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 |
|---|---|---|---|---|---|---|---|---|---|---|
eqbF_neg b : (b == false) = ~~ b. | Proof. by case: b. Qed. | Lemma | eqbF_neg | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqb_negLR b1 b2 : (~~ b1 == b2) = (b1 == ~~ b2). | Proof. by case: b1; case: b2. Qed. | Lemma | eqb_negLR | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpred1 | := (fun a1 x => x == a1). | Notation | xpred1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1"
] | Equality-based predicates. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
xpred2 | := (fun a1 a2 x => (x == a1) || (x == a2)). | Notation | xpred2 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpred3 | := (fun a1 a2 a3 x => [|| x == a1, x == a2 | x == a3]). | Notation | xpred3 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpred4 | :=
(fun a1 a2 a3 a4 x => [|| x == a1, x == a2, x == a3 | x == a4]). | Notation | xpred4 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2",
"a4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpredU1 | := (fun a1 (p : pred _) x => (x == a1) || p x). | Notation | xpredU1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpredC1 | := (fun a1 x => x != a1). | Notation | xpredC1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
xpredD1 | := (fun (p : pred _) a1 x => (x != a1) && p x). | Notation | xpredD1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred1 (a1 : T) | := SimplPred (xpred1 a1). | Definition | pred1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"xpred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred2 (a1 a2 : T) | := SimplPred (xpred2 a1 a2). | Definition | pred2 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2",
"xpred2"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred3 (a1 a2 a3 : T) | := SimplPred (xpred3 a1 a2 a3). | Definition | pred3 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2",
"xpred3"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred4 (a1 a2 a3 a4 : T) | := SimplPred (xpred4 a1 a2 a3 a4). | Definition | pred4 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"a2",
"a4",
"xpred4"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predU1 (a1 : T) p | := SimplPred (xpredU1 a1 p). | Definition | predU1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"xpredU1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predC1 (a1 : T) | := SimplPred (xpredC1 a1). | Definition | predC1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"xpredC1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predD1 p (a1 : T) | := SimplPred (xpredD1 p a1). | Definition | predD1 | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"a1",
"xpredD1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred1E : pred1 =2 eq_op. | Proof. by move=> x y; apply: eq_sym. Qed. | Lemma | pred1E | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"eq_sym",
"pred1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predU1P : reflect (x = y \/ b) ((x == y) || b). | Proof. by apply: (iffP orP); do [case=> [/eqP|]; [left | right]]. Qed. | Lemma | predU1P | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pred2P : reflect (x = y \/ z = u) ((x == y) || (z == u)). | Proof. by apply: (iffP orP); do [case=> /eqP; [left | right]]. Qed. | Lemma | pred2P | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predD1P : reflect (x <> y /\ b) ((x != y) && b). | Proof. by apply: (iffP andP)=> [] [] // /eqP. Qed. | Lemma | predD1P | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predU1l : x = y -> (x == y) || b. | Proof. by move->; rewrite eqxx. Qed. | Lemma | predU1l | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eqxx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
predU1r : b -> (x == y) || b. | Proof. by move->; rewrite orbT. Qed. | Lemma | predU1r | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'predU1' x & A ]" | := (predU1 x [in A])
(format "[ 'predU1' x & A ]") : function_scope. | Notation | [ 'predU1' x & A ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"predU1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'predD1' A & x ]" | := (predD1 [in A] x)
(format "[ 'predD1' A & x ]") : function_scope. | Notation | [ 'predD1' A & x ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"predD1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_eq : injective f -> forall x y, (f x == f y) = (x == y). | Proof. by move=> inj_f x y; apply/eqP/eqP=> [|-> //]; apply: inj_f. Qed. | Lemma | inj_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can_eq : cancel f g -> forall x y, (f x == f y) = (x == y). | Proof. by move/can_inj; apply: inj_eq. Qed. | Lemma | can_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
bij_eq : bijective f -> forall x y, (f x == f y) = (x == y). | Proof. by move/bij_inj; apply: inj_eq. Qed. | Lemma | bij_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can2_eq : cancel f g -> cancel g f -> forall x y, (f x == y) = (x == g y). | Proof. by move=> fK gK x y; rewrite -[y in LHS]gK; apply: can_eq. Qed. | Lemma | can2_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"can_eq",
"fK",
"gK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inj_in_eq :
{in D &, injective f} -> {in D &, forall x y, (f x == f y) = (x == y)}. | Proof. by move=> inj_f x y Dx Dy; apply/eqP/eqP=> [|-> //]; apply: inj_f. Qed. | Lemma | inj_in_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Dx",
"apply",
"inj_f"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
can_in_eq :
{in D, cancel f g} -> {in D &, forall x y, (f x == f y) = (x == y)}. | Proof. by move/can_in_inj; apply: inj_in_eq. Qed. | Lemma | can_in_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_in_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
frel f | := [rel x y : T | f x == y]. | Definition | frel | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"rel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inv_eq f : involutive f -> forall x y : T, (f x == y) = (x == f y). | Proof. by move=> fK; apply: can2_eq. Qed. | Lemma | inv_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"can2_eq",
"fK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_frel f f' : f =1 f' -> frel f =2 frel f'. | Proof. by move=> eq_f x y; rewrite /= eq_f. Qed. | Lemma | eq_frel | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"eq_f",
"frel"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invariant (rT : eqType) f (k : aT -> rT) | :=
[pred x | k (f x) == k x]. | Definition | invariant | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aT"
] | that have the same projection as their image. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
invariant_comp : subpred (invariant f k) (invariant f (h \o k)). | Proof. by move=> x eq_kfx; rewrite /= (eqP eq_kfx). Qed. | Lemma | invariant_comp | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"invariant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invariant_inj : injective h -> invariant f (h \o k) =1 invariant f k. | Proof. by move=> inj_h x; apply: (inj_eq inj_h). Qed. | Lemma | invariant_inj | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"inj_eq",
"invariant"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
coerced_frel f | := (rel_of_simpl (frel f)) (only parsing). | Notation | coerced_frel | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"frel"
] | The coercion to rel must be explicit for derived Notations to unparse. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
fun_delta : Type | := FunDelta of aT & rT. | Variant | fun_delta | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
fwith x y (f : aT -> rT) | := [fun z => if z == x then y else f z]. | Definition | fwith | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"aT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
app_fdelta df f z | :=
let: FunDelta x y := df in if z == x then y else f z. | Definition | app_fdelta | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x |-> y" | := (FunDelta x y)
(at level 190, no associativity,
format "'[hv' x '/ ' |-> y ']'") : fun_delta_scope. | Notation | x |-> y | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fun' z : T => F 'with' d1 , .. , dn ]" | :=
(SimplFunDelta (fun z : T =>
app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA (fun _ => F)) ..))
(z name, only parsing) : function_scope. | Notation | [ 'fun' z : T => F 'with' d1 , .. , dn ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"app_fdelta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'fun' z => F 'with' d1 , .. , dn ]" | :=
(SimplFunDelta (fun z =>
app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA (fun _ => F)) ..))
(z name, format
"'[hv' [ '[' 'fun' z => '/ ' F ']' '/' 'with' '[' d1 , '/' .. , '/' dn ']' ] ']'"
) : function_scope. | Notation | [ 'fun' z => F 'with' d1 , .. , dn ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"app_fdelta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'eta' f 'with' d1 , .. , dn ]" | :=
(SimplFunDelta (fun _ =>
app_fdelta d1%FUN_DELTA .. (app_fdelta dn%FUN_DELTA f) ..))
(format
"'[hv' [ '[' 'eta' '/ ' f ']' '/' 'with' '[' d1 , '/' .. , '/' dn ']' ] ']'"
) : function_scope. | Notation | [ 'eta' f 'with' d1 , .. , dn ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"app_fdelta"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfwith i (x : T i) (j : I) : T j | :=
if (i =P j) is ReflectT ij then ecast j (T j) ij x else f j. | Definition | dfwith | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfwith_in i x : dfwith x i = x. | Proof. by rewrite /dfwith; case: eqP => // ii; rewrite eq_axiomK. Qed. | Lemma | dfwith_in | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"dfwith",
"eq_axiomK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfwith_out i (x : T i) j : i != j -> dfwith x j = f j. | Proof. by rewrite /dfwith; case: eqP. Qed. | Lemma | dfwith_out | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"dfwith"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfwith_spec i (x : T i) : forall j, T j -> Type | :=
| DFunWithIn : dfwith_spec x x
| DFunWithOut j : i != j -> dfwith_spec x (f j). | Variant | dfwith_spec | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dfwithP i (x : T i) (j : I) : dfwith_spec x (dfwith x j). | Proof.
by case: (eqVneq i j) => [<-|nij];
[rewrite dfwith_in|rewrite dfwith_out//]; constructor.
Qed. | Lemma | dfwithP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"dfwith",
"dfwith_in",
"dfwith_out",
"dfwith_spec",
"eqVneq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparable | := forall x y : T, decidable (x = y). | Definition | comparable | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"decidable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
compare_T : comparable. | Hypothesis | compare_T | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | ||
compareb x y : bool | := compare_T x y. | Definition | compareb | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"compare_T"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
compareP : Equality.axiom compareb. | Proof. by move=> x y; apply: sumboolP. Qed. | Lemma | compareP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"axiom",
"compareb"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
comparableMixin | := hasDecEq.Build T compareP. | Definition | comparableMixin | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Build",
"compareP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eq_comparable (T : eqType) : comparable T | :=
fun x y => decP (x =P y). | Definition | eq_comparable | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"comparable"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val | := (isSub.val_subdef (SubType.on _)). | Notation | val | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\val" | := (isSub.val_subdef (SubType.on _)) (only parsing). | Notation | \val | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"on"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"\val" | := (isSub.val_subdef _) (only printing). | Notation | \val | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
vrefl : forall x, P x -> x = x. | Proof. by []. Qed. | Lemma | vrefl | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | The vrefl_rect alias is used to flag generic proofs of the first property. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
vrefl_rect | := vrefl. | Definition | vrefl_rect | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"vrefl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val | := (isSub.val_subdef (SubType.on sT)). | Notation | val | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"on",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sub | := (@Sub _ _ sT). | Notation | Sub | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SubK x Px : val (@Sub x Px) = x. | Proof. exact: SubK_subproof. Qed. | Lemma | SubK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"Sub",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Sub_spec : sT -> Type | := subSpec x Px : Sub_spec (Sub x Px). | Variant | Sub_spec | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"Sub",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
SubP u : Sub_spec u. | Proof. by elim/(@Sub_rect _ _ sT) : u. Qed. | Lemma | SubP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Sub_spec",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub x | := if idP is ReflectT Px then Some (Sub x Px) else None. | Definition | insub | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"Sub"
] | BUG in elim? sT could be inferred from u | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
insubd u0 x | := odflt u0 (insub x). | Definition | insubd | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub_spec x : option sT -> Type | :=
| InsubSome u of P x & val u = x : insub_spec x (Some u)
| InsubNone of ~~ P x : insub_spec x None. | Variant | insub_spec | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"sT",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubP x : insub_spec x (insub x). | Proof.
by rewrite /insub; case: {-}_ / idP; [left; rewrite ?SubK | right; apply/negP].
Qed. | Lemma | insubP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"SubK",
"apply",
"insub",
"insub_spec"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubT x Px : insub x = Some (Sub x Px). | Proof.
do [case: insubP => [/SubP[y Py] _ <- | /negP// ]; rewrite SubK] in Px *.
by rewrite (bool_irrelevance Px Py).
Qed. | Lemma | insubT | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"Sub",
"SubK",
"SubP",
"bool_irrelevance",
"insub",
"insubP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubF x : P x = false -> insub x = None. | Proof. by move/idP; case: insubP. Qed. | Lemma | insubF | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub",
"insubP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubN x : ~~ P x -> insub x = None. | Proof. by move/negPf/insubF. Qed. | Lemma | insubN | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub",
"insubF"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
isSome_insub : ([eta insub] : pred T) =1 P. | Proof. by apply: fsym => x; case: insubP => // /negPf. Qed. | Lemma | isSome_insub | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"apply",
"insub",
"insubP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubK : ocancel insub val. | Proof. by move=> x; case: insubP. Qed. | Lemma | insubK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub",
"insubP",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valP u : P (val u). | Proof. by case/SubP: u => x Px; rewrite SubK. Qed. | Lemma | valP | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"SubK",
"SubP",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valK : pcancel val insub. | Proof. by case/SubP=> x Px; rewrite SubK; apply: insubT. Qed. | Lemma | valK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"SubK",
"SubP",
"apply",
"insub",
"insubT",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_inj : injective val. | Proof. exact: pcan_inj valK. Qed. | Lemma | val_inj | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"val",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
valKd u0 : cancel val (insubd u0). | Proof. by move=> u; rewrite /insubd valK. Qed. | Lemma | valKd | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insubd",
"val",
"valK"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
val_insubd u0 x : val (insubd u0 x) = if P x then x else val u0. | Proof. by rewrite /insubd; case: insubP => [u -> | /negPf->]. Qed. | Lemma | val_insubd | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insubP",
"insubd",
"val"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insubdK u0 : {in P, cancel (insubd u0) val}. | Proof. by move=> x Px; rewrite val_insubd [P x]Px. Qed. | Lemma | insubdK | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"insubd",
"val",
"val_insubd"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub_eq_aux x isPx : P x = isPx -> option sT | :=
if isPx as b return _ = b -> _ then fun Px => Some (Sub x Px) else fun=> None. | Let | insub_eq_aux | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px",
"Sub",
"sT"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub_eq x | := insub_eq_aux (erefl (P x)). | Definition | insub_eq | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub_eq_aux"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
insub_eqE : insub_eq =1 insub. | Proof.
rewrite /insub_eq => x; set b := P x; rewrite [in LHS]/b in (Db := erefl b) *.
by case: b in Db *; [rewrite insubT | rewrite insubF].
Qed. | Lemma | insub_eqE | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"insub",
"insubF",
"insubT",
"insub_eq"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inlined_sub_rect | :=
(fun K K_S u => let (x, Px) as u return K u := u in K_S x Px). | Notation | inlined_sub_rect | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Px"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
inlined_new_rect | :=
(fun K K_S u => let (x) as u return K u := u in K_S x). | Notation | inlined_new_rect | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isSub' 'for' v ]" | :=
(@isSub.phant_Build _ _ _ v _ inlined_sub_rect vrefl_rect)
(only parsing) : form_scope. | Notation | [ 'isSub' 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"inlined_sub_rect",
"vrefl_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isSub' 'of' T 'for' v ]" | :=
(@isSub.phant_Build _ _ T v _ inlined_sub_rect vrefl_rect)
(only parsing) : form_scope. | Notation | [ 'isSub' 'of' T 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"inlined_sub_rect",
"vrefl_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isSub' 'for' v 'by' rec ]" | :=
(@isSub.phant_Build _ _ _ v _ rec vrefl)
(format "[ 'isSub' 'for' v 'by' rec ]") : form_scope. | Notation | [ 'isSub' 'for' v 'by' rec ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"vrefl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isSub' 'for' v ]" | := (@isSub.phant_Build _ _ _ v _ _ _)
(only printing, format "[ 'isSub' 'for' v ]") : form_scope. | Notation | [ 'isSub' 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
NewMixin T U v c Urec sk | :=
let Urec' P IH := Urec P (fun x : T => IH x isT : P _) in
@isSub.phant_Build _ _ U v (fun x _ => c x) Urec' sk. | Definition | NewMixin | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isNew' 'for' v ]" | :=
(@NewMixin _ _ v _ inlined_new_rect vrefl_rect) (only parsing) : form_scope. | Notation | [ 'isNew' 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"NewMixin",
"inlined_new_rect",
"vrefl_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isNew' 'for' v ]" | := (@NewMixin _ _ v _ _ _)
(only printing, format "[ 'isNew' 'for' v ]") : form_scope. | Notation | [ 'isNew' 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"NewMixin"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"[ 'isNew' 'of' T 'for' v ]" | :=
(@NewMixin _ T v _ inlined_new_rect vrefl_rect) (only parsing) : form_scope. | Notation | [ 'isNew' 'of' T 'for' v ] | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"NewMixin",
"inlined_new_rect",
"vrefl_rect"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
innew T nT x | := @Sub T predT nT x (erefl true). | Definition | innew | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"Sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
innew_val T nT : cancel val (@innew T nT). | Proof. by move=> u; apply: val_inj; apply: SubK. Qed. | Lemma | innew_val | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [
"SubK",
"apply",
"innew",
"val",
"val_inj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ x 'in' A }" | := {x | x \in A}
(x at level 99, format "{ x 'in' A }") : type_scope. | Notation | { x 'in' A } | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | Shorthand for sigma types over collective predicates. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"{ 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 | |
"{ ? x : T | P }" | := (option {x : T | is_true P})
(x at level 99, only parsing) : type_scope. | Notation | { ? x : T | P } | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | Shorthand for the return type of insub. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"{ ? x | P }" | := {? x : _ | P}
(x at level 99, format "{ ? x | P }") : type_scope. | Notation | { ? x | P } | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"{ ? x 'in' A }" | := {? x | x \in A}
(x at level 99, format "{ ? x 'in' A }") : type_scope. | Notation | { ? x 'in' A } | boot | boot/eqtype.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool"
] | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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