Veri edComputer-Aided Mathematics[Introduction to Interval Analysis - R. E. Moore, R. B. Kearfott,...

Verified Computer-AidedMathematics

Assia Mahboubi

Assia Mahboubi – Verified Computer-Aided Mathematics 1

What is a proof assistant?

Mathematica: A System for Doing Mathematics by Computer,Stephen Wolfram, Addison-Wesley 1988.

Mathematica:

A System for Doing Mathematics by Computer

,Stephen Wolfram, Addison-Wesley 1988.

Mathematica: A System for Doing Mathematics by Computer,Stephen Wolfram, Addison-Wesley 1988.

Proof assistants

Representing mathematics in a fixed formal language:

• Define mathematical objects, statements, proofs;

• Verify the correctness of proofs by a routine process.

And use a proof assistant to make this possible in practice.

Proof assistants

Representing mathematics in a fixed formal language:

• Define mathematical objects, statements, proofs;

• Verify the correctness of proofs by a routine process.

And use a proof assistant to make this possible in practice.

Proof Assistants

Formal Logic

Proof Assistant

ProofChecker

Libraries

Assistance for building proofs

• Tactic language;

• Formal-proof producing decision procedures.

Assistance for writing statements

Variables gT (G : {group gT}).

Lemma cfCauchySchwarz (phi psi : ’CF(G)) :

|[phi, psi]| 2 <= [phi] * [psi] , eq_iff (~~ free (phi :: psi)).

Assistance for writing statements

Variables gT (G : {group gT}).

Lemma cfCauchySchwarz (phi psi : ’CF(G)) :

|[phi, psi]| 2 <= [phi] * [psi] , eq_iff (~~ free (phi :: psi)).

• Program verification:- Compcert compiler (Leroy et al., 2006),- sel4 micro-kernel (Klein et al., 2008),- etc.

• Mathematics:- Four color theorem (Gonthier, 2004),- Odd Order Theorem (Gonthier et al, 2012),- Flyspeck (Hales et al. 2014),- etc.

• Program verification:- Compcert compiler (Leroy et al., 2006),- sel4 micro-kernel (Klein et al., 2008),- etc.

• Mathematics:- Four color theorem (Gonthier, 2004),- Odd Order Theorem (Gonthier et al, 2012),- Flyspeck (Hales et al. 2014),- etc.

In the Age of the Turing Machine

The Misfortunes of a Trio of Mathematicians Using ComputerAlgebra Systems. Can We Trust in Them?

Antonio J. Duran, Mario Perez, and Juan L. Varona, Notices of the AMS, Nov. 2014

In the Age of the Turing Machine

The Misfortunes of a Trio of Mathematicians Using ComputerAlgebra Systems. Can We Trust in Them?

Antonio J. Duran, Mario Perez, and Juan L. Varona, Notices of the AMS, Nov. 2014

Computing Integrals

m ≤∫ b

f ≤ M

sin(t + exp(t))dt,

ln(t)dt,

∫ +∞

cos(t)ln(t)

Joint work with Guillaume Melquiond and Thomas Sibut-Pinote, JAR 2018

Computing Integrals

m ≤∫ b

f ≤ M

sin(t + exp(t))dt,

ln(t)dt,

∫ +∞

cos(t)ln(t)

Joint work with Guillaume Melquiond and Thomas Sibut-Pinote, JAR 2018

Example

Ternary Goldbach Conjecture

Integral computations

Real numbers in a computer: Floats

• Compromise between precision and range• IEEE standards• Efficient algorithms

• Compromise between precision and range• Notoriously difficult to specify

Credit: Dominique ThiebautAssia Mahboubi – Verified Computer-Aided Mathematics 14

Real numbers in a computer: intervals

I = {[a, b] | a, b ∈ R ∪ {+∞} ∪ {−∞}}

• Examples:

[0, 0] [3.14, 3.15] [17,+∞] [−∞,+∞]

• Operations:

[−1, 2]× [−3, 1] = [−6, 2] [0, 100]− [0, 100] = [−100, 100]

• Computations: floating-point numbers as bounds.

• Reasoning: interval analysis.

Interval extensions

F : In → I is an interval extension of f : Rn → R, if:

∀x ∈ Rn,∀x ∈ In, x ∈ x⇒ f (x) ∈ F(x)

An interval function F : In → I is isotonic if:

yi ⊂ xi ⇒ F(y1, . . . , yn) ⊂ F(x1, . . . , xn)

[Introduction to Interval Analysis - R. E. Moore, R. B. Kearfott, M. J. Cloud]

[Validated Numerics: A Short Introduction to Rigorous Computations - W. Tucker]

Interval extensions

F : In → I is an interval extension of f : Rn → R, if:

∀x ∈ Rn,∀x ∈ In, x ∈ x⇒ f (x) ∈ F(x)

An interval function F : In → I is isotonic if:

yi ⊂ xi ⇒ F(y1, . . . , yn) ⊂ F(x1, . . . , xn)

[Introduction to Interval Analysis - R. E. Moore, R. B. Kearfott, M. J. Cloud]

[Validated Numerics: A Short Introduction to Rigorous Computations - W. Tucker]

Interval Riemann integration

If f is continuous on [a, b], then:∫ b

f ∈ (b − a)[m,M]

If F extends f , and a ∈ a, b ∈ b:∫ b

f ∈ (b− a)F(hull(a,b))

Combine with dichotomy and higher order methods.

f ∈ (b − a)[m,M]

If F extends f , and a ∈ a, b ∈ b:

f ∈ (b − a)[m,M]

Correctness theorem

• Program: cos : I→ I

• Mathematical definition: cos : R→ R• Specification:

For any x ∈ R and any i ∈ I, x ∈ i⇒ cos(x) ∈ cos(i)

⇒ A formal proof per integrand.

Correctness theorem

• Program: cos : I→ I• Mathematical definition: cos : R→ R

• Specification:

Correctness theorem

• Program: cos : I→ I• Mathematical definition: cos : R→ R• Specification:

Correctness theorem

• Program: cos : I→ I• Mathematical definition: cos : R→ R• Specification:

Catalogue

Based on an abstract syntax of (univariate) expressions:

E := x | F| π |E + E | E − E | E × E | E ÷ E | − E | ‖E‖ |√E | Ek |

And on an arithmetic of interval extensions.[Proving bounds on real-valued functions with computations - G. Melquiond, Proc. of IJCAR 2008]

Computations of extensions

Expressions are equipped with several evaluations schemes:

• [e]R : R→ R• [e]R⊥ : R⊥ → R⊥• [e]I⊥ : I⊥ → I⊥

• Value ⊥R models the complement of the definition domain.

• Interval ⊥I is the only one containing ⊥R.

• A correctness theorem ensures:

∀e ∈ E ,∀i ∈ I⊥,∀x ∈ i, [e]R⊥(x) ∈ [e]I⊥(i)

• [e]R : R→ R• [e]R⊥ : R⊥ → R⊥• [e]I⊥ : I⊥ → I⊥

∀e ∈ E ,∀i ∈ I⊥,∀x ∈ i, [e]R⊥(x) ∈ [e]I⊥(i)

• [e]R : R→ R• [e]R⊥ : R⊥ → R⊥• [e]I⊥ : I⊥ → I⊥

∀e ∈ E ,∀i ∈ I⊥,∀x ∈ i, [e]R⊥(x) ∈ [e]I⊥(i)

• [e]R : R→ R• [e]R⊥ : R⊥ → R⊥• [e]I⊥ : I⊥ → I⊥

∀e ∈ E ,∀i ∈ I⊥,∀x ∈ i, [e]R⊥(x) ∈ [e]I⊥(i)

Meaning

In fact, for any expression e ∈ E and interval i:

[e]I⊥(i) 6= ⊥I ⇒ f is continuous on i

⇒ If∫ b

af can be approximated, then f is integrable on [a, b].

Meaning

⇒ If∫ b

Meaning

⇒ If∫ b

Benchmarks

∣∣(x4 + 10x3 + 19x2 − 6x − 6)ex∣∣ dx ' 11.14731055005714

Error Time Accuracy Degree Depth Precision10−3 0.7 14 5 8 3010−6 0.9 24 6 13 4010−9 1.3 34 8 18 5010−12 1.9 44 10 22 6010−15 2.7 54 12 28 70

Benchmarks

∣∣(x4 + 10x3 + 19x2 − 6x − 6)ex∣∣ dx ' 11.14731055005714

• Octave’s quad/quadgk: only 10/9 correct digits;

• INTLAB verifyquad: false answer without warning;

• VNODE-LP: cannot be used because of the absolute value.

Benchmarks

∫ +∞

100000

0.5·ln(1+2.25/τ2)+4.1396+lnπln τ

1 + 0.25/τ2·

ln2 τ

τ2dτ ' −3.2555895745 · 10−6

Error Time Accuracy Degree Depth Precision

10−3 0.6 2 3 0 3010−4 0.8 5 5 2 3010−5 1.5 8 7 6 3010−6 3.1 11 9 11 3010−7 5.6 15 12 12 3010−8 11.2 18 15 15 30

Thus: ∫ +∞

−∞

0.5·ln(1+2.25/τ2)+4.1396+lnπln τ

1 + 0.25/τ2·

ln2 τ

τ2dτ ∈ [226.849; 226.850]

The upper bound 226.844 given in the preprint is incorrect.

Benchmarks

∫ +∞

100000

0.5·ln(1+2.25/τ2)+4.1396+lnπln τ

1 + 0.25/τ2·

ln2 τ

τ2dτ ' −3.2555895745 · 10−6

Error Time Accuracy Degree Depth Precision

10−3 0.6 2 3 0 3010−4 0.8 5 5 2 3010−5 1.5 8 7 6 3010−6 3.1 11 9 11 3010−7 5.6 15 12 12 3010−8 11.2 18 15 15 30

Thus: ∫ +∞

−∞

0.5·ln(1+2.25/τ2)+4.1396+lnπln τ

1 + 0.25/τ2·

ln2 τ

τ2dτ ∈ [226.849; 226.850]

The upper bound 226.844 given in the preprint is incorrect.

• Formal Verification of Nonlinear Inequalities with TaylorInterval Approximations

Thomas Hales and Alexey Solovyev, Proc. of NFM 2013.

• A Verified ODE Solver and the Lorenz AttractorFabian Immler, JAR 2018.

• Formal Verification of Nonlinear Inequalities with TaylorInterval Approximations

Thomas Hales and Alexey Solovyev, Proc. of NFM 2013.

• A Verified ODE Solver and the Lorenz AttractorFabian Immler, JAR 2018.

Special functions and sequences

Abramowitz and Stegun Plouffe and Sloane

∂-finiteness

• A function f : K→ K is D-finite if for all x :

pr (x)f (r)(x) + · · ·+ p1(x)f ′(x) + p0(x)f (x) = 0

• A sequence u ∈ KN is P-recursive if for all n ∈ N

pr (n)un+r + · · ·+ p0(n)un = 0

where pi ∈ C[n] are polynomial coefficients and pr 6= 0.

⇒ Extends to systems and multivariates sequences/functions.

∂-finiteness

pr (x)f (r)(x) + · · ·+ p1(x)f ′(x) + p0(x)f (x) = 0

pr (n)un+r + · · ·+ p0(n)un = 0

∂-finiteness

pr (x)f (r)(x) + · · ·+ p1(x)f ′(x) + p0(x)f (x) = 0

pr (n)un+r + · · ·+ p0(n)un = 0

Examples

• Bessel functions:

dx+ (x2 − a2)f = 0

• Binomial coefficients(nm

un+1,m+1, = un+1,m + un,m+1 with u0,0 = un,n = 1

Closure properties

P-recursive sequences on a field K form a K-algebra.

If u and v are P-recursive it is possible to:

• compute recurrences canceling (u + v)

• compute recurrences canceling (u ∗ v)

For (un,k) P-recursive, it is possible to:

• Compute a recurrence canceling Un :=∑n

k=0 un,k

Closure properties

P-recursive sequences on a field K form a K-algebra.

If u and v are P-recursive it is possible to:

• compute recurrences canceling (u + v)

• compute recurrences canceling (u ∗ v)

For (un,k) P-recursive, it is possible to:

• Compute a recurrence canceling Un :=∑n

k=0 un,k

Irrationality of ζ(3)

Let ζ(3) be the real number∑+∞

k=11k3 .

Theorem (Apery, 1978): The constant ζ(3) is irrational.

See:“A proof that Euler missed: Apery’s proof of the irrationality ofζ(3). An informal report.” by A. van der Poorten, 1979.

for the tale of this proof, and of its verification.

The crux in Apery’s proof is to verify that (an)n∈N and (bn)n∈Nverify the recurrence:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

an =n∑

)2(n+kk

)2, bn = an

n∑k=1

k∑m=1

(−1)m+1(nk

)2(n+kk

2m3(nm

)(n+mm

The crux in Apery’s proof is to verify that (an)n∈N and (bn)n∈Nverify the recurrence:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

an =n∑

)2(n+kk

)2, bn = an

n∑k=1

k∑m=1

(−1)m+1(nk

)2(n+kk

2m3(nm

)(n+mm

Checking the recurrence: telescopes

To prove that an :=∑

k vn,k with vn,k :=(nk

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

Cohen and Zagier constructed wn,k such that:

n3vn,k−(34n3−51n2+27n−5)vn−1,k+(n−1)3vn−2,k = wn,k+1 − wn,k

And then they sum over k both hand sides:∑k

(n3vn,k−(34n3−51n2+27n−5)vn−1,k+(n−1)3vn−2,k) =∑k

(wn,k+1 − wn,k)

n3an − (34n3 − 51n2 + 27n − 5)an−1 + (n − 1)3an−2 = wn,∞ − wn,0 = 0

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

(wn,k+1 − wn,k)

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

And then they sum over k both hand sides:

(wn,k+1 − wn,k)

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

(wn,k+1 − wn,k)

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

(wn,k+1 − wn,k)

n3an − (34n3 − 51n2 + 27n − 5)an−1 + (n − 1)3an−2

wn,∞ − wn,0 = 0

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

(wn,k+1 − wn,k)

n3an − (34n3 − 51n2 + 27n − 5)an−1 + (n − 1)3an−2 =

wn,∞ − wn,0 = 0

)2(n+kk

)2verifies:

n3yn − (34n3 − 51n2 + 27n − 5)yn−1 + (n − 1)3yn−2 = 0

(wn,k+1 − wn,k)

Alternate proofs

• Paper proofs:Beukers (1979, 1987), Nesterenko (1996), Rajkumar(2012), . . .

• Algorithmic proofs:Zeilberger (1993), Zudilin (2002, 2009), Salvy (2003),Schneider (2007), . . .

Formal recurrence operators

We denote A = Q(n, k)〈Sn, Sk〉 the ring of skew polynomials:

• in the (commuting) indeterminates Sn and Sk

• with coefficients in Q(n, k)

• with the commutation rule:

jkc(n, k) = c(n + i , k + j)S i

nSjk .

Elements of A can be viewed as algebraic representations forlinear recurrence operators with rational function coefficients.

Formal recurrence operators

Consider

• P =∑

(i ,j)∈I pi ,j(n, k)S inS

jk ∈ A,

• f a sequence.

The operator P acts on f by:

(P · f )n,k =∑

(i ,j)∈I

pi ,j(n, k)fn+i ,k+j

Example: binomial coefficient

Recurrences for the binomial coefficient(nk

)− n + 1

n + 1− k

)− n − k

can be represented using the operators:

P1 = Sn −n + 1

n + 1− k1, P2 = Sk −

n − k

k + 11

as:P1 · f = 0, P2 · f = 0

Example: binomial coefficient

Recurrences for the binomial coefficient(nk

)− n + 1

n + 1− k

)− n − k

can be represented using the operators:

P1 = Sn −n + 1

n + 1− k, P2 = Sk −

n − k

k + 1as:

P1 · f = 0, P2 · f = 0

Creative telescoping

The aim is:

• starting from sequence (fn,k),

• to compute annihilating operators for Fn =∑n

k=0 fn,k .

For this:

• An algorithm finds a pair (P ,Q) satisfying

P · f = (Sk − 1)Q · f ;

• With Q ∈ A and P ∈ Q(n, k)〈Sn〉.

Creative telescoping

The aim is:

• starting from sequence (fn,k),

• to compute annihilating operators for Fn =∑n

k=0 fn,k .

For this:

• An algorithm finds a pair (P ,Q) satisfying

P · f = (Sk − 1)Q · f ;

• With Q ∈ A and P ∈ Q(n, k)〈Sn〉.

Annihilating ideal and Grobner basis

Let f a (germ of) sequence, cancelled by a certain Pf ∈ A.

• {P ∈ A : P · f = 0} is a left ideal of A;

• Algorithms exist to compute Grobner bases of such an ideal.

• Such a Grobner basis is a good representation for f itself.

Checking a candidate recurrence

Grobner bases are excellent tools to check the validity of acandidate recurrence.

Check: (n + 1

)−(n

Using:(n + 1

n + 1− k

n − k

Computing Apery’s recurrence

an =n∑

)2(n+kk

)2, bn =

n∑k=1

)2(n+kk

n∑m=1

k∑m=1

(−1)m+1

2m3(nm

)(n+mm

step explicit form system operation input(s)

1 cn,k =(nk

)2(n+kk

)2C direct

2 an =∑n

k=1 cn,k A Σ C

3 dn,m = (−1)m+1

2m3(nm)(n+m

m )D direct

4 sn,k =∑k

m=1 dn,m S Σ D

5 zn =∑n

m=11m3 Z direct

6 un,k = zn + sn,k U + Z and S

7 vn,k = cn,kun,k V × C and U

8 bn =∑n

k=1 vn,k B Σ V

Computing Apery’s recurrence

an =n∑

)2(n+kk

)2, bn =

n∑k=1

)2(n+kk

n∑m=1

k∑m=1

(−1)m+1

2m3(nm

)(n+mm

step explicit form system operation input(s)

1 cn,k =(nk

)2(n+kk

)2C direct

2 an =∑n

k=1 cn,k A Σ C

3 dn,m = (−1)m+1

2m3(nm)(n+m

m )D direct

4 sn,k =∑k

m=1 dn,m S Σ D

5 zn =∑n

m=11m3 Z direct

6 un,k = zn + sn,k U + Z and S

7 vn,k = cn,kun,k V × C and U

8 bn =∑n

k=1 vn,k B Σ V

Checking Apery’s recurrence

an =n∑

)2(n+kk

)2, bn =

n∑k=1

)2(n+kk

n∑m=1

k∑m=1

(−1)m+1

2m3(nm

)(n+mm

step explicit form GB operation input(s)8 bn =

∑nk=1 vn,k B creative telescoping V

7 vn,k = cn,kun,k V product C and U

6 un,k = zn + sn,k U addition Z and S

m=11m3

direct

4 sn,k =∑k

m=1 dn,m S creative telescoping D

3 dn,m

= (−1)m+1

2m3(nm)(n+m

direct

2 an =∑n

k=1 cn,k A creative telescoping C

1 cn,k

)2(n+kk

direct

Checking Apery’s recurrence

an =n∑

)2(n+kk

)2, bn =

n∑k=1

)2(n+kk

n∑m=1

k∑m=1

(−1)m+1

2m3(nm

)(n+mm

step explicit form GB operation input(s)8 bn =

∑nk=1 vn,k B creative telescoping V

7 vn,k = cn,kun,k V product C and U

6 un,k = zn + sn,k U addition Z and S

5 zn =∑n

m=11m3 Z direct

4 sn,k =∑k

m=1 dn,m S creative telescoping D

3 dn,m = (−1)m+1

2m3(nm)(n+m

m )D direct

2 an =∑n

k=1 cn,k A creative telescoping C

1 cn,k =(nk

)2(n+kk

)2C direct

Sound creative telescoping

From the guarded creative telescoping identity for (fn,k)

(n, k) /∈ ∆⇒ (P · f ,k)n = (Q · f )n,k+1 − (Q · f )n,k ,

We can prove for Fn =∑n+β

k=α fn,k that:

(P · F )n =(

(Q · f )n,n+β+1 − (Q · f )n,α)

i∑j=1

pi(n) fn+i ,n+β+j

α≤k≤n+β ∧ (n,k)∈∆

(P · f ,k)n − (Q · f )n,k+1 + (Q · f )n,k ,

Sound creative telescoping

From the guarded creative telescoping identity for (fn,k)

(n, k) /∈ ∆⇒ (P · f ,k)n = (Q · f )n,k+1 − (Q · f )n,k ,

We can prove for Fn =∑n+β

k=α fn,k that:

(P · F )n =(

(Q · f )n,n+β+1 − (Q · f )n,α)

i∑j=1

pi(n) fn+i ,n+β+j

α≤k≤n+β ∧ (n,k)∈∆

(P · f ,k)n − (Q · f )n,k+1 + (Q · f )n,k ,

A formal, computer-algebra aided, proof

Joint work with Frederic Chyzak, Thomas Sibut-Pinote, ITP 2014

• A Maple session computes the recurrence operators;

• It generates Coq statements of candidate properties;

• We annotate the generated recurrences with provisos;

• We prove lemmas by interactive formal proofs.

See also: Formal proofs of hypergeometric sums, John Harrison, Journal of Automated Reasoning, 2015

First order theories

Inductive term : Type :=

| Var (n : N)| Add (t1 t2 : term)

| Opp (t : term)

| NatConst (n : N)| NatMul (t : term)(n : N)| Mul (t1 t2 : term)

Inductive formula : Type :=

| Bool (b : B)| Equal (t1 t2 : term)

| Leq (t1 t2 : term)

| And (f1 f2 : formula)

| Or (f1 f2 : formula)

| Implies (f1 f2 : formula)

| Not (f : formula)

| Exists (n : N) (f : formula)

| Forall (n : N) (f : formula)

Definition qfree_form : formula -> B

Definition eval (R : ringType) : seq R -> term -> R

Definition holds (R : ringType) : seq R -> formula -> Prop

⇒ (holds R e f) is the (yet unproved) statement “R |=e f ”.

| Opp (t : term)

| Not (f : formula)

| Opp (t : term)

| Not (f : formula)

| Opp (t : term)

| Not (f : formula)

| Opp (t : term)

| Not (f : formula)

Quantifier elimination: real closed fields

⇒ A function:

Definition rcf_qelim : formula -> formula

such that:

• forall f : formula, qfree (qelim f)= true

• forall (R : realClosedField) (e : seq R) (f : formula),

holds e f <-> holds e (rcf_qelim e f)

Joint work with Cyril Cohen, LMCS 2012.

⇒ A (constructive) theory of semi-algebraic sets and functions.Boris Djalal, Proceedings of CPP 2018.

Quantifier elimination: real closed fields

⇒ A function:

Definition rcf_qelim : formula -> formula

such that:

• forall f : formula, qfree (qelim f)= true

• forall (R : realClosedField) (e : seq R) (f : formula),

holds e f <-> holds e (rcf_qelim e f)

Joint work with Cyril Cohen, LMCS 2012.

⇒ A (constructive) theory of semi-algebraic sets and functions.Boris Djalal, Proceedings of CPP 2018.

Formal abstracts?

Uniform p-adic cell decomposition and local zeta functions,Johan Pas, J. reine angew. Math. 399 (1989), 137—172.

Define:

qelim : formula -> formula

Kq_free : formula -> B

Such that:

forall f : formula, Kq_free (qelim f)

forall f : formula,

exists out : {finset N}, forall (p : N), prime p -> p /∈ out ->

forall (e : seq Qp), Qp, holds Qp e f <-> holds Qp e (qelim f).

Formal abstracts?

Define:

Such that:

forall f : formula,

Formal abstracts?

Define:

Such that:

forall f : formula,

Formal abstracts?

Define:

Such that:

forall f : formula,

Formal abstracts?

Define:

Such that:

forall f : formula,

Conclusion

Happy Birthday Tom

Conclusion

Happy Birthday Tom

Veri edComputer-Aided Mathematics[Introduction to Interval Analysis - R. E. Moore, R. B. Kearfott,...

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