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phanerozoic
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Coq-Infotheo

statement stringlengths 1 1.01k	proof stringlengths 0 10k	type stringclasses 20 values	symbolic_name stringlengths 1 38	library stringclasses 7 values	filename stringclasses 87 values	imports listlengths 2 49	deps listlengths 0 59	docstring stringclasses 295 values	source_url stringclasses 1 value	commit stringclasses 1 value
location_polynomial_points	:= \row_i \prod_(j < n \| j != i) (a ``_ i - a ``_ j).	Definition	location_polynomial_points	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]	the values of the location polynomial at points a``_i, they determine uniquely the location polynomial of size n (i.e., deg <= n.-1)	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
b	:= \row_(i < n) (((location_polynomial_points a) ``_ i)^-1 * g.[a ``_ i]).	Let	b	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "location_polynomial_points" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
GRS_PCM_polynomial	:= @GRS.PCM _ F a b r.	Definition	GRS_PCM_polynomial	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "PCM" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
ext_inj : {rmorphism F0 -> F1}	:= [the {rmorphism F0 -> F1} of @GRing.in_alg _ _].	Definition	ext_inj	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
ext_inj_tmp : {rmorphism F0 -> (FinFieldExtType F1)}	:= ext_inj.	Definition	ext_inj_tmp	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "ext_inj" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
ext_inj_rV : 'rV[F0]_n -> 'rV[F1]_n	:= @map_mx _ _ ext_inj 1 n.	Definition	ext_inj_rV	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "ext_inj" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
u	:= u'.+1.	Let	u	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
primep : prime p.		Hypothesis	primep	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
Fq : finFieldType	:= GF u primep.	Let	Fq	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "GF", "primep" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
q	:= p ^ u.	Let	q	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
p_char : p \in [pchar Fq].	Proof. apply: char_GFqm. Qed.	Let	p_char	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "Fq", "char_GFqm", "pchar" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
m	:= m'.+1.	Let	m	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "m'" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
card_Fqm : #\| FinFieldExtType Fqm \| = q ^ m.		Hypothesis	card_Fqm	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
alternant_PCM : 'M_(k, n)	:= @GRS_PCM_polynomial n (FinFieldExtType Fqm) a g k.	Definition	alternant_PCM	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "GRS_PCM_polynomial" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
alternant_code	:= Rcode.t (@ext_inj_tmp Fq Fqm) (kernel alternant_PCM).	Definition	alternant_code	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "Fq", "alternant_PCM", "ext_inj_tmp", "kernel" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
goppa_code_condition	:= size g = (n - k).+1.	Definition	goppa_code_condition	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]	Goppa codes are a special case of alternant codes	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
n : nat	:= (q^m).-1.	Let	n	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]	we are talking about narrow-sense Goppa codes	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
e_prim : n.-primitive_root e.		Hypothesis	e_prim	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
a : 'rV[Fqm]_n	:= rVexp e n.	Let	a	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "rVexp" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
g : {poly (FinFieldExtType Fqm)}	:= 'X^(n - t).	Let	g	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[]	we have to instantiate Goppa codes with a monomial to recover BCH codes	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
goppa_code_condition_check : goppa_code_condition n g t.	Proof. by rewrite /goppa_code_condition size_polyXn. Qed.	Let	goppa_code_condition_check	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "goppa_code_condition" ]	from the Goppa code condition, we have only one choice for its degree	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
narrow_sense_BCH_are_Goppa : @BCH.PCM (FinFieldExtType _) _ a t = @alternant_PCM _ u' primep Fqm _ a g t(?).	Proof. rewrite /BCH.code /alternant_code. Abort.	Lemma	narrow_sense_BCH_are_Goppa	ecc_classic	ecc_classic/alternant.v	[ "mathcomp", "all_ssreflect", "ssralg", "finalg", "poly", "polydiv", "cyclic", "perm", "matrix", "mxpoly", "vector", "mxalgebra", "zmodp", "finfield", "falgebra", "fieldext", "ssr_ext", "ssralg_ext", "linearcode", "dft", "poly_decoding", "grs", "bch", "GRing.Theory" ]	[ "PCM", "alternant_PCM", "alternant_code", "code", "primep" ]	wip	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
PCM : 'M_(t, n)	:= \matrix_(i < t, j < n) (a ``_ j) ^+ i.*2.+1.	Definition	PCM	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
PCM_alt : 'M[F]_(t.*2, n)	:= \matrix_(i < t.*2, j < n) (a ``_ j) ^+ i.+1.	Definition	PCM_alt	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
PCM_alt_GRS : PCM_alt = GRS.PCM a a t.*2.	Proof. apply/matrixP => i j. rewrite !mxE (bigD1 j) //= !mxE eqxx mulr1n exprS mulrC. rewrite (eq_bigr (fun=> 0)) ?big_const ?iter_addr0 ?addr0 // => k kj. by rewrite !mxE (negbTE kj) mulr0n mulr0. Qed.	Lemma	PCM_alt_GRS	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "PCM", "PCM_alt", "iter_addr0", "mulrC" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
F	:= GF2 m.	Let	F	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "GF2" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_PCM_altP2 (x : 'rV[F]_n) : (forall i : 'I_t.2, \sum_j x ``_ j a ``_ j ^+ i.+1 = 0) -> (forall i : 'I_t,\sum_j x ``_ j * a ``_ j ^+ i.*2.+1 = 0).	Proof. move=> H i. have @j : 'I_t.2 by refine (@Ordinal _ i.2 _); rewrite -!muln2 ltn_pmul2r. rewrite -[RHS](H j); by apply: eq_bigr => /= k _. Qed.	Lemma	BCH_PCM_altP2	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_PCM_altP1 (x : 'rV['F_2]_n) : (forall i : 'I_t, \sum_j GF2_of_F2 x ``_ j * a ``_ j ^+ i.2.+1 = 0) -> (forall i : 'I_t.2, \sum_j GF2_of_F2 x ``_ j * a ``_ j ^+ i.+1 = 0).	Proof. move=> H [i]. elim: i {-2}i (leqnn i) => [\|i IH j ji i1]. move=> i; rewrite leqn0 => /eqP -> i0. destruct t. exfalso; by rewrite -muln2 mul0n ltnn in i0. by rewrite -[RHS](H ord0); apply: eq_bigr. case/boolP : (odd j) => [odd_j\|even_j]; last first. have j2t : j./2 < t by rewrite -divn2 ltn_divLR // m...	Lemma	BCH_PCM_altP1	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "GF2_of_F2", "char_GFqm", "expr2_char2", "mulrC", "sum_sqr" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
PCM_altP (x : 'rV_n) : (syndrome (BCH.PCM a t) (F2_to_GF2 m x) == 0) = (syndrome (BCH.PCM_alt a t) (F2_to_GF2 m x) == 0).	Proof. apply/idP/idP. - rewrite /BCH.PCM /BCH.PCM_alt /syndrome => H. suff H' : forall j : 'I_t.2, \sum_j0 (GF2_of_F2 (x ``_ j0) (a ``_ j0) ^+ j.+1) = 0. apply/eqP/rowP => j; rewrite !mxE. rewrite -[RHS](H' j). apply: eq_bigr => /= i _. by rewrite !mxE mulrC. move=> j. apply: BCH_PCM_altP1 => i...	Lemma	PCM_altP	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_PCM_altP1", "F2_to_GF2", "GF2_of_F2", "PCM", "PCM_alt", "mulrC", "syndrome" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
code (a : 'rV_n) t	:= Rcode.t (@GF2_of_F2 m) (kernel (PCM a t)).	Definition	code	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "GF2_of_F2", "PCM", "kernel" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
n	:= n'.+1.	Let	n	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
F : fieldType	:= GF2 m.	Let	F	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "GF2" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
syndromep a y t	:= syndromep a (F2_to_GF2 m y) t.*2.	Definition	syndromep	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_to_GF2" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"'\BCHsynp_(' a , e , t )"	:= (BCH.syndromep a e t) : bch_scope.	Notation	'\BCHsynp_(' a , e , t )	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "syndromep" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
tn : t.*2 < n.		Hypothesis	tn	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
fdcoor_syndrome_coord (y : 'rV['F_2]_n) i (it : i < t.*2) : fdcoor (rVexp a n) (F2_to_GF2 m y) (inord i.+1) = GRS.syndrome_coord (rVexp a n) (rVexp a n) i (F2_to_GF2 m y).	Proof. rewrite /fdcoor /GRS.syndrome_coord horner_poly; apply: eq_bigr => j _. rewrite insubT // => jn. rewrite !mxE -mulrA; congr (GF2_of_F2 (y _ _) * _); first by apply: val_inj. by rewrite -exprS -exprM mulnC exprM inordK // ltnS (leq_trans it). Qed.	Lemma	fdcoor_syndrome_coord	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_to_GF2", "GF2_of_F2", "fdcoor", "mulrA", "rVexp", "syndrome_coord" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_syndromep_is_GRS_syndromep y : \BCHsynp_(rVexp a n, y, t) = GRS.syndromep (rVexp a n) (rVexp a n) t.*2 (F2_to_GF2 m y).	Proof. apply/polyP => i. by rewrite !coef_poly; case: ifPn => // it; rewrite fdcoor_syndrome_coord. Qed.	Lemma	BCH_syndromep_is_GRS_syndromep	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_to_GF2", "fdcoor_syndrome_coord", "rVexp", "syndromep" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
tn : t <= n.-1./2.		Hypothesis	tn	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_codebook (c : 'rV['F_2]_n) : (c \in BCHcode) = [forall j : 'I_t, (map_mx (@GF2_of_F2 m) c) ^`_(rVexp a n, inord j.*2.+1) == 0].	Proof. apply/idP/idP. - case: BCHcode => code condition /= /condition. rewrite mem_kernel_syndrome0 -trmx0 => /eqP/trmx_inj/matrixP Hc. apply/forallP => /= i; apply/eqP; rewrite /fdcoor horner_poly. move: (Hc i ord0); rewrite !mxE => H1; rewrite -[RHS]H1. apply/eq_bigr => /= j _; rewrite insubT // => jn. rewr...	Lemma	BCH_codebook	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "GF2_of_F2", "Hc", "code", "fdcoor", "mem_kernel_syndrome0", "mulrC", "rVexp", "tn" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
t	:= t'.+1.	Let	t	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_syndrome_synp y : t.*2 < n -> (syndrome (BCH.PCM (rVexp a n) t) (F2_to_GF2 m y) == 0) = (\BCHsynp_(rVexp a n, y, t) == 0).	Proof. move=> tn. apply/idP/idP. - move/eqP/eqP. rewrite BCH.PCM_altP // /BCH.PCM_alt. move/eqP => K. rewrite /BCH.syndromep /syndromep. rewrite poly_def (eq_bigr (fun=> 0)) ?big_const ?iter_addr0 // => /= i _. apply/eqP; rewrite scaler_eq0. rewrite fdcoorE -size_poly_eq0 size_polyXn /= orbF. move/rowP : ...	Lemma	BCH_syndrome_synp	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_to_GF2", "PCM", "PCM_alt", "PCM_altP", "fdcoorE", "iter_addr0", "mulr1", "mulrC", "rVexp", "syndrome", "syndromep", "tn" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_det_mlinear t r' (f : 'I_r'.+1 -> 'I_n) (rt : r' < t.2) : let B := \matrix_(i, j) BCH.PCM_alt a t (widen_ord rt i) (f j) in let V := Vandermonde r'.+1 (row 0 B) in \det B = \prod_(i < r'.+1) BCH.PCM_alt a t (widen_ord rt 0) (f i) \det V.	Proof. move=> B V. set h := Vandermonde r'.+1 (row 0 B). set g := fun i => BCH.PCM_alt a t (widen_ord (rt) 0) (f i). transitivity (\det (\matrix_(i, j) (h i j * g j))). congr (\det _). apply/matrixP => i j. by rewrite !mxE /h /g /BCH.PCM_alt !mxE -!exprD /= -exprM mul1n addn1. rewrite det_mlinear; congr (_ * _). ...	Lemma	BCH_det_mlinear	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "PCM_alt", "det_mlinear" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
a_neq0 : distinct_non_zero a.		Hypothesis	a_neq0	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "distinct_non_zero" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
det_B_neq0 t r f (Hr' : r.+1 < t.*2) (Hinj : injective f) : let B := \matrix_(i < r.+1, j < r.+1) BCH.PCM_alt a t (widen_ord (ltnW Hr') i) (f j) in \det B != 0.	Proof. rewrite /=; set B := \matrix_(_, _) _. rewrite BCH_det_mlinear det_Vandermonde mulf_neq0 //. - rewrite prodf_seq_neq0 /=; apply/allP => /= j _. by rewrite !mxE /= expr1 (proj2 a_neq0). - rewrite prodf_seq_neq0 /=; apply/allP => /= j _. rewrite prodf_seq_neq0 /=; apply/allP => /= k _. apply/implyP => jk; re...	Lemma	det_B_neq0	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_det_mlinear", "PCM_alt", "a_neq0" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_min_dist1 t x (x0 : x != 0) (t0 : 0 < t) (C : BCH.code a t) : x \in C -> t < wH x.	Proof. move/(proj1 (Rcode.P C _)). rewrite mem_kernel_syndrome0 BCH.PCM_altP // => /eqP H. (* there are r <= t columns in the PCM s.t. their linear comb. w.r.t. is 0 *) rewrite leqNgt; apply/negP => abs. have /eqP : #\| wH_supp x \| = (wH x).-1.+1. by rewrite card_wH_supp prednK // lt0n wH_eq0. rewrite cardE => supp_w...	Lemma	BCH_min_dist1	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_eq1", "GF2_of_F2", "Hf", "PCM_alt", "PCM_altP", "card_wH_supp", "code", "det_B_neq0", "enum", "f'", "iter_addr0", "max_wH", "mem_kernel_syndrome0", "mem_wH_supp", "mulrC", "n0", "t0", "tmp", "wH", "wH_eq0", "wH_supp" ]	parity-check matrix of a binary (n, k) code capable ot correcting all error patterns of weight <= t with dimension k >= n - m * t (see [McEliece 2002], thm 9.1)	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_erreval	:= erreval (const_mx 1) a.	Definition	BCH_erreval	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "erreval" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"'\BCHomega_(' a , e )"	:= (BCH_erreval a e) : bch_scope.	Notation	'\BCHomega_(' a , e )	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_erreval" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_key_equation_old (y : 'rV[F]_n) : a ^+ n = 1 -> \sigma_(rVexp a n, y) * rVpoly (dft (rVexp a n) n y) = \BCHomega_(rVexp a n, y) * (1 - 'X^n).	Proof. move=> an1. rewrite dftE big_distrr /=. under eq_bigr. move=> i ie. rewrite -scalerAr. rewrite (_ : \sigma_(rVexp a n, y) = \sigma_(rVexp a n, y, i) * (1 - ((rVexp a n) ``_ i) : 'X)); last first. rewrite /errloc (bigD1 i) //= mulrC; congr (_ _). by apply: eq_bigl => ij; rewrite in_setD1 andb...	Lemma	BCH_key_equation_old	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_erreval", "a0", "an1", "dft", "dftE", "errloc", "mul1r", "mulr1", "mulrA", "mulrC", "mulrDl", "rVexp", "supp" ]	see [McEliece 2002], thm 9.4	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
H	:= BCH.PCM (rVexp a n) t.	Let	H	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "PCM", "rVexp" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
a_neq0 : a != 0.		Hypothesis	a_neq0	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_mod (y : 'rV[F]_n) : {poly F}	:= \sum_(i in supp y) y ``_ i : (\prod_(j in supp y :\ i) (1 - a ^+ j : 'X) * - (a ^+ (t.2.+1 i))%:P).	Definition	BCH_mod	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "supp" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_mod_is_GRS_mod y : BCH_mod y = GRS_mod y (rVexp a n) (rVexp a n) t.*2.	Proof. rewrite /BCH_mod /GRS_mod; apply/eq_bigr => i iy. rewrite !mulrN scalerN; congr (- _). rewrite -!scalerAl; congr (_ : _). rewrite mxE -mulrA; congr (_ _). apply/eq_big. by move=> j; rewrite in_setD1 andbC. move=> j _; by rewrite mxE. by rewrite -polyCM -exprSr -exprM mulnC. Qed.	Lemma	BCH_mod_is_GRS_mod	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_mod", "GRS_mod", "mulrA", "rVexp" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
(tn : t.*2 < n).		Hypothesis	tn	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_key_equation y : \sigma_(rVexp a n, F2_to_GF2 m y) * \BCHsynp_(rVexp a n, y, t) = \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m y)) + BCH_mod (F2_to_GF2 m y) * 'X^(t.*2).	Proof. move: (@GRS_key_equation F n (F2_to_GF2 m y) (rVexp a n) (rVexp a n) t.2) => H0. rewrite BCH_syndromep_is_GRS_syndromep // H0; congr (_ + _ _). rewrite /Omega /BCH_erreval; apply/polyP => i. rewrite !coef_sum supp_twisted //. apply: eq_bigr => /= j jy. by rewrite !mxE mulr1. by rewrite -BCH_mod_is_GRS...	Lemma	BCH_key_equation	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_erreval", "BCH_mod", "BCH_mod_is_GRS_mod", "BCH_syndromep_is_GRS_syndromep", "F2_to_GF2", "GRS_key_equation", "Omega", "mulr1", "rVexp", "supp_twisted", "twisted" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
errloc_twisted (y : 'rV[F]_n) : errloc (rVexp a n) (supp y) = errloc (rVexp a n) (supp (twisted a y)).	Proof. by rewrite /errloc supp_twisted. Qed.	Lemma	errloc_twisted	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "errloc", "rVexp", "supp", "supp_twisted", "twisted" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_erreval_twisted (e : 'rV_n) : #\|supp (F2_to_GF2 m e)\| <= t -> size (\omega_(const_mx 1, rVexp a n, twisted a (F2_to_GF2 m e))) <= t.	Proof. move=> et. have et' : #\|supp (twisted a (F2_to_GF2 m e))\| <= t by rewrite supp_twisted. by apply: (size_erreval (rVexp a n) (Errvec et') (const_mx 1)). Qed.	Lemma	size_erreval_twisted	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "F2_to_GF2", "rVexp", "size_erreval", "supp", "supp_twisted", "twisted" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"'v'"	:= (Euclid.v).	Notation	'v'	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_err y : {poly 'F_2}	:= let r0 : {poly F} := 'X^(t.2) in let r1 := \BCHsynp_(rVexp a n, y, t) in let vstop := v r0 r1 (stop t r0 r1) in let s := vstop.[0]^-1 : vstop in \poly_(i < n) (if s.[a^- i] == 0 then 1 else 0).	Definition	BCH_err	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "rVexp", "stop" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_repair : repairT 'F_2 'F_2 n	:= [ffun y => if \BCHsynp_(rVexp a n, y, t) == 0 then Some y else let ret := y + poly_rV (BCH_err y) in if \BCHsynp_(rVexp a n, ret, t) == 0 then Some ret else None].	Definition	BCH_repair	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_err", "rVexp", "repairT" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_err_is_correct (a_not_uroot : not_uroot_on a n) l (e y : 'rV_n) : let r0 := 'X^(t.2) : {poly F} in let r1 := \BCHsynp_(rVexp a n, y, t) in let vj := Euclid.v r0 r1 (stop t r0 r1) in l <> 0 -> vj = l : \sigma_(rVexp a n, F2_to_GF2 m e) -> e = poly_rV (BCH_err y).	Proof. have H1 := distinct_non_zero_rVexp a_neq0 a_not_uroot. move=> r0 r1 vj /eqP l0 Hvj; apply/rowP => i. set r_ := \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m e)). rewrite !mxE coef_poly ltn_ord; case: ifPn. rewrite -/r0 -/r1 -/r_ -/vj Hvj. rewrite !hornerZ !mulf_eq0 invrM; last 2 first. by rewrite unitfE. ...	Lemma	BCH_err_is_correct	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_err", "F2_eq1", "F2_to_GF2", "a_neq0", "distinct_non_zero_rVexp", "errloc_zero", "horner_errloc_0", "l0", "not_uroot_on", "rVexp", "stop", "supp", "supp_F2_to_GF2", "twisted" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
BCH_repair_is_correct (C : BCH.code (rVexp a n) t) : not_uroot_on a n -> t.-BDD (C, BCH_repair).	Proof. move=> a_not_uroot c e. set y := c + e. set r0 : {poly F} := 'X^(t.*2). set r1 := \BCHsynp_(rVexp a n, y, t). set vj := Euclid.v r0 r1 (stop t r0 r1). move=> Hc et. set r : {poly F} := \BCHomega_(rVexp a n, twisted a (F2_to_GF2 m e)). have H1 := distinct_non_zero_rVexp a_neq0 a_not_uroot. rewrite /BCH_repair. ha...	Lemma	BCH_repair_is_correct	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_err_is_correct", "BCH_key_equation", "BCH_min_dist1", "BCH_mod", "BCH_repair", "BCH_syndrome_synp", "F2_addmx", "F2_to_GF2", "Hc", "a_neq0", "aclosed", "code", "coprime_errloc_erreval", "distinct_non_zero_rVexp", "e0", "errloc_neq0", "errloc_twisted", "l0", "mem_kernel_syndr...		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcsP_BCH_cyclic (C : BCH.code (rVexp a n) t) : a ^+ n = 1 -> rcsP [set cw in C].	Proof. move=> a1 x. rewrite inE (BCH_codebook C tn) => /forallP /= Hx. rewrite inE (BCH_codebook C tn); apply/forallP => /= i. rewrite map_mx_rcs rcs_rcs_poly /rcs_poly. set x' := map_mx _ _. have [M HM] : exists M, `[ 'X * rVpoly x' ]_ n = 'X * rVpoly x' + M * ('X^n - 1). exists (- ('X * rVpoly x') %/ ('X^n - 1)). ...	Lemma	rcsP_BCH_cyclic	ecc_classic	ecc_classic/bch.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "fieldext", "finfield", "ssralg_ext", "hamming", "linearcode", "decoding", "cyclic_code", "poly_decoding", "dft", "euclid", "grs", "f2", "GRing.Theory...	[ "BCH_codebook", "HM", "Hx", "code", "fdcoor", "map_mx_rcs", "rVexp", "rcsP", "rcs_poly", "rcs_rcs_poly", "tn" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_perm_ffun n : {ffun 'I_n.+1 -> 'I_n.+1}	:= [ffun x : 'I_n.+1 => if x == ord0 then ord_max else inord x.-1].	Definition	rcs_perm_ffun	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_perm_ffun_injectiveb n : injectiveb (@rcs_perm_ffun n).	Proof. apply/injectiveP => x y. rewrite /rcs_perm_ffun 2!ffunE. case: ifPn => [/eqP ->\|x0]. case: ifPn => [/eqP -> //\|y0]. move/eqP; rewrite -val_eqE /= => /eqP n0y. exfalso; move: (ltn_ord y). rewrite ltnNge => /negP; apply. rewrite [in X in X < _]n0y /= inordK //; first by rewrite prednK // lt0n. by apply...	Lemma	rcs_perm_ffun_injectiveb	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs_perm_ffun" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_perm n : {perm 'I_n}	:= if n is n.+1 then Perm (rcs_perm_ffun_injectiveb n) else 1.	Definition	rcs_perm	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs_perm_ffun_injectiveb" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs (R : idomainType) n (x : 'rV[R]_n)	:= col_perm (rcs_perm n) x.	Definition	rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs_perm" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : size (rVpoly (rcs x)) < size ('X^(n.+1) - 1%:P : {poly R}).	Proof. by rewrite size_XnsubC // ltnS /rVpoly size_poly. Qed.	Lemma	size_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
map_mx_rcs (R0 R1 : idomainType) n (x : 'rV[R0]_n.+1) (f : R0 -> R1) : map_mx f (rcs x) = rcs (map_mx f x).	Proof. by rewrite /rcs map_col_perm. Qed.	Lemma	map_mx_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs' (R : idomainType) n (x : 'rV[R]_n.+1)	:= \row_(i < n.+1) (if i == ord0 then x ord0 (inord n) else x ord0 (inord i.-1)).	Definition	rcs'	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs'0 (R : idomainType) n (x : 'rV[R]_n.+1) : (rcs' x == 0) = (x == 0).	Proof. apply/idP/idP => [\|/eqP ->]. move/eqP/rowP => H; apply/eqP/rowP => i; rewrite !mxE. have ni : i.+1 %% n.+1 < n.+1 by apply: ltn_pmod. move: (H (Ordinal ni)); rewrite !mxE -val_eqE /=. case: ifPn => [in' Hx\|in' Hx]. have ? : n = i. case/dvdnP : in' => -[//\|[\|k abs]] /=; last first. exfal...	Lemma	rcs'0	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "Hx", "rcs'" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs'_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : rcs' x = rcs x.	Proof. rewrite /rcs' /rcs; apply/rowP => i; rewrite !mxE. case: ifPn => [/eqP -> \|i0]. congr (x _ _). rewrite /rcs_perm/= unlock/= ffunE eqxx. apply/val_inj => /=. by rewrite inordK. congr (x _ _); apply: val_inj => /=. rewrite unlock ffunE /= inordK. by rewrite (negPf i0) inordK // (ltn_trans _ (ltn_ord i)) ...	Lemma	rcs'_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs", "rcs'", "rcs_perm" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
sub_XrVpoly_rcs (R : idomainType) n (x : 'rV[R]_n.+1) : 'X * rVpoly x - rVpoly (rcs x) = (x 0 (inord n))%:P * ('X^(n.+1) - 1).	Proof. apply/polyP => i. rewrite coef_add_poly coefXM coefCM coef_add_poly coefXn 2!coef_opp_poly coefC. case: ifPn => [/eqP ->\|i0]; last rewrite subr0. rewrite 2!add0r mulrN1 coef_rVpoly /=. case: insubP => //= j _ j0. rewrite mxE unlock ffunE /= -val_eqE j0 /= j0 eqxx; congr (- x _ _). by apply: val_inj => /=...	Lemma	sub_XrVpoly_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "mulr1", "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_poly (R : idomainType) (p : {poly R}) n	:= ('X * p) %% ('X^n - 1).	Definition	rcs_poly	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_rcs_poly (R : idomainType) n (x : {poly R}) (xn : size x <= n) : 0 < n -> size (rcs_poly x n) <= n.	Proof. move=> n0. rewrite /rcs_poly. set xn1 := _ - _. apply: (@leq_trans (size xn1).-1); last by rewrite /xn1 size_XnsubC. rewrite -ltnS prednK; last by rewrite size_XnsubC. have : xn1 != 0 by apply/monic_neq0/monicXnsubC. by move/ltn_modpN0; exact. Qed.	Lemma	size_rcs_poly	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "n0", "rcs_poly" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_rcs_poly_old (R : idomainType) n (x : 'rV[R]_n) : size (rcs_poly (rVpoly x) n) <= n.	Proof. destruct n as [\|n']. rewrite /rcs_poly subrr modp0. have -> : x = 0 by apply/matrixP => i []. by rewrite linear0 mulr0 size_poly0. by rewrite size_rcs_poly // size_poly. Qed.	Lemma	size_rcs_poly_old	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs_poly", "size_rcs_poly" ]	TODO: not used?	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_rcs_poly (F : fieldType) n0 (x : 'rV[F]_n0) : rcs x = poly_rV (rcs_poly (rVpoly x) n0).	Proof. destruct n0 as [\|n0]. rewrite /rcs_poly (_ : 'X^0 = 1); last first. by apply/polyP => i; rewrite coefXn. rewrite subrr modp0 (_ : rVpoly x = 0); last first. apply/polyP => i. rewrite coef_rVpoly. by case: insubP => //; rewrite coef0. rewrite mulr0 /rcs /= col_perm1. apply/rowP. case; by...	Lemma	rcs_rcs_poly	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "n0", "rcs", "rcs'", "rcs'_rcs", "rcs_poly", "size_rcs", "sub_XrVpoly_rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcs_poly_rcs (F : fieldType) n0 (x : {poly F}) (xn0 : size x <= n0.+1) : rcs_poly x n0.+1 = rVpoly (@rcs _ n0.+1 (poly_rV x)).	Proof. by rewrite rcs_rcs_poly poly_rV_K // poly_rV_K // size_rcs_poly. Qed.	Lemma	rcs_poly_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "n0", "rcs", "rcs_poly", "rcs_rcs_poly", "size_rcs_poly" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
rcsP (F: finFieldType) n (C : {set 'rV[F]_n})	:= forall x, x \in C -> rcs x \in C.	Definition	rcsP	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"'`[' P ']_' n"	:= (P %% ('X^n - 1)) : cyclic_code_scope.	Notation	'`[' P ']_' n	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"v ^`_ i"	:= (fdcoor (rVexp a n) v i) (at level 9).	Notation	v ^`_ i	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "fdcoor", "rVexp" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
fdcoor_rcs' (i : 'I_n) x : a ^+ n = 1 -> a ^+ i * x ^`_ i = 0 -> (rcs x) ^`_ i = 0.	Proof. move=> an1. rewrite /fdcoor (@horner_coef_wide _ n); last by rewrite /rVpoly size_poly. rewrite big_distrr /=. rewrite (eq_bigr (fun i1 : 'I_ _ => (rVpoly x)`_i1 * a ^+ i ^+ i1.+1)); last first. move=> i1 _; rewrite mulrC -mulrA; congr (_ * _). by rewrite mxE -!exprM -exprD mulnS addnC. move=> x_RS. rewrite ...	Lemma	fdcoor_rcs'	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "an1", "fdcoor", "mulr1", "mulrA", "mulrC", "rcs", "rcs_perm", "rcs_poly_rcs", "rcs_rcs_poly", "size_rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
fdcoor_rcs (i : 'I_n) x : a ^+ n = 1 -> x ^`_ i = 0 -> (rcs x) ^`_ i = 0.	Proof. move=> H1 H2; by rewrite (fdcoor_rcs' H1) // H2 mulr0. Qed.	Lemma	fdcoor_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "H2", "fdcoor_rcs'", "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
loop (T : Type) (f : T -> T) (n : nat) (x : T)	:= if n is m.+1 then loop f m (f x) else x.	Fixpoint	loop	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
fdcoor_loop_rcs k (i : 'I_n) (x : 'rV_n) : a ^+ n = 1 -> x ^`_ i = 0 -> (loop (@rcs F n) k x) ^`_ i = 0.	Proof. move: i x. elim: k => // k IH i x an1 H /=. by rewrite IH // fdcoor_rcs. Qed.	Lemma	fdcoor_loop_rcs	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "an1", "fdcoor_rcs", "loop", "rcs" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
is_pgen	:= [pred g \| [forall x, (x \in C) == (g %\| rVpoly x)]].	Definition	is_pgen	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"''pgen[' C ]"	:= (is_pgen C) : cyclic_code_scope.	Notation	''pgen[' C ]	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "is_pgen" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
t	:= mk { lcode0 :> Lcode0.t F n ; gen : {poly F} ; size_gen : size gen < n ; P : gen \in 'pgen[[set cw in lcode0]] }.	Record	t	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "mk" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
pcode_coercion (F : finFieldType) (n : nat) (c : Pcode.t F n) : {vspace 'rV[F]_n}	:= let: Pcode.mk v _ _ _ := c in v.	Coercion	pcode_coercion	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "mk" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
t	:= mk { lcode0 :> Lcode0.t F n ; P : rcsP [set cw in lcode0] }.	Record	t	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "mk", "rcsP" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
ccode_coercion (F : finFieldType) (n : nat) (c : Ccode.t F n) : {vspace 'rV[F]_n}	:= let: Ccode.mk v _ := c in v.	Coercion	ccode_coercion	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "mk" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
C_not_trivial : not_trivial C.		Hypothesis	C_not_trivial	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "not_trivial" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
is_cgen	:= [pred x \| non0_codeword_lowest_deg C x].	Definition	is_cgen	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "non0_codeword_lowest_deg" ]	A nonzero polynomial of lowest degree in the code is called a generator polynomial:	https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
"''cgen'"	:= (is_cgen).	Notation	''cgen'	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "is_cgen" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
is_cgenE g : g \in 'cgen = non0_codeword_lowest_deg C g.	Proof. by []. Qed.	Lemma	is_cgenE	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "non0_codeword_lowest_deg" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_is_cgen g : g \in 'cgen -> size (rVpoly g) <= n.	Proof. by rewrite size_poly. Qed.	Lemma	size_is_cgen	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
canonical_cgen : 'rV[F]_n	:= val (exists_non0_codeword_lowest_deg C_not_trivial).	Definition	canonical_cgen	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "C_not_trivial", "exists_non0_codeword_lowest_deg" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
canonical_cgenP : non0_codeword_lowest_deg C canonical_cgen.	Proof. rewrite /canonical_cgen; by case: exists_non0_codeword_lowest_deg. Qed.	Lemma	canonical_cgenP	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "canonical_cgen", "exists_non0_codeword_lowest_deg", "non0_codeword_lowest_deg" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
size_canonical_cgen : size (rVpoly canonical_cgen) <= n.	Proof. by rewrite size_poly. Qed.	Lemma	size_canonical_cgen	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "canonical_cgen" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
canonical_cgen_lowest_size : size (rVpoly canonical_cgen) = lowest_size C_not_trivial.	Proof. by []. Qed.	Lemma	canonical_cgen_lowest_size	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "C_not_trivial", "canonical_cgen", "lowest_size" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5
parity_check : {poly F}	:= ('X^n - 1) %/ rVpoly canonical_cgen.	Definition	parity_check	ecc_classic	ecc_classic/cyclic_code.v	[ "mathcomp", "all_ssreflect", "ssralg", "poly", "polydiv", "fingroup", "perm", "finalg", "zmodp", "matrix", "mxalgebra", "mxpoly", "vector", "ssralg_ext", "poly_ext", "f2", "hamming", "linearcode", "dft", "Prenex", "Implicits", "GRing.Theory" ]	[ "canonical_cgen" ]		https://github.com/affeldt-aist/infotheo	529fbd82d60639bddf2faef8d85d6f9dd0580dd5

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Coq-Infotheo

Structured dataset from Infotheo — Information theory and error-correcting codes.

Source

Repository: https://github.com/affeldt-aist/infotheo
Commit: 529fbd82d60639bddf2faef8d85d6f9dd0580dd5
Files: 87
License: lgpl-2.1

Schema

Column	Type	Description
statement	string	Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof
proof	string	Verbatim proof/body, empty if the declaration has none
type	string	Declaration keyword
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, empty if absent
source_url	string	Upstream repository
commit	string	Upstream commit extracted

Statistics

Entries: 4,953
With proof: 4,746 (95.8%)
With docstring: 339 (6.8%)
Libraries: 7

By type

Type	Count
Lemma	2,926
Definition	910
Let	405
Notation	338
Hypothesis	188
Fixpoint	39
Record	31
Theorem	25
Coercion	20
Canonical	14
Fact	13
Corollary	10
Inductive	9
Hypotheses	9
Example	5
CoInductive	4
Ltac	3
Proposition	2
Variant	1
Function	1

Example

goppa_code_condition_check : goppa_code_condition n g t.
Proof. by rewrite /goppa_code_condition size_polyXn. Qed.

type: Let | symbolic_name: goppa_code_condition_check | ecc_classic/alternant.v

Use

Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint and together form the complete declaration, for proof modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_infotheo_dataset,
  title  = {Coq-Infotheo},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/affeldt-aist/infotheo, commit 529fbd82d606},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-Infotheo}
}

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