fact stringlengths 9 12.1k | type stringclasses 25 values | library stringclasses 6 values | imports listlengths 0 15 | filename stringclasses 105 values | symbolic_name stringlengths 1 50 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
forallPNP T (P Q : T -> Prop) : (forall x, P x -> ~ Q x) <-> ~ (exists2 x, P x & Q x). Proof. split=> [PQ [t Pt Qt]|PQ t]; first by have [] := PQ t. by move=> Pt Qt; apply: PQ; exists t. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forallPNP | |
existsPNP T (P Q : T -> Prop) : (exists2 x, P x & ~ Q x) <-> ~ (forall x, P x -> Q x). Proof. split=> [[x Px NQx] /(_ x Px)//|]; apply: contra_notP => + x Px. by apply: contra_notP => NQx; exists x. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | existsPNP | |
forallp_asboolPn2 {T} {P Q : T -> Prop} : reflect (forall x : T, ~ P x \/ ~ Q x) (~~ `[<exists2 x : T, P x & Q x>]). Proof. apply: (iffP idP)=> [/asboolPn NP x|NP]. by move/forallPNP : NP => /(_ x)/and_rec/not_andP. by apply/asboolP=> -[x Px Qx]; have /not_andP := NP x; exact. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | forallp_asboolPn2 | |
bigmax_geP : reflect (m <= x \/ exists2 i, P i & m <= F i) (m <= \big[max/x]_(i | P i) F i). Proof. apply: (iffP idP) => [|[mx|[i Pi mFi]]]. - rewrite leNgt => /bigmax_ltP /not_andP[/negP|]; first by rewrite -leNgt; left. by move=> /existsNP[i /not_implyP[Pi /negP]]; rewrite -leNgt; right; exists i. - by rewrite bigmax_idl le_max mx. - by rewrite (bigmaxD1 i)// le_max mFi. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | bigmax_geP | |
bigmax_gtP : reflect (m < x \/ exists2 i, P i & m < F i) (m < \big[max/x]_(i | P i) F i). Proof. apply: (iffP idP) => [|[mx|[i Pi mFi]]]. - rewrite ltNge => /bigmax_leP /not_andP[/negP|]; first by rewrite -ltNge; left. by move=> /existsNP[i /not_implyP[Pi /negP]]; rewrite -ltNge; right; exists i. - by rewrite bigmax_idl lt_max mx. - by rewrite (bigmaxD1 i)// lt_max mFi. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | bigmax_gtP | |
bigmin_leP : reflect (x <= m \/ exists2 i, P i & F i <= m) (\big[min/x]_(i | P i) F i <= m). Proof. apply: (iffP idP) => [|[xm|[i Pi Fim]]]. - rewrite leNgt => /bigmin_gtP /not_andP[/negP|]; first by rewrite -leNgt; left. by move=> /existsNP[i /not_implyP[Pi /negP]]; rewrite -leNgt; right; exists i. - by rewrite bigmin_idl ge_min xm. - by rewrite (bigminD1 i)// ge_min Fim. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | bigmin_leP | |
bigmin_ltP : reflect (x < m \/ exists2 i, P i & F i < m) (\big[min/x]_(i | P i) F i < m). Proof. apply: (iffP idP) => [|[xm|[i Pi Fim]]]. - rewrite ltNge => /bigmin_geP /not_andP[/negP|]; first by rewrite -ltNge; left. by move=> /existsNP[i /not_implyP[Pi /negP]]; rewrite -ltNge; right; exists i. - by rewrite bigmin_idl gt_min xm. - by rewrite (bigminD1 _ _ _ Pi) gt_min Fim. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | bigmin_ltP | |
fun_display : Order.disp_t. Proof. exact: Order.default_display. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | fun_display | |
lef f g := `[< forall x, (f x <= g x)%O >]. Local Notation "f <= g" := (lef f g). | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lef | |
ltf f g := `[< (forall x, (f x <= g x)%O) /\ exists x, f x != g x >]. Local Notation "f < g" := (ltf f g). | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | ltf | |
ltf_def f g : (f < g) = (g != f) && (f <= g). Proof. apply/idP/andP => [fg|[gf fg]]; [split|apply/asboolP; split; [exact/asboolP|]]. - by apply/eqP => gf; move: fg => /asboolP[fg] [x /eqP]; apply; rewrite gf. - apply/asboolP => x; rewrite le_eqVlt; move/asboolP : fg => [fg [y gfy]]. by have [//|gfx /=] := boolP (f x == g x); rewrite lt_neqAle gfx /= fg. - apply/not_existsP => h. have : f =1 g by move=> x; have /negP/negPn/eqP := h x. by rewrite -funeqE; apply/nesym/eqP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | ltf_def | |
lef_refl : reflexive lef. Proof. by move=> f; apply/asboolP => x. Qed. | Fact | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lef_refl | |
lef_anti : antisymmetric lef. Proof. move=> f g => /andP[/asboolP fg /asboolP gf]; rewrite funeqE => x. by apply/eqP; rewrite eq_le fg gf. Qed. | Fact | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lef_anti | |
lef_trans : transitive lef. Proof. move=> g f h /asboolP fg /asboolP gh; apply/asboolP => x. by rewrite (le_trans (fg x)). Qed. #[export] HB.instance Definition _ := @Order.isPOrder.Build fun_display (aT -> T) lef ltf ltf_def lef_refl lef_anti lef_trans. | Fact | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lef_trans | |
meetf f g := fun x => Order.meet (f x) (g x). | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | meetf | |
joinf f g := fun x => Order.join (f x) (g x). | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | joinf | |
meetfC : commutative meetf. Proof. move=> f g; apply/funext => x; exact: meetC. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | meetfC | |
joinfC : commutative joinf. Proof. move=> f g; apply/funext => x; exact: joinC. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | joinfC | |
meetfA : associative meetf. Proof. move=> f g h; apply/funext => x; exact: meetA. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | meetfA | |
joinfA : associative joinf. Proof. move=> f g h; apply/funext => x; exact: joinA. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | joinfA | |
joinfKI g f : meetf f (joinf f g) = f. Proof. apply/funext => x; exact: joinKI. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | joinfKI | |
meetfKU g f : joinf f (meetf f g) = f. Proof. apply/funext => x; exact: meetKU. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | meetfKU | |
lef_meet f g : (f <= g)%O = (meetf f g == f). Proof. apply/idP/idP => [/asboolP f_le_g|/eqP <-]. - apply/eqP/funext => x; exact/meet_l/f_le_g. - apply/asboolP => x; exact: leIr. Qed. #[export] HB.instance Definition _ := Order.POrder_isLattice.Build _ (aT -> T) meetfC joinfC meetfA joinfA joinfKI meetfKU lef_meet. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lef_meet | |
lefP (aT : Type) d (T : porderType d) (f g : aT -> T) : reflect (forall x, (f x <= g x)%O) (f <= g)%O. Proof. by apply: (iffP idP) => [fg|fg]; [exact/asboolP | exact/asboolP]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | lefP | |
meetfE (aT : Type) d (T : latticeType d) (f g : aT -> T) x : ((f `&` g) x = f x `&` g x)%O. Proof. by []. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | meetfE | |
joinfE (aT : Type) d (T : latticeType d) (f g : aT -> T) x : ((f `|` g) x = f x `|` g x)%O. Proof. by []. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | joinfE | |
iterfS {T} (f : T -> T) (n : nat) : iter n.+1 f = f \o iter n f. Proof. by []. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | iterfS | |
iterfSr {T} (f : T -> T) (n : nat) : iter n.+1 f = iter n f \o f. Proof. by apply/funeqP => ?; rewrite iterSr. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | iterfSr | |
iter0 {T} (f : T -> T) : iter 0 f = id. Proof. by []. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | iter0 | |
inhabitedE : inhabited T = exists x : T, True. Proof. by eqProp; case. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | inhabitedE | |
inhabited_witness : inhabited T -> T. Proof. by rewrite inhabitedE => /cid[]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | inhabited_witness | |
uncurryK {X Y Z : Type} : cancel (@uncurry X Y Z) curry. Proof. by move=> f; rewrite funeq2E. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | uncurryK | |
curryK {X Y Z : Type} : cancel curry (@uncurry X Y Z). Proof. by move=> f; rewrite funeqE=> -[]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect.",
"From mathcomp Require Import mathcomp_extra.",
"From mathcomp Require internal_Eqdep_dec."
] | classical/boolp.v | curryK | |
card_le T U (A : set T) (B : set U) := `[< $|{injfun [set: A] >-> [set: B]}| >]. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le | |
card_eq T U (A : set T) (B : set U) := `[< $|{bij [set: A] >-> [set: B]}| >]. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq | |
finite_set {T} (A : set T) := exists n, A #= `I_n. | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | finite_set | |
infinite_set A := (~ finite_set A). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | infinite_set | |
cofinite_set A := (finite_set (~` A)). | Notation | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | cofinite_set | |
injPex {T U} {A : set T} : $|{inj A >-> U}| <-> exists f : T -> U, set_inj A f. Proof. by split=> [[f]|[_ /Pinj[f _]]]; first by exists f. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | injPex | |
surjPex {T U} {A : set T} {B : set U} : $|{surj A >-> B}| <-> exists f, set_surj A B f. Proof. by split=> [[f]|[_ /Psurj[f _]]]; first by exists f. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | surjPex | |
bijPex {T U} {A : set T} {B : set U} : $|{bij A >-> B}| <-> exists f, set_bij A B f. Proof. by split=> [[f]|[_ /Pbij[f _]]]; first by exists f. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | bijPex | |
surjfunPex {T U} {A : set T} {B : set U} : $|{surjfun A >-> B}| <-> exists f, B = f @` A. Proof. split=> [[f]|[f ->]]; last by squash [fun f in A]. by exists f; apply/seteqP; split=> //; apply: surj. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | surjfunPex | |
injfunPex {T U} {A : set T} {B : set U}: $|{injfun A >-> B}| <-> exists2 f : T -> U, set_fun A B f & set_inj A f. Proof. by split=> [[f]|[_ /Pfun[? ->] /funPinj[f]]]; [exists f | squash f]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | injfunPex | |
card_leP {T U} {A : set T} {B : set U} : reflect $|{injfun [set: A] >-> [set: B]}| (A #<= B). Proof. exact: asboolP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_leP | |
inj_card_le {T U} {A : set T} {B : set U} : {injfun A >-> B} -> (A #<= B). Proof. by move=> f; apply/card_leP; squash (sigLR f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | inj_card_le | |
pcard_leP {T} {U : pointedType} {A : set T} {B : set U} : reflect $|{injfun A >-> B}| (A #<= B). Proof. by apply: (iffP card_leP) => -[f]; [squash (valLR point f) | squash (sigLR f)]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | pcard_leP | |
pcard_leTP {T} {U : pointedType} {A : set T} : reflect $|{inj A >-> U}| (A #<= [set: U]). Proof. by apply: (iffP pcard_leP) => -[f]; [squash f | squash ('totalfun_A f)]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | pcard_leTP | |
pcard_injP {T} {U : pointedType} {A : set T} : reflect (exists f : T -> U, {in A &, injective f}) (A #<= [set: U]). Proof. by apply: (iffP pcard_leTP); rewrite injPex. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | pcard_injP | |
ppcard_leP {T U : pointedType} {A : set T} {B : set U} : reflect $|{splitinjfun A >-> B}| (A #<= B). Proof. by apply: (iffP pcard_leP) => -[f]; squash (split f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | ppcard_leP | |
card_ge0 T U (S : set U) : @set0 T #<= S. Proof. by apply/card_leP; squash set0fun. Qed. #[global] Hint Resolve card_ge0 : core. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_ge0 | |
card_le0P T U (A : set T) : reflect (A = set0) (A #<= @set0 U). Proof. apply: (iffP idP) => [/card_leP[f]|->//]. by rewrite -subset0 => a /mem_set aA; have [x /set_mem] := f (SigSub aA). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le0P | |
card_le0 T U (A : set T) : (A #<= @set0 U) = (A == set0). Proof. exact/card_le0P/eqP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le0 | |
card_eqP {T U} {A : set T} {B : set U} : reflect $|{bij [set: A] >-> [set: B]}| (A #= B). Proof. exact: asboolP. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eqP | |
pcard_eq {T U} {A : set T} {B : set U} : {bij A >-> B} -> A #= B. Proof. by move=> f; apply/card_eqP; squash (sigLR f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | pcard_eq | |
pcard_eqP {T} {U : pointedType} {A : set T} {B : set U} : reflect $| {bij A >-> B} | (A #= B). Proof. by apply: (iffP card_eqP) => -[f]; [squash (valLR point f) | squash (sigLR f)]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | pcard_eqP | |
card_bijP {T U} {A : set T} {B : set U} : reflect (exists f : A -> B, bijective f) (A #= B). Proof. by apply: (iffP card_eqP) => [[f]|[_ /PbijTT[f _]]]; [exists f|squash f]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_bijP | |
card_eqVP {T U} {A : set T} {B : set U} : reflect $|{splitbij [set: A] >-> [set: B]}| (A #= B). Proof. by apply: (iffP card_bijP) => [[_ /PbijTT[f _]]//|[f]]; exists f. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eqVP | |
card_set_bijP {T} {U : pointedType} {A : set T} {B : set U} : reflect (exists f, set_bij A B f) (A #= B). Proof. by apply: (iffP pcard_eqP) => [[f]|[_ /Pbij[f _]]]; [exists f|squash f]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_set_bijP | |
ppcard_eqP {T U : pointedType} {A : set T} {B : set U} : reflect $| {splitbij A >-> B} | (A #= B). Proof. by apply: (iffP pcard_eqP) => -[f]; [squash (split f)|squash f]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | ppcard_eqP | |
card_eqxx T (A : set T) : A #= A. Proof. by apply/card_eqP; squash idfun. Qed. #[global] Hint Resolve card_eqxx : core. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eqxx | |
card_eq00 T U : @set0 T #= @set0 U. Proof. apply/card_eqP/squash; apply: @bijection_of_bijective set0fun _. by exists set0fun => -[x x0]; have := set_mem x0. Qed. #[global] Hint Resolve card_eq00 : core. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq00 | |
empty_eq0 T : all_equal_to (set0 : set T). Proof. by move=> X; apply/setF_eq0/no. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | empty_eq0 | |
card_le_emptyl T U (A : set T) (B : set U) : A #<= B. Proof. by rewrite empty_eq0. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_emptyl | |
card_le_emptyr T U (A : set T) (B : set U) : (B #<= A) = (B == set0). Proof. by rewrite empty_eq0; apply/idP/eqP=> [/card_le0P|->//]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_emptyr | |
emptyE_subdef := (empty_eq0, card_le_emptyl, card_le_emptyr, eq_opE). | Definition | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | emptyE_subdef | |
Cantor_Bernstein T U (A : set T) (B : set U) : A #<= B -> B #<= A -> A #= B. Proof. elim/Ppointed: T => T in A *; first by rewrite !emptyE_subdef => _ ->. elim/Ppointed: U => U in B *; first by rewrite !emptyE_subdef => ->. suff {A B} card_eq (A B : set U) : B `<=` A -> A #<= B -> A #= B. move=> /ppcard_leP[f] /ppcard_leP[g]. have /(_ _)/ppcard_eqP[|h] := card_eq _ _ (fun_image_sub f). by apply/pcard_leP; squash ([fun f in A] \o g). by apply/pcard_eqP; squash ((split h)^-1 \o [fun f in A]). move=> BA /ppcard_leP[u]; have uAB := 'funS_u. pose C_ := fix C n := if n is n.+1 then u @` C n else A `\` B. pose C := \bigcup_n C_ n; have CA : C `<=` A. by move=> + [] => /[swap]; elim=> [|i IH] y _ []// x /IH/uAB/BA + <-; apply. have uC: {homo u : x / x \in C}. by move=> x; rewrite !inE => -[i _ Cix]; exists i.+1 => //; exists x. apply/card_set_bijP; exists (fun x => if x \in C then u x else x); split. - move=> x Ax; case: ifPn; first by move=> _; apply: uAB. by move/negP; apply: contra_notP => NBx; rewrite inE; exists 0%N. - move=> x y xA yA; have := 'inj_u xA yA. have [xC|] := boolP (x \in C); have [yC|] := boolP (y \in C) => // + _. by move=> /[swap]<-; rewrite uC// xC. by move=> /[swap]->; rewrite uC// yC. - move=> y /[dup] By /BA Ay/=. case: (boolP (y \in C)); last by exists y; rewrite // ifN. rewrite inE => -[[|i]/= _ []// x Cix <-]; have Cx : C x by exists i. by exists x; [exact: CA|rewrite ifT// inE]. Qed. | Theorem | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | Cantor_Bernstein | |
card_esym T U (A : set T) (B : set U) : A #= B -> B #= A. Proof. by move=> /card_eqVP[f]; apply/card_eqP; squash f^-1. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_esym | |
card_eq_le T U (A : set T) (B : set U) : (A #= B) = (A #<= B) && (B #<= A). Proof. apply/idP/andP => [/card_eqVP[f]|[]]; last exact: Cantor_Bernstein. by split; apply/card_leP; [squash f|squash f^-1]. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_le | |
card_eqPle T U (A : set T) (B : set U) : (A #= B) <-> (A #<= B) /\ (B #<= A). Proof. by rewrite card_eq_le (rwP andP). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eqPle | |
card_lexx T (A : set T) : A #<= A. Proof. by apply/card_leP; squash idfun. Qed. #[global] Hint Resolve card_lexx : core. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_lexx | |
card_leT T (S : set T) : S #<= [set: T]. Proof. by apply/card_leP; squash (to_setT \o inclT _ \o val). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_leT | |
subset_card_le T (A B : set T) : A `<=` B -> A #<= B. Proof. by move=> AB; apply/card_leP; squash (inclT _ \o subfun AB). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | subset_card_le | |
card_image_le {T U} (f : T -> U) (A : set T) : f @` A #<= A. Proof. elim/Ppointed: T => T in A f *; first by rewrite !emptyE_subdef image_set0. by apply/pcard_leP; squash (pinv A f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_image_le | |
inj_card_eq {T U} {A} {f : T -> U} : {in A &, injective f} -> f @` A #= A. Proof. by move=> /inj_bij/pcard_eq/card_esym. Qed. Arguments inj_card_eq {T U A f}. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | inj_card_eq | |
card_some {T} {A : set T} : some @` A #= A. Proof. exact: inj_card_eq. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_some | |
card_image {T U} {A : set T} (f : {inj A >-> U}) : f @` A #= A. Proof. exact: inj_card_eq. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_image | |
card_imsub {T U} (A : set T) (f : {inj A >-> U}) X : X `<=` A -> f @` X #= X. Proof. by move=> XA; rewrite (card_image [inj of f \o incl XA]). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_imsub | |
card_le_trans (T U V : Type) (B : set U) (A : set T) (C : set V) : A #<= B -> B #<= C -> A #<= C. Proof. by move=> /card_leP[f]/card_leP[g]; apply/card_leP; squash (g \o f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_trans | |
card_eq_sym T U (A : set T) (B : set U) : (A #= B) = (B #= A). Proof. by rewrite !card_eq_le andbC. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_sym | |
card_eq_trans T U V (A : set T) (B : set U) (C : set V) : A #= B -> B #= C -> A #= C. Proof. by move=> /card_eqP[f]/card_eqP[g]; apply/card_eqP; squash (g \o f). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_trans | |
card_le_eql T T' T'' (A : set T) (B : set T') [C : set T''] : A #= B -> (A #<= C) = (B #<= C). Proof. by move=> /card_eqPle[*]; apply/idP/idP; apply: card_le_trans. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_eql | |
card_le_eqr T T' T'' (A : set T) (B : set T') [C : set T''] : A #= B -> (C #<= A) = (C #<= B). Proof. by move=> /card_eqPle[*]; apply/idP/idP => /card_le_trans; apply. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_eqr | |
card_eql T T' T'' (A : set T) (B : set T') [C : set T''] : A #= B -> (A #= C) = (B #= C). Proof. by move=> e; rewrite !card_eq_le (card_le_eql e) (card_le_eqr e). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eql | |
card_eqr T T' T'' (A : set T) (B : set T') [C : set T''] : A #= B -> (C #= A) = (C #= B). Proof. by move=> e; rewrite !card_eq_le (card_le_eql e) (card_le_eqr e). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eqr | |
card_ge_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) : X `<=` A -> (f @` X #<= Y) = (X #<= Y). Proof. by move=> XA; rewrite (card_le_eql (card_imsub _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_ge_image | |
card_le_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) : X `<=` A -> (Y #<= f @` X) = (Y #<= X). Proof. by move=> XA; rewrite (card_le_eqr (card_imsub _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_image | |
card_le_image2 {T U} (A : set T) (f : {inj A >-> U}) X Y : X `<=` A -> Y `<=` A -> (f @` X #<= f @` Y) = (X #<= Y). Proof. by move=> *; rewrite card_ge_image// card_le_image. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_image2 | |
card_eq_image {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) : X `<=` A -> (f @` X #= Y) = (X #= Y). Proof. by move=> XA; rewrite (card_eql (card_imsub _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_image | |
card_eq_imager {T U V} {A : set T} (f : {inj A >-> U}) X (Y : set V) : X `<=` A -> (Y #= f @` X) = (Y #= X). Proof. by move=> XA; rewrite (card_eqr (card_imsub _ _)). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_imager | |
card_eq_image2 {T U} (A : set T) (f : {inj A >-> U}) X Y : X `<=` A -> Y `<=` A -> (f @` X #= f @` Y) = (X #= Y). Proof. by move=> *; rewrite card_eq_image// card_eq_imager. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_image2 | |
card_ge_some {T T'} {A : set T} {B : set T'} : (some @` A #<= B) = (A #<= B). Proof. by rewrite (card_le_eql card_some). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_ge_some | |
card_le_some {T T'} {A : set T} {B : set T'} : (A #<= some @` B) = (A #<= B). Proof. by rewrite (card_le_eqr card_some). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_some | |
card_le_some2 {T T'} {A : set T} {B : set T'} : (some @` A #<= some @` B) = (A #<= B). Proof. by rewrite card_ge_some card_le_some. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_le_some2 | |
card_eq_somel {T T'} {A : set T} {B : set T'} : (some @` A #= B) = (A #= B). Proof. by rewrite (card_eql card_some). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_somel | |
card_eq_somer {T T'} {A : set T} {B : set T'} : (A #= some @` B) = (A #= B). Proof. by rewrite (card_eqr card_some). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_somer | |
card_eq_some2 {T T'} {A : set T} {B : set T'} : (some @` A #= some @` B) = (A #= B). Proof. by rewrite card_eq_somel card_eq_somer. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_some2 | |
card_eq0 {T U} {A : set T} : (A #= @set0 U) = (A == set0). Proof. by rewrite card_eq_le card_le0 card_ge0 andbT. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq0 | |
card_set1 {T} {x : T} : [set x] #= `I_1. Proof. apply/pcard_eqP; suff /Pbij[f]: set_bij [set x] `I_1 (fun=> 0%N) by squash f. by split=> [//|y z /[!in_setE]-> ->//|[]//]; exists x. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_set1 | |
eq_card1 {T U} (x : T) (y : U) : [set x] #= [set y]. Proof. by rewrite (card_eql card_set1) (card_eqr card_set1). Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | eq_card1 | |
card_eq_emptyr (T : emptyType) U (A : set T) (B : set U) : (B #= A) = (B == set0). Proof. by rewrite empty_eq0; exact: card_eq0. Qed. | Lemma | classical | [
"From HB Require Import structures.",
"From mathcomp Require Import all_ssreflect finmap ssralg ssrnum ssrint rat.",
"From mathcomp Require Import mathcomp_extra boolp classical_sets functions."
] | classical/cardinality.v | card_eq_emptyr |
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