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 |
|---|---|---|---|---|---|---|---|---|---|---|
dffun_aT rT rS | := {dffun forall x : aT, rT x : rS}. | Notation | dffun_aT | 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 | |
tuple_of_finfun (f : T ^ n) : n.-tuple T | := [tuple f i | i < n]. | Definition | tuple_of_finfun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_of_tuple (t : n.-tuple T) : (T ^ n) | := [ffun i => tnth t i]. | Definition | finfun_of_tuple | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
finfun_of_tupleK : cancel finfun_of_tuple tuple_of_finfun. | Proof.
by move=> t; apply: eq_from_tnth => i; rewrite tnth_map ffunE tnth_ord_tuple.
Qed. | Lemma | finfun_of_tupleK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"eq_from_tnth",
"ffunE",
"finfun_of_tuple",
"tnth_map",
"tnth_ord_tuple",
"tuple_of_finfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tuple_of_finfunK : cancel tuple_of_finfun finfun_of_tuple. | Proof.
by move=> f; apply/ffunP => i; rewrite ffunE tnth_map tnth_ord_tuple.
Qed. | Lemma | tuple_of_finfunK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"ffunE",
"ffunP",
"finfun_of_tuple",
"tnth_map",
"tnth_ord_tuple",
"tuple_of_finfun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcat (f : T ^ n) (g : T ^ m) : T ^ (n + m) | :=
[ffun i => match split i with inl j => f j | inr j => g j end]. | Definition | fcat | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcat_lshift f g j : fcat f g (lshift m j) = f j. | Proof. by rewrite ffunE (unsplitK (inl _ _)). Qed. | Lemma | fcat_lshift | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fcat",
"ffunE",
"lshift",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fcat_rshift f g j : fcat f g (rshift n j) = g j. | Proof. by rewrite ffunE (unsplitK (inr _ _)). Qed. | Lemma | fcat_rshift | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fcat",
"ffunE",
"rshift",
"unsplitK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fT | := {ffun aT -> rT}. | Notation | fT | 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 | |
fgraph f | := codom_tuple f. | Definition | fgraph | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"codom_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Finfun (G : #|aT|.-tuple rT) | := [ffun x => tnth G (enum_rank x)]. | Definition | Finfun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"enum_rank",
"tnth",
"tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tnth_fgraph f i : tnth (fgraph f) i = f (enum_val i). | Proof. by rewrite tnth_map /tnth -enum_val_nth. Qed. | Lemma | tnth_fgraph | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"enum_val",
"enum_val_nth",
"fgraph",
"tnth",
"tnth_map"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
FinfunK : cancel Finfun fgraph. | Proof.
by move=> G; apply/eq_from_tnth=> i; rewrite tnth_fgraph ffunE enum_valK.
Qed. | Lemma | FinfunK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"Finfun",
"apply",
"enum_valK",
"eq_from_tnth",
"ffunE",
"fgraph",
"tnth_fgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fgraphK : cancel fgraph Finfun. | Proof. by move=> f; apply/ffunP=> x; rewrite ffunE tnth_fgraph enum_rankK. Qed. | Lemma | fgraphK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"Finfun",
"apply",
"enum_rankK",
"ffunE",
"ffunP",
"fgraph",
"tnth_fgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fgraph_ffun0 aT0 : fgraph (ffun0 aT0) = nil :> seq rT. | Proof. by apply/nilP/eqP; rewrite size_tuple. Qed. | Lemma | fgraph_ffun0 | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"ffun0",
"fgraph",
"nilP",
"seq",
"size_tuple"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
codom_ffun f : codom f = fgraph f. | Proof. by []. Qed. | Lemma | codom_ffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"codom",
"fgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
tagged_tfgraph f : @map _ rT tagged (tfgraph f) = fgraph f. | Proof. by rewrite -map_comp. Qed. | Lemma | tagged_tfgraph | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fgraph",
"map",
"map_comp",
"tfgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_ffun (g1 g2 : aT -> rT) : g1 =1 g2 -> finfun g1 = finfun g2. | Proof. exact: eq_dffun. Qed. | Lemma | eq_ffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"eq_dffun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fgraph_codom f : fgraph f = codom_tuple f. | Proof. exact/esym/val_inj/codom_ffun. Qed. | Lemma | fgraph_codom | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"codom_ffun",
"codom_tuple",
"fgraph",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_on_mem (mR : mem_pred rT) | := family_mem (fun _ : aT => mR). | Definition | ffun_on_mem | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"family_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_onP R f : reflect (forall x, f x \in R) (f \in ffun_on_mem (mem R)). | Proof. exact: forallP. Qed. | Lemma | ffun_onP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"ffun_on_mem",
"forallP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ffun_on R | := (ffun_on_mem _ (mem R)). | Notation | ffun_on | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"ffun_on_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"@ 'ffun_on' aT R" | :=
(ffun_on R : simpl_pred (finfun_of (Phant (aT -> id _))))
(at level 10, aT, R at level 9). | Notation | @ 'ffun_on' aT R | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"ffun_on",
"finfun_of",
"id"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
nth_fgraph_ord T n (x0 : T) (i : 'I_n) f : nth x0 (fgraph f) i = f i. | Proof.
by rewrite -[i in RHS]enum_rankK -tnth_fgraph (tnth_nth x0) enum_rank_ord.
Qed. | Lemma | nth_fgraph_ord | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"enum_rankK",
"enum_rank_ord",
"fgraph",
"nth",
"tnth_fgraph",
"tnth_nth"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
support_for y (f : aT -> rT) | := [pred x | f x != y]. | Definition | support_for | 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 | |
supportE x y f : (x \in support_for y f) = (f x != y). | Proof. by []. Qed. | Lemma | supportE | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"support_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"y .-support" | := (support_for y)
(at level 1, format "y .-support") : function_scope. | Notation | y .-support | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"support_for"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
supportP y D g :
reflect (forall x, x \notin D -> g x = y) (y.-support g \subset D). | Proof.
by (apply: (iffP subsetP) => Dg x; [apply: contraNeq|apply: contraR]) => /Dg->.
Qed. | Lemma | supportP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"contraNeq",
"subsetP",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pfamily_mem y mD (mF : aT -> mem_pred rT) | :=
family (fun i : aT => if in_mem i mD then pred_of_simpl (mF i) else pred1 y). | Definition | pfamily_mem | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"family",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pfamilyP (pT : predType rT) y D (F : aT -> pT) f :
reflect (y.-support f \subset D /\ {in D, forall x, f x \in F x})
(f \in pfamily_mem y (mem D) (fmem F)). | Proof.
apply: (iffP familyP) => [/= f_pfam | [/supportP f_supp f_fam] x].
split=> [|x Ax]; last by have:= f_pfam x; rewrite Ax.
by apply/subsetP=> x; case: ifP (f_pfam x) => //= _ fx0 /negP[].
by case: ifPn => Ax /=; rewrite inE /= (f_fam, f_supp).
Qed. | Lemma | pfamilyP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"familyP",
"fmem",
"inE",
"last",
"pfamily_mem",
"split",
"subsetP",
"support",
"supportP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pffun_on_mem y mD mR | := pfamily_mem y mD (fun _ => mR). | Definition | pffun_on_mem | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"pfamily_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pffun_onP y D R f :
reflect (y.-support f \subset D /\ {subset image f D <= R})
(f \in pffun_on_mem y (mem D) (mem R)). | Proof.
apply: (iffP (pfamilyP y D (fun _ => R) f)) => [] [-> f_fam]; split=> //.
by move=> _ /imageP[x Ax ->]; apply: f_fam.
by move=> x Ax; apply: f_fam; apply/imageP; exists x.
Qed. | Lemma | pffun_onP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"image",
"imageP",
"pfamilyP",
"pffun_on_mem",
"split",
"support"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pfamily y D F | := (pfamily_mem y (mem D) (fmem F)). | Notation | pfamily | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fmem",
"pfamily_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pffun_on y D R | := (pffun_on_mem y (mem D) (mem R)). | Notation | pffun_on | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"pffun_on_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fT | := {dffun forall x : aT, rT x}. | Notation | fT | 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 | |
card_family (F : forall x, pred (rT x)) :
#|(family F : simpl_pred fT)| = foldr muln 1 [seq #|F x| | x : aT]. | Proof.
rewrite /image_mem; set E := enum aT in (uniqE := enum_uniq aT) *.
have trivF x: x \notin E -> #|F x| = 1 by rewrite mem_enum.
elim: E uniqE => /= [_ | x0 E IH_E /andP[E'x0 uniqE]] in F trivF *.
have /fin_all_exists[f0 Ff0] x: exists y0, F x =i pred1 y0.
have /pred0Pn[y Fy]: #|F x| != 0 by rewrite trivF.
... | Lemma | card_family | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"Dx",
"F1",
"aT",
"apply",
"card1",
"cardX",
"card_image",
"enum",
"enum_uniq",
"eq_axiomK",
"eq_card",
"eq_card0",
"eq_card1",
"eq_in_map",
"eqxx",
"f1",
"fT",
"family",
"familyP",
"ffunE",
"ffunP",
"fin_all_exists",
"foldr",
"gK",
"imageP",
"image_mem",
"inE",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_dep_ffun : #|fT| = foldr muln 1 [seq #|rT x| | x : aT]. | Proof. by rewrite -card_family; apply/esym/eq_card=> f; apply/familyP. Qed. | Lemma | card_dep_ffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"card_family",
"eq_card",
"fT",
"familyP",
"foldr",
"muln",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_pfamily y0 D F :
#|pfamily y0 D F| = foldr muln 1 [seq #|F x| | x in D]. | Proof.
rewrite card_family !/(image _ _) /(enum D) -enumT /=.
by elim: (enum aT) => //= x E ->; have [// | D'x] := ifP; rewrite card1 mul1n.
Qed. | Lemma | card_pfamily | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"card1",
"card_family",
"enum",
"enumT",
"foldr",
"image",
"mul1n",
"muln",
"pfamily",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_pffun_on y0 D R : #|pffun_on y0 D R| = #|R| ^ #|D|. | Proof.
rewrite (cardE D) card_pfamily /image_mem.
by elim: (enum D) => //= _ e ->; rewrite expnS.
Qed. | Lemma | card_pffun_on | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"cardE",
"card_pfamily",
"enum",
"expnS",
"image_mem",
"pffun_on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_ffun_on R : #|@ffun_on aT R| = #|R| ^ #|aT|. | Proof.
rewrite card_family /image_mem cardT.
by elim: (enum aT) => //= _ e ->; rewrite expnS.
Qed. | Lemma | card_ffun_on | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"cardT",
"card_family",
"enum",
"expnS",
"ffun_on",
"image_mem"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
card_ffun : #|fT| = #|rT| ^ #|aT|. | Proof. by rewrite -card_ffun_on; apply/esym/eq_card=> f; apply/forallP. Qed. | Lemma | card_ffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"aT",
"apply",
"card_ffun_on",
"eq_card",
"fT",
"forallP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_type | := (forall i : I, T_ i) (only parsing). | Notation | fprod_type | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod : predArgType | := FProd
{ fprod_fun : {ffun I -> {i : I & T_ i}} ;
fprod_prop : [forall i : I, tag (fprod_fun i) == i] }. | Record | fprod | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [] | Definition of [fprod] := dependent product of finTypes | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
tag_fprod_fun (f : fprod) i : tag (fprod_fun f i) = i. | Proof. by have /'forall_eqP/(_ i) := fprod_prop f. Qed. | Lemma | tag_fprod_fun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_fprod (f : fprod) : fprod_type | :=
fun i => etagged ('forall_eqP (fprod_prop f) i). | Definition | fun_of_fprod | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"etagged",
"fprod",
"fprod_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fun_of_fprod : fprod >-> Funclass. | Coercion | fun_of_fprod | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
fprod_of_prod_type_subproof (f : fprod_type) :
[forall i : I, tag ([ffun i => Tagged T_ (f i)] i) == i]. | Proof. by apply/'forall_eqP => i /=; rewrite ffunE. Qed. | Lemma | fprod_of_prod_type_subproof | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"ffunE",
"fprod_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_of_fun (f : fprod_type) : fprod | :=
FProd (fprod_of_prod_type_subproof f). | Definition | fprod_of_fun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod",
"fprod_of_prod_type_subproof",
"fprod_type"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprodK : cancel fun_of_fprod fprod_of_fun. | Proof.
rewrite /fun_of_fprod /fprod_of_fun; case=> f fP.
by apply/val_inj/ffunP => i /=; rewrite !ffunE etaggedK.
Qed. | Lemma | fprodK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"etaggedK",
"fP",
"ffunE",
"ffunP",
"fprod_of_fun",
"fun_of_fprod",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprodE g i : fprod_of_fun g i = g i. | Proof.
rewrite /fprod_of_fun /fun_of_fprod/=.
by move: ('forall_eqP _ _); rewrite ffunE/= => e; rewrite eq_axiomK.
Qed. | Lemma | fprodE | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"eq_axiomK",
"ffunE",
"fprod_of_fun",
"fun_of_fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprodP (f1 f2 : fprod) : (forall x, f1 x = f2 x) <-> f1 = f2. | Proof.
split=> [eq_f12|->//]; rewrite -[f1]fprodK -[f2]fprodK.
by apply/val_inj/ffunP => i; rewrite !ffunE eq_f12.
Qed. | Lemma | fprodP | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"f1",
"f2",
"ffunE",
"ffunP",
"fprod",
"fprodK",
"split",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dffun_of_fprod (f : fprod) : {dffun forall i : I, T_ i} | :=
[ffun x => f x]. | Definition | dffun_of_fprod | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_of_dffun (f : {dffun forall i : I, T_ i}) : fprod | :=
fprod_of_fun f. | Definition | fprod_of_dffun | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod",
"fprod_of_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dffun_of_fprodK : cancel dffun_of_fprod fprod_of_dffun. | Proof. by move=> f; apply/fprodP=> i; rewrite fprodE ffunE. Qed. | Lemma | dffun_of_fprodK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"dffun_of_fprod",
"ffunE",
"fprodE",
"fprodP",
"fprod_of_dffun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_of_dffunK : cancel fprod_of_dffun dffun_of_fprod. | Proof. by move=> f; apply/ffunP => i; rewrite !ffunE fprodE. Qed. | Lemma | fprod_of_dffunK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"dffun_of_fprod",
"ffunE",
"ffunP",
"fprodE",
"fprod_of_dffun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dffun_of_fprod_bij : bijective dffun_of_fprod. | Proof. by exists fprod_of_dffun. Qed. | Lemma | dffun_of_fprod_bij | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"dffun_of_fprod",
"fprod_of_dffun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fprod_of_dffun_bij : bijective fprod_of_dffun. | Proof. by exists dffun_of_fprod. Qed. | Lemma | fprod_of_dffun_bij | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"dffun_of_fprod",
"fprod_of_dffun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
to_family_tagged_with (f : fprod) : {x in family (tagged_with T_)} | :=
exist _ (fprod_fun f) (fprod_prop f). | Definition | to_family_tagged_with | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"family",
"fprod",
"tagged_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
of_family_tagged_with (f : {x in family (tagged_with T_)}) : fprod | :=
FProd (valP f). | Definition | of_family_tagged_with | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"family",
"fprod",
"tagged_with",
"valP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
to_family_tagged_withK :
cancel to_family_tagged_with of_family_tagged_with. | Proof. by case=> f fP; apply/val_inj. Qed. | Lemma | to_family_tagged_withK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"fP",
"of_family_tagged_with",
"to_family_tagged_with",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
of_family_tagged_withK :
cancel of_family_tagged_with to_family_tagged_with. | Proof. by case=> f fP; apply/val_inj. Qed. | Lemma | of_family_tagged_withK | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"fP",
"of_family_tagged_with",
"to_family_tagged_with",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
to_family_tagged_with_bij : bijective to_family_tagged_with. | Proof. by exists of_family_tagged_with. Qed. | Lemma | to_family_tagged_with_bij | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"of_family_tagged_with",
"to_family_tagged_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
of_family_tagged_with_bij : bijective of_family_tagged_with. | Proof. by exists to_family_tagged_with. Qed. | Lemma | of_family_tagged_with_bij | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"of_family_tagged_with",
"to_family_tagged_with"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
etaggedE (a : fprod) (i : I) (e : tag (fprod_fun a i) = i) :
etagged e = a i. | Proof. by case: a e => //= f fP e; congr etagged; apply: eq_irrelevance. Qed. | Lemma | etaggedE | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"apply",
"eq_irrelevance",
"etagged",
"fP",
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fprod' i : I => F ]" | := (fprod_of_fun (fun i : I => F))
(i name, only parsing) : function_scope. | Notation | [ 'fprod' i : I => F ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod_of_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fprod' : I => F ]" | := (fprod_of_fun (fun _ : I => F))
(only parsing) : function_scope. | Notation | [ 'fprod' : I => F ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod_of_fun"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fprod' i => F ]" | := [fprod i : _ => F]
(i name, format "[ 'fprod' i => F ]") : function_scope. | Notation | [ 'fprod' i => F ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fprod' => F ]" | := [fprod : _ => F]
(format "[ 'fprod' => F ]") : function_scope. | Notation | [ 'fprod' => F ] | boot | boot/finfun.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"tuple"
] | [
"fprod"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
grel (T : eqType) (g : T -> seq T) | := [rel x y | y \in g x]. | Definition | grel | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"rel",
"seq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfs n v x | :=
if x \in v then v else
if n is n'.+1 then foldl (dfs n') (x :: v) (g x) else v. | Fixpoint | dfs | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"foldl",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
subset_dfs n v a : v \subset foldl (dfs n) v a. | Proof.
elim: n a v => [|n IHn]; first by elim=> //= *; rewrite if_same.
elim=> //= x a IHa v; apply: subset_trans {IHa}(IHa _); case: ifP => // _.
by apply: subset_trans (IHn _ _); apply/subsetP=> y; apply: predU1r.
Qed. | Lemma | subset_dfs | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"dfs",
"foldl",
"predU1r",
"subsetP",
"subset_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfs_path v x y : Prop | :=
DfsPath p of path (grel g) x p & y = last x p & [disjoint x :: p & v]. | Inductive | dfs_path | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"disjoint",
"grel",
"last",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfs_pathP n x y v :
#|T| <= #|v| + n -> y \notin v -> reflect (dfs_path v x y) (y \in dfs n v x). | Proof.
have dfs_id w z: z \notin w -> dfs_path w z z.
by exists [::]; rewrite ?disjoint_has //= orbF.
elim: n => [|n IHn] /= in x y v * => le_v'_n not_vy.
rewrite addn0 (geq_leqif (subset_leqif_card (subset_predT _))) in le_v'_n.
by rewrite predT_subset in not_vy.
have [v_x | not_vx] := ifPn.
by rewrite (negPf ... | Lemma | dfs_pathP | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"addSnnS",
"addn0",
"apply",
"cardU1",
"cat_cons",
"cat_path",
"cat_rcons",
"dfs",
"dfs_path",
"disjoint",
"disjointWl",
"disjoint_cat",
"disjoint_cons",
"disjoint_has",
"disjoint_sym",
"eqVneq",
"geq_leqif",
"has_sym",
"last",
"lastI",
"last_cat",
"leq_add2r",
"leq_trans... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfsP x y :
reflect (exists2 p, path (grel g) x p & y = last x p) (y \in dfs #|T| [::] x). | Proof.
apply: (iffP (dfs_pathP _ _ _)); rewrite ?card0 // => [] [p]; exists p => //.
by rewrite disjoint_sym disjoint0.
Qed. | Lemma | dfsP | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"card0",
"dfs",
"dfs_pathP",
"disjoint0",
"disjoint_sym",
"grel",
"last",
"path"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rgraph x | := enum (e x). | Definition | rgraph | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"enum"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rgraphK : grel rgraph =2 e. | Proof. by move=> x y; rewrite /= mem_enum. Qed. | Lemma | rgraphK | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"grel",
"mem_enum",
"rgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect : rel T | := [rel x y | y \in dfs rgraph #|T| [::] x]. | Definition | connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"dfs",
"rel",
"rgraph"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_app_pred x | := ApplicativePred (connect x). | Canonical | connect_app_pred | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connectP x y :
reflect (exists2 p, path e x p & y = last x p) (connect x y). | Proof.
apply: (equivP (dfsP _ x y)).
by split=> [] [p e_p ->]; exists p => //; rewrite (eq_path rgraphK) in e_p *.
Qed. | Lemma | connectP | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"dfsP",
"eq_path",
"last",
"path",
"rgraphK",
"split"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_trans : transitive connect. | Proof.
move=> x y z /connectP[p e_p ->] /connectP[q e_q ->]; apply/connectP.
by exists (p ++ q); rewrite ?cat_path ?e_p ?last_cat.
Qed. | Lemma | connect_trans | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cat_path",
"connect",
"connectP",
"last_cat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect0 x : connect x x. | Proof. by apply/connectP; exists [::]. Qed. | Lemma | connect0 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_connect0 x y : x = y -> connect x y. | Proof. by move->; apply: connect0. Qed. | Lemma | eq_connect0 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connect0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect1 x y : e x y -> connect x y. | Proof. by move=> e_xy; apply/connectP; exists [:: y]; rewrite /= ?e_xy. Qed. | Lemma | connect1 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connectP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
path_connect x p : path e x p -> subpred [in x :: p] (connect x). | Proof.
move=> e_p y p_y; case/splitPl: p / p_y e_p => p q <-.
by rewrite cat_path => /andP[e_p _]; apply/connectP; exists p.
Qed. | Lemma | path_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cat_path",
"connect",
"connectP",
"path",
"splitPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_cycle p : cycle e p -> {in p &, forall x y, connect x y}. | Proof.
move=> e_p x y /rot_to[i q rip]; rewrite -(mem_rot i) rip => yqx.
have /= : cycle e (x :: q) by rewrite -rip rot_cycle.
case/splitPl: yqx => r s lxr; rewrite rcons_cat cat_path => /andP[xr _].
by apply/connectP; exists r.
Qed. | Lemma | connect_cycle | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"cat_path",
"connect",
"connectP",
"cycle",
"mem_rot",
"rcons_cat",
"rot_cycle",
"rot_to",
"splitPl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root x | := odflt x (pick (connect x)). | Definition | root | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"pick"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
roots : pred T | := fun x => root x == x. | Definition | roots | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
roots_pred | := ApplicativePred roots. | Canonical | roots_pred | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
n_comp_mem (m_a : mem_pred T) | := #|predI roots m_a|. | Definition | n_comp_mem | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_root x : connect x (root x). | Proof. by rewrite /root; case: pickP; rewrite ?connect0. Qed. | Lemma | connect_root | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"connect0",
"pickP",
"root"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
connect_sym | := symmetric connect. | Definition | connect_sym | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sym_e : connect_sym. | Hypothesis | sym_e | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
same_connect : left_transitive connect. | Proof. exact: sym_left_transitive connect_trans. Qed. | Lemma | same_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"connect_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
same_connect_r : right_transitive connect. | Proof. exact: sym_right_transitive connect_trans. Qed. | Lemma | same_connect_r | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"connect_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
same_connect1 x y : e x y -> connect x =1 connect y. | Proof. by move/connect1; apply: same_connect. Qed. | Lemma | same_connect1 | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connect1",
"same_connect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
same_connect1r x y : e x y -> connect^~ x =1 connect^~ y. | Proof. by move/connect1; apply: same_connect_r. Qed. | Lemma | same_connect1r | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connect1",
"same_connect_r"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rootP x y : reflect (root x = root y) (connect x y). | Proof.
apply: (iffP idP) => e_xy.
by rewrite /root -(eq_pick (same_connect e_xy)); case: pickP e_xy => // ->.
by apply: (connect_trans (connect_root x)); rewrite e_xy sym_e connect_root.
Qed. | Lemma | rootP | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"apply",
"connect",
"connect_root",
"connect_trans",
"eq_pick",
"pickP",
"root",
"same_connect",
"sym_e"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_root x : root (root x) = root x. | Proof. exact/esym/rootP/connect_root. Qed. | Lemma | root_root | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect_root",
"root",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
roots_root x : roots (root x). | Proof. exact/eqP/root_root. Qed. | Lemma | roots_root | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"root",
"root_root",
"roots"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
root_connect x y : (root x == root y) = connect x y. | Proof. exact: sameP eqP (rootP x y). Qed. | Lemma | root_connect | boot | boot/fingraph.v | [
"mathcomp",
"ssreflect",
"ssrbool",
"ssrfun",
"eqtype",
"ssrnat",
"seq",
"path",
"fintype"
] | [
"connect",
"root",
"rootP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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