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	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
card_void : #\|{: void}\| = 0.	Proof. by rewrite cardT enumT unlock. Qed.	Lemma	card_void	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardT", "enumT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enumF T	:= (Finite.enum T).	Notation	enumF	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
option_enum	:= None :: map some (enumF T).	Definition	option_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumF", "map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
option_enumP : Finite.axiom option_enum.	Proof. by case=> [x\|]; rewrite /= count_map (count_pred0, enumP). Qed.	Lemma	option_enumP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "axiom", "count_map", "count_pred0", "enumP", "option_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_option : #\|{: option T}\| = #\|T\|.+1.	Proof. by rewrite !cardT !enumT [in LHS]unlock /= !size_map. Qed.	Lemma	card_option	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardT", "enumT", "size_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcan_enumP g : pcancel f g -> Finite.axiom (undup (pmap g (enumF fT))).	Proof. move=> fK x; rewrite count_uniq_mem ?undup_uniq // mem_undup. by rewrite mem_pmap -fK map_f // -enumT mem_enum. Qed.	Lemma	pcan_enumP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "axiom", "count_uniq_mem", "enumF", "enumT", "fK", "fT", "map_f", "mem_enum", "mem_pmap", "mem_undup", "pmap", "undup", "undup_uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCanIsFinite g fK	:= @isFinite.Build _ _ (@pcan_enumP g fK).	Definition	PCanIsFinite	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Build", "fK", "pcan_enumP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
CanIsFinite g (fK : cancel f g)	:= PCanIsFinite (can_pcan fK).	Definition	CanIsFinite	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "PCanIsFinite", "fK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
codom_val sT x : (x \in codom (val : sT -> T)) = P x.	Proof. by apply/codomP/idP=> [[u ->]\|Px]; last exists (Sub x Px); rewrite ?valP ?SubK. Qed.	Lemma	codom_val	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Px", "Sub", "SubK", "apply", "codom", "codomP", "last", "sT", "val", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_enum : seq sT	:= pmap insub (enumF T).	Definition	sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumF", "insub", "pmap", "sT", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_sub_enum u : u \in sub_enum.	Proof. by rewrite mem_pmap_sub -enumT mem_enum. Qed.	Lemma	mem_sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumT", "mem_enum", "mem_pmap_sub", "sub_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_enum_uniq : uniq sub_enum.	Proof. by rewrite pmap_sub_uniq // -enumT enum_uniq. Qed.	Lemma	sub_enum_uniq	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumT", "enum_uniq", "pmap_sub_uniq", "sub_enum", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_sub_enum : map val sub_enum = enum P.	Proof. rewrite pmap_filter; first exact: insubK. by apply: eq_filter => x; apply: isSome_insub. Qed.	Lemma	val_sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "eq_filter", "insubK", "isSome_insub", "map", "pmap_filter", "sub_enum", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Finite' 'of' T 'by' <: ]"	:= (Finite.copy T%type (sub_type T%type)) (format "[ 'Finite' 'of' T 'by' <: ]") : form_scope.	Notation	[ 'Finite' 'of' T 'by' <: ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "copy", "sub_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_sub : #\|sfT\| = #\|[pred x \| P x]\|.	Proof. by rewrite -(eq_card (codom_val sfT)) (card_image val_inj). Qed.	Lemma	card_sub	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card_image", "codom_val", "eq_card", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card_sub (A : {pred sfT}) : A =i predT -> #\|A\| = #\|[pred x \| P x]\|.	Proof. exact: eq_card_trans card_sub. Qed.	Lemma	eq_card_sub	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card_sub", "eq_card_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_sig : #\|{: {x \| P x}}\| = #\|[pred x \| P x]\|.	Proof. exact: card_sub. Qed.	Lemma	card_sig	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub : Type	:= SeqSub {ssval : T; ssvalP : in_mem ssval (@mem T _ s)}.	Record	seq_sub	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_enum : seq seq_sub	:= undup (pmap insub s).	Definition	seq_sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "insub", "pmap", "seq", "seq_sub", "undup" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_seq_sub_enum x : x \in seq_sub_enum.	Proof. by rewrite mem_undup mem_pmap -valK map_f ?ssvalP. Qed.	Lemma	mem_seq_sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "map_f", "mem_pmap", "mem_undup", "seq_sub_enum", "valK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_seq_sub_enum : uniq s -> map val seq_sub_enum = s.	Proof. move=> Us; rewrite /seq_sub_enum undup_id ?pmap_sub_uniq //. rewrite (pmap_filter (insubK _)); apply/all_filterP. by apply/allP => x; rewrite isSome_insub. Qed.	Lemma	val_seq_sub_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "allP", "all_filterP", "apply", "insubK", "isSome_insub", "map", "pmap_filter", "pmap_sub_uniq", "seq_sub_enum", "undup_id", "uniq", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_pickle x	:= index x seq_sub_enum.	Definition	seq_sub_pickle	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "index", "seq_sub_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_unpickle n	:= nth None (map some seq_sub_enum) n.	Definition	seq_sub_unpickle	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "map", "nth", "seq_sub_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_pickleK : pcancel seq_sub_pickle seq_sub_unpickle.	Proof. rewrite /seq_sub_unpickle => x. by rewrite (nth_map x) ?nth_index ?index_mem ?mem_seq_sub_enum. Qed.	Lemma	seq_sub_pickleK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "index_mem", "mem_seq_sub_enum", "nth_index", "nth_map", "seq_sub_pickle", "seq_sub_unpickle" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_isCountable	:= isCountable.Build seq_sub seq_sub_pickleK.	Definition	seq_sub_isCountable	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Build", "isCountable", "seq_sub", "seq_sub_pickleK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_axiom : Finite.axiom seq_sub_enum.	Proof. exact: Finite.uniq_enumP (undup_uniq _) mem_seq_sub_enum. Qed.	Fact	seq_sub_axiom	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "axiom", "mem_seq_sub_enum", "seq_sub_enum", "undup_uniq", "uniq_enumP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_isFinite	:= isFinite.Build seq_sub seq_sub_axiom.	Definition	seq_sub_isFinite	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Build", "seq_sub", "seq_sub_axiom" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adhoc_seq_sub_choiceType : choiceType	:= pcan_type seq_sub_pickleK.	Definition	adhoc_seq_sub_choiceType	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "pcan_type", "seq_sub_pickleK" ]	with the generic sub_choiceType canonical instance.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adhoc_seq_sub_countType	:= HB.pack_for countType seq_sub seq_sub_isCountable (Choice.class adhoc_seq_sub_choiceType).	Definition	adhoc_seq_sub_countType	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "adhoc_seq_sub_choiceType", "class", "seq_sub", "seq_sub_isCountable" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
adhoc_seq_sub_finType	:= HB.pack_for finType seq_sub seq_sub_isFinite seq_sub_isCountable (Choice.class adhoc_seq_sub_choiceType).	Definition	adhoc_seq_sub_finType	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "adhoc_seq_sub_choiceType", "class", "seq_sub", "seq_sub_isCountable", "seq_sub_isFinite" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_sub_default s : size s > 0 -> seq_sub s.	Proof. by case: s => // x s _; exists x; rewrite mem_head. Qed.	Lemma	seq_sub_default	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "mem_head", "seq_sub", "size" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
seq_subE s (s_gt0 : size s > 0) : s = map val (map (insubd (seq_sub_default s_gt0)) s : seq (seq_sub s)).	Proof. by rewrite -map_comp map_id_in// => x x_in_s /=; rewrite insubdK. Qed.	Lemma	seq_subE	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "insubd", "insubdK", "map", "map_comp", "map_id_in", "seq", "seq_sub", "seq_sub_default", "size", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
in_sub_seq s_gt0	:= (insubd (seq_sub_default s_gt0)).	Notation	in_sub_seq	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "insubd", "seq_sub_default" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sT	:= (seq_sub s).	Notation	sT	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "seq_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_seq_sub : uniq s -> #\|{:sT}\| = size s.	Proof. by move=> Us; rewrite cardE enumT -(size_map val) unlock val_seq_sub_enum. Qed.	Lemma	card_seq_sub	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "enumT", "sT", "size", "size_map", "uniq", "val", "val_seq_sub_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extremum_spec {T : eqType} (ord : rel T) {I : finType} (P : pred I) (F : I -> T) : I -> Type	:= ExtremumSpec (i : I) of P i & (forall j : I, P j -> ord (F i) (F j)) : extremum_spec ord P F i.	Variant	extremum_spec	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_pred {T : eqType} ord {I : finType} (P : pred I) (F : I -> T)	:= [pred i \| P i & [forall (j \| P j), ord (F i) (F j)]].	Let	arg_pred	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extremum	:= odflt i0 (pick (arg_pred ord P F)).	Definition	extremum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "arg_pred", "i0", "pick" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_refl : reflexive ord.		Hypothesis	ord_refl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_trans : transitive ord.		Hypothesis	ord_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_total : total ord.		Hypothesis	ord_total	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "total" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Pi0 : P i0.		Hypothesis	Pi0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extremumP : extremum_spec ord P F extremum.	Proof. rewrite /extremum; case: pickP => [i /andP[Pi /'forall_implyP/= min_i] \| no_i]. by split=> // j; apply/implyP. have := sort_sorted ord_total [seq F i \| i <- enum P]. set s := sort _ _ => ss; have s_gt0 : size s > 0 by rewrite size_sort size_map -cardE; apply/card_gt0P; exists i0. pose t0 := nth (F i0) s 0; ...	Lemma	extremumP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardE", "card_gt0P", "enum", "extremum", "extremum_spec", "i0", "mapP", "map_f", "mem_enum", "mem_nth", "mem_sort", "nth", "nthP", "ord_total", "pickP", "seq", "sig2_eqW", "size", "size_map", "size_sort", "sort", "sort_sorted", "sorted_leq_nth", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_refl : {in P, reflexive (relpre F ord)}.		Hypothesis	ord_refl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_trans : {in P & P & P, transitive (relpre F ord)}.		Hypothesis	ord_trans	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_total : {in P &, total (relpre F ord)}.		Hypothesis	ord_total	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "total" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
extremum_inP : extremum_spec ord P F (extremum ord i0 P F).	Proof. rewrite /extremum; case: pickP => [i /andP[Pi /'forall_implyP/= min_i] \| no_i]. by split=> // j; apply/implyP. pose TP := seq_sub [seq F i \| i <- enum P]. have FPP (iP : {i \| P i}) : F (proj1_sig iP) \in [seq F i \| i <- enum P]. by rewrite map_f// mem_enum; apply: valP. pose FP := SeqSub (FPP _). have []//= ...	Lemma	extremum_inP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Pi0", "apply", "enum", "existsP", "extremum", "extremumP", "extremum_spec", "i0", "mapP", "map_f", "mem_enum", "negb_forall", "ord_refl", "ord_total", "ord_trans", "pickP", "seq", "seq_sub", "sigW", "split", "val", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg[' ord ]_( i < i0 \| P ) F ]"	:= (extremum ord i0 (fun i => P%B) (fun i => F)) (ord, i, i0 at level 10, format "[ 'arg[' ord ]_( i < i0 \| P ) F ]") : nat_scope.	Notation	[ 'arg[' ord ]_( i < i0 \| P ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "extremum", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg[' ord ]_( i < i0 'in' A ) F ]"	:= [arg[ord]_(i < i0 \| i \in A) F] (format "[ 'arg[' ord ]_( i < i0 'in' A ) F ]") : nat_scope.	Notation	[ 'arg[' ord ]_( i < i0 'in' A ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg[' ord ]_( i < i0 ) F ]"	:= [arg[ord]_(i < i0 \| true) F] (format "[ 'arg[' ord ]_( i < i0 ) F ]") : nat_scope.	Notation	[ 'arg[' ord ]_( i < i0 ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_min	:= extremum leq i0 P F.	Definition	arg_min	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "extremum", "i0", "leq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_max	:= extremum geq i0 P F.	Definition	arg_max	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "extremum", "geq", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_minnP : extremum_spec leq P F arg_min.	Proof. by apply: extremumP => //; [apply: leq_trans\|apply: leq_total]. Qed.	Lemma	arg_minnP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "arg_min", "extremumP", "extremum_spec", "leq", "leq_total", "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
arg_maxnP : extremum_spec geq P F arg_max.	Proof. apply: extremumP => //; first exact: leqnn. by move=> n m p mn np; apply: leq_trans mn. by move=> ??; apply: leq_total. Qed.	Lemma	arg_maxnP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "arg_max", "extremumP", "extremum_spec", "geq", "leq_total", "leq_trans", "leqnn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'min_' ( i < i0 \| P ) F ]"	:= (arg_min i0 (fun i => P%B) (fun i => F)) (i, i0 at level 10, format "[ 'arg' 'min_' ( i < i0 \| P ) F ]") : nat_scope.	Notation	[ 'arg' 'min_' ( i < i0 \| P ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "arg_min", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'min_' ( i < i0 'in' A ) F ]"	:= [arg min_(i < i0 \| i \in A) F] (format "[ 'arg' 'min_' ( i < i0 'in' A ) F ]") : nat_scope.	Notation	[ 'arg' 'min_' ( i < i0 'in' A ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'min_' ( i < i0 ) F ]"	:= [arg min_(i < i0 \| true) F] (format "[ 'arg' 'min_' ( i < i0 ) F ]") : nat_scope.	Notation	[ 'arg' 'min_' ( i < i0 ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'max_' ( i > i0 \| P ) F ]"	:= (arg_max i0 (fun i => P%B) (fun i => F)) (i, i0 at level 10, format "[ 'arg' 'max_' ( i > i0 \| P ) F ]") : nat_scope.	Notation	[ 'arg' 'max_' ( i > i0 \| P ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "arg_max", "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'max_' ( i > i0 'in' A ) F ]"	:= [arg max_(i > i0 \| i \in A) F] (format "[ 'arg' 'max_' ( i > i0 'in' A ) F ]") : nat_scope.	Notation	[ 'arg' 'max_' ( i > i0 'in' A ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'arg' 'max_' ( i > i0 ) F ]"	:= [arg max_(i > i0 \| true) F] (format "[ 'arg' 'max_' ( i > i0 ) F ]") : nat_scope.	Notation	[ 'arg' 'max_' ( i > i0 ) F ]	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "i0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordinal : predArgType	:= Ordinal m of m < n.	Inductive	ordinal	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat_of_ord i	:= let: Ordinal m _ := i in m.	Coercion	nat_of_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ltn_ord (i : ordinal) : i < n.	Proof. exact: valP i. Qed.	Lemma	ltn_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordinal", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_inj : injective nat_of_ord.	Proof. exact: val_inj. Qed.	Lemma	ord_inj	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "nat_of_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_enum : seq ordinal	:= pmap insub (iota 0 n).	Definition	ord_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "insub", "iota", "ordinal", "pmap", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_ord_enum : map val ord_enum = iota 0 n.	Proof. rewrite pmap_filter; first exact: insubK. by apply/all_filterP; apply/allP=> i; rewrite mem_iota isSome_insub. Qed.	Lemma	val_ord_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "allP", "all_filterP", "apply", "insubK", "iota", "isSome_insub", "map", "mem_iota", "ord_enum", "pmap_filter", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_enum_uniq : uniq ord_enum.	Proof. by rewrite pmap_sub_uniq ?iota_uniq. Qed.	Lemma	ord_enum_uniq	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "iota_uniq", "ord_enum", "pmap_sub_uniq", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_ord_enum i : i \in ord_enum.	Proof. by rewrite -(mem_map ord_inj) val_ord_enum mem_iota ltn_ord. Qed.	Lemma	mem_ord_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ltn_ord", "mem_iota", "mem_map", "ord_enum", "ord_inj", "val_ord_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"''I_' n"	:= (ordinal n) (at level 0, n at level 2, format "''I_' n").	Notation	''I_' n	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordinal" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
val_enum_ord : map val (enum 'I_n) = iota 0 n.	Proof. by rewrite enumT unlock val_ord_enum. Qed.	Lemma	val_enum_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "enumT", "iota", "map", "val", "val_ord_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
size_enum_ord : size (enum 'I_n) = n.	Proof. by rewrite -(size_map val) val_enum_ord size_iota. Qed.	Lemma	size_enum_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "size", "size_iota", "size_map", "val", "val_enum_ord" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_ord : #\|'I_n\| = n.	Proof. by rewrite cardE size_enum_ord. Qed.	Lemma	card_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "size_enum_ord" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_enum_ord i0 m : m < n -> nth i0 (enum 'I_n) m = m :> nat.	Proof. by move=> ?; rewrite -(nth_map _ 0) (size_enum_ord, val_enum_ord) // nth_iota. Qed.	Lemma	nth_enum_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "i0", "nat", "nth", "nth_iota", "nth_map", "size_enum_ord", "val_enum_ord" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nth_ord_enum (i0 i : 'I_n) : nth i0 (enum 'I_n) i = i.	Proof. by apply: val_inj; apply: nth_enum_ord. Qed.	Lemma	nth_ord_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "i0", "nth", "nth_enum_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
index_enum_ord (i : 'I_n) : index i (enum 'I_n) = i.	Proof. by rewrite -[in LHS](nth_ord_enum i i) index_uniq ?(enum_uniq, size_enum_ord). Qed.	Lemma	index_enum_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "enum_uniq", "index", "index_uniq", "nth_ord_enum", "size_enum_ord" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mask_enum_ord m : mask m (enum 'I_n) = [seq i <- enum 'I_n \| nth false m (val i)].	Proof. rewrite mask_filter ?enum_uniq//; apply: eq_filter => i. by rewrite in_mask ?enum_uniq ?mem_enum// index_enum_ord. Qed.	Lemma	mask_enum_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "enum_uniq", "eq_filter", "in_mask", "index_enum_ord", "mask", "mask_filter", "mem_enum", "nth", "seq", "val" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
enum_ord0 : enum 'I_0 = [::].	Proof. by apply/eqP; rewrite -size_eq0 size_enum_ord. Qed.	Lemma	enum_ord0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "enum", "size_enum_ord", "size_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_ord_proof n m (i : 'I_n) : n <= m -> i < m.	Proof. exact: leq_trans. Qed.	Lemma	widen_ord_proof	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "leq_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
widen_ord n m le_n_m i	:= Ordinal (@widen_ord_proof n m i le_n_m).	Definition	widen_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "widen_ord_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord_proof n m (i : 'I_n) : n = m -> i < m.	Proof. by move <-. Qed.	Lemma	cast_ord_proof	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord n m eq_n_m i	:= Ordinal (@cast_ord_proof n m i eq_n_m).	Definition	cast_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cast_ord_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord_id n eq_n i : cast_ord eq_n i = i :> 'I_n.	Proof. exact: val_inj. Qed.	Lemma	cast_ord_id	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cast_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord_comp n1 n2 n3 eq_n2 eq_n3 i : @cast_ord n2 n3 eq_n3 (@cast_ord n1 n2 eq_n2 i) = cast_ord (etrans eq_n2 eq_n3) i.	Proof. exact: val_inj. Qed.	Lemma	cast_ord_comp	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cast_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ordK n1 n2 eq_n : cancel (@cast_ord n1 n2 eq_n) (cast_ord (esym eq_n)).	Proof. by move=> i; apply: val_inj. Qed.	Lemma	cast_ordK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cast_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ordKV n1 n2 eq_n : cancel (cast_ord (esym eq_n)) (@cast_ord n1 n2 eq_n).	Proof. by move=> i; apply: val_inj. Qed.	Lemma	cast_ordKV	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cast_ord", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cast_ord_inj n1 n2 eq_n : injective (@cast_ord n1 n2 eq_n).	Proof. exact: can_inj (cast_ordK eq_n). Qed.	Lemma	cast_ord_inj	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cast_ord", "cast_ordK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordS_subproof n (i : 'I_n) : i.+1 %% n < n.	Proof. by case: n i => [\|n] [m m_lt]//=; rewrite ltn_pmod. Qed.	Fact	ordS_subproof	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ltn_pmod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordS n (i : 'I_n)	:= Ordinal (ordS_subproof i).	Definition	ordS	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordS_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_pred_subproof n (i : 'I_n) : (i + n).-1 %% n < n.	Proof. by case: n i => [\|n] [m m_lt]//=; rewrite ltn_pmod. Qed.	Fact	ord_pred_subproof	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ltn_pmod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_pred n (i : 'I_n)	:= Ordinal (ord_pred_subproof i).	Definition	ord_pred	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ord_pred_subproof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordSK n : cancel (@ordS n) (@ord_pred n).	Proof. move=> [i ilt]; apply/val_inj => /=. case: (ltngtP i.+1) (ilt) => // [Silt\|<-]; last by rewrite modnn/= modn_small. by rewrite [i.+1 %% n]modn_small// addSn/= modnDr modn_small. Qed.	Lemma	ordSK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "addSn", "apply", "last", "ltngtP", "modnDr", "modn_small", "modnn", "ordS", "ord_pred", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_predK n : cancel (@ord_pred n) (@ordS n).	Proof. move=> [[\|i] ilt]; apply/val_inj => /=. by rewrite [n.-1 %% n]modn_small// prednK// modnn. by rewrite modnDr [i %% n]modn_small ?modn_small// ltnW. Qed.	Lemma	ord_predK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "ltnW", "modnDr", "modn_small", "modnn", "ordS", "ord_pred", "prednK", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordS_bij n : bijective (@ordS n).	Proof. exact: (Bijective (@ordSK n) (@ord_predK n)). Qed.	Lemma	ordS_bij	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordS", "ordSK", "ord_predK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ordS_inj n : injective (@ordS n).	Proof. exact: (bij_inj (ordS_bij n)). Qed.	Lemma	ordS_inj	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordS", "ordS_bij" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_pred_bij n : bijective (@ord_pred n).	Proof. exact (Bijective (@ord_predK n) (@ordSK n)). Qed.	Lemma	ord_pred_bij	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ordSK", "ord_pred", "ord_predK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ord_pred_inj n : injective (@ord_pred n).	Proof. exact: (bij_inj (ord_pred_bij n)). Qed.	Lemma	ord_pred_inj	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ord_pred", "ord_pred_bij" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_ord_proof n (i : 'I_n) : n - i.+1 < n.	Proof. by case: n i => [\|n] [i lt_i_n] //; rewrite ltnS subSS leq_subr. Qed.	Lemma	rev_ord_proof	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "leq_subr", "ltnS", "subSS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_ord n i	:= Ordinal (@rev_ord_proof n i).	Definition	rev_ord	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "rev_ord_proof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_ordK {n} : involutive (@rev_ord n).	Proof. by case: n => [\|n] [i lti] //; apply: val_inj; rewrite /= !subSS subKn. Qed.	Lemma	rev_ordK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "rev_ord", "subKn", "subSS", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_ord_inj {n} : injective (@rev_ord n).	Proof. exact: inv_inj rev_ordK. Qed.	Lemma	rev_ord_inj	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "rev_ord", "rev_ordK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_void : #|{: void}| = 0.

Proof. by rewrite cardT enumT unlock. Qed.

Lemma

card_void

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardT", "enumT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enumF T

:= (Finite.enum T).

Notation

enumF

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

option_enum

:= None :: map some (enumF T).

Definition

option_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumF", "map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

option_enumP : Finite.axiom option_enum.

Proof. by case=> [x|]; rewrite /= count_map (count_pred0, enumP). Qed.

Lemma

option_enumP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "axiom", "count_map", "count_pred0", "enumP", "option_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_option : #|{: option T}| = #|T|.+1.

Proof. by rewrite !cardT !enumT [in LHS]unlock /= !size_map. Qed.

Lemma

card_option

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardT", "enumT", "size_map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcan_enumP g : pcancel f g -> Finite.axiom (undup (pmap g (enumF fT))).

Proof. move=> fK x; rewrite count_uniq_mem ?undup_uniq // mem_undup. by rewrite mem_pmap -fK map_f // -enumT mem_enum. Qed.

Lemma

pcan_enumP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "axiom", "count_uniq_mem", "enumF", "enumT", "fK", "fT", "map_f", "mem_enum", "mem_pmap", "mem_undup", "pmap", "undup", "undup_uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PCanIsFinite g fK

:= @isFinite.Build _ _ (@pcan_enumP g fK).

Definition

PCanIsFinite

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Build", "fK", "pcan_enumP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

CanIsFinite g (fK : cancel f g)

:= PCanIsFinite (can_pcan fK).

Definition

CanIsFinite

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "PCanIsFinite", "fK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

codom_val sT x : (x \in codom (val : sT -> T)) = P x.

Proof. by apply/codomP/idP=> [[u ->]|Px]; last exists (Sub x Px); rewrite ?valP ?SubK. Qed.

Lemma

codom_val

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Px", "Sub", "SubK", "apply", "codom", "codomP", "last", "sT", "val", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_enum : seq sT

:= pmap insub (enumF T).

Definition

sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumF", "insub", "pmap", "sT", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_sub_enum u : u \in sub_enum.

Proof. by rewrite mem_pmap_sub -enumT mem_enum. Qed.

Lemma

mem_sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumT", "mem_enum", "mem_pmap_sub", "sub_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_enum_uniq : uniq sub_enum.

Proof. by rewrite pmap_sub_uniq // -enumT enum_uniq. Qed.

Lemma

sub_enum_uniq

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumT", "enum_uniq", "pmap_sub_uniq", "sub_enum", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_sub_enum : map val sub_enum = enum P.

Proof. rewrite pmap_filter; first exact: insubK. by apply: eq_filter => x; apply: isSome_insub. Qed.

Lemma

val_sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "eq_filter", "insubK", "isSome_insub", "map", "pmap_filter", "sub_enum", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Finite' 'of' T 'by' <: ]"

:= (Finite.copy T%type (sub_type T%type)) (format "[ 'Finite' 'of' T 'by' <: ]") : form_scope.

Notation

[ 'Finite' 'of' T 'by' <: ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "copy", "sub_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_sub : #|sfT| = #|[pred x | P x]|.

Proof. by rewrite -(eq_card (codom_val sfT)) (card_image val_inj). Qed.

Lemma

card_sub

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card_image", "codom_val", "eq_card", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card_sub (A : {pred sfT}) : A =i predT -> #|A| = #|[pred x | P x]|.

Proof. exact: eq_card_trans card_sub. Qed.

Lemma

eq_card_sub

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card_sub", "eq_card_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_sig : #|{: {x | P x}}| = #|[pred x | P x]|.

Proof. exact: card_sub. Qed.

Lemma

card_sig

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub : Type

:= SeqSub {ssval : T; ssvalP : in_mem ssval (@mem T _ s)}.

Record

seq_sub

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_enum : seq seq_sub

:= undup (pmap insub s).

Definition

seq_sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "insub", "pmap", "seq", "seq_sub", "undup" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_seq_sub_enum x : x \in seq_sub_enum.

Proof. by rewrite mem_undup mem_pmap -valK map_f ?ssvalP. Qed.

Lemma

mem_seq_sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "map_f", "mem_pmap", "mem_undup", "seq_sub_enum", "valK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_seq_sub_enum : uniq s -> map val seq_sub_enum = s.

Proof. move=> Us; rewrite /seq_sub_enum undup_id ?pmap_sub_uniq //. rewrite (pmap_filter (insubK _)); apply/all_filterP. by apply/allP => x; rewrite isSome_insub. Qed.

Lemma

val_seq_sub_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "allP", "all_filterP", "apply", "insubK", "isSome_insub", "map", "pmap_filter", "pmap_sub_uniq", "seq_sub_enum", "undup_id", "uniq", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_pickle x

:= index x seq_sub_enum.

Definition

seq_sub_pickle

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "index", "seq_sub_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_unpickle n

:= nth None (map some seq_sub_enum) n.

Definition

seq_sub_unpickle

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "map", "nth", "seq_sub_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_pickleK : pcancel seq_sub_pickle seq_sub_unpickle.

Proof. rewrite /seq_sub_unpickle => x. by rewrite (nth_map x) ?nth_index ?index_mem ?mem_seq_sub_enum. Qed.

Lemma

seq_sub_pickleK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "index_mem", "mem_seq_sub_enum", "nth_index", "nth_map", "seq_sub_pickle", "seq_sub_unpickle" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_isCountable

:= isCountable.Build seq_sub seq_sub_pickleK.

Definition

seq_sub_isCountable

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Build", "isCountable", "seq_sub", "seq_sub_pickleK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_axiom : Finite.axiom seq_sub_enum.

Proof. exact: Finite.uniq_enumP (undup_uniq _) mem_seq_sub_enum. Qed.

Fact

seq_sub_axiom

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "axiom", "mem_seq_sub_enum", "seq_sub_enum", "undup_uniq", "uniq_enumP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_isFinite

:= isFinite.Build seq_sub seq_sub_axiom.

Definition

seq_sub_isFinite

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Build", "seq_sub", "seq_sub_axiom" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adhoc_seq_sub_choiceType : choiceType

:= pcan_type seq_sub_pickleK.

Definition

adhoc_seq_sub_choiceType

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "pcan_type", "seq_sub_pickleK" ]

with the generic sub_choiceType canonical instance.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adhoc_seq_sub_countType

:= HB.pack_for countType seq_sub seq_sub_isCountable (Choice.class adhoc_seq_sub_choiceType).

Definition

adhoc_seq_sub_countType

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "adhoc_seq_sub_choiceType", "class", "seq_sub", "seq_sub_isCountable" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

adhoc_seq_sub_finType

:= HB.pack_for finType seq_sub seq_sub_isFinite seq_sub_isCountable (Choice.class adhoc_seq_sub_choiceType).

Definition

adhoc_seq_sub_finType

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "adhoc_seq_sub_choiceType", "class", "seq_sub", "seq_sub_isCountable", "seq_sub_isFinite" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_sub_default s : size s > 0 -> seq_sub s.

Proof. by case: s => // x s _; exists x; rewrite mem_head. Qed.

Lemma

seq_sub_default

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "mem_head", "seq_sub", "size" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

seq_subE s (s_gt0 : size s > 0) : s = map val (map (insubd (seq_sub_default s_gt0)) s : seq (seq_sub s)).

Proof. by rewrite -map_comp map_id_in// => x x_in_s /=; rewrite insubdK. Qed.

Lemma

seq_subE

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "insubd", "insubdK", "map", "map_comp", "map_id_in", "seq", "seq_sub", "seq_sub_default", "size", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

in_sub_seq s_gt0

:= (insubd (seq_sub_default s_gt0)).

Notation

in_sub_seq

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "insubd", "seq_sub_default" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sT

:= (seq_sub s).

Notation

sT

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "seq_sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_seq_sub : uniq s -> #|{:sT}| = size s.

Proof. by move=> Us; rewrite cardE enumT -(size_map val) unlock val_seq_sub_enum. Qed.

Lemma

card_seq_sub

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "enumT", "sT", "size", "size_map", "uniq", "val", "val_seq_sub_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extremum_spec {T : eqType} (ord : rel T) {I : finType} (P : pred I) (F : I -> T) : I -> Type

:= ExtremumSpec (i : I) of P i & (forall j : I, P j -> ord (F i) (F j)) : extremum_spec ord P F i.

Variant

extremum_spec

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_pred {T : eqType} ord {I : finType} (P : pred I) (F : I -> T)

:= [pred i | P i & [forall (j | P j), ord (F i) (F j)]].

Let

arg_pred

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extremum

:= odflt i0 (pick (arg_pred ord P F)).

Definition

extremum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "arg_pred", "i0", "pick" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_refl : reflexive ord.

Hypothesis

ord_refl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_trans : transitive ord.

Hypothesis

ord_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_total : total ord.

Hypothesis

ord_total

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "total" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Pi0 : P i0.

Hypothesis

Pi0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extremumP : extremum_spec ord P F extremum.

Proof. rewrite /extremum; case: pickP => [i /andP[Pi /'forall_implyP/= min_i] | no_i]. by split=> // j; apply/implyP. have := sort_sorted ord_total [seq F i | i <- enum P]. set s := sort _ _ => ss; have s_gt0 : size s > 0 by rewrite size_sort size_map -cardE; apply/card_gt0P; exists i0. pose t0 := nth (F i0) s 0; ...

Lemma

extremumP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardE", "card_gt0P", "enum", "extremum", "extremum_spec", "i0", "mapP", "map_f", "mem_enum", "mem_nth", "mem_sort", "nth", "nthP", "ord_total", "pickP", "seq", "sig2_eqW", "size", "size_map", "size_sort", "sort", "sort_sorted", "sorted_leq_nth", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_refl : {in P, reflexive (relpre F ord)}.

Hypothesis

ord_refl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_trans : {in P & P & P, transitive (relpre F ord)}.

Hypothesis

ord_trans

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_total : {in P &, total (relpre F ord)}.

Hypothesis

ord_total

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "total" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

extremum_inP : extremum_spec ord P F (extremum ord i0 P F).

Proof. rewrite /extremum; case: pickP => [i /andP[Pi /'forall_implyP/= min_i] | no_i]. by split=> // j; apply/implyP. pose TP := seq_sub [seq F i | i <- enum P]. have FPP (iP : {i | P i}) : F (proj1_sig iP) \in [seq F i | i <- enum P]. by rewrite map_f// mem_enum; apply: valP. pose FP := SeqSub (FPP _). have []//= ...

Lemma

extremum_inP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Pi0", "apply", "enum", "existsP", "extremum", "extremumP", "extremum_spec", "i0", "mapP", "map_f", "mem_enum", "negb_forall", "ord_refl", "ord_total", "ord_trans", "pickP", "seq", "seq_sub", "sigW", "split", "val", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg[' ord ]_( i < i0 | P ) F ]"

:= (extremum ord i0 (fun i => P%B) (fun i => F)) (ord, i, i0 at level 10, format "[ 'arg[' ord ]_( i < i0 | P ) F ]") : nat_scope.

Notation

[ 'arg[' ord ]_( i < i0 | P ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "extremum", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg[' ord ]_( i < i0 'in' A ) F ]"

:= [arg[ord]_(i < i0 | i \in A) F] (format "[ 'arg[' ord ]_( i < i0 'in' A ) F ]") : nat_scope.

Notation

[ 'arg[' ord ]_( i < i0 'in' A ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg[' ord ]_( i < i0 ) F ]"

:= [arg[ord]_(i < i0 | true) F] (format "[ 'arg[' ord ]_( i < i0 ) F ]") : nat_scope.

Notation

[ 'arg[' ord ]_( i < i0 ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_min

:= extremum leq i0 P F.

Definition

arg_min

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "extremum", "i0", "leq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_max

:= extremum geq i0 P F.

Definition

arg_max

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "extremum", "geq", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_minnP : extremum_spec leq P F arg_min.

Proof. by apply: extremumP => //; [apply: leq_trans|apply: leq_total]. Qed.

Lemma

arg_minnP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "arg_min", "extremumP", "extremum_spec", "leq", "leq_total", "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

arg_maxnP : extremum_spec geq P F arg_max.

Proof. apply: extremumP => //; first exact: leqnn. by move=> n m p mn np; apply: leq_trans mn. by move=> ??; apply: leq_total. Qed.

Lemma

arg_maxnP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "arg_max", "extremumP", "extremum_spec", "geq", "leq_total", "leq_trans", "leqnn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'min_' ( i < i0 | P ) F ]"

:= (arg_min i0 (fun i => P%B) (fun i => F)) (i, i0 at level 10, format "[ 'arg' 'min_' ( i < i0 | P ) F ]") : nat_scope.

Notation

[ 'arg' 'min_' ( i < i0 | P ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "arg_min", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'min_' ( i < i0 'in' A ) F ]"

:= [arg min_(i < i0 | i \in A) F] (format "[ 'arg' 'min_' ( i < i0 'in' A ) F ]") : nat_scope.

Notation

[ 'arg' 'min_' ( i < i0 'in' A ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'min_' ( i < i0 ) F ]"

:= [arg min_(i < i0 | true) F] (format "[ 'arg' 'min_' ( i < i0 ) F ]") : nat_scope.

Notation

[ 'arg' 'min_' ( i < i0 ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'max_' ( i > i0 | P ) F ]"

:= (arg_max i0 (fun i => P%B) (fun i => F)) (i, i0 at level 10, format "[ 'arg' 'max_' ( i > i0 | P ) F ]") : nat_scope.

Notation

[ 'arg' 'max_' ( i > i0 | P ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "arg_max", "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'max_' ( i > i0 'in' A ) F ]"

:= [arg max_(i > i0 | i \in A) F] (format "[ 'arg' 'max_' ( i > i0 'in' A ) F ]") : nat_scope.

Notation

[ 'arg' 'max_' ( i > i0 'in' A ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'arg' 'max_' ( i > i0 ) F ]"

:= [arg max_(i > i0 | true) F] (format "[ 'arg' 'max_' ( i > i0 ) F ]") : nat_scope.

Notation

[ 'arg' 'max_' ( i > i0 ) F ]

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "i0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordinal : predArgType

:= Ordinal m of m < n.

Inductive

ordinal

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat_of_ord i

:= let: Ordinal m _ := i in m.

Coercion

nat_of_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ltn_ord (i : ordinal) : i < n.

Proof. exact: valP i. Qed.

Lemma

ltn_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordinal", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_inj : injective nat_of_ord.

Proof. exact: val_inj. Qed.

Lemma

ord_inj

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "nat_of_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_enum : seq ordinal

:= pmap insub (iota 0 n).

Definition

ord_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "insub", "iota", "ordinal", "pmap", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_ord_enum : map val ord_enum = iota 0 n.

Proof. rewrite pmap_filter; first exact: insubK. by apply/all_filterP; apply/allP=> i; rewrite mem_iota isSome_insub. Qed.

Lemma

val_ord_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "allP", "all_filterP", "apply", "insubK", "iota", "isSome_insub", "map", "mem_iota", "ord_enum", "pmap_filter", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_enum_uniq : uniq ord_enum.

Proof. by rewrite pmap_sub_uniq ?iota_uniq. Qed.

Lemma

ord_enum_uniq

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "iota_uniq", "ord_enum", "pmap_sub_uniq", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_ord_enum i : i \in ord_enum.

Proof. by rewrite -(mem_map ord_inj) val_ord_enum mem_iota ltn_ord. Qed.

Lemma

mem_ord_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ltn_ord", "mem_iota", "mem_map", "ord_enum", "ord_inj", "val_ord_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"''I_' n"

:= (ordinal n) (at level 0, n at level 2, format "''I_' n").

Notation

''I_' n

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordinal" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

val_enum_ord : map val (enum 'I_n) = iota 0 n.

Proof. by rewrite enumT unlock val_ord_enum. Qed.

Lemma

val_enum_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "enumT", "iota", "map", "val", "val_ord_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

size_enum_ord : size (enum 'I_n) = n.

Proof. by rewrite -(size_map val) val_enum_ord size_iota. Qed.

Lemma

size_enum_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "size", "size_iota", "size_map", "val", "val_enum_ord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_ord : #|'I_n| = n.

Proof. by rewrite cardE size_enum_ord. Qed.

Lemma

card_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "size_enum_ord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by move=> ?; rewrite -(nth_map _ 0) (size_enum_ord, val_enum_ord) // nth_iota. Qed.

Lemma

nth_enum_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "i0", "nat", "nth", "nth_iota", "nth_map", "size_enum_ord", "val_enum_ord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by apply: val_inj; apply: nth_enum_ord. Qed.

Lemma

nth_ord_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "i0", "nth", "nth_enum_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite -[in LHS](nth_ord_enum i i) index_uniq ?(enum_uniq, size_enum_ord). Qed.

Lemma

index_enum_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "enum_uniq", "index", "index_uniq", "nth_ord_enum", "size_enum_ord" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mask_enum_ord m : mask m (enum 'I_n) = [seq i <- enum 'I_n | nth false m (val i)].

Proof. rewrite mask_filter ?enum_uniq//; apply: eq_filter => i. by rewrite in_mask ?enum_uniq ?mem_enum// index_enum_ord. Qed.

Lemma

mask_enum_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "enum_uniq", "eq_filter", "in_mask", "index_enum_ord", "mask", "mask_filter", "mem_enum", "nth", "seq", "val" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

enum_ord0 : enum 'I_0 = [::].

Proof. by apply/eqP; rewrite -size_eq0 size_enum_ord. Qed.

Lemma

enum_ord0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "enum", "size_enum_ord", "size_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_ord_proof n m (i : 'I_n) : n <= m -> i < m.

Proof. exact: leq_trans. Qed.

Lemma

widen_ord_proof

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "leq_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

widen_ord n m le_n_m i

:= Ordinal (@widen_ord_proof n m i le_n_m).

Definition

widen_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "widen_ord_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord_proof n m (i : 'I_n) : n = m -> i < m.

Proof. by move <-. Qed.

Lemma

cast_ord_proof

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord n m eq_n_m i

:= Ordinal (@cast_ord_proof n m i eq_n_m).

Definition

cast_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cast_ord_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord_id n eq_n i : cast_ord eq_n i = i :> 'I_n.

Proof. exact: val_inj. Qed.

Lemma

cast_ord_id

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cast_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord_comp n1 n2 n3 eq_n2 eq_n3 i : @cast_ord n2 n3 eq_n3 (@cast_ord n1 n2 eq_n2 i) = cast_ord (etrans eq_n2 eq_n3) i.

Proof. exact: val_inj. Qed.

Lemma

cast_ord_comp

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cast_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ordK n1 n2 eq_n : cancel (@cast_ord n1 n2 eq_n) (cast_ord (esym eq_n)).

Proof. by move=> i; apply: val_inj. Qed.

Lemma

cast_ordK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cast_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ordKV n1 n2 eq_n : cancel (cast_ord (esym eq_n)) (@cast_ord n1 n2 eq_n).

Proof. by move=> i; apply: val_inj. Qed.

Lemma

cast_ordKV

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cast_ord", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cast_ord_inj n1 n2 eq_n : injective (@cast_ord n1 n2 eq_n).

Proof. exact: can_inj (cast_ordK eq_n). Qed.

Lemma

cast_ord_inj

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cast_ord", "cast_ordK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordS_subproof n (i : 'I_n) : i.+1 %% n < n.

Proof. by case: n i => [|n] [m m_lt]//=; rewrite ltn_pmod. Qed.

Fact

ordS_subproof

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ltn_pmod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordS n (i : 'I_n)

:= Ordinal (ordS_subproof i).

Definition

ordS

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordS_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_pred_subproof n (i : 'I_n) : (i + n).-1 %% n < n.

Proof. by case: n i => [|n] [m m_lt]//=; rewrite ltn_pmod. Qed.

Fact

ord_pred_subproof

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ltn_pmod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_pred n (i : 'I_n)

:= Ordinal (ord_pred_subproof i).

Definition

ord_pred

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ord_pred_subproof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordSK n : cancel (@ordS n) (@ord_pred n).

Proof. move=> [i ilt]; apply/val_inj => /=. case: (ltngtP i.+1) (ilt) => // [Silt|<-]; last by rewrite modnn/= modn_small. by rewrite [i.+1 %% n]modn_small// addSn/= modnDr modn_small. Qed.

Lemma

ordSK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "addSn", "apply", "last", "ltngtP", "modnDr", "modn_small", "modnn", "ordS", "ord_pred", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_predK n : cancel (@ord_pred n) (@ordS n).

Proof. move=> [[|i] ilt]; apply/val_inj => /=. by rewrite [n.-1 %% n]modn_small// prednK// modnn. by rewrite modnDr [i %% n]modn_small ?modn_small// ltnW. Qed.

Lemma

ord_predK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "ltnW", "modnDr", "modn_small", "modnn", "ordS", "ord_pred", "prednK", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordS_bij n : bijective (@ordS n).

Proof. exact: (Bijective (@ordSK n) (@ord_predK n)). Qed.

Lemma

ordS_bij

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordS", "ordSK", "ord_predK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ordS_inj n : injective (@ordS n).

Proof. exact: (bij_inj (ordS_bij n)). Qed.

Lemma

ordS_inj

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordS", "ordS_bij" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_pred_bij n : bijective (@ord_pred n).

Proof. exact (Bijective (@ord_predK n) (@ordSK n)). Qed.

Lemma

ord_pred_bij

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ordSK", "ord_pred", "ord_predK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ord_pred_inj n : injective (@ord_pred n).

Proof. exact: (bij_inj (ord_pred_bij n)). Qed.

Lemma

ord_pred_inj

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ord_pred", "ord_pred_bij" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_ord_proof n (i : 'I_n) : n - i.+1 < n.

Proof. by case: n i => [|n] [i lt_i_n] //; rewrite ltnS subSS leq_subr. Qed.

Lemma

rev_ord_proof

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "leq_subr", "ltnS", "subSS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_ord n i

:= Ordinal (@rev_ord_proof n i).

Definition

rev_ord

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "rev_ord_proof" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_ordK {n} : involutive (@rev_ord n).

Proof. by case: n => [|n] [i lti] //; apply: val_inj; rewrite /= !subSS subKn. Qed.

Lemma

rev_ordK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "rev_ord", "subKn", "subSS", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_ord_inj {n} : injective (@rev_ord n).

Proof. exact: inv_inj rev_ordK. Qed.

Lemma

rev_ord_inj

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "rev_ord", "rev_ordK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d