fact stringlengths 6 2.88k | type stringclasses 17
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values | imports listlengths 0 16 | filename stringclasses 89
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rel_of_simplT (sr : simpl_rel T) : rel T := fun x : T => sr x.
Arguments rel_of_simpl {T} sr x /.
Abbreviation xrelU := (fun (r1 r2 : rel _) x y => r1 x y || r2 x y).
Abbreviation xrelpre := (fun f (r : rel _) x y => r (f x) (f y)). | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rel_of_simpl | |
SimplRel{T} (r : rel T) : simpl_rel T := fun x => SimplPred (r x). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | SimplRel | |
relU{T} (r1 r2 : rel T) := SimplRel (xrelU r1 r2). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | relU | |
relpre{aT rT} (f : aT -> rT) (r : rel rT) := SimplRel (xrelpre f r). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | relpre | |
subrelUlT (r1 r2 : rel T) : subrel r1 (relU r1 r2).
Proof. by move=> x y r1xy; apply/orP; left. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subrelUl | |
subrelUrT (r1 r2 : rel T) : subrel r2 (relU r1 r2).
Proof. by move=> x y r2xy; apply/orP; right. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subrelUr | |
mem_predT := Mem of pred T. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mem_pred | |
pred_of_mem{T} mp : {pred T} := let: Mem p := mp in [eta p].
Canonical memPredType T := PredType (@pred_of_mem T). | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred_of_mem | |
in_mem{T} (x : T) mp := pred_of_mem mp x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_mem | |
eq_mem{T} mp1 mp2 := forall x : T, in_mem x mp1 = in_mem x mp2. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | eq_mem | |
sub_mem{T} mp1 mp2 := forall x : T, in_mem x mp1 -> in_mem x mp2.
Arguments in_mem {T} x mp : simpl never.
Global Typeclasses Opaque eq_mem sub_mem. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_mem | |
simpl_of_mem{T} mp := SimplPred (fun x : T => in_mem x mp). | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | simpl_of_mem | |
sub_reflT (mp : mem_pred T) : sub_mem mp mp. Proof. by []. Qed.
Arguments sub_refl {T mp} [x] mp_x. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_refl | |
memT (pT : predType T) : pT -> mem_pred T :=
let: PredType toP := pT in fun A => Mem [eta toP A].
Arguments mem {T pT} A : rename, simpl never. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mem | |
applicative_predT := pred T. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | applicative_pred | |
collective_predT := pred T. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | collective_pred | |
applicative_pred_of_simplT (sp : simpl_pred T) : applicative_pred T :=
fun_of_simpl sp. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | applicative_pred_of_simpl | |
collective_pred_of_simplT (sp : simpl_pred T) : collective_pred T :=
let: SimplFun p := sp in p. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | collective_pred_of_simpl | |
Apred: applicative_pred T := [pred x | ...] idiom.
This instance is mainly intended for the in_applicative component of inE,
in conjunction with manifest_mem_pred and applicative_mem_pred.
- manifest_simpl_pred: the only instance of this structure matches manifest
simpl_pred values of the form SimplPred p... | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | Apred | |
Acoll: collective_pred T := [pred x | ...].
as the collective_pred_of_simpl is _not_ convertible to pred_of_simpl. **) | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | Acoll | |
registered_applicative_predp := RegisteredApplicativePred {
applicative_pred_value :> pred T;
_ : applicative_pred_value = p
}. | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | registered_applicative_pred | |
ApplicativePredp := RegisteredApplicativePred (erefl p).
Canonical applicative_pred_applicative sp :=
ApplicativePred (applicative_pred_of_simpl sp). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ApplicativePred | |
manifest_simpl_predp := ManifestSimplPred {
simpl_pred_value :> simpl_pred T;
_ : simpl_pred_value = SimplPred p
}.
Canonical expose_simpl_pred p := ManifestSimplPred (erefl (SimplPred p)). | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | manifest_simpl_pred | |
manifest_mem_predp := ManifestMemPred {
mem_pred_value :> mem_pred T;
_ : mem_pred_value = Mem [eta p]
}.
Canonical expose_mem_pred p := ManifestMemPred (erefl (Mem [eta p])). | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | manifest_mem_pred | |
applicative_mem_predp :=
ApplicativeMemPred {applicative_mem_pred_value :> manifest_mem_pred p}.
Canonical check_applicative_mem_pred p (ap : registered_applicative_pred p) :=
[eta @ApplicativeMemPred ap]. | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | applicative_mem_pred | |
mem_topredpT (pp : pT) : mem (topred pp) = mem pp.
Proof. by case: pT pp. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mem_topred | |
topredEpT x (pp : pT) : topred pp x = (x \in pp).
Proof. by rewrite -mem_topred. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | topredE | |
app_predEx p (ap : registered_applicative_pred p) : ap x = (x \in p).
Proof. by case: ap => _ /= ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | app_predE | |
in_applicativex p (amp : applicative_mem_pred p) : in_mem x amp = p x.
Proof. by case: amp => -[_ /= ->]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_applicative | |
in_collectivex p (msp : manifest_simpl_pred p) :
(x \in collective_pred_of_simpl msp) = p x.
Proof. by case: msp => _ /= ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_collective | |
in_simplx p (msp : manifest_simpl_pred p) :
in_mem x (Mem [eta pred_of_simpl msp]) = p x.
Proof. by case: msp => _ /= ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_simpl | |
unfold_inx p : (x \in ([eta p] : pred T)) = p x.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | unfold_in | |
simpl_predEp : SimplPred p =1 p.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | simpl_predE | |
inE:= (in_applicative, in_simpl, simpl_predE). (* to be extended *) | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inE | |
mem_simplsp : mem sp = sp :> pred T.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mem_simpl | |
memE:= mem_simpl. (* could be extended *) | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | memE | |
mem_memmp :
(mem mp = mp) * (mem (mp : simpl_pred T) = mp) * (mem (mp : pred T) = mp).
Proof. by case: mp. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mem_mem | |
qualifier(q : nat) T := Qualifier of {pred T}. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | qualifier | |
has_qualityn T (q : qualifier n T) : {pred T} :=
fun x => let: Qualifier _ p := q in p x.
Arguments has_quality n {T}. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | has_quality | |
qualifEn T p x : (x \in @Qualifier n T p) = p x. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | qualifE | |
pred_key(p : {pred T}) : Prop := DefaultPredKey.
Variable p : {pred T}. | Variant | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred_key | |
keyed_pred(k : pred_key p) :=
PackKeyedPred {unkey_pred :> {pred T}; _ : unkey_pred =i p}.
Variable k : pred_key p. | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | keyed_pred | |
KeyedPred:= @PackKeyedPred k p (frefl _).
Variable k_p : keyed_pred k. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | KeyedPred | |
keyed_predE: k_p =i p. Proof. by case: k_p. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | keyed_predE | |
keyed_qualifier(k : pred_key q) :=
PackKeyedQualifier {unkey_qualifier; _ : unkey_qualifier = q}. | Structure | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | keyed_qualifier | |
KeyedQualifierk := PackKeyedQualifier k (erefl q).
Variables (k : pred_key q) (k_q : keyed_qualifier k).
Fact keyed_qualifier_suproof : unkey_qualifier k_q =i q.
Proof. by case: k_q => /= _ ->. Qed.
Canonical keyed_qualifier_keyed := PackKeyedPred k keyed_qualifier_suproof. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | KeyedQualifier | |
all_tag_cond_depI T (C : pred I) U :
(forall x, T x) -> (forall x, C x -> {y : T x & U x y}) ->
{f : forall x, T x & forall x, C x -> U x (f x)}.
Proof.
move=> f0 fP; apply: all_tag (fun x y => C x -> U x y) _ => x.
by case Cx: (C x); [case/fP: Cx => y; exists y | exists (f0 x)].
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_tag_cond_dep | |
all_tag_condI T (C : pred I) U :
T -> (forall x, C x -> {y : T & U x y}) ->
{f : I -> T & forall x, C x -> U x (f x)}.
Proof. by move=> y0; apply: all_tag_cond_dep. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_tag_cond | |
all_sig_cond_depI T (C : pred I) P :
(forall x, T x) -> (forall x, C x -> {y : T x | P x y}) ->
{f : forall x, T x | forall x, C x -> P x (f x)}.
Proof. by move=> f0 /(all_tag_cond_dep f0)[f]; exists f. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_sig_cond_dep | |
all_sig_condI T (C : pred I) P :
T -> (forall x, C x -> {y : T | P x y}) ->
{f : I -> T | forall x, C x -> P x (f x)}.
Proof. by move=> y0; apply: all_sig_cond_dep. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_sig_cond | |
all_sig2_cond{I T} (C : pred I) P Q :
T -> (forall x, C x -> {y : T | P x y & Q x y}) ->
{f : I -> T | forall x, C x -> P x (f x) & forall x, C x -> Q x (f x)}.
Proof.
by move=> /all_sig_cond/[apply]-[f Pf]; exists f => i Di; have [] := Pf i Di.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | all_sig2_cond | |
total:= forall x y, R x y || R y x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | total | |
transitive:= forall y x z, R x y -> R y z -> R x z. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | transitive | |
symmetric:= forall x y, R x y = R y x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | symmetric | |
antisymmetric:= forall x y, R x y && R y x -> x = y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | antisymmetric | |
pre_symmetric:= forall x y, R x y -> R y x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pre_symmetric | |
symmetric_from_pre: pre_symmetric -> symmetric.
Proof. by move=> symR x y; apply/idP/idP; apply: symR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | symmetric_from_pre | |
reflexive:= forall x, R x x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | reflexive | |
irreflexive:= forall x, R x x = false. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | irreflexive | |
left_transitive:= forall x y, R x y -> R x =1 R y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | left_transitive | |
right_transitive:= forall x y, R x y -> R^~ x =1 R^~ y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | right_transitive | |
sym_left_transitive: left_transitive.
Proof. by move=> x y Rxy z; apply/idP/idP; apply: trR; rewrite // symR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sym_left_transitive | |
sym_right_transitive: right_transitive.
Proof. by move=> x y /sym_left_transitive Rxy z; rewrite !(symR z) Rxy. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sym_right_transitive | |
equivalence_rel:= forall x y z, R z z * (R x y -> R x z = R y z). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivalence_rel | |
equivalence_relP: equivalence_rel <-> reflexive /\ left_transitive.
Proof.
split=> [eqiR | [Rxx trR] x y z]; last by split=> [|/trR->].
by split=> [x | x y Rxy z]; [rewrite (eqiR x x x) | rewrite (eqiR x y z)].
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivalence_relP | |
rev_transT (R : rel T) : transitive R -> transitive (fun x y => R y x).
Proof. by move=> trR x y z Ryx Rzy; apply: trR Rzy Ryx. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rev_trans | |
prop_for(x : T1) P & ph {all1 P} := P x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_for | |
forEx P phP : @prop_for x P phP = P x. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | forE | |
prop_in1P & ph {all1 P} :=
forall x, in_mem x d1 -> P x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in1 | |
prop_in11P & ph {all2 P} :=
forall x y, in_mem x d1 -> in_mem y d2 -> P x y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in11 | |
prop_in2P & ph {all2 P} :=
forall x y, in_mem x d1 -> in_mem y d1 -> P x y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in2 | |
prop_in111P & ph {all3 P} :=
forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d3 -> P x y z. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in111 | |
prop_in12P & ph {all3 P} :=
forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d2 -> P x y z. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in12 | |
prop_in21P & ph {all3 P} :=
forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d2 -> P x y z. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in21 | |
prop_in3P & ph {all3 P} :=
forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d1 -> P x y z.
Variable f : T1 -> T2. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_in3 | |
prop_on1Pf P & phantom T3 (Pf f) & ph {all1 P} :=
forall x, in_mem (f x) d2 -> P x. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_on1 | |
prop_on2Pf P & phantom T3 (Pf f) & ph {all2 P} :=
forall x y, in_mem (f x) d2 -> in_mem (f y) d2 -> P x y. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_on2 | |
inPhantom:= Phantom Prop. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inPhantom | |
onPhantom{T} P (x : T) := Phantom Prop (P x). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onPhantom | |
bijective_inaT rT (d : mem_pred aT) (f : aT -> rT) :=
exists2 g, prop_in1 d (inPhantom (cancel f g))
& prop_on1 d (Phantom _ (cancel g)) (onPhantom (cancel g) f). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | bijective_in | |
bijective_onaT rT (cd : mem_pred rT) (f : aT -> rT) :=
exists2 g, prop_on1 cd (Phantom _ (cancel f)) (onPhantom (cancel f) g)
& prop_in1 cd (inPhantom (cancel g f)). | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | bijective_on | |
in1W: {all1 P1} -> {in D1, {all1 P1}}.
Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in1W | |
in2W: {all2 P2} -> {in D1 & D2, {all2 P2}}.
Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in2W | |
in3W: {all3 P3} -> {in D1 & D2 & D3, {all3 P3}}.
Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in3W | |
in1T: {in T1, {all1 P1}} -> {all1 P1}.
Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in1T | |
in2T: {in T1 & T2, {all2 P2}} -> {all2 P2}.
Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in2T | |
in3T: {in T1 & T2 & T3, {all3 P3}} -> {all3 P3}.
Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in3T | |
sub_in1(Ph : ph {all1 P1}) : prop_in1 d1' Ph -> prop_in1 d1 Ph.
Proof. by move=> allP x /sub1; apply: allP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in1 | |
sub_in11(Ph : ph {all2 P2}) : prop_in11 d1' d2' Ph -> prop_in11 d1 d2 Ph.
Proof. by move=> allP x1 x2 /sub1 d1x1 /sub2; apply: allP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in11 | |
sub_in111(Ph : ph {all3 P3}) :
prop_in111 d1' d2' d3' Ph -> prop_in111 d1 d2 d3 Ph.
Proof. by move=> allP x1 x2 x3 /sub1 d1x1 /sub2 d2x2 /sub3; apply: allP. Qed.
Let allQ1 f'' := {all1 Q1 f''}.
Let allQ1l f'' h' := {all1 Q1l f'' h'}.
Let allQ2 f'' := {all2 Q2 f''}. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in111 | |
on1W: allQ1 f -> {on D2, allQ1 f}. Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1W | |
on1lW: allQ1l f h -> {on D2, allQ1l f & h}. Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1lW | |
on2W: allQ2 f -> {on D2 &, allQ2 f}. Proof. by move=> ? ?. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on2W | |
on1T: {on T2, allQ1 f} -> allQ1 f. Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1T | |
on1lT: {on T2, allQ1l f & h} -> allQ1l f h.
Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1lT | |
on2T: {on T2 &, allQ2 f} -> allQ2 f.
Proof. by move=> ? ?; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on2T | |
subon1(Phf : ph (allQ1 f)) (Ph : ph (allQ1 f)) :
prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph.
Proof. by move=> allQ x /sub2; apply: allQ. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subon1 | |
subon1l(Phf : ph (allQ1l f)) (Ph : ph (allQ1l f h)) :
prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph.
Proof. by move=> allQ x /sub2; apply: allQ. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subon1l | |
subon2(Phf : ph (allQ2 f)) (Ph : ph (allQ2 f)) :
prop_on2 d2' Phf Ph -> prop_on2 d2 Phf Ph.
Proof. by move=> allQ x y /sub2=> d2fx /sub2; apply: allQ. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subon2 | |
can_in_inj: {in D1, cancel f g} -> {in D1 &, injective f}.
Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | can_in_inj |
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