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
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|---|---|---|---|---|---|---|---|---|---|---|
"{ ? x 'in' A | P }" | := {? x | (x \in A) && P}
(x at level 99, format "{ ? x 'in' A | P }") : type_scope. | Notation | { ? x 'in' A | P } | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insigd T (A : mem_pred T) x (Ax : in_mem x A) | :=
insubd (exist [eta A] x Ax). | Definition | insigd | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insubd"
] | from the membership proof for the default value. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
inj_type & injective f : Type | := T. | Definition | inj_type | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pcan_type g & pcancel f g : Type | := T. | Definition | pcan_type | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can_type g & cancel f g : Type | := T. | Definition | can_type | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_eqAxiom : injective f -> Equality.axiom (fun x y => f x == f y). | Proof. by move=> f_inj x y; apply: (iffP eqP) => [|-> //]; apply: f_inj. Qed. | Lemma | inj_eqAxiom | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"f_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deprecated_InjEqMixin f_inj | := hasDecEq.Build T (inj_eqAxiom f_inj). | Definition | deprecated_InjEqMixin | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Build",
"f_inj",
"inj_eqAxiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deprecated_PcanEqMixin g (fK : pcancel f g) | :=
deprecated_InjEqMixin (pcan_inj fK). | Definition | deprecated_PcanEqMixin | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"deprecated_InjEqMixin",
"fK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
deprecated_CanEqMixin g (fK : cancel f g) | :=
deprecated_InjEqMixin (can_inj fK). | Definition | deprecated_CanEqMixin | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"deprecated_InjEqMixin",
"fK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_type T (P : pred T) (sT : subType P) : Type | := sT. | Definition | sub_type | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ev_ax | := (fun T v => @Equality.axiom T (fun x y => v x == v y)). | Notation | ev_ax | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"axiom"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_eqP : ev_ax sT val. | Proof. exact: inj_eqAxiom val_inj. Qed. | Lemma | val_eqP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"ev_ax",
"inj_eqAxiom",
"sT",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_eqE (T : eqType) (P : pred T) (sT : subEqType P)
(u v : sT) : (val u == val v) = (u == v). | Proof. exact/val_eqP/eqP. Qed. | Lemma | val_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"sT",
"val",
"val_eqP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'Equality' 'of' T 'by' <: ]" | := (Equality.copy T%type (sub_type T%type))
(format "[ 'Equality' 'of' T 'by' <: ]") : form_scope. | Notation | [ 'Equality' 'of' T 'by' <: ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"copy",
"sub_type",
"type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_eq : rel (T1 * T2) | := fun u v => (u.1 == v.1) && (u.2 == v.2). | Definition | pair_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_eqP : Equality.axiom pair_eq. | Proof.
move=> [x1 x2] [y1 y2] /=; apply: (iffP andP) => [[]|[<- <-]] //=.
by do 2!move/eqP->.
Qed. | Lemma | pair_eqP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"pair_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_eqE : pair_eq = eq_op :> rel _. | Proof. by []. Qed. | Lemma | pair_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"pair_eq",
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpair_eqE (x1 y1 : T1) (x2 y2 : T2) :
((x1, x2) == (y1, y2)) = ((x1 == y1) && (x2 == y2)). | Proof. by []. Qed. | Lemma | xpair_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_eq1 (u v : T1 * T2) : u == v -> u.1 == v.1. | Proof. by case/andP. Qed. | Lemma | pair_eq1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pair_eq2 (u v : T1 * T2) : u == v -> u.2 == v.2. | Proof. by case/andP. Qed. | Lemma | pair_eq2 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predX T1 T2 (p1 : pred T1) (p2 : pred T2) | :=
[pred z | p1 z.1 & p2 z.2]. | Definition | predX | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'predX' A1 & A2 ]" | := (predX [in A1] [in A2])
(format "[ 'predX' A1 & A2 ]") : function_scope. | Notation | [ 'predX' A1 & A2 ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"predX"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opt_eq (u v : option T) : bool | :=
oapp (fun x => oapp (eq_op x) false v) (~~ v) u. | Definition | opt_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
opt_eqP : Equality.axiom opt_eq. | Proof.
case=> [x|] [y|] /=; by [constructor | apply: (iffP eqP) => [|[]] ->].
Qed. | Lemma | opt_eqP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"opt_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_as u v | :=
if tag u =P tag v is ReflectT eq_uv then
eq_rect_r T_ (tagged v) eq_uv
else tagged u. | Definition | tagged_as | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_asE u x : tagged_as u (Tagged T_ x) = x. | Proof.
by rewrite /tagged_as /=; case: eqP => // eq_uu; rewrite [eq_uu]eq_axiomK.
Qed. | Lemma | tagged_asE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eq_axiomK",
"tagged_as"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
T | := {i : I & T_ i}. | Notation | T | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
etagged i u (p : tag u = i) | := ecast i (T_ i) p (tagged u). | Definition | etagged | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_from_Tagged i (t s : T_ i) : Tagged T_ t = Tagged T_ s -> t = s. | Proof. by move=> /(congr1 (tagged_as (Tagged T_ t))); rewrite !tagged_asE. Qed. | Lemma | eq_from_Tagged | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"tagged_as",
"tagged_asE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
etaggedK i u (p : tag u = i) : Tagged T_ (etagged p) = u. | Proof. by case: _ / p; apply: taggedK. Qed. | Lemma | etaggedK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"etagged",
"taggedK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_with i : pred {i : I & T_ i} | := [pred j | tag j == i]. | Definition | tagged_with | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag_with i (x : {x in tagged_with i}) : T_ i | :=
etagged (eqP (valP x)). | Definition | untag_with | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"etagged",
"tagged_with",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_with i (t : T_ i) : {x in tagged_with i} | :=
exist _ (Tagged T_ t) (eq_refl i). | Definition | tag_with | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eq_refl",
"tagged_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag_withK i : cancel (@untag_with i) (@tag_with i). | Proof. by case=> -[j /= x eq_ji]; apply/val_inj=> /=; rewrite etaggedK. Qed. | Lemma | untag_withK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"etaggedK",
"tag_with",
"untag_with",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_withK i : cancel (@tag_with i) (@untag_with i). | Proof. by move=> x; rewrite /untag_with/= eq_axiomK. Qed. | Lemma | tag_withK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eq_axiomK",
"tag_with",
"untag_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_with_bij i : bijective (@tag_with i). | Proof. by exists (@untag_with i). Qed. | Lemma | tag_with_bij | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"tag_with",
"untag_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag_with_bij i : bijective (@untag_with i). | Proof. by exists (@tag_with i). Qed. | Lemma | untag_with_bij | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"tag_with",
"untag_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag (R : Type) (idx : R) (i : I) (F : T_ i -> R) u | :=
if tag u =P i is ReflectT e then F (etagged e) else idx. | Definition | untag | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"etagged"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untagE (R : Type) (idx : R) (i : I) (F : T_ i -> R) u (e : tag u = i):
untag idx F u = F (etagged e). | Proof. by rewrite /untag; case: eqP => // p; rewrite (eq_irrelevance p e). Qed. | Lemma | untagE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eq_irrelevance",
"etagged",
"untag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag_dflt (R : Type) (idx : R) (i : I) (F : T_ i -> R) u : tag u != i ->
untag idx F u = idx. | Proof. by rewrite /untag; case: eqP. Qed. | Lemma | untag_dflt | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"untag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
untag_cst (R : Type) (idx : R) (i : I) u :
untag idx (fun _ : T_ i => idx) u = idx. | Proof. by rewrite /untag; case: eqP. Qed. | Lemma | untag_cst | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"untag"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_eq u v | := (tag u == tag v) && (tagged u == tagged_as u v). | Definition | tag_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"tagged_as"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_eqP : Equality.axiom tag_eq. | Proof.
rewrite /tag_eq => [] [i x] [j] /=.
case: eqP => [<-|Hij] y; last by right; case.
by apply: (iffP eqP) => [->|<-]; rewrite tagged_asE.
Qed. | Lemma | tag_eqP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"last",
"tag_eq",
"tagged_asE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tag_eqE : tag_eq = eq_op. | Proof. by []. Qed. | Lemma | tag_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"tag_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_tag u v : u == v -> tag u = tag v. | Proof. by move/eqP->. Qed. | Lemma | eq_tag | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_Tagged u x :(u == Tagged _ x) = (tagged u == x). | Proof. by rewrite -tag_eqE /tag_eq eqxx tagged_asE. Qed. | Lemma | eq_Tagged | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eqxx",
"tag_eq",
"tag_eqE",
"tagged_asE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_eq u v | :=
match u, v with
| inl x, inl y | inr x, inr y => x == y
| _, _ => false
end. | Definition | sum_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_eqP : Equality.axiom sum_eq. | Proof. case=> x [] y /=; by [right | apply: (iffP eqP) => [->|[->]]]. Qed. | Lemma | sum_eqP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"sum_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sum_eqE : sum_eq = eq_op. | Proof. by []. Qed. | Lemma | sum_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"sum_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aR_refl : reflexive aR. | Hypothesis | aR_refl | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rR_refl : reflexive rR. | Hypothesis | rR_refl | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
aR'E : forall x y, aR' x y = (x != y) && (aR x y). | Hypothesis | aR'E | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rR'E : forall x y, rR' x y = (x != y) && (rR x y). | Hypothesis | rR'E | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
aRE x y : aR x y = (x == y) || (aR' x y). | Proof. by rewrite aR'E; case: eqVneq => //= ->; apply: aR_refl. Qed. | Let | aRE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR'E",
"aR_refl",
"apply",
"eqVneq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rRE x y : rR x y = (x == y) || (rR' x y). | Proof. by rewrite rR'E; case: eqVneq => //= ->; apply: rR_refl. Qed. | Let | rRE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"eqVneq",
"rR'E",
"rR_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homoW_in : {in D & D', {homo f : x y / aR' x y >-> rR' x y}} ->
{in D & D', {homo f : x y / aR x y >-> rR x y}}. | Proof.
by move=> mf x y xD yD /[!aRE]/orP[/eqP->|/mf]; rewrite rRE ?eqxx// orbC => ->.
Qed. | Lemma | homoW_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aRE",
"eqxx",
"rRE"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_homo_in : {in D & D', injective f} ->
{in D & D', {homo f : x y / aR x y >-> rR x y}} ->
{in D & D', {homo f : x y / aR' x y >-> rR' x y}}. | Proof.
move=> fI mf x y xD yD /[!(aR'E, rR'E)] /andP[neq_xy xy].
by rewrite mf ?andbT//; apply: contra_neq neq_xy; apply: fI.
Qed. | Lemma | inj_homo_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR'E",
"apply",
"contra_neq",
"rR'E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
aR_anti : antisymmetric aR. | Hypothesis | aR_anti | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
rR_anti : antisymmetric rR. | Hypothesis | rR_anti | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
mono_inj_in : {in D &, {mono f : x y / aR x y >-> rR x y}} ->
{in D &, injective f}. | Proof. by move=> mf x y ?? eqf; apply/aR_anti; rewrite -!mf// eqf rR_refl. Qed. | Lemma | mono_inj_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR_anti",
"apply",
"rR_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
anti_mono_in : {in D &, {mono f : x y / aR x y >-> rR x y}} ->
{in D &, {mono f : x y / aR' x y >-> rR' x y}}. | Proof.
move=> mf x y ??; rewrite rR'E aR'E mf// (@inj_in_eq _ _ D)//.
exact: mono_inj_in.
Qed. | Lemma | anti_mono_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR'E",
"inj_in_eq",
"mono_inj_in",
"rR'E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
total_homo_mono_in : total aR ->
{in D &, {homo f : x y / aR' x y >-> rR' x y}} ->
{in D &, {mono f : x y / aR x y >-> rR x y}}. | Proof.
move=> aR_tot mf x y xD yD.
have [->|neq_xy] := eqVneq x y; first by rewrite ?eqxx ?aR_refl ?rR_refl.
have [xy|] := (boolP (aR x y)); first by rewrite rRE mf ?orbT// aR'E neq_xy.
have /orP [->//|] := aR_tot x y.
rewrite aRE eq_sym (negPf neq_xy) /= => /mf -/(_ yD xD).
rewrite rR'E => /andP[Nfxfy fyfx] _; apply: ... | Lemma | total_homo_mono_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR'E",
"aRE",
"aR_refl",
"apply",
"contra_neqF",
"eqVneq",
"eq_sym",
"eqxx",
"rR'E",
"rRE",
"rR_anti",
"rR_refl",
"total"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
D | := @predT aT. | Let | D | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
homoW : {homo f : x y / aR' x y >-> rR' x y} ->
{homo f : x y / aR x y >-> rR x y}. | Proof. by move=> mf ???; apply: (@homoW_in D D) => // ????; apply: mf. Qed. | Lemma | homoW | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"homoW_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_homo : injective f ->
{homo f : x y / aR x y >-> rR x y} ->
{homo f : x y / aR' x y >-> rR' x y}. | Proof.
by move=> fI mf ???; apply: (@inj_homo_in D D) => //????; [apply: fI|apply: mf].
Qed. | Lemma | inj_homo | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_homo_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mono_inj : {mono f : x y / aR x y >-> rR x y} -> injective f. | Proof. by move=> mf x y eqf; apply/aR_anti; rewrite -!mf eqf rR_refl. Qed. | Lemma | mono_inj | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR_anti",
"apply",
"rR_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
anti_mono : {mono f : x y / aR x y >-> rR x y} ->
{mono f : x y / aR' x y >-> rR' x y}. | Proof. by move=> mf x y; rewrite rR'E aR'E mf inj_eq //; apply: mono_inj. Qed. | Lemma | anti_mono | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aR'E",
"apply",
"inj_eq",
"mono_inj",
"rR'E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
total_homo_mono : total aR ->
{homo f : x y / aR' x y >-> rR' x y} ->
{mono f : x y / aR x y >-> rR x y}. | Proof.
move=> /(@total_homo_mono_in D rR_anti) hmf hf => x y.
by apply: hmf => // ?? _ _; apply: hf.
Qed. | Lemma | total_homo_mono | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"rR_anti",
"total",
"total_homo_mono_in"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_on : seq aT -> Type | :=
| finfun_nil : finfun_on [::]
| finfun_cons x s of rT x & finfun_on s : finfun_on (x :: s). | Inductive | finfun_on | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_rec (g : forall x, rT x) s : finfun_on s | :=
if s is x1 :: s1 then finfun_cons (g x1) (finfun_rec g s1) else finfun_nil. | Fixpoint | finfun_rec | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"finfun_on",
"s1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_fin_rec x s (f_s : finfun_on s) : x \in s -> rT x | :=
if f_s is finfun_cons x1 s1 y1 f_s1 then
if eqP is ReflectT Dx in reflect _ Dxb return Dxb || (x \in s1) -> rT x then
fun=> ecast x (rT x) (esym Dx) y1
else fun_of_fin_rec f_s1
else fun isF => False_rect (rT x) (notF isF). | Fixpoint | fun_of_fin_rec | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"Dx",
"finfun_on",
"s1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_of (ph : phant (forall x, rT x)) : predArgType | :=
FinfunOf of finfun_on (enum aT). | Variant | finfun_of | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"enum",
"finfun_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfinfun_of ph | := finfun_of ph. | Definition | dfinfun_of | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"finfun_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_fin ph (f : finfun_of ph) x | :=
let: FinfunOf f_aT := f in fun_of_fin_rec f_aT (mem_enum aT x). | Definition | fun_of_fin | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"finfun_of",
"fun_of_fin_rec",
"mem_enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_fin : finfun_of >-> Funclass. | Coercion | fun_of_fin | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"finfun_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
"{ 'ffun' fT }" | := (finfun_of (Phant fT))
(format "{ 'ffun' '[hv' fT ']' }") : type_scope. | Notation | { 'ffun' fT } | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fT",
"finfun_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ 'dffun' fT }" | := (dfinfun_of (Phant fT))
(format "{ 'dffun' '[hv' fT ']' }") : type_scope. | Notation | { 'dffun' fT } | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"dfinfun_of",
"fT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
exp_finIndexType n : finType | := 'I_n. | Definition | exp_finIndexType | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"T ^ n" | :=
(@finfun_of (exp_finIndexType n) (fun=> T) (Phant _)) : type_scope. | Notation | T ^ n | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"exp_finIndexType",
"finfun_of"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finPi aT rT | := (forall x : Finite.sort aT, rT x) (only parsing). | Notation | finPi | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"sort"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_unlock | := Unlockable finfun.unlock. | Canonical | finfun_unlock | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'ffun' x : aT => E ]" | := (finfun (fun x : aT => E))
(x name) : function_scope. | Notation | [ 'ffun' x : aT => E ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'ffun' x => E ]" | := (@finfun _ (fun=> _) (fun x => E))
(x name, format "[ 'ffun' x => E ]") : function_scope. | Notation | [ 'ffun' x => E ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'ffun' => E ]" | := [ffun _ => E]
(format "[ 'ffun' => E ]") : function_scope. | Notation | [ 'ffun' => E ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fT | := {ffun finPi aT rT}. | Notation | fT | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"finPi"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun0 (aT0 : #|aT| = 0) : fT. | Proof. by apply/finfun=> x; have:= card0_eq aT0 x. Qed. | Fact | ffun0 | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"card0_eq",
"fT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffunE g x : (finfun g : fT) x = g x. | Proof.
rewrite unlock /=; set s := enum aT; set s_x : mem_seq s x := mem_enum _ _.
by elim: s s_x => //= x1 s IHs; case: eqP => [|_]; [case: x1 / | apply: IHs].
Qed. | Lemma | ffunE | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"enum",
"fT",
"mem_enum",
"mem_seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffunP (f1 f2 : fT) : (forall x, f1 x = f2 x) <-> f1 = f2. | Proof.
suffices ffunK f g: (forall x, f x = g x) -> f = finfun g.
by split=> [/ffunK|] -> //; apply/esym/ffunK.
case: f => f Dg; rewrite unlock; congr FinfunOf.
have{} Dg x (aTx : mem_seq (enum aT) x): g x = fun_of_fin_rec f aTx.
by rewrite -Dg /= (bool_irrelevance (mem_enum _ _) aTx).
elim: (enum aT) / f (enum_uni... | Lemma | ffunP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"bool_irrelevance",
"enum",
"enum_uniq",
"eq_axiomK",
"eqxx",
"f1",
"f2",
"fT",
"ffunK",
"fun_of_fin_rec",
"memPn",
"mem_enum",
"mem_seq",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffunK : @cancel (finPi aT rT) fT fun_of_fin finfun. | Proof. by move=> f; apply/ffunP=> x; rewrite ffunE. Qed. | Lemma | ffunK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"fT",
"ffunE",
"ffunP",
"finPi",
"fun_of_fin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_dffun (g1 g2 : forall x, rT x) :
(forall x, g1 x = g2 x) -> finfun g1 = finfun g2. | Proof. by move=> eq_g; apply/ffunP => x; rewrite !ffunE eq_g. Qed. | Lemma | eq_dffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"ffunE",
"ffunP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
total_fun g x | := Tagged rT (g x : rT x). | Definition | total_fun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tfgraph f | := codom_tuple (total_fun f). | Definition | tfgraph | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"codom_tuple",
"total_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codom_tffun f : codom (total_fun f) = tfgraph f. | Proof. by []. Qed. | Lemma | codom_tffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"codom",
"tfgraph",
"total_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tfgraph_inv (G : #|aT|.-tuple {x : aT & rT x}) : option fT | :=
if eqfunP isn't ReflectT Dtg then None else
Some [ffun x => ecast x (rT x) (Dtg x) (tagged (tnth G (enum_rank x)))]. | Definition | tfgraph_inv | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"enum_rank",
"eqfunP",
"fT",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tfgraphK : pcancel tfgraph tfgraph_inv. | Proof.
move=> f; have Dg x: tnth (tfgraph f) (enum_rank x) = total_fun f x.
by rewrite tnth_map -[tnth _ _]enum_val_nth enum_rankK.
rewrite /tfgraph_inv; case: eqfunP => /= [Dtg | [] x]; last by rewrite Dg.
congr (Some _); apply/ffunP=> x; rewrite ffunE.
by rewrite Dg in (Dx := Dtg x) *; rewrite eq_axiomK.
Qed. | Lemma | tfgraphK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"Dx",
"apply",
"enum_rank",
"enum_rankK",
"enum_val_nth",
"eq_axiomK",
"eqfunP",
"ffunE",
"ffunP",
"last",
"tfgraph",
"tfgraph_inv",
"tnth",
"tnth_map",
"total_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tfgraph_inj : injective tfgraph. | Proof. exact: pcan_inj tfgraphK. Qed. | Lemma | tfgraph_inj | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"tfgraph",
"tfgraphK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
family_mem mF | := [pred f : fT | [forall x, in_mem (f x) (mF x)]]. | Definition | family_mem | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fmem F x | := mem (F x : pT x). | Definition | fmem | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | Helper for defining notation for function families. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
familyP f : reflect (forall x, f x \in F x) (f \in family_mem (fmem F)). | Proof. exact: forallP. Qed. | Lemma | familyP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"family_mem",
"fmem",
"forallP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
family F | := (family_mem (fmem F)). | Notation | family | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"family_mem",
"fmem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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