Semigroups PhDs in Logic IX, Bochum - - PowerPoint PPT Presentation

Proof Mining for Nonexpansive Semigroups

PhDs in Logic IX, Bochum 2017 Angeliki Koutsoukou-Argyraki Research Group Logic, Department of Mathematics TU Darmstadt, Germany

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ?

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨

• del (1931) : Impossible to prove the consistency of a theory

T within T .

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨

• del (1931) : Impossible to prove the consistency of a theory

T within T . Let theories T1, T2 with languages L(T1), L(T2) . T2 is consistent relative to T1 if it can be proved that if T1 is consistent then T2 is consistent.

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨

• del (1931) : Impossible to prove the consistency of a theory

T within T . Let theories T1, T2 with languages L(T1), L(T2) . T2 is consistent relative to T1 if it can be proved that if T1 is consistent then T2 is consistent. Proof Interpretations originally developed for relative consistency proofs.

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨

• del (1931) : Impossible to prove the consistency of a theory

T within T . Let theories T1, T2 with languages L(T1), L(T2) . T2 is consistent relative to T1 if it can be proved that if T1 is consistent then T2 is consistent. Proof Interpretations originally developed for relative consistency proofs. G¨

• del’s motivation: obtain a relative consistency proof for HA

(and hence for PA).

Origin of proof interpretations

Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨

• del (1931) : Impossible to prove the consistency of a theory

T within T . Let theories T1, T2 with languages L(T1), L(T2) . T2 is consistent relative to T1 if it can be proved that if T1 is consistent then T2 is consistent. Proof Interpretations originally developed for relative consistency proofs. G¨

• del’s motivation: obtain a relative consistency proof for HA

(and hence for PA). G¨

• del’s functional ”Dialectica” Interpretation (1958):

consistency of PA reduced to a quantifier-free calculus of primitive recursive functionals of finite type.

Proof Mining

Shift of focus :

• G. Kreisel (1950’s): Unwinding of proofs

”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?”

Proof Mining

Shift of focus :

• G. Kreisel (1950’s): Unwinding of proofs

”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?” Possible to obtain new quantitative/ qualitative information by logical analysis of proofs of statements of certain logical form.

Proof Mining

Shift of focus :

• G. Kreisel (1950’s): Unwinding of proofs

”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?” Possible to obtain new quantitative/ qualitative information by logical analysis of proofs of statements of certain logical form. Extraction of constructive information from non-constructive proofs.

Proof Mining

T1 transformed into T2 by transforming every theorem φ ∈ L(T1) into φI ∈ L(T2) via the proof interpretation I so that T1 ⊢ φ ⇒ T2 ⊢ φI holds.

Proof Mining

T1 transformed into T2 by transforming every theorem φ ∈ L(T1) into φI ∈ L(T2) via the proof interpretation I so that T1 ⊢ φ ⇒ T2 ⊢ φI holds. Then a given proof p of φ in T1 is transformed into a proof pI of φI in T2 by a simple recursion over φ in T1.

Proof Mining

T1 transformed into T2 by transforming every theorem φ ∈ L(T1) into φI ∈ L(T2) via the proof interpretation I so that T1 ⊢ φ ⇒ T2 ⊢ φI holds. Then a given proof p of φ in T1 is transformed into a proof pI of φI in T2 by a simple recursion over φ in T1. This gives new quantitative information.

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free.

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x))

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x))

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x)) AS ↔ A.

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x)) AS ↔ A. Idea :

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x)) AS ↔ A. Idea : S extracts from a given proof p: p ⊢ ∀x ∃y A(x, y) an explicit effective functional that realizes AS :

G¨

• del ’s functional ”Dialectica” interpretation (with

negative translation)

To every formula A in L(WE-HAω) we assign AS ≡ ∀x ∃y AS(x, y) where AS is quantifier-free. By classical logic and QF-AC : ∀x ∃y F0(x, y) → ∃B ∀x F0(x, B(x)) AS ↔ A. Idea : S extracts from a given proof p: p ⊢ ∀x ∃y A(x, y) an explicit effective functional that realizes AS : ∀x AS(x, Φ(x)).

Proof Mining

Monotone functional interpretation (and negative translation) extracts a Φ∗ such that:

Proof Mining

Monotone functional interpretation (and negative translation) extracts a Φ∗ such that: ∃Y (Φ∗ Y ∧ ∀x AS(x, Y (x))).

Proof Mining

Monotone functional interpretation (and negative translation) extracts a Φ∗ such that: ∃Y (Φ∗ Y ∧ ∀x AS(x, Y (x))). Majorizability

Proof Mining

Monotone functional interpretation (and negative translation) extracts a Φ∗ such that: ∃Y (Φ∗ Y ∧ ∀x AS(x, Y (x))). Majorizability x∗ N x :≡ x∗ ≥ x, x∗ ρ→τ x :≡ ∀y∗, y (y∗ ρ y → x∗(y∗) τ x(y)).

Proof Mining

General logical metatheorems by Kohlenbach et al use Dialectica and its variations (within specific formal frameworks).

Proof Mining

General logical metatheorems by Kohlenbach et al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀x ∃y A∃(x, y).

Proof Mining

General logical metatheorems by Kohlenbach et al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀x ∃y A∃(x, y). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof.

Proof Mining

General logical metatheorems by Kohlenbach et al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀x ∃y A∃(x, y). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof. Bounds are highly uniform : depend only on bounding information

• n the input data (majorants).

Proof Mining

General logical metatheorems by Kohlenbach et al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀x ∃y A∃(x, y). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof. Bounds are highly uniform : depend only on bounding information

• n the input data (majorants).

We will see examples of metatheorems adapted for specific mathematical situations.

Proof Mining

By the logical metatheorems we cannot know a priori:

Proof Mining

By the logical metatheorems we cannot know a priori: difficulty of extraction

Proof Mining

By the logical metatheorems we cannot know a priori: difficulty of extraction complexity (but possible estimation by looking at proof)

Proof Mining

By the logical metatheorems we cannot know a priori: difficulty of extraction complexity (but possible estimation by looking at proof) method of extraction (not automated but not ad hoc)

Proof Mining

How is the quantitative information(bound) extracted from the proof?

The purpose of a metatheorem is to serve only as a guideline to guarantee the extractability and uniformity of a computable bound from the proof of a mathematical statement that can be written in a certain logical form (∀∃).

Proof Mining

How is the quantitative information(bound) extracted from the proof?

The purpose of a metatheorem is to serve only as a guideline to guarantee the extractability and uniformity of a computable bound from the proof of a mathematical statement that can be written in a certain logical form (∀∃). The precise method of extracting the bound is not known a priori. Typically, this is done in three stages :

Proof Mining

How is the quantitative information(bound) extracted from the proof?

The purpose of a metatheorem is to serve only as a guideline to guarantee the extractability and uniformity of a computable bound from the proof of a mathematical statement that can be written in a certain logical form (∀∃). The precise method of extracting the bound is not known a priori. Typically, this is done in three stages : (Important: following process not automated and (even though not completely ad hoc) is open to the manipulations of the mathematician(s) performing proof mining on a given proof.)

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers.

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers. (ii) The mathematical objects involved must have the correct uniformity.

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers. (ii) The mathematical objects involved must have the correct

• uniformity. So:

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers. (ii) The mathematical objects involved must have the correct

• uniformity. So: we make explicit the quantitative content of their

properties

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers. (ii) The mathematical objects involved must have the correct

• uniformity. So: we make explicit the quantitative content of their

properties (i.e. modulus of continuity for uniform continuity, modulus of accretivity for uniform accretivity, modulus of convexity for uniform convexity, effective irrationality measure for irrationality etc). In that way we obtain quantitative versions of the statements/ lemmas involved.

Proof Mining

How is the quantitative information(bound) extracted from the proof?

(i) Write all the statements involved in a formal version using quantifiers. (ii) The mathematical objects involved must have the correct

• uniformity. So: we make explicit the quantitative content of their

properties (i.e. modulus of continuity for uniform continuity, modulus of accretivity for uniform accretivity, modulus of convexity for uniform convexity, effective irrationality measure for irrationality etc). In that way we obtain quantitative versions of the statements/ lemmas involved. (iii) Put everything together in a deduction schema just like the

• ne of the original proof, i.e. the structure of the original proof is

typically preserved.

Proof Mining

Proof Mining

Within past ≈ 15 years, U. Kohlenbach et al have applied proof mining to : approximation theory, ergodic theory, fixed point theory, nonlinear analysis, and (recently) PDE theory.

Proof Mining

Within past ≈ 15 years, U. Kohlenbach et al have applied proof mining to : approximation theory, ergodic theory, fixed point theory, nonlinear analysis, and (recently) PDE theory. Applications described as instances of logical phenomena by the general logical metatheorems.

Work presented here included in : K.-A., A.: Proof Mining for Nonlinear Operator Theory: Four Case Studies on Accretive Operators, the Cauchy Problem and Nonexpansive Semigroups, PhD thesis, Technische Universit¨ at Darmstadt, available online on http : //tuprints.ulb.tu − darmstadt.de/6101 (2017). Kohlenbach, U. and K.-A., A.: Effective asymptotic regularity for one-parameter nonexpansive semigroups, J. Math. Anal.

• Appl. vol. 433, 18831903 (2016).

K.-A., A.: New effective bounds for the approximate common fixed points and asymptotic regularity of nonexpansive semigroups, submitted preprint (2017).

One-parameter Nonexpansive Semigroups

Definition Given a Banach space X and C ⊆ X, a mapping T on C is nonexpansive if ∀x, y ∈ C Tx − Ty ≤ x − y.

One-parameter Nonexpansive Semigroups

Definition Given a Banach space X and C ⊆ X, a mapping T on C is nonexpansive if ∀x, y ∈ C Tx − Ty ≤ x − y. Definition A family {T(t) : t ≥ 0} of T(t) : C → C is called a one-parameter nonexpansive semigroup on C ⊆ X if :

1 for all t ≥ 0, T(t) is a nonexpansive mapping on C, 2 T(s + t) = T(s) ◦ T(t), 3 for each x ∈ C, the mapping t → T(t)x from [0, ∞) into C is

continuous.

Question 1

How to find the set of all common fixed points of {T(t) : t ≥ 0} ?

Question 1

How to find the set of all common fixed points of {T(t) : t ≥ 0} ? Two answers given by T. Suzuki :

Answer 1. A

Theorem (Suzuki (2006)) a Let {T(t) : t ≥ 0} a nonexpansive semigroup on C ⊆ X. Let F(T(t)) the set of fixed points of T(t). Let α, β ∈ R+, α/β ∈ R+ \ Q+. Let λ ∈ (0, 1). Then :

• t≥0

F(T(t)) = F(λT(α) + (1 − λ)T(β)).

aSuzuki, T. : Common fixed points of one-parameter nonexpansive

semigroups, Bull. London Math. Soc. 38 1009-1018(2006).

Answer 1. B

Theorem (Suzuki (2005) a Let {T(t) : t ≥ 0} a nonexpansive semigroup on C ⊆ X. Let F(T(t)) the set of fixed points of T(t). Let α, β ∈ R+, α/β ∈ R+ \ Q+. Then :

• t≥0

F(T(t)) = F(T(α)) ∩ F(T(β).

aSuzuki, T. : The set of common fixed points of a one-parameter continuous

semigroup of mappings is F(T(1)) ∩ F(T( √ 2)), Proceedings of the American Mathematical Society 134, No 3, 673-681(2005).

Question 2

Can we find a computable bound for the computation of the approximate common fixed points of {T(t) : t ≥ 0} ?

Question 2

Can we find a computable bound for the computation of the approximate common fixed points of {T(t) : t ≥ 0} ? Yes

Question 2

Can we find a computable bound for the computation of the approximate common fixed points of {T(t) : t ≥ 0} ? Yes (for proof-theoretic reasons to be sketched in following slides.. )

Two answers:

Answer 2.A : Proof mining on Suzuki’s theorem of Answer 1.A : Kohlenbach, U. and K.-A., A. : Effective asymptotic regularity for

• ne-parameter nonexpansive semigroups , J. Math. Anal. Appl.

433, 1883-1903 (2016).

Two answers:

Answer 2.A : Proof mining on Suzuki’s theorem of Answer 1.A : Kohlenbach, U. and K.-A., A. : Effective asymptotic regularity for

• ne-parameter nonexpansive semigroups , J. Math. Anal. Appl.

433, 1883-1903 (2016). Answer 2.B : Proof mining on Suzuki’s theorem of Answer 1.B : K.-A., A. New effective bounds for the approximate common fixed points and asymptotic regularity of nonexpansive semigroups , submitted preprint (2017).

Main Idea

here written for F(T(α)) ∩ F(T(β)) and analogously for F(λT(α) + (1 − λ)T(β))

• t≥0 F(T(t)) ⊆ F(T(α)) ∩ F(T(β)) is trivial.

Main Idea

here written for F(T(α)) ∩ F(T(β)) and analogously for F(λT(α) + (1 − λ)T(β))

• t≥0 F(T(t)) ⊆ F(T(α)) ∩ F(T(β)) is trivial.

We will extract a bound from (the proof of)

• t≥0

F(T(t)) ⊇ F(T(α)) ∩ F(T(β));

Main Idea

here written for F(T(α)) ∩ F(T(β)) and analogously for F(λT(α) + (1 − λ)T(β))

• t≥0 F(T(t)) ⊆ F(T(α)) ∩ F(T(β)) is trivial.

We will extract a bound from (the proof of)

• t≥0

F(T(t)) ⊇ F(T(α)) ∩ F(T(β)); The above gives for q ∈ C T(α)q = q ∧ T(β)q = q → ∀t ≥ 0 T(t)q = q

Main Idea

here written for F(T(α)) ∩ F(T(β)) and analogously for F(λT(α) + (1 − λ)T(β))

• t≥0 F(T(t)) ⊆ F(T(α)) ∩ F(T(β)) is trivial.

We will extract a bound from (the proof of)

• t≥0

F(T(t)) ⊇ F(T(α)) ∩ F(T(β)); The above gives for q ∈ C T(α)q = q ∧ T(β)q = q → ∀t ≥ 0 T(t)q = q i.e. ∀m ∈ N ∀M ∈ N ∀t ∈ [0, M] ∃k ∈ N (T(α)q−q ≤ 2−k ∧T(β)q−q ≤ 2−k → T(t)q−q < 2−m), which is a ∀∃ statement .

Main Idea

here written for F(T(α)) ∩ F(T(β)) and analogously for F(λT(α) + (1 − λ)T(β))

By proof mining on the proof of Suzuki’s theorem we will extract a computable bound X > 0 depending on bounds on the input data so that (where for given b ∈ N let Cb := {q ∈ C : q ≤ b}), ∀b ∈ N ∀q ∈ Cb ∀M ∈ N ∀m ∈ N (T(α)q − q ≤ X ∧ T(β)q − q ≤ X → ∀t ∈ [0, M] T(t)q − q < 2−m).

Uniform Equicontinuity for Nonexpansive Semigroups

Definition {T(t) : t ≥ 0} on C ⊆ X is uniformly equicontinuous if t → T(t)q is uniformly continuous on each [0, K] for all K ∈ N with a common modulus of uniform continuity for all q ∈ Cb. i.e. if there exists ω : N × N × N → N so that ∀b ∈ N ∀q ∈ Cb ∀m ∈ N ∀K ∈ N ∀t, t′ ∈ [0, K] (|t − t′| < 2−ωK,b(m) → T(t)q − T(t′)q < 2−m). We call ω a modulus of uniform equicontinuity for {T(t) : t ≥ 0}.

’Quantifying’ Irrationality

Let γ ∈ R+ \ Q+. Then ∀p ∈ N ∀p′ ∈ Z ∃z ∈ N (|γ − p′ p | ≥ 1 z ).

’Quantifying’ Irrationality

Let γ ∈ R+ \ Q+. Then ∀p ∈ N ∀p′ ∈ Z ∃z ∈ N (|γ − p′ p | ≥ 1 z ). The Skolem normal form of the above is ∃fγ : N × Z → N ∀p ∈ N ∀p′ ∈ Z (|γ − p′ p | ≥ 1 fγ(p′, p))

’Quantifying’ Irrationality

Let γ ∈ R+ \ Q+. Then ∀p ∈ N ∀p′ ∈ Z ∃z ∈ N (|γ − p′ p | ≥ 1 z ). The Skolem normal form of the above is ∃fγ : N × Z → N ∀p ∈ N ∀p′ ∈ Z (|γ − p′ p | ≥ 1 fγ(p′, p)) and fγ is the corresponding Skolem function.

’Quantifying’ Irrationality

Let γ ∈ R+ \ Q+. Then ∀p ∈ N ∀p′ ∈ Z ∃z ∈ N (|γ − p′ p | ≥ 1 z ). The Skolem normal form of the above is ∃fγ : N × Z → N ∀p ∈ N ∀p′ ∈ Z (|γ − p′ p | ≥ 1 fγ(p′, p)) and fγ is the corresponding Skolem function. Definition The function fγ as above is called an effective irrationality measure for γ.

Answer 2.A

Theorem (Kohlenbach, K.-A.(2016)) In addition to Suzuki’s assumptions, assume that {T(t) : t ≥ 0} is uniformly equicontinuous with a modulus ω. Let fγ be the effective irrationality measure for γ := α/β, Λ ∈ N so that 1/Λ ≤ λ, 1 − λ, N ∈ N so that β ≥ 1/N , N ∋ D ≥ β. Then ∀b ∈ N ∀q ∈ Cb ∀M ∈ N ∀t ∈ [0, M] ∀m ∈ N ((λT(α) + (1 − λ)T(β))q − q ≤ Ψ → T(t)q − q < 2−m)

Theorem with Ψ(m, M, N, Λ, D, b, fγ, ω) = 2−m 8(φ(k,fγ)−1

i=1

Λi + 1)(1 + MN) where φ(k, f ) := max{2f (i − j) + 6, 0 ≤ j < i ≤ k + 1} ∈ N k := D2ωD,b(3+[log2(1+MN)]+m)+1 ∈ N.

extraction guaranteed by... Metatheorem Assume that we have a proof of a sentence in Aω[X, · , C]−b ∀α, β, t ∈ R+ ∀N ∈ N ∀λ ∈ (0, 1) ∀z ∈ C ∀T ∈ C × R+ → C ∀ω ∈ N × N × N → N ∀f α

β ∈ N → N ∀m ∈ N ∃k ∈ N

• (∀t ∈ R+ ∀x, y ∈ C T(t)x − T(t)y ≤R x − y)

∧(∀x ∈ C ∀t, s ∈ R+ T(s) ◦ T(t)(x) =X T(s + t)(x)) ∧ ( 1 N ≤R β) ∧(∀p ∈ N∀p′ ∈ Z+ (|α β − p′ p | ≥R 1 f α

β (p)))

∧(∀b ∈ N ∀q ∈ C ∀m ∈ N ∀K ∈ N ∀t, t′ ∈ [0, K] (q <R b ∧ |t − t′| <R 2−ωK,b(m) → T(t)q − T(t′)q ≤R 2−m)) ∧((λT(α)+(1−λ)T(β))z−z ≤R 2−k) → T(t)z−z <R 2−m . Then one can extract from the proof a computable functional Φ so that: ∀D ∈ N ∀α, β ∈ [0, D] ∀N ∈ N ∀M ∈ N ∀t ∈ [0, M] ∀λ ∈ (0, 1) ∀Λ ∈ N ∀B, B′ ∈ N ∀z ∈ CB ∀T ∈ C × R+ → C ∀ω ∈ N × N × N → N ∀f α

β ∈ N → N ∀m ∈ N ∃k ≤ Φ(B, D, M, Λ, N, m, f ′ α β , ω′)

• (∀t ∈ R+ ∀x, y ∈ C T(t)x − T(t)y ≤R x − y)

∧(∀x ∈ C ∀t, s ∈ R+ T(s) ◦ T(t)(x) =X T(s + t)(x)) ∧ ( 1 N ≤R β) ∧( 1 Λ ≤R λ) ∧ ( 1 Λ ≤R 1 − λ) ∧(∀p ∈ N ∀p′ ∈ Z+ (|α β − p′ p | ≥R 1 f α

β (p)))

∧(∀b ∈ N ∀q ∈ C ∀m ∈ N ∀K ∈ N ∀t, t′ ∈ [0, K] (q <R b ∧ |t − t′| <R 2−ωK,b(m) → T(t)q − T(t′)q ≤R 2−m)) ∧z ≤R B ∧ T(τ)z − z ≤R B′ ∧ τ ≤R M ∧((λT(α)+(1−λ)T(β))z−z ≤R 2−k) → T(t)z−z ≤R 2−m holds for any nontrivial normed space X with a nonempty C ⊆ X.

Answer 2.B

Theorem In addition to Suzuki’s assumptions, assume that {T(t) : t ≥ 0} is uniformly equicontinuous with a modulus ω. Let α, β ∈ R+ with 2−G < α < β for some G ∈ N and satisfying β/α ∈ R \ Q with an effective irrationality measure (with domain restricted to N × N) f β

α . Then

∀k ∈ N ∀M ∈ N ∀b ∈ N ∀z ∈ Cb (T(α)z − z ≤ X ∧ T(β)z − z ≤ X → ∀t ∈ [0, M] T(t)z − z < 2−k)

Answer 2.B

with X = X(f β

α , ⌈β⌉, G, b, M, k, Φ, Ψ, ω, ˜

W ) = √ 5

2−(k+1) 6M Φ(ωb,M+1(k+1))

i=1

2Ψ(i)

(( 1+

√ 5 2

) ˜

W −1 − ( 1− √ 5 2

) ˜

W −1) ˜ W −2 i=1 ⌈β⌉2Ψ(i+1)

where Ψ(1) := G, Ψ(2) := G and for n > 2 Ψ(n) :=

n−2

• i=2

⌈log2( max

l≤⌈β⌉2Ψ(i+1){f

αi αi+1 (l, 1)})⌉ + G

Answer 2.B

with f α1

α2 (p, q) := f β α (p, q)

f αn+1

αn+2

(p, q) := max

k≤⌈β⌉2Ψ(n+1){f

αn αn+1 (kp + q, p)}|⌈q

p ⌉|, where {αn} defined by α1 := β, α2 := α αn+2 := αn − [ αn

αn+1 ]αn+1,

Φ(k) := ⌈β⌉2k + 2 ˜ W = ˜ W (k, b, M, ⌈β⌉, Φ, Ψ, ω) = max{Φ(ωb,M+1(k+1)), Φ(ωb,⌈β⌉(k+1+⌈log2(3M

Φ(ωb,M+1(k+1))

• i=1

2Ψ(i))⌉))}.

Remark

In both works it would be possible to remove the equicontinuity assumption for the semigroup. (In principle needed to guarantee majorizability for the metatheorem)) Then, the bound would depend on z ∈ C, (instead of b ∈ N as Cb := {z ∈ C : z ≤ b} In fact, that would be a quantitative version of Suzuki’s result. But, there would be disadvantages: bound less uniform not possible to have Corollary on asymptotic regularity.

An intermediate result used to obtain Answer 2.B

Theorem (Quantitative version of a result by Suzuki) Let X be a Banach space and let {T(t) : t ≥ 0} be a one-parameter uniformly equicontinuous semigroup of nonexpansive mappings on a subset C

• f X, with a modulus of uniform equicontinuity ω. Let {αn} be a

sequence of reals in [0, ∞) converging to α∞ ∈ [0, ∞) with a rate

• f convergence Φ : N → N, and so that

∀n ∈ N(|αn − α∞| > 2−Ψ(n)) where Ψ : N → N. Let L ∈ N be such that for all n ∈ N {αn}, α∞ ∈ [0, L]. Then ∀k ∈ N ∀b ∈ N ∀z ∈ Cb ∀M ∈ N ∀L ∈ N ∃n ≤ ˜ W (T(αn)z − z ≤ W → ∀t ∈ [0, M] T(t)z − z < 2−k) with

˜ W = ˜ W (k, b, M, L, Φ, Ψ, ω) = max{Φ(ωb,M+1(k+1)), Φ(ωb,L(k+1+⌈log2(3M

Φ(ωb,M+1(k+1))

• i=1

2Ψ(i))⌉))} and W = W (k, b, M, Φ, Ψ, ω) = 2−(k+1) 3M Φ(ωb,M+1(k+1))

i=1

2Ψ(i) .

An intermediate result used to obtain Answer 2.B

By a proposition in Suzuki, if z ∈ C is a fixed point of T(αn) for all n ∈ N, then z ∈ C is a fixed point of T(t) for all t ∈ [0, ∞). Formalised version: ∀z ∈ C(∀δ > 0 ∀n ∈ N T(αn)z − z ≤ δ → ∀k ∈ N ∀t ∈ [0, ∞)T(t)z − z < 2−k). By prenexing ( setting Cb := {z ∈ C : z ≤ b}) ∀b ∈ N ∀z ∈ Cb ∀k ∈ N ∀M ∈ N ∀t ∈ [0, M] ∃δ > 0 ∃n ∈ N (T(αn)z − z ≤ δ → T(t)z − z < 2−k).

Metatheorem extraction guaranteed by... Assume that we have a proof of a sentence in Aω[X, · , C]−b ∀t ∈ R+ ∀z ∈ C ∀T ∈ C × R+ → C ∀{αn} ⊆ R+ ∀α∞ ∈ R+ ∀ω ∈ N × N × N → N ∀Φ, Ψ ∈ N → N ∀m ∈ N ∃k ∈ N ∃n ∈ N

• (∀t ∈ R+ ∀x, y ∈ C T(t)x − T(t)y ≤R x − y)

∧(∀x ∈ C ∀t, s ∈ R+ T(s) ◦ T(t)(x) =X T(s + t)(x))

∧(∀b ∈ N ∀q ∈ C ∀m ∈ N ∀K ∈ N ∀t, t′ ∈ [0, K] (q <R b ∧ |t − t′| <R 2−ωK,b(m) → T(t)q − T(t′)q ≤R 2−m)) ∧(∀n ∈ N |αn − α∞| ≥R 2−Ψ(n)) ∧(∀k ∈ N ∀n ≥ Φ(k) |αn − α∞| ≤R 2−k) ∧(T(αn)z − z ≤R 2−k) → T(t)z − z <R 2−m . Then one can extract from the proof computable functionals W , ˜ W so that

∀M ∈ N ∀t ∈ [0, M] ∀L ∈ N ∀{αn} ∈ [0, L]N ∀α∞ ∈ [0, L] ∀B ∈ N ∀z ∈ CB ∀T ∈ C ×R+ → C ∀ω ∈ N×N×N → N ∀Φ, Ψ ∈ N → N ∀m ∈ N ∃k ≤ W (B, M, L, Ψ′, Φ′, m, ω′) ∃n ≤ ˜ W (B, M, L, Ψ′, Φ′, m, ω′)

• ((∀t ∈ R+ ∀x, y ∈ C T(t)x − T(t)y ≤R x − y)

∧(∀x ∈ C ∀t, s ∈ R+ T(s) ◦ T(t)(x) =X T(s + t)(x)) ∧(∀b ∈ N ∀q ∈ C ∀m ∈ N ∀K ∈ N ∀t, t′ ∈ [0, K] (q <R b ∧ |t − t′| <R 2−ωK,b(m) → T(t)q − T(t′)q ≤R 2−m)) ∧(∀n ∈ N |αn − α∞| ≥R 2−Ψ(n)) ∧(∀k ∈ N ∀n ≥ Φ(k) |αn − α∞| ≤R 2−k) ∧(T(αn)z − z ≤R 2−k) → T(t)z − z <R 2−m holds for any nontrivial normed space X with a nonempty C ⊆ X.

THANK YOU !