Mathematical Components

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Mathematical Components

• Georges Gonthier
• Y. Bertot, F. Garillot, A. Mahboubi, S. Ould
  Biha,
• I. Pasca, L. Rideau, L. Théry, B. Werner

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Why use a computer for math?

• To prove calculations
• To improve refereeing
• To explore structure

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Building computer proofs
Proof text
Proof script
(ssreflect)
tactic interpreter
Typed ?-calculus
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Finite Group Theory

• Theory of composable, reversible operators
• Puzzles
• Solving polynomials x5 3x2 7
• Gauge theory
• Cristallography
• Cryptography

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Simple Group Classification

• Theorem (Jordan-Hölder)
• Every finite group factors uniquely into simple
  groups.
• Theorem (Classification)
• All finite simple groups belong to 12 general
  classes, except for 26 sporadic exceptions.

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The Finite Group Challenge
Frobenius groups Thompson factorisation character
theory linear representation Galois theory linear
algebra polynomials
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Interfaces and objects
bool if b then
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Big Operators
\bigcap_H lt G \atop H \rm\ maximal H
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Example determinants

• Leibnitz formula
• Cauchy theorem AB A B

(?j Ai,j Bj,is)
(AB)i,is
?i Bi?,is
Bi?,is
? ?i Ai,i? ? (-1)s ?j Bj,j?-1s
i j?-1
?
s?Sn
??Sn
??Sn
A B
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Interfacing big ops
bool
Equality
nat
Finite
I_n
seq
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A proof
\begineqnarray \det AB \sum_\sigma \in
S_n (-1)\sigma \prod_i
\left(\sum_j A_ij B_j\sigma(i)\right)\\
\sum_\phi \ 1, n \rightarrow 1,
n \sum_\sigma \in S_n
(-1)\sigma \prod_i A_i\phi(i)
B_\phi(i)\sigma(i)\\ \sum_\phi
\notin S_n \sum_\sigma \in S_n
(-1)\sigma \prod_i A_i\phi(i)
B_\phi(i)\sigma(i)
\sum_\phi \in S_n \sum_\sigma \in S_n
(-1)\sigma \prod_i A_i\phi(i)
B_\phi(i)\sigma(i)\\ \sum_\phi
\notin S_n \left(\prod_i A_i\phi(i)\right)?
\sum_\sigma \in S_n
(-1)\sigma \prod_i B_\phi(i)\sigma(i)
\\ \sum_\phi \in S_n
(-1)\phi \left(\prod_i
A_i\phi(i)\right)?
\sum_\sigma \in S_n
(-1)\phi-1\sigma
\prod_k B_k\sigma(\phi-1(k))\\
\sum_\phi \notin S_n \left(\prod_i
A_i\phi(i)\right)? \det
\left(B_\phi(i)j\right)_ij
(\det A)\sum_\tau \in S_n(-1)\tau \prod_k
B_k\tau(k)\\ 0 (\det
A)(\det B)\\ \endeqnarray The first step swaps
the iterated product of the Leibnitz formula
with the sum in the general term of the matrix
product, generating a sum over all functions from
indices to indices. This is split into a sum
over non-injective functions and a sum over
permutations. The former is rearranged into a
weighted sum of determinants of matrices with
repeated rows, while the latter is reindexed,
using the group properties of permutations, to
become the desired product of determinants.
Lemma determinantM forall n (A B M_(n)), \det
(A m B) \det A \det B. Proof. movegt n A B
rewrite big_distrl /. pose AB (f ffun _) (s
'S_n) i A i (f i) B (f i) (s
i). transitivity (\sum_f \sum_(s 'S_n) (-1)
s \prod_i AB f s i). rewrite exchange_big
apply eq_bigr gt / s _. rewrite -big_distrr
/ congr (_ _). pose F i j A i j B j (s
i) rewrite -(bigA_distr_bigA F) /. by apply
eq_bigr gt x _ rewrite mxK. rewrite (bigID (fun
f ffun _ gt injectiveb f)) / addrC big1
?simp gt f Uf. rewrite (reindex (fun s gt
pval s)) last first. have s0 'S_n 1g
pose uf (f F_(n)) uniq (val f). exists
(insubd s0) gt / f Uf first apply val_inj
exact insubdK. apply eq_big gt / ss _
rewrite ?(valP s) // big_distrr /. rewrite
(reindex (mulg s)) last first. by exists
(mulg s-1) gt t _ rewrite ?mulKVg ?mulKg.
apply eq_bigr gt t _ rewrite big_split / mulrA
mulrCA mulrA mulrCA mulrA. rewrite -signr_addb
odd_permM !pvalE congr (_ _). rewrite
(reindex s-1) last first. by exists (s _
-gt _) gt i _ rewrite ?permK ?permKV. by apply
eq_bigr gt i _ rewrite permM permKV ?eqxx //
-3i(permKV s). transitivity (\det
(\matrix_(i, j) B (f i) j) \prod_i A i (f i)).
rewrite mulrC big_distrr / apply eq_bigr gt s
_. rewrite mulrCA big_split // congr (_ (_
_)). by apply eq_bigr gt x _ rewrite
mxK. case/injectivePn Uf gt i1 i2 Di12
Ef12. by rewrite (alternate_determinant Di12)
?simp // gt j rewrite !mxK Ef12. Qed.
rewrite (reindex (mulg s)) last first. by
exists (mulg s-1) gt t _ rewrite ?mulKVg
?mulKg.
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Interfacing groups
bool
Equality
Morphism
Finite
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Some conclusions

• Type theory provides a rich language for
  expressing the intended use of theories.
• Investing in theory infrastructure is most
  productive.
• Computer mathematics is becoming a reality

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