phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
"n .-tuplelexi"	:= n.-tuplelexi[seqlexi_display disp].	Notation	n .-tuplelexi	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "seqlexi_display" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subset_display : disp_t.	Proof. exact. Qed.	Fact	subset_display	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "disp_t" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
type (disp : disp_t) (T : finType)	:= {set T}.	Definition	type	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "disp_t" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
le_def A B : (A \subset B) = (A :&: B == A).	Proof. exact/setIidPl/eqP. Qed.	Lemma	le_def	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "setIidPl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEsubset A B : (A <= B) = (A \subset B).	Proof. by []. Qed.	Lemma	leEsubset	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'subset' [ d ] T }"	:= (type d T) (format "{ 'subset' [ d ] T }") : type_scope.	Notation	{ 'subset' [ d ] T }	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'subset' T }"	:= {subset[subset_display] T} (format "{ 'subset' T }") : type_scope.	Notation	{ 'subset' T }	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "subset_display" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leEsubset	:= @leEsubset.	Definition	leEsubset	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum A	:= (sort <=%O (enum A)).	Notation	enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardE (A : {pred T}) : #\|A\| = size (enum A).	Proof. by rewrite size_sort cardE. Qed.	Lemma	cardE	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "size", "size_sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_enum (A : {pred T}) : enum A =i A.	Proof. by move=> x; rewrite mem_sort mem_enum. Qed.	Lemma	mem_enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "mem_sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_uniq (A : {pred T}) : uniq (enum A).	Proof. by rewrite sort_uniq enum_uniq. Qed.	Lemma	enum_uniq	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "sort_uniq", "uniq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardT : #\|T\| = size (enum T).	Proof. by rewrite cardT size_sort. Qed.	Lemma	cardT	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "size", "size_sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enumT : enum T = sort <=%O (Finite.enum T).	Proof. by rewrite enumT. Qed.	Lemma	enumT	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum0 : enum (pred0 : {pred T}) = [::].	Proof. by rewrite enum0. Qed.	Lemma	enum0	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum1 (x : T) : enum (pred1 x) = [:: x].	Proof. by rewrite enum1. Qed.	Lemma	enum1	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "pred1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_enum (A B : {pred T}) : A =i B -> enum A = enum B.	Proof. by move=> /eq_enum->. Qed.	Lemma	eq_enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_cardT (A : {pred T}) : A =i predT -> #\|A\| = size (enum T).	Proof. by move=> /eq_enum<-; rewrite cardE. Qed.	Lemma	eq_cardT	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "cardE", "enum", "eq_enum", "size" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
set_enum (A : {set T}) : [set x in enum A] = A.	Proof. by apply/setP => x; rewrite inE mem_enum. Qed.	Lemma	set_enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "apply", "enum", "inE", "mem_enum", "setP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_set0 : enum (set0 : {set T}) = [::].	Proof. by rewrite enum_set0. Qed.	Lemma	enum_set0	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "set0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_setT : enum [set: T] = sort <=%O (Finite.enum T).	Proof. by rewrite enum_setT. Qed.	Lemma	enum_setT	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "sort" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_set1 (a : T) : enum [set a] = [:: a].	Proof. by rewrite enum_set1. Qed.	Lemma	enum_set1	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_ord n : enum 'I_n = fintype.enum 'I_n.	Proof. rewrite (sorted_sort le_trans)// -(@sorted_map _ _ (val : 'I_n -> nat))/=. by rewrite val_enum_ord iota_sorted. Qed.	Lemma	enum_ord	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "iota_sorted", "le_trans", "nat", "sorted_map", "sorted_sort", "val", "val_enum_ord" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_enum_ord n : [seq val i \| i <- enum 'I_n] = iota 0 n.	Proof. by rewrite enum_ord val_enum_ord. Qed.	Lemma	val_enum_ord	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_ord", "iota", "seq", "val" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_enum_ord n : size (enum 'I_n) = n.	Proof. by rewrite -cardE card_ord. Qed.	Lemma	size_enum_ord	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "cardE", "card_ord", "enum", "size" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_enum_ord (n : nat) (i0 : 'I_n) (m : nat) : (m < n)%N -> nth i0 (enum 'I_n) m = m :> nat.	Proof. by move=> lemn; rewrite enum_ord nth_enum_ord. Qed.	Lemma	nth_enum_ord	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_ord", "i0", "nat", "nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_ord_enum (n : nat) (i0 i : 'I_n) : nth i0 (enum 'I_n) i = i.	Proof. by rewrite enum_ord nth_ord_enum. Qed.	Lemma	nth_ord_enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_ord", "i0", "nat", "nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_enum_ord (n : nat) (i : 'I_n) : index i (enum 'I_n) = i.	Proof. by rewrite enum_ord index_enum_ord. Qed.	Lemma	index_enum_ord	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_ord", "index", "nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mono_sorted_enum d d' (T : finPreorderType d) (T' : preorderType d') (f : T -> T') : total (<=%O : rel T) -> {mono f : x y / (x <= y)%O} -> sorted <=%O [seq f x \| x <- enum T].	Proof. move=> /sort_sorted ss_sorted lef; wlog [x0 x'0] : / (T * T')%type. by case: (enum T) => // x ? => /(_ (x, f x)). rewrite (sorted_pairwise le_trans). apply/(pairwiseP x'0) => i j; rewrite !inE !size_map -!cardT. move=> ilt jlt ij; rewrite !(nth_map x0) -?cardT// lef. by rewrite (sorted_leq_nth le_trans le_refl...	Lemma	mono_sorted_enum	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "T'", "apply", "cardT", "enum", "inE", "le_refl", "le_trans", "ltnW", "nth_map", "pairwiseP", "rel", "seq", "size_map", "sort_sorted", "sorted", "sorted_leq_nth", "sorted_pairwise", "total", "type" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank_in x0 A (Ax0 : x0 \in A) x	:= insubd (Ordinal (@enum_rank_subproof _ x0 A Ax0)) (index x (enum A)).	Definition	enum_rank_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_rank_subproof", "index", "insubd" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank x	:= @enum_rank_in x T (erefl true) x.	Definition	enum_rank	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum_rank_in" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val A i	:= nth (@enum_default _ A i) (enum A) i.	Definition	enum_val	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "enum", "enum_default", "nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_valP A i : @enum_val A i \in A.	Proof. suff: enum_val i \in enum A by rewrite mem_enum. by apply: mem_nth; rewrite -cardE. Qed.	Lemma	enum_valP	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "apply", "cardE", "enum", "enum_val", "mem_enum", "mem_nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val_nth A x i : @enum_val A i = nth x (enum A) i.	Proof. by apply: set_nth_default; rewrite cardE in i *. Qed.	Lemma	enum_val_nth	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "apply", "cardE", "enum", "enum_val", "nth", "set_nth_default" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_enum_rank_in x00 x0 A Ax0 : {in A, cancel (@enum_rank_in x0 A Ax0) (nth x00 (enum A))}.	Proof. move=> x Ax; rewrite /= insubdK ?nth_index ?mem_enum //. by rewrite cardE [_ \in _]index_mem mem_enum. Qed.	Lemma	nth_enum_rank_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "cardE", "enum", "enum_rank_in", "index_mem", "insubdK", "mem_enum", "nth", "nth_index" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rankK_in x0 A Ax0 : {in A, cancel (@enum_rank_in x0 A Ax0) enum_val}.	Proof. by move=> x; apply: nth_enum_rank_in. Qed.	Lemma	enum_rankK_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "apply", "enum_rank_in", "enum_val", "nth_enum_rank_in" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_valK_in x0 A Ax0 : cancel enum_val (@enum_rank_in x0 A Ax0).	Proof. move=> x; apply: ord_inj; rewrite insubdK. by rewrite cardE [_ \in _]index_mem mem_nth // -cardE. by rewrite index_uniq ?enum_uniq // -cardE. Qed.	Lemma	enum_valK_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[ "apply", "cardE", "enum_rank_in", "enum_uniq", "enum_val", "index_mem", "index_uniq", "insubdK", "mem_nth", "ord_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val	:= enum_val.	Notation	enum_val	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank_in	:= enum_rank_in.	Notation	enum_rank_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank	:= enum_rank.	Notation	enum_rank	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_valP	:= enum_valP.	Notation	enum_valP	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val_nth	:= enum_val_nth.	Notation	enum_val_nth	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_enum_rank_in	:= nth_enum_rank_in.	Notation	nth_enum_rank_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_enum_rank	:= nth_enum_rank.	Notation	nth_enum_rank	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rankK_in	:= enum_rankK_in.	Notation	enum_rankK_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rankK	:= enum_rankK.	Notation	enum_rankK	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_valK_in	:= enum_valK_in.	Notation	enum_valK_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_valK	:= enum_valK.	Notation	enum_valK	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank_inj	:= enum_rank_inj.	Notation	enum_rank_inj	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val_inj	:= enum_val_inj.	Notation	enum_val_inj	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val_bij_in	:= enum_val_bij_in.	Notation	enum_val_bij_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_enum_rank_in	:= eq_enum_rank_in.	Notation	eq_enum_rank_in	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank_in_inj	:= enum_rank_in_inj.	Notation	enum_rank_in_inj	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_rank_bij	:= enum_rank_bij.	Notation	enum_rank_bij	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_val_bij	:= enum_val_bij.	Notation	enum_val_bij	order	order/preorder.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ldiv n	:= [set x : gT \| x ^+ n == 1].	Definition	Ldiv	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "gT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent A	:= \big[lcmn/1%N]_(x in A) #[x].	Definition	exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "lcmn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
abelem p A	:= [&& p.-group A, abelian A & exponent A %\| p].	Definition	abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelian", "exponent", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
is_abelem A	:= abelem (pdiv #\|A\|) A.	Definition	is_abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pElem p A	:= [set E : {group gT} \| E \subset A & abelem p E].	Definition	pElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem", "gT", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnElem p n A	:= [set E in pElem p A \| logn p #\|E\| == n].	Definition	pnElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "logn", "pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nElem n A	:= \bigcup_(0 <= p < #\|A\|.+1) pnElem p n A.	Definition	nElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pnElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pmaxElem p A	:= [set E \| [max E \| E \in pElem p A]].	Definition	pmaxElem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "max", "pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p_rank p A	:= \max_(E in pElem p A) logn p #\|E\|.	Definition	p_rank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "logn", "pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rank A	:= \max_(0 <= p < #\|A\|.+1) p_rank p A.	Definition	rank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "p_rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
gen_rank A	:= #\|[arg min_(B < A \| <<B>> == A) #\|B\|]\|.	Definition	gen_rank	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Ldiv_' n ()"	:= (Ldiv _ n) (n at level 2, format "''Ldiv_' n ()") : group_scope.	Notation	''Ldiv_' n ()	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Ldiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Ldiv_' n ( G )"	:= (G :&: 'Ldiv_n()) (format "''Ldiv_' n ( G )") : group_scope.	Notation	''Ldiv_' n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"p .-abelem"	:= (abelem p) (format "p .-abelem") : group_scope.	Notation	p .-abelem	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "abelem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''E_' p ( G )"	:= (pElem p G) (p at level 2, format "''E_' p ( G )") : group_scope.	Notation	''E_' p ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''E_' p ^ n ( G )"	:= (pnElem p n G) (n at level 2, format "''E_' p ^ n ( G )") : group_scope.	Notation	''E_' p ^ n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pnElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''E' ^ n ( G )"	:= (nElem n G) (n at level 2, format "''E' ^ n ( G )") : group_scope.	Notation	''E' ^ n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "nElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''E*_' p ( G )"	:= (pmaxElem p G) (p at level 2, format "''E*_' p ( G )") : group_scope.	Notation	''E*_' p ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pmaxElem" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''m' ( A )"	:= (gen_rank A) (format "''m' ( A )") : group_scope.	Notation	''m' ( A )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "gen_rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''r' ( A )"	:= (rank A) (format "''r' ( A )") : group_scope.	Notation	''r' ( A )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''r_' p ( A )"	:= (p_rank p A) (p at level 2, format "''r_' p ( A )") : group_scope.	Notation	''r_' p ( A )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "p_rank" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm	:= <<[set x in A \| x ^+ (pdiv #[x] ^ n) == 1]>>.	Definition	Ohm	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho	:= <<[set x ^+ (pdiv #[x] ^ n) \| x in A & (pdiv #[x]).-elt x]>>.	Definition	Mho	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ohm_group : {group gT}	:= Eval hnf in [group of Ohm].	Canonical	Ohm_group	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Ohm", "gT", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_group : {group gT}	:= Eval hnf in [group of Mho].	Canonical	Mho_group	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Mho", "gT", "group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pdiv_p_elt (p : nat) (x : gT) : p.-elt x -> x != 1 -> pdiv #[x] = p.	Proof. move=> p_x; rewrite /order -cycle_eq1. by case/(pgroup_pdiv p_x)=> p_pr _ [k ->]; rewrite pdiv_pfactor. Qed.	Lemma	pdiv_p_elt	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "cycle_eq1", "gT", "nat", "order", "p_pr", "pdiv", "pdiv_pfactor", "pgroup_pdiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
OhmPredP (x : gT) : reflect (exists2 p, prime p & x ^+ (p ^ n) = 1) (x ^+ (pdiv #[x] ^ n) == 1).	Proof. have [-> \| nt_x] := eqVneq x 1. by rewrite expg1n eqxx; left; exists 2; rewrite ?expg1n. apply: (iffP idP) => [/eqP \| [p p_pr /eqP x_pn]]. by exists (pdiv #[x]); rewrite ?pdiv_prime ?order_gt1. rewrite (@pdiv_p_elt p) //; rewrite -order_dvdn in x_pn. by rewrite [p_elt _ _](pnat_dvd x_pn) // pnatX pnat_id. Qe...	Lemma	OhmPredP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "eqVneq", "eqxx", "expg1n", "gT", "order_dvdn", "order_gt1", "p_elt", "p_pr", "pdiv", "pdiv_p_elt", "pdiv_prime", "pnatX", "pnat_dvd", "pnat_id", "prime" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Mho_p_elt (p : nat) x : x \in A -> p.-elt x -> x ^+ (p ^ n) \in Mho.	Proof. move=> Ax p_x; have [-> \| ntx] := eqVneq x 1; first by rewrite groupX. by apply/mem_gen/imsetP; exists x; rewrite ?inE ?Ax (pdiv_p_elt p_x). Qed.	Lemma	Mho_p_elt	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Mho", "apply", "eqVneq", "groupX", "imsetP", "inE", "mem_gen", "nat", "pdiv_p_elt" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Ohm_' n ( G )"	:= (Ohm n G) (n at level 2, format "''Ohm_' n ( G )") : group_scope.	Notation	''Ohm_' n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Ohm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Ohm_' n ( G )"	:= (Ohm_group n G) : Group_scope.	Notation	''Ohm_' n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Ohm_group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Mho^' n ( G )"	:= (Mho n G) (n at level 2, format "''Mho^' n ( G )") : group_scope.	Notation	''Mho^' n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Mho" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''Mho^' n ( G )"	:= (Mho_group n G) : Group_scope.	Notation	''Mho^' n ( G )	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Mho_group" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
LdivP A n x : reflect (x \in A /\ x ^+ n = 1) (x \in 'Ldiv_n(A)).	Proof. by rewrite !inE; apply: (iffP andP) => [] [-> /eqP]. Qed.	Lemma	LdivP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "inE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_exponent x A : x \in A -> #[x] %\| exponent A.	Proof. by move=> Ax; rewrite (biglcmn_sup x). Qed.	Lemma	dvdn_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "biglcmn_sup", "exponent" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expg_exponent x A : x \in A -> x ^+ exponent A = 1.	Proof. by move=> Ax; apply/eqP; rewrite -order_dvdn dvdn_exponent. Qed.	Lemma	expg_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "dvdn_exponent", "exponent", "order_dvdn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponentS A B : A \subset B -> exponent A %\| exponent B.	Proof. by move=> sAB; apply/dvdn_biglcmP=> x Ax; rewrite dvdn_exponent ?(subsetP sAB). Qed.	Lemma	exponentS	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "dvdn_biglcmP", "dvdn_exponent", "exponent", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponentP A n : reflect (forall x, x \in A -> x ^+ n = 1) (exponent A %\| n).	Proof. apply: (iffP (dvdn_biglcmP _ _ _)) => eAn x Ax. by apply/eqP; rewrite -order_dvdn eAn. by rewrite order_dvdn eAn. Qed.	Lemma	exponentP	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "dvdn_biglcmP", "exponent", "order_dvdn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
trivg_exponent G : (G :==: 1) = (exponent G %\| 1).	Proof. rewrite -subG1. by apply/subsetP/exponentP=> trG x /trG; rewrite expg1 => /set1P. Qed.	Lemma	trivg_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "expg1", "exponent", "exponentP", "set1P", "subG1", "subsetP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent1 : exponent [1 gT] = 1%N.	Proof. by apply/eqP; rewrite -dvdn1 -trivg_exponent eqxx. Qed.	Lemma	exponent1	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "dvdn1", "eqxx", "exponent", "gT", "trivg_exponent" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_dvdn G : exponent G %\| #\|G\|.	Proof. by apply/dvdn_biglcmP=> x Gx; apply: order_dvdG. Qed.	Lemma	exponent_dvdn	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "dvdn_biglcmP", "exponent", "order_dvdG" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_gt0 G : 0 < exponent G.	Proof. exact: dvdn_gt0 (exponent_dvdn G). Qed.	Lemma	exponent_gt0	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "dvdn_gt0", "exponent", "exponent_dvdn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pnat_exponent pi G : pi.-nat (exponent G) = pi.-group G.	Proof. congr (_ && _); first by rewrite cardG_gt0 exponent_gt0. apply: eq_all_r => p; rewrite !mem_primes cardG_gt0 exponent_gt0 /=. apply: andb_id2l => p_pr; apply/idP/idP=> pG. exact: dvdn_trans pG (exponent_dvdn G). by case/Cauchy: pG => // x Gx <-; apply: dvdn_exponent. Qed.	Lemma	pnat_exponent	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "Cauchy", "apply", "cardG_gt0", "dvdn_exponent", "dvdn_trans", "eq_all_r", "exponent", "exponent_dvdn", "exponent_gt0", "group", "mem_primes", "nat", "pG", "p_pr", "pi" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponentJ A x : exponent (A :^ x) = exponent A.	Proof. rewrite /exponent (reindex_inj (conjg_inj x)). by apply: eq_big => [y \| y _]; rewrite ?orderJ ?memJ_conjg. Qed.	Lemma	exponentJ	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "conjg_inj", "eq_big", "exponent", "memJ_conjg", "orderJ", "reindex_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_witness G : nilpotent G -> {x \| x \in G & exponent G = #[x]}.	Proof. move=> nilG; have [//=\| /= x Gx max_x] := @arg_maxnP _ 1 [in G] order. exists x => //; apply/eqP; rewrite eqn_dvd dvdn_exponent // andbT. apply/dvdn_biglcmP=> y Gy; apply/dvdn_partP=> //= p. rewrite mem_primes => /andP[p_pr _]; have p_gt1: p > 1 := prime_gt1 p_pr. rewrite p_part pfactor_dvdn // -(leq_exp2l _ _ p...	Lemma	exponent_witness	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "arg_maxnP", "centsP", "coprime_partC", "dprodP", "dvdn_biglcmP", "dvdn_exponent", "dvdn_partP", "eqn_dvd", "exponent", "groupM", "groupX", "leq_exp2l", "leq_pmul2r", "mem_normal_Hall", "mem_primes", "nilpotent", "nilpotent_pcoreC", "nilpotent_pcore_Hall", "order", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exponent_cycle x : exponent <[x]> = #[x].	Proof. by apply/eqP; rewrite eqn_dvd exponent_dvdn dvdn_exponent ?cycle_id. Qed.	Lemma	exponent_cycle	solvable	solvable/abelian.v	[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...	[ "apply", "cycle_id", "dvdn_exponent", "eqn_dvd", "exponent", "exponent_dvdn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"n .-tuplelexi"

:= n.-tuplelexi[seqlexi_display disp].

Notation

n .-tuplelexi

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "seqlexi_display" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subset_display : disp_t.

Proof. exact. Qed.

Fact

subset_display

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "disp_t" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

type (disp : disp_t) (T : finType)

:= {set T}.

Definition

type

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "disp_t" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

le_def A B : (A \subset B) = (A :&: B == A).

Proof. exact/setIidPl/eqP. Qed.

Lemma

le_def

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "setIidPl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEsubset A B : (A <= B) = (A \subset B).

Proof. by []. Qed.

Lemma

leEsubset

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'subset' [ d ] T }"

:= (type d T) (format "{ 'subset' [ d ] T }") : type_scope.

Notation

{ 'subset' [ d ] T }

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "type" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'subset' T }"

:= {subset[subset_display] T} (format "{ 'subset' T }") : type_scope.

Notation

{ 'subset' T }

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "subset_display" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leEsubset

:= @leEsubset.

Definition

leEsubset

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum A

:= (sort <=%O (enum A)).

Notation

enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardE (A : {pred T}) : #|A| = size (enum A).

Proof. by rewrite size_sort cardE. Qed.

Lemma

cardE

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "size", "size_sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_enum (A : {pred T}) : enum A =i A.

Proof. by move=> x; rewrite mem_sort mem_enum. Qed.

Lemma

mem_enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "mem_sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_uniq (A : {pred T}) : uniq (enum A).

Proof. by rewrite sort_uniq enum_uniq. Qed.

Lemma

enum_uniq

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "sort_uniq", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardT : #|T| = size (enum T).

Proof. by rewrite cardT size_sort. Qed.

Lemma

cardT

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "size", "size_sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enumT : enum T = sort <=%O (Finite.enum T).

Proof. by rewrite enumT. Qed.

Lemma

enumT

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum0 : enum (pred0 : {pred T}) = [::].

Proof. by rewrite enum0. Qed.

Lemma

enum0

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum1 (x : T) : enum (pred1 x) = [:: x].

Proof. by rewrite enum1. Qed.

Lemma

enum1

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "pred1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_enum (A B : {pred T}) : A =i B -> enum A = enum B.

Proof. by move=> /eq_enum->. Qed.

Lemma

eq_enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_cardT (A : {pred T}) : A =i predT -> #|A| = size (enum T).

Proof. by move=> /eq_enum<-; rewrite cardE. Qed.

Lemma

eq_cardT

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "cardE", "enum", "eq_enum", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

set_enum (A : {set T}) : [set x in enum A] = A.

Proof. by apply/setP => x; rewrite inE mem_enum. Qed.

Lemma

set_enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "apply", "enum", "inE", "mem_enum", "setP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_set0 : enum (set0 : {set T}) = [::].

Proof. by rewrite enum_set0. Qed.

Lemma

enum_set0

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "set0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_setT : enum [set: T] = sort <=%O (Finite.enum T).

Proof. by rewrite enum_setT. Qed.

Lemma

enum_setT

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "sort" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_set1 (a : T) : enum [set a] = [:: a].

Proof. by rewrite enum_set1. Qed.

Lemma

enum_set1

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_ord n : enum 'I_n = fintype.enum 'I_n.

Proof. rewrite (sorted_sort le_trans)// -(@sorted_map _ _ (val : 'I_n -> nat))/=. by rewrite val_enum_ord iota_sorted. Qed.

Lemma

enum_ord

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "iota_sorted", "le_trans", "nat", "sorted_map", "sorted_sort", "val", "val_enum_ord" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_enum_ord n : [seq val i | i <- enum 'I_n] = iota 0 n.

Proof. by rewrite enum_ord val_enum_ord. Qed.

Lemma

val_enum_ord

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_ord", "iota", "seq", "val" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_enum_ord n : size (enum 'I_n) = n.

Proof. by rewrite -cardE card_ord. Qed.

Lemma

size_enum_ord

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "cardE", "card_ord", "enum", "size" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_enum_ord (n : nat) (i0 : 'I_n) (m : nat) : (m < n)%N -> nth i0 (enum 'I_n) m = m :> nat.

Proof. by move=> lemn; rewrite enum_ord nth_enum_ord. Qed.

Lemma

nth_enum_ord

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_ord", "i0", "nat", "nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_ord_enum (n : nat) (i0 i : 'I_n) : nth i0 (enum 'I_n) i = i.

Proof. by rewrite enum_ord nth_ord_enum. Qed.

Lemma

nth_ord_enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_ord", "i0", "nat", "nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

index_enum_ord (n : nat) (i : 'I_n) : index i (enum 'I_n) = i.

Proof. by rewrite enum_ord index_enum_ord. Qed.

Lemma

index_enum_ord

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_ord", "index", "nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mono_sorted_enum d d' (T : finPreorderType d) (T' : preorderType d') (f : T -> T') : total (<=%O : rel T) -> {mono f : x y / (x <= y)%O} -> sorted <=%O [seq f x | x <- enum T].

Proof. move=> /sort_sorted ss_sorted lef; wlog [x0 x'0] : / (T * T')%type. by case: (enum T) => // x ? => /(_ (x, f x)). rewrite (sorted_pairwise le_trans). apply/(pairwiseP x'0) => i j; rewrite !inE !size_map -!cardT. move=> ilt jlt ij; rewrite !(nth_map x0) -?cardT// lef. by rewrite (sorted_leq_nth le_trans le_refl...

Lemma

mono_sorted_enum

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "T'", "apply", "cardT", "enum", "inE", "le_refl", "le_trans", "ltnW", "nth_map", "pairwiseP", "rel", "seq", "size_map", "sort_sorted", "sorted", "sorted_leq_nth", "sorted_pairwise", "total", "type" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank_in x0 A (Ax0 : x0 \in A) x

:= insubd (Ordinal (@enum_rank_subproof _ x0 A Ax0)) (index x (enum A)).

Definition

enum_rank_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_rank_subproof", "index", "insubd" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank x

:= @enum_rank_in x T (erefl true) x.

Definition

enum_rank

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum_rank_in" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val A i

:= nth (@enum_default _ A i) (enum A) i.

Definition

enum_val

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "enum", "enum_default", "nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_valP A i : @enum_val A i \in A.

Proof. suff: enum_val i \in enum A by rewrite mem_enum. by apply: mem_nth; rewrite -cardE. Qed.

Lemma

enum_valP

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "apply", "cardE", "enum", "enum_val", "mem_enum", "mem_nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val_nth A x i : @enum_val A i = nth x (enum A) i.

Proof. by apply: set_nth_default; rewrite cardE in i *. Qed.

Lemma

enum_val_nth

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "apply", "cardE", "enum", "enum_val", "nth", "set_nth_default" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_enum_rank_in x00 x0 A Ax0 : {in A, cancel (@enum_rank_in x0 A Ax0) (nth x00 (enum A))}.

Proof. move=> x Ax; rewrite /= insubdK ?nth_index ?mem_enum //. by rewrite cardE [_ \in _]index_mem mem_enum. Qed.

Lemma

nth_enum_rank_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "cardE", "enum", "enum_rank_in", "index_mem", "insubdK", "mem_enum", "nth", "nth_index" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rankK_in x0 A Ax0 : {in A, cancel (@enum_rank_in x0 A Ax0) enum_val}.

Proof. by move=> x; apply: nth_enum_rank_in. Qed.

Lemma

enum_rankK_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "apply", "enum_rank_in", "enum_val", "nth_enum_rank_in" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_valK_in x0 A Ax0 : cancel enum_val (@enum_rank_in x0 A Ax0).

Proof. move=> x; apply: ord_inj; rewrite insubdK. by rewrite cardE [_ \in _]index_mem mem_nth // -cardE. by rewrite index_uniq ?enum_uniq // -cardE. Qed.

Lemma

enum_valK_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[ "apply", "cardE", "enum_rank_in", "enum_uniq", "enum_val", "index_mem", "index_uniq", "insubdK", "mem_nth", "ord_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val

:= enum_val.

Notation

enum_val

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank_in

:= enum_rank_in.

Notation

enum_rank_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank

:= enum_rank.

Notation

enum_rank

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_valP

:= enum_valP.

Notation

enum_valP

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val_nth

:= enum_val_nth.

Notation

enum_val_nth

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_enum_rank_in

:= nth_enum_rank_in.

Notation

nth_enum_rank_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nth_enum_rank

:= nth_enum_rank.

Notation

nth_enum_rank

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rankK_in

:= enum_rankK_in.

Notation

enum_rankK_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rankK

:= enum_rankK.

Notation

enum_rankK

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_valK_in

:= enum_valK_in.

Notation

enum_valK_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_valK

:= enum_valK.

Notation

enum_valK

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank_inj

:= enum_rank_inj.

Notation

enum_rank_inj

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val_inj

:= enum_val_inj.

Notation

enum_val_inj

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val_bij_in

:= enum_val_bij_in.

Notation

enum_val_bij_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_enum_rank_in

:= eq_enum_rank_in.

Notation

eq_enum_rank_in

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank_in_inj

:= enum_rank_in_inj.

Notation

enum_rank_in_inj

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_rank_bij

:= enum_rank_bij.

Notation

enum_rank_bij

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_val_bij

:= enum_val_bij.

Notation

enum_val_bij

order

order/preorder.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "path", "fintype", "tuple", "bigop", "finset", "div", "prime", "finfun", "DvdSyntax", "DefaultSeqProdOrder", "DefaultSeqLexiOrder", "SetSubsetOrder.Exports", "OrdinalOrd...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ldiv n

:= [set x : gT | x ^+ n == 1].

Definition

Ldiv

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "gT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent A

:= \big[lcmn/1%N]_(x in A) #[x].

Definition

exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "lcmn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

abelem p A

:= [&& p.-group A, abelian A & exponent A %| p].

Definition

abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelian", "exponent", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

is_abelem A

:= abelem (pdiv #|A|) A.

Definition

is_abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pElem p A

:= [set E : {group gT} | E \subset A & abelem p E].

Definition

pElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnElem p n A

:= [set E in pElem p A | logn p #|E| == n].

Definition

pnElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "logn", "pElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nElem n A

:= \bigcup_(0 <= p < #|A|.+1) pnElem p n A.

Definition

nElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pnElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pmaxElem p A

:= [set E | [max E | E \in pElem p A]].

Definition

pmaxElem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "max", "pElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p_rank p A

:= \max_(E in pElem p A) logn p #|E|.

Definition

p_rank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "logn", "pElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rank A

:= \max_(0 <= p < #|A|.+1) p_rank p A.

Definition

rank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "p_rank" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

gen_rank A

:= #|[arg min_(B < A | <<B>> == A) #|B|]|.

Definition

gen_rank

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Ldiv_' n ()"

:= (Ldiv _ n) (n at level 2, format "''Ldiv_' n ()") : group_scope.

Notation

''Ldiv_' n ()

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Ldiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Ldiv_' n ( G )"

:= (G :&: 'Ldiv_n()) (format "''Ldiv_' n ( G )") : group_scope.

Notation

''Ldiv_' n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"p .-abelem"

:= (abelem p) (format "p .-abelem") : group_scope.

Notation

p .-abelem

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "abelem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''E_' p ( G )"

:= (pElem p G) (p at level 2, format "''E_' p ( G )") : group_scope.

Notation

''E_' p ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''E_' p ^ n ( G )"

:= (pnElem p n G) (n at level 2, format "''E_' p ^ n ( G )") : group_scope.

Notation

''E_' p ^ n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pnElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''E' ^ n ( G )"

:= (nElem n G) (n at level 2, format "''E' ^ n ( G )") : group_scope.

Notation

''E' ^ n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "nElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''E*_' p ( G )"

:= (pmaxElem p G) (p at level 2, format "''E*_' p ( G )") : group_scope.

Notation

''E*_' p ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pmaxElem" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''m' ( A )"

:= (gen_rank A) (format "''m' ( A )") : group_scope.

Notation

''m' ( A )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "gen_rank" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''r' ( A )"

:= (rank A) (format "''r' ( A )") : group_scope.

Notation

''r' ( A )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "rank" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''r_' p ( A )"

:= (p_rank p A) (p at level 2, format "''r_' p ( A )") : group_scope.

Notation

''r_' p ( A )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "p_rank" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm

:= <<[set x in A | x ^+ (pdiv #[x] ^ n) == 1]>>.

Definition

Ohm

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho

:= <<[set x ^+ (pdiv #[x] ^ n) | x in A & (pdiv #[x]).-elt x]>>.

Definition

Mho

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ohm_group : {group gT}

:= Eval hnf in [group of Ohm].

Canonical

Ohm_group

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Ohm", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_group : {group gT}

:= Eval hnf in [group of Mho].

Canonical

Mho_group

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Mho", "gT", "group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pdiv_p_elt (p : nat) (x : gT) : p.-elt x -> x != 1 -> pdiv #[x] = p.

Proof. move=> p_x; rewrite /order -cycle_eq1. by case/(pgroup_pdiv p_x)=> p_pr _ [k ->]; rewrite pdiv_pfactor. Qed.

Lemma

pdiv_p_elt

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "cycle_eq1", "gT", "nat", "order", "p_pr", "pdiv", "pdiv_pfactor", "pgroup_pdiv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

OhmPredP (x : gT) : reflect (exists2 p, prime p & x ^+ (p ^ n) = 1) (x ^+ (pdiv #[x] ^ n) == 1).

Proof. have [-> | nt_x] := eqVneq x 1. by rewrite expg1n eqxx; left; exists 2; rewrite ?expg1n. apply: (iffP idP) => [/eqP | [p p_pr /eqP x_pn]]. by exists (pdiv #[x]); rewrite ?pdiv_prime ?order_gt1. rewrite (@pdiv_p_elt p) //; rewrite -order_dvdn in x_pn. by rewrite [p_elt _ _](pnat_dvd x_pn) // pnatX pnat_id. Qe...

Lemma

OhmPredP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "eqVneq", "eqxx", "expg1n", "gT", "order_dvdn", "order_gt1", "p_elt", "p_pr", "pdiv", "pdiv_p_elt", "pdiv_prime", "pnatX", "pnat_dvd", "pnat_id", "prime" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Mho_p_elt (p : nat) x : x \in A -> p.-elt x -> x ^+ (p ^ n) \in Mho.

Proof. move=> Ax p_x; have [-> | ntx] := eqVneq x 1; first by rewrite groupX. by apply/mem_gen/imsetP; exists x; rewrite ?inE ?Ax (pdiv_p_elt p_x). Qed.

Lemma

Mho_p_elt

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Mho", "apply", "eqVneq", "groupX", "imsetP", "inE", "mem_gen", "nat", "pdiv_p_elt" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Ohm_' n ( G )"

:= (Ohm n G) (n at level 2, format "''Ohm_' n ( G )") : group_scope.

Notation

''Ohm_' n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Ohm" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Ohm_' n ( G )"

:= (Ohm_group n G) : Group_scope.

Notation

''Ohm_' n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Ohm_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Mho^' n ( G )"

:= (Mho n G) (n at level 2, format "''Mho^' n ( G )") : group_scope.

Notation

''Mho^' n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Mho" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''Mho^' n ( G )"

:= (Mho_group n G) : Group_scope.

Notation

''Mho^' n ( G )

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Mho_group" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

LdivP A n x : reflect (x \in A /\ x ^+ n = 1) (x \in 'Ldiv_n(A)).

Proof. by rewrite !inE; apply: (iffP andP) => [] [-> /eqP]. Qed.

Lemma

LdivP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "inE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_exponent x A : x \in A -> #[x] %| exponent A.

Proof. by move=> Ax; rewrite (biglcmn_sup x). Qed.

Lemma

dvdn_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "biglcmn_sup", "exponent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expg_exponent x A : x \in A -> x ^+ exponent A = 1.

Proof. by move=> Ax; apply/eqP; rewrite -order_dvdn dvdn_exponent. Qed.

Lemma

expg_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "dvdn_exponent", "exponent", "order_dvdn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponentS A B : A \subset B -> exponent A %| exponent B.

Proof. by move=> sAB; apply/dvdn_biglcmP=> x Ax; rewrite dvdn_exponent ?(subsetP sAB). Qed.

Lemma

exponentS

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "dvdn_biglcmP", "dvdn_exponent", "exponent", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponentP A n : reflect (forall x, x \in A -> x ^+ n = 1) (exponent A %| n).

Proof. apply: (iffP (dvdn_biglcmP _ _ _)) => eAn x Ax. by apply/eqP; rewrite -order_dvdn eAn. by rewrite order_dvdn eAn. Qed.

Lemma

exponentP

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "dvdn_biglcmP", "exponent", "order_dvdn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

trivg_exponent G : (G :==: 1) = (exponent G %| 1).

Proof. rewrite -subG1. by apply/subsetP/exponentP=> trG x /trG; rewrite expg1 => /set1P. Qed.

Lemma

trivg_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "expg1", "exponent", "exponentP", "set1P", "subG1", "subsetP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent1 : exponent [1 gT] = 1%N.

Proof. by apply/eqP; rewrite -dvdn1 -trivg_exponent eqxx. Qed.

Lemma

exponent1

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "dvdn1", "eqxx", "exponent", "gT", "trivg_exponent" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_dvdn G : exponent G %| #|G|.

Proof. by apply/dvdn_biglcmP=> x Gx; apply: order_dvdG. Qed.

Lemma

exponent_dvdn

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "dvdn_biglcmP", "exponent", "order_dvdG" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_gt0 G : 0 < exponent G.

Proof. exact: dvdn_gt0 (exponent_dvdn G). Qed.

Lemma

exponent_gt0

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "dvdn_gt0", "exponent", "exponent_dvdn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pnat_exponent pi G : pi.-nat (exponent G) = pi.-group G.

Proof. congr (_ && _); first by rewrite cardG_gt0 exponent_gt0. apply: eq_all_r => p; rewrite !mem_primes cardG_gt0 exponent_gt0 /=. apply: andb_id2l => p_pr; apply/idP/idP=> pG. exact: dvdn_trans pG (exponent_dvdn G). by case/Cauchy: pG => // x Gx <-; apply: dvdn_exponent. Qed.

Lemma

pnat_exponent

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "Cauchy", "apply", "cardG_gt0", "dvdn_exponent", "dvdn_trans", "eq_all_r", "exponent", "exponent_dvdn", "exponent_gt0", "group", "mem_primes", "nat", "pG", "p_pr", "pi" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponentJ A x : exponent (A :^ x) = exponent A.

Proof. rewrite /exponent (reindex_inj (conjg_inj x)). by apply: eq_big => [y | y _]; rewrite ?orderJ ?memJ_conjg. Qed.

Lemma

exponentJ

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "conjg_inj", "eq_big", "exponent", "memJ_conjg", "orderJ", "reindex_inj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_witness G : nilpotent G -> {x | x \in G & exponent G = #[x]}.

Proof. move=> nilG; have [//=| /= x Gx max_x] := @arg_maxnP _ 1 [in G] order. exists x => //; apply/eqP; rewrite eqn_dvd dvdn_exponent // andbT. apply/dvdn_biglcmP=> y Gy; apply/dvdn_partP=> //= p. rewrite mem_primes => /andP[p_pr _]; have p_gt1: p > 1 := prime_gt1 p_pr. rewrite p_part pfactor_dvdn // -(leq_exp2l _ _ p...

Lemma

exponent_witness

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "arg_maxnP", "centsP", "coprime_partC", "dprodP", "dvdn_biglcmP", "dvdn_exponent", "dvdn_partP", "eqn_dvd", "exponent", "groupM", "groupX", "leq_exp2l", "leq_pmul2r", "mem_normal_Hall", "mem_primes", "nilpotent", "nilpotent_pcoreC", "nilpotent_pcore_Hall", "order", "...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exponent_cycle x : exponent <[x]> = #[x].

Proof. by apply/eqP; rewrite eqn_dvd exponent_dvdn dvdn_exponent ?cycle_id. Qed.

Lemma

exponent_cycle

solvable

solvable/abelian.v

[ "mathcomp", "ssreflect", "ssrbool", "ssrfun", "eqtype", "ssrnat", "seq", "path", "choice", "div", "fintype", "finfun", "bigop", "finset", "prime", "binomial", "fingroup", "morphism", "perm", "automorphism", "action", "quotient", "gfunctor", "gproduct", "ssralg", "co...

[ "apply", "cycle_id", "dvdn_exponent", "eqn_dvd", "exponent", "exponent_dvdn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d