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<!-- This file is automatically generated; please do not modify it manually. -->
<!DOCTYPE html>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="description" content="A list of 100 famous math theorems, some of which have been formalized in the Rocq prover. This page keeps track of those.">
<title>Famous theorems proven in Rocq</title>
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<div class="middle">
<h1>Formalizing 100 theorems in Rocq</h1>
<p>
This is an appendix to Freek
Wiedijk's <a href="http://www.cs.ru.nl/~freek/100/">webpage</a> webpage
on the "top 100" mathematical theorems, to keep track of the statements
of the 80 theorems that are formalised in Rocq (formerly Coq).
</p>
<p>
Drop me or Freek an email or
<a href="https://github.com/coq-community/coq-100-theorems">
make a pull request
</a> if you have updates. Preferably edit
<a href="https://github.com/coq-community/coq-100-theorems/blob/master/statements.yml">this file</a>,
from which this page has been generated.
</p>
<p>
Options:<br>
<label><input type="checkbox" checked="checked" onclick="javascript:toggle(this, 'solved');"> Show theorems that have been formalized in Rocq</label><br>
<label><input type="checkbox" checked="checked" onclick="javascript:toggle(this, 'unsolved');"> Show theorems that have not been formalized in Rocq</label><br>
<label><input type="checkbox" onclick="javascript:toggle(this, 'existing');"> Show existing formalizations</label><br>
<label><input type="checkbox" onclick="javascript:toggle(this, 'axioms');"> Show axioms used</label><br>
<label><input type="checkbox" onclick="javascript:toggle(this, 'repro');"> Show some reproducibility information</label><br>
</p>
<h3 id='1' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#1'>1. The Irrationality of the Square Root of 2</a></h3>
<div>
<p class='author'><strong>many versions</strong>
(in <a href="https://github.com/coq-community/qarith-stern-brocot/blob/master/theories/sqrt2.v">coq-community/qarith-stern-brocot</a>):
</p>
<p class='comment'>One of many proofs</p>
<pre class='statement'>Theorem sqrt2_not_rational : forall p q : nat, q <> 0 -> p * p = 2 * (q * q) -> False.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, Naproche, NuPRL (constructive), PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='2' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#2'>2. Fundamental Theorem of Algebra</a></h3>
<div>
<p class='author'><strong>Herman Geuvers et al.</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/fta/FTA.v#L188">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma FTA : forall f : CCX, nonConst _ f -> {z : CC | f ! z [=] [0]}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='3' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#3'>3. The Denumerability of the Rational Numbers</a></h3>
<div>
<p class='author'><strong>Milad Niqui</strong>
(in <a href="https://github.com/coq-community/qarith-stern-brocot/blob/master/theories/Q_denumerable.v">coq-community/qarith-stern-brocot</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Definition same_cardinality (A:Type) (B:Type) :=
{ f : A -> B & { g : B -> A |
forall b,(compose _ _ _ f g) b = (identity B) b /\
forall a,(compose _ _ _ g f) a = (identity A) a } }.
Definition is_denumerable A := same_cardinality A nat.
Theorem Q_is_denumerable: is_denumerable Q.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='author'><strong>Daniel Schepler</strong>
(in <a href="https://github.com/coq-community/topology/blob/master/theories/ZornsLemma/CountableTypes.v">coq-community/topology</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Inductive CountableT (X:Type) : Prop :=
| intro_nat_injection: forall f:X->nat, injective f -> CountableT X.
Lemma Q_countable: CountableT Q.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, NuPRL (constructive), PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='4' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#4'>4. Pythagorean Theorem</a></h3>
<div>
<p class='author'><strong>Frédérique Guilhot</strong>
(in <a href="https://github.com/coq-community/HighSchoolGeometry/blob/master/theories/euclidien_classiques.v">coq-community/HighSchoolGeometry</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Pythagore :
forall A B C : PO,
orthogonal (vec A B) (vec A C) <->
Rsqr (distance B C) = Rsqr (distance A B) + Rsqr (distance A C) :>R.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>FunctionalExtensionality.functional_extensionality_dep</li>
<li>classic</li>
<li>other geometry axioms</li>
</ul>
<p class='author'><strong>Gabriel Braun</strong>
(in <a href="https://github.com/GeoCoq/GeoCoq/blob/master/Tarski_dev/Ch15_lengths.v">https://github.com/GeoCoq/GeoCoq/blob/master/Tarski_dev/Ch15_lengths.v</a>):
</p>
<p class='comment'>A synthetic proof in the context of Tarski's axioms.</p>
<pre class='statement'>Lemma pythagoras :
forall
(Tn : Tarski_neutral_dimensionless)
(TnEQD : Tarski_neutral_dimensionless_with_decidable_point_equality Tn),
Tarski_2D TnEQD ->
Tarski_euclidean TnEQD ->
forall
O E E' A B C AC BC AB AC2 BC2
AB2 : Tpoint,
O <> E ->
Per A C B ->
Length O E E' A B AB ->
Length O E E' A C AC ->
Length O E E' B C BC ->
Prod O E E' AC AC AC2 ->
Prod O E E' BC BC BC2 ->
Prod O E E' AB AB AB2 ->
Sum O E E' AC2 BC2 AB2.
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: hypotheses partially classical</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>Eqdep.Eq_rect_eq.eq_rect_eq</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq</p></div>
<h3 id='5' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#5'>5. Prime Number Theorem</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath</p></div>
<h3 id='6' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#6'>6. Gödel’s Incompleteness Theorem</a></h3>
<div>
<p class='author'><strong>Russell O'Connor</strong>
(in <a href="https://github.com/coq-community/hydra-battles/blob/master/theories/goedel/">coq-community/hydra-battles</a>):
</p>
<p class='comment'>The second theorem assumes the last two Hilbert-Bernays-Loeb derivability conditions.</p>
<pre class='statement'>(* in goedel1.v *)
Theorem Goedel'sIncompleteness1st :
wConsistent T ->
exists f : Formula,
~ SysPrf T f /\
~ SysPrf T (notH f) /\ (forall v : nat, ~ In v (freeVarFormula LNN f)).
(* in goedel2.v *)
Hypothesis HBL2 : forall f, SysPrf T (impH (box f) (box (box f))).
Hypothesis HBL3 : forall f g, SysPrf T (impH (box (impH f g)) (impH (box f) (box g))).
Theorem SecondIncompletness :
SysPrf T Con -> Inconsistent LNT T.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Rocq (constructive), nqthm</p></div>
<h3 id='7' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#7'>7. Law of Quadratic Reciprocity</a></h3>
<div>
<p class='author'><strong>Nathanaëlle Courant</strong>
(in <a href="https://github.com/Ekdohibs/coq-proofs/blob/master/Reciprocity/Reciprocity.v#L1560">https://github.com/Ekdohibs/coq-proofs/blob/master/Reciprocity/Reciprocity.v#L1560</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Quadratic_reciprocity :
(legendre p q) * (legendre q p) = (-1) ^ (((p - 1) / 2) * ((q - 1) / 2)).
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>proof_irrelevance</li>
<li>constructive_definite_description</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq, nqthm</p></div>
<h3 id='8' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#8'>8. The Impossibility of Trisecting the Angle and Doubling the Cube</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle</p></div>
<h3 id='9' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#9'>9. The Area of a Circle</a></h3>
<div>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, ProofPower</p></div>
<h3 id='10' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#10'>10. Euler’s Generalization of Fermat’s Little Theorem</a></h3>
<div>
<p class='author'><strong>Laurent Théry</strong>
(in <a href="https://github.com/math-comp/math-comp/blob/master/mathcomp/solvable/cyclic.v">mathcomp</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Euler_exp_totient a n : coprime a n -> a ^ totient n = 1 %[mod n].
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq (constructive)</p></div>
<h3 id='11' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#11'>11. The Infinitude of Primes</a></h3>
<div>
<p class='author'><strong>Russell O'Connor</strong>
(in <a href="https://web.archive.org/web/20151210214004/coq.inria.fr/cocorico/NotFinitePrimes">https://web.archive.org/web/20151210214004/coq.inria.fr/cocorico/NotFinitePrimes</a>):
</p>
<p class='comment'>This statement was formerly in cocorico, compatible with Coq 7.3</p>
<pre class='statement'>Theorem ManyPrimes : ~(EX l:(list Prime) | (p:Prime)(In p l)).
</pre>
<p class='repro'>Reproducibility: old proof compatible with Coq 7.3</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='author'><strong>Georges Gonthier</strong>
(in <a href="https://x80.org/rhino-coq/">https://x80.org/rhino-coq/</a>):
</p>
<p class='comment'>There are other proofs of this theorem but you can browse this one in your browser with JsCoq.</p>
<pre class='statement'>Lemma prime_above m : {p | m < p & prime p}.
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, Naproche, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='12' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#12'>12. The Independence of the Parallel Postulate</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle</p></div>
<h3 id='13' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#13'>13. Polyhedron Formula</a></h3>
<div>
<p class='author'><strong>Jean-François Dufourd</strong>
(in <a href="https://github.com/coq-contribs/euler-formula/blob/master/Euler3.v">coq-contribs/euler-formula</a>):
</p>
<p class='comment'></p>
<pre class='statement'>(* nc : number of connected components
nv : number of vertices
ne : number of edges
nf : number of faces
nd : number of darts
plf m : m is a planar formation
inv_qhmap : see Euler1.v *)
Theorem Euler_Poincare_criterion: forall m:fmap,
inv_qhmap m -> (plf m <-> (nv m + ne m + nf m - nd m) / 2 = nc m).
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.13.2, failure with 8.14.1 (using git tag v8.11.0)</p>
<p class='axioms'>Constructive: to be determined</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>Euler3.expf_clos_symm</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Mizar, Rocq (constructive)</p></div>
<h3 id='14' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#14'>14. Euler’s Summation of 1 + (1/2)^2 + (1/3)^2 + ….</a></h3>
<div>
<p class='author'><strong>Jean-Marie Madiot</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Reals/Rzeta2.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem zeta2_pi_2_6 : Rser_cv (fun n => 1 / (n + 1) ^ 2) (PI ^ 2 / 6).
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq</p></div>
<h3 id='15' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#15'>15. Fundamental Theorem of Integral Calculus</a></h3>
<div>
<p class='author'><strong>Luís Cruz-Filipe</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/ftc/FTC.v#L181">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem FTC1 : Derivative J pJ G F.
Theorem FTC2 : {c : IR | Feq J (G{-}G0) [-C-]c}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='author'><strong>The Coq development team</strong>
(in <a href="https://rocq-prover.org/doc/V9.0.0/stdlib/Stdlib.Reals.RiemannInt.html">coq's standard library</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma FTC_Riemann :
forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b),
RiemannInt pr = f b - f a.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='16' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#16'>16. Insolvability of General Higher Degree Equations</a></h3>
<div>
<p class='author'><strong>Sophie Bernard, Cyril Cohen, Assia Mahboubi, Pierre-Yves Strub</strong>
(in <a href="https://github.com/math-comp/Abel/blob/7a85e03fcd1b8d772d592fd53afe753b49f8fe96/theories/abel.v#L1454-L1455">mathcomp</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma example_not_solvable_by_radicals :
~ solvable_by_radical_poly ('X^5 - 4 *: 'X + 2 : {poly rat}).
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: Lean, Rocq</p></div>
<h3 id='17' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#17'>17. DeMoivre’s Theorem</a></h3>
<div>
<p class='author'><strong>Coqtail team</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Complex/Cexp.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'>De Moivre's formula is easily deduced from Euler's formula thanks to e^(n*z)=(e^z)^n.</p>
<pre class='statement'>Lemma Cexp_trigo_compat : forall a, Cexp (0 +i a) = cos a +i sin a.
Lemma Cexp_mult : forall a n, Cexp (INC n * a) = (Cexp a) ^ n.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, ProofPower, Rocq</p></div>
<h3 id='18' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#18'>18. Liouville’s Theorem and the Construction of Transcendental Numbers</a></h3>
<div>
<p class='author'><strong>Valentin Blot</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/liouville/Liouville.v">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Liouville_theorem2 :
{n : nat | {C : IR | Zero [<] C | forall (x : Q),
(C[*]inj_Q IR (1#Qden x)%Q[^]n) [<=] AbsIR (inj_Q _ x [-] a)}}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq (constructive)</p></div>
<h3 id='19' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#19'>19. Four Squares Theorem</a></h3>
<div>
<p class='author'><strong>Guillaume Allais, Jean-Marie Madiot</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Arith/Lagrange_four_square.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'>Coq extraction produces a program that indeed computes a, b, c, d</p>
<pre class='statement'>Theorem lagrange_4_square_theorem : forall n, 0 <= n ->
{ a : _ & { b: _ & { c : _ & { d |
n = a * a + b * b + c * c + d * d } } } }.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, Rocq (constructive)</p></div>
<h3 id='20' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#20'>20. All Primes (= 1 mod 4) Equal the Sum of Two Squares</a></h3>
<div>
<p class='author'><strong>Laurent Théry</strong>
(in <a href="https://github.com/coq-contribs/sum-of-two-square/blob/master/TwoSquares.v#L641">coq-contribs/sum-of-two-square</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Definition sum_of_two_squares :=
fun p => exists a , exists b , p = a * a + b * b.
Theorem two_squares_exists:
forall p, prime p -> p = 2 \/ Zis_mod p 1 4 -> sum_of_two_squares p.
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.10.2, failure with 8.11.2 (using git tag v8.10.0)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='author'><strong>Laurent Théry</strong>
(in <a href="https://github.com/thery/twoSquare/blob/master/fermat2.v">https://github.com/thery/twoSquare/blob/master/fermat2.v</a>):
</p>
<p class='comment'>proof using gaussian integers and mathcomp</p>
<pre class='statement'>Lemma sum2sprime p :
odd p -> prime p -> p \is a sum_of_two_square = (p %% 4 == 1).
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='21' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#21'>21. Green’s Theorem</a></h3>
<div>
<p class='existing'>Formalized in: Isabelle</p></div>
<h3 id='22' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#22'>22. The Non-Denumerability of the Continuum</a></h3>
<div>
<p class='author'><strong>Pierre-Marie Pédrot</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Reals/Logic/Runcountable.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem R_uncountable : forall (f : nat -> R),
{l : R | forall n, l <> f n}.
Theorem R_uncountable_strong : forall (f : nat -> R) (x y : R),
x < y -> {l : R | forall n, l <> f n & x <= l <= y}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='author'><strong>C-CoRN team</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/reals/RealCount.v#L362">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem reals_not_countable :
forall (f : nat -> IR), {x :IR | forall n : nat, x [#] (f n)}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='23' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#23'>23. Formula for Pythagorean Triples</a></h3>
<div>
<p class='author'><strong>David Delahaye</strong>
(in <a href="https://github.com/coq-contribs/fermat4/blob/master/Pythagorean.v">coq-contribs/fermat4</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma pytha_thm3 : forall a b c : Z,
is_pytha a b c -> Zodd a ->
exists p : Z, exists q : Z, exists m : Z,
a = m * (q * q - p * p) /\ b = 2 * m * (p * q) /\
c = m * (p * p + q * q) /\ m >= 0 /\
p >= 0 /\ q > 0 /\ p <= q /\ (rel_prime p q) /\
(distinct_parity p q).
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.11.2, failure with 8.12.2 (using git tag v8.10.0)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>Coq's axioms for real numbers</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, ProofPower, Rocq (constructive)</p></div>
<h3 id='24' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#24'>24. The Undecidability of the Continuum Hypothesis</a></h3>
<div>
<p class='existing'>Formalized in: Isabelle, Lean</p></div>
<h3 id='25' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#25'>25. Schroeder-Bernstein Theorem</a></h3>
<div>
<p class='author'><strong>Hugo Herbelin</strong>
(in <a href="https://github.com/coq-contribs/schroeder/blob/master/Schroeder.v">coq-contribs/schroeder</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Schroeder : A <=_card B -> B <=_card A -> A =_card B.
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.16 (using git tag v8.11.0)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>classic</li>
<li>Extensionality_Ensembles</li>
</ul>
<p class='author'><strong>Daniel Schepler</strong>
(in <a href="https://github.com/coq-community/topology/blob/master/theories/ZornsLemma/CSB.v">coq-community/topology</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem CSB (X Y : Type) (f : X -> Y) (g : Y -> X) :
injective f ->
injective g ->
exists h : X -> Y, bijective h.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>constructive_definite_description</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, Naproche, ProofPower, Rocq</p></div>
<h3 id='26' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#26'>26. Leibniz’s Series for Pi</a></h3>
<div>
<p class='author'><strong>Guillaume Allais</strong>
(in <a href="https://rocq-prover.org/doc/V9.0.0/stdlib/Stdlib.Reals.Ratan.html">coq's standard library</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma PI_2_aux : {z : R | 7 / 8 <= z <= 7 / 4 /\ - cos z = 0}.
Definition PI := 2 * proj1_sig PI_2_aux.
Definition PI_tg n := / INR (2 * n + 1).
Lemma exists_PI : {l:R | Un_cv (fun N => sum_f_R0 (tg_alt PI_tg) N) l}.
Definition Alt_PI : R := 4 * (let (a,_) := exist_PI in a).
Theorem Alt_PI_eq : Alt_PI = PI.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
</ul>
<p class='author'><strong>Luís Cruz-Filipe</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/transc/MoreArcTan.v#L553">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma ArcTan_one : ArcTan One[=]Pi[/]FourNZ.
Lemma arctan_series : forall c : IR,
forall (Hs :
fun_series_convergent_IR
(olor ([--]One) One)
(fun i =>
(([--]One)[^]i[/]nring (S (2*i))
[//] nringS_ap_zero _ (2*i)){**}Fid IR{^}(2*i+1)))
Hc,
FSeries_Sum Hs c Hc[=]ArcTan c.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, Rocq (constructive)</p></div>
<h3 id='27' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#27'>27. Sum of the Angles of a Triangle</a></h3>
<div>
<p class='author'><strong>Frédérique Guilhot</strong>
(in <a href="https://github.com/coq-community/HighSchoolGeometry/blob/master/theories/angles_vecteurs.v">coq-community/HighSchoolGeometry</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem somme_triangle :
forall A B C : PO,
A <> B :>PO ->
A <> C :>PO ->
B <> C :>PO ->
plus (cons_AV (vec A B) (vec A C))
(plus (cons_AV (vec B C) (vec B A)) (cons_AV (vec C A) (vec C B))) =
image_angle pi :>AV.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
<li>many geometry axioms</li>
</ul>
<p class='author'><strong>Boutry, Gries, Narboux</strong>
(in <a href="https://github.com/GeoCoq/GeoCoq/blob/master/Meta_theory/Parallel_postulates/parallel_postulates.v">https://github.com/GeoCoq/GeoCoq/blob/master/Meta_theory/Parallel_postulates/parallel_postulates.v</a>):
</p>
<p class='comment'>It is shown that the sum of angles is two rights is equivalent to other versions of the parallel postulate</p>
<pre class='statement'>Theorem equivalent_postulates_assuming_greenberg_s_axiom :
greenberg_s_axiom ->
all_equiv
(alternate_interior_angles_postulate::
alternative_playfair_s_postulate::
alternative_proclus_postulate::
alternative_strong_parallel_postulate::
consecutive_interior_angles_postulate::
euclid_5::
euclid_s_parallel_postulate::
existential_playfair_s_postulate::
existential_thales_postulate::
inverse_projection_postulate::
midpoint_converse_postulate::
perpendicular_transversal_postulate::
postulate_of_transitivity_of_parallelism::
playfair_s_postulate::
posidonius_postulate::
universal_posidonius_postulate::
postulate_of_existence_of_a_right_lambert_quadrilateral::
postulate_of_existence_of_a_right_saccheri_quadrilateral::
postulate_of_existence_of_a_triangle_whose_angles_sum_to_two_rights::
postulate_of_existence_of_similar_triangles::
postulate_of_parallelism_of_perpendicular_transversals::
postulate_of_right_lambert_quadrilaterals::
postulate_of_right_saccheri_quadrilaterals::
postulate_of_transitivity_of_parallelism::
proclus_postulate::
strong_parallel_postulate::
tarski_s_parallel_postulate::
thales_postulate::
thales_converse_postulate::
triangle_circumscription_principle::
triangle_postulate::
nil).
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>Eqdep.Eq_rect_eq.eq_rect_eq</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq</p></div>
<h3 id='28' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#28'>28. Pascal’s Hexagon Theorem</a></h3>
<div>
<p class='author'><strong>Magaud and Narboux</strong>
(in <a href="https://github.com/coq-contribs/projective-geometry/blob/master/Plane/hexamys.v">coq-contribs/projective-geometry</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Definition is_hexamy A B C D E F :=
(A<>B / A<>C / A<>D / A<>E / A<>F /
B<>C / B<>D / B<>E / B<>F /
C<>D / C<>E / C<>F /
D<>E / D<>F /
E<>F) /
let a:= inter (line B C) (line E F) in
let b:= inter (line C D) (line F A) in
let c:= inter (line A B) (line D E) in
Col a b c.
Lemma hexamy_prop: pappus_strong -> forall A B C D E F,
(line C D) <> (line A F) ->
(line B C) <> (line E F) ->
(line A B) <> (line D E) ->
(line A C) <> (line E F) ->
(line B F) <> (line C D) ->
is_hexamy A B C D E F -> is_hexamy B A C D E F.
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.13.2, failure with 8.14.1 (using git tag v8.10.0)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>many projective geometry axioms</li>
<li>including decidability of incidence</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Mizar, Rocq</p></div>
<h3 id='29' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#29'>29. Feuerbach’s Theorem</a></h3>
<div>
<p class='author'><strong>Benjamin Grégoire, Loïc Pottier, and Laurent Théry</strong>
(in <a href="https://github.com/coq/coq/blob/master/test-suite/success/Nsatz.v#L376">Coq's test suite</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma Feuerbach: forall A B C A1 B1 C1 O A2 B2 C2 O2:point,
forall r r2:R,
X A = 0 -> Y A = 0 -> X B = 1 -> Y B = 0->
middle A B C1 -> middle B C A1 -> middle C A B1 ->
distance2 O A1 = distance2 O B1 ->
distance2 O A1 = distance2 O C1 ->
collinear A B C2 -> orthogonal A B O2 C2 ->
collinear B C A2 -> orthogonal B C O2 A2 ->
collinear A C B2 -> orthogonal A C O2 B2 ->
distance2 O2 A2 = distance2 O2 B2 ->
distance2 O2 A2 = distance2 O2 C2 ->
r^2%Z = distance2 O A1 ->
r2^2%Z = distance2 O2 A2 ->
distance2 O O2 = (r + r2)^2%Z
\/ distance2 O O2 = (r - r2)^2%Z
\/ collinear A B C.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Rocq (constructive)</p></div>
<h3 id='30' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#30'>30. The Ballot Problem</a></h3>
<div>
<p class='author'><strong>Jean-Marie Madiot</strong>
(in <a href="https://github.com/coq-community/coq-100-theorems/blob/master/ballot.v">coq-community/coq-100-theorems</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem bertrand_ballot p q :
let l := filter (fun votes => count_votes votes "A" =? p) (picks (p + q) ["A"; "B"]) in
p >= q ->
(p + q) * List.length (filter (throughout (wins "A" "B")) l) =
(p - q) * List.length (filter (wins "A" "B") l).
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, ProofPower, Rocq</p></div>
<h3 id='31' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#31'>31. Ramsey’s Theorem</a></h3>
<div>
<p class='author'><strong>Frédéric Blanqui</strong>
(in <a href="https://github.com/fblanqui/color/blob/master/Util/Set_/Ramsey.v">color</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem ramsey A (W : set A) n (P : Pinf W) B : forall (C : Pf B)
(f : Pcard P (S n) -> elts C), Proper (Pcard_equiv ==> elts_eq) f ->
exists c (Q : Pinf P), forall X : Pcard Q (S n), f (Pcard_subset Q X) = c.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>CoLoR.Util.Logic.DepChoice.dep_choice</li>
<li>ClassicalEpsilon.constructive_indefinite_description</li>
<li>Description.constructive_definite_description</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, NuPRL (constructive), PVS, ProofPower, Rocq (constructive), nqthm</p></div>
<h3 id='32' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#32'>32. The Four Color Problem</a></h3>
<div>
<p class='author'><strong>Georges Gonthier, Benjamin Werner</strong>
(in <a href="https://github.com/coq-community/fourcolor/blob/master/theories/fourcolor.v">coq-community/fourcolor</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem four_color m : simple_map m -> colorable_with 4 m.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: Rocq</p></div>
<h3 id='33' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#33'>33. Fermat’s Last Theorem</a></h3>
<div>
<p class='existing'>Formalized in: (none)</p></div>
<h3 id='34' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#34'>34. Divergence of the Harmonic Series</a></h3>
<div>
<p class='author'><strong>Coqtail</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Reals/Rseries/Rseries_RiemannInt.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma harmonic_series_equiv : (sum_f_R0 (fun n ⇒ / (S n))) ~ (fun n ⇒ ln (S (S n))).
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='35' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#35'>35. Taylor’s Theorem</a></h3>
<div>
<p class='author'><strong>Luís Cruz-Filipe</strong>
(in <a href="https://github.com/coq-community/corn/blob/master/ftc/Taylor.v#L355">coq-community/corn</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem Taylor : forall e, Zero [<] e -> forall Hb',
{c : IR | Compact (Min_leEq_Max a b) c |
forall Hc, AbsIR (F b Hb'[-]Part _ _ Taylor_aux[-]deriv_Sn c Hc[*] (b[-]a)) [<=] e}.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, PVS, ProofPower, Rocq (constructive)</p></div>
<h3 id='36' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#36'>36. Brouwer Fixed Point Theorem</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Mizar</p></div>
<h3 id='37' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#37'>37. The Solution of a Cubic</a></h3>
<div>
<p class='author'><strong>Frédéric Chardard</strong>
(in <a href="https://github.com/coq-community/coq-100-theorems/blob/master/cardan_ferrari.v">coq-community/coq-100-theorems</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Definition Cardan_Tartaglia_formula:=fun (a1:C) (a2:C) (a3:C) (n:nat) =>
let s:=-a1/3 in
let p:=a2+2*s*a1+3*Cpow s 2 in
let q:=a3+a2*s+a1*Cpow s 2+Cpow s 3 in
let Delta:=(Cpow (q/2) 2)+(Cpow (p/3) 3) in
let alpha : C :=if(Ceq_dec p 0) then (RtoC 0) else (cubicroot (-(q/2)+Csqrt Delta)) in
let beta:=if(Ceq_dec p 0) then -cubicroot q else -(p/3)/alpha in
s+(alpha*Cpow CJ n+beta*Cpow CJ (n+n)).
Theorem Cardan_Tartaglia : forall a1 a2 a3 :C,
let u1:=(Cardan_Tartaglia_formula a1 a2 a3 0) in
let u2:=(Cardan_Tartaglia_formula a1 a2 a3 1) in
let u3:=(Cardan_Tartaglia_formula a1 a2 a3 2) in
forall u:C, (u-u1)*(u-u2)*(u-u3)=Cpow u 3+a1*Cpow u 2+a2*u+a3.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
<li>classic</li>
</ul>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar, Rocq</p></div>
<h3 id='38' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#38'>38. Arithmetic Mean/Geometric Mean</a></h3>
<div>
<p class='author'><strong>Jean-Marie Madiot</strong>
(in <a href="https://github.com/coq-community/coq-100-theorems/blob/master/mean.v">coq-community/coq-100-theorems</a>):
</p>
<p class='comment'>using forward-backward induction</p>
<pre class='statement'>Theorem geometric_arithmetic_mean (a : nat -> R) (n : nat) :
n <> O ->
(forall i, (i < n)%nat -> 0 <= a i) ->
prod n a <= (sum n a / INR n) ^ n
/\
(prod n a = (sum n a / INR n) ^ n -> forall i, (i < n)%nat -> a i = a O).
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_not_dec</li>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, ProofPower, Rocq</p></div>
<h3 id='39' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#39'>39. Solutions to Pell’s Equation</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, Metamath, Mizar</p></div>
<h3 id='40' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#40'>40. Minkowski’s Fundamental Theorem</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light, Isabelle, Lean, ProofPower</p></div>
<h3 id='41' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#41'>41. Puiseux’s Theorem</a></h3>
<div>
<p class='author'><strong>Daniel de Rauglaudre</strong>
(in <a href="https://github.com/roglo/puiseuxth/blob/master/coq/Puiseux.v">https://github.com/roglo/puiseuxth/blob/master/coq/Puiseux.v</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Theorem puiseux_series_algeb_closed : ∀ (α : Type) (R : ring α) (K : field R),
algeb_closed_field K
→ ∀ pol : polynomial (puiseux_series α),
degree (ps_zerop K) pol ≥ 1
→ ∃ s : puiseux_series α, (ps_pol_apply pol s = 0)%ps.
</pre>
<p class='repro'>Reproducibility: as of December 2022, build success with Coq 8.16</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>Puiseux_series.LPO (equivalent to sig_forall_dec)</li>
</ul>
<p class='existing'>Formalized in: Isabelle, Rocq</p></div>
<h3 id='42' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#42'>42. Sum of the Reciprocals of the Triangular Numbers</a></h3>
<div>
<p class='author'><strong>Coqtail</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Reals/Triangular.v">coq-community/coqtail-math</a>):
</p>
<p class='comment'></p>
<pre class='statement'>Lemma sum_reciprocal_triangular : Rser_cv (fun n => / triangle (S n)) 2.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, ProofPower, Rocq (constructive)</p></div>
<h3 id='43' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#43'>43. The Isoperimetric Theorem</a></h3>
<div>
<p class='existing'>Formalized in: HOL Light</p></div>
<h3 id='44' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#44'>44. The Binomial Theorem</a></h3>
<div>
<p class='author'><strong>The Coq development team</strong>
(in <a href="https://rocq-prover.org/doc/V9.0.0/stdlib/Stdlib.Reals.Binomial.html">coq's standard library</a>):
</p>
<p class='comment'>For the non-constructive type R</p>
<pre class='statement'>Lemma binomial :
forall (x y:R) (n:nat),
(x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: no</p>
<p class='axioms'>Axioms used:</p>
<ul class='axioms'>
<li>ClassicalDedekindReals.sig_forall_dec</li>
<li>functional_extensionality_dep</li>
</ul>
<p class='author'><strong>Laurent Thery & Jose C. Almeida</strong>
(in <a href="https://github.com/coq-contribs/rsa/blob/master/Binomials.v">coq-contribs/rsa</a>):
</p>
<p class='comment'>For the type nat</p>
<pre class='statement'>Theorem exp_Pascal :
forall a b n : nat,
power (a + b) n =
sum_nm n 0 (fun k : nat => binomial n k * (power a (n - k) * power b k)).
</pre>
<p class='repro'>Reproducibility: last checked December 2022 with Coq 8.16</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='author'><strong>Jean-Marie Madiot</strong>
(in <a href="https://github.com/coq-community/coqtail-math/blob/master/Hierarchy/Commutative_ring_binomial.v#L271">coq-community/coqtail-math</a>):
</p>
<p class='comment'>For any commutative ring X.</p>
<pre class='statement'>Definition newton_sum n a b : X :=
CRsum (fun k => (Nbinomial n k) ** (a ^ k) * (b ^ (n - k))) n.
Theorem Newton : forall n a b, (a + b) ^ n == newton_sum n a b.
</pre>
<p class='repro'>Reproducibility: Good (CI)</p>
<p class='axioms'>Constructive: yes</p>
<p class='axioms'>Axioms used: (none)</p>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar, NuPRL (constructive), ProofPower, Rocq (constructive)</p></div>
<h3 id='45' class='notformalizedyet'><a href='http://www.cs.ru.nl/~freek/100/#45'>45. The Partition Theorem</a></h3>
<div>
<p class='existing'>Formalized in: ACL2, HOL Light, Isabelle, Lean, Metamath, Mizar</p></div>
<h3 id='46' class='formalized'><a href='http://www.cs.ru.nl/~freek/100/#46'>46. The Solution of the General Quartic Equation</a></h3>
<div>
<p class='author'><strong>Frédéric Chardard</strong>
(in <a href="https://github.com/coq-community/coq-100-theorems/blob/master/cardan_ferrari.v">coq-community/coq-100-theorems</a>):
</p>
<p class='comment'>For the non-constructive type R</p>
<pre class='statement'>Theorem Ferrari_formula: forall (a:C) (b:C) (c:C) (d:C),
let s:=-a/4 in
let p:= b+3*s*a+6*Cpow s 2 in
let q:= c+2*b*s+3*a*Cpow s 2+4*Cpow s 3 in
let r:= d+c*s+b*Cpow s 2+a*Cpow s 3+Cpow s 4 in
let lambda:=Cardan_Tartaglia_formula (-p/2) (-r) (r*p/2-/8*Cpow q 2) 0 in
let A:=Csqrt(2*lambda-p) in
let cond:=(Ceq_dec (2*lambda) p) in