fact stringlengths 6 2.88k | type stringclasses 17
values | library stringclasses 2
values | imports listlengths 0 16 | filename stringclasses 89
values | symbolic_name stringlengths 1 36 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
canLR_inx y : {in D1, cancel f g} -> y \in D1 -> x = f y -> g x = y.
Proof. by move=> fK D1y ->; rewrite fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | canLR_in | |
canRL_inx y : {in D1, cancel f g} -> x \in D1 -> f x = y -> x = g y.
Proof. by move=> fK D1x <-; rewrite fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | canRL_in | |
on_can_inj: {on D2, cancel f & g} -> {on D2 &, injective f}.
Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on_can_inj | |
canLR_onx y : {on D2, cancel f & g} -> f y \in D2 -> x = f y -> g x = y.
Proof. by move=> fK D2fy ->; rewrite fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | canLR_on | |
canRL_onx y : {on D2, cancel f & g} -> f x \in D2 -> f x = y -> x = g y.
Proof. by move=> fK D2fx <-; rewrite fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | canRL_on | |
inW_bij: bijective f -> {in D1, bijective f}.
Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inW_bij | |
onW_bij: bijective f -> {on D2, bijective f}.
Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onW_bij | |
inT_bij: {in T1, bijective f} -> bijective f.
Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inT_bij | |
onT_bij: {on T2, bijective f} -> bijective f.
Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onT_bij | |
sub_in_bij(D1' : pred T1) :
{subset D1 <= D1'} -> {in D1', bijective f} -> {in D1, bijective f}.
Proof.
by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K].
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in_bij | |
subon_bij(D2' : pred T2) :
{subset D2 <= D2'} -> {on D2', bijective f} -> {on D2, bijective f}.
Proof.
by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K].
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | subon_bij | |
in_on1P: {in D1, {on D2, allQ1 f}} <->
{in [pred x in D1 | f x \in D2], allQ1 f}.
Proof.
split => allf x; have := allf x; rewrite inE => Q1f; first by case/andP.
by move=> ? ?; apply: Q1f; apply/andP.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1P | |
in_on1lP: {in D1, {on D2, allQ1l f & h}} <->
{in [pred x in D1 | f x \in D2], allQ1l f h}.
Proof.
split => allf x; have := allf x; rewrite inE => Q1f; first by case/andP.
by move=> ? ?; apply: Q1f; apply/andP.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1lP | |
in_on2P: {in D1 &, {on D2 &, allQ2 f}} <->
{in [pred x in D1 | f x \in D2] &, allQ2 f}.
Proof.
split => allf x y; have := allf x y; rewrite !inE => Q2f.
by move=> /andP[? ?] /andP[? ?]; apply: Q2f.
by move=> ? ? ? ?; apply: Q2f; apply/andP.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on2P | |
on1W_in: {in D1, allQ1 f} -> {in D1, {on D2, allQ1 f}}.
Proof. by move=> D1f ? /D1f. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1W_in | |
on1lW_in: {in D1, allQ1l f h} -> {in D1, {on D2, allQ1l f & h}}.
Proof. by move=> D1f ? /D1f. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1lW_in | |
on2W_in: {in D1 &, allQ2 f} -> {in D1 &, {on D2 &, allQ2 f}}.
Proof. by move=> D1f ? ? ? ? ? ?; apply: D1f. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on2W_in | |
in_on1W: allQ1 f -> {in D1, {on D2, allQ1 f}}.
Proof. by move=> allf ? ? ?; apply: allf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1W | |
in_on1lW: allQ1l f h -> {in D1, {on D2, allQ1l f & h}}.
Proof. by move=> allf ? ? ?; apply: allf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1lW | |
in_on2W: allQ2 f -> {in D1 &, {on D2 &, allQ2 f}}.
Proof. by move=> allf ? ? ? ? ? ?; apply: allf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on2W | |
on1S: (forall x, f x \in D2) -> {on D2, allQ1 f} -> allQ1 f.
Proof. by move=> ? fD1 ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1S | |
on1lS: (forall x, f x \in D2) -> {on D2, allQ1l f & h} -> allQ1l f h.
Proof. by move=> ? fD1 ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1lS | |
on2S: (forall x, f x \in D2) -> {on D2 &, allQ2 f} -> allQ2 f.
Proof. by move=> ? fD1 ? ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on2S | |
on1S_in: {homo f : x / x \in D1 >-> x \in D2} ->
{in D1, {on D2, allQ1 f}} -> {in D1, allQ1 f}.
Proof. by move=> fD fD1 ? ?; apply/fD1/fD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1S_in | |
on1lS_in: {homo f : x / x \in D1 >-> x \in D2} ->
{in D1, {on D2, allQ1l f & h}} -> {in D1, allQ1l f h}.
Proof. by move=> fD fD1 ? ?; apply/fD1/fD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on1lS_in | |
on2S_in: {homo f : x / x \in D1 >-> x \in D2} ->
{in D1 &, {on D2 &, allQ2 f}} -> {in D1 &, allQ2 f}.
Proof. by move=> fD fD1 ? ? ? ?; apply: fD1 => //; apply: fD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | on2S_in | |
in_on1S: (forall x, f x \in D2) -> {in T1, {on D2, allQ1 f}} -> allQ1 f.
Proof. by move=> fD2 fD1 ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1S | |
in_on1lS: (forall x, f x \in D2) ->
{in T1, {on D2, allQ1l f & h}} -> allQ1l f h.
Proof. by move=> fD2 fD1 ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on1lS | |
in_on2S: (forall x, f x \in D2) ->
{in T1 &, {on D2 &, allQ2 f}} -> allQ2 f.
Proof. by move=> fD2 fD1 ? ?; apply: fD1. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_on2S | |
can_in_pcan[rT aT : Type] (A : {pred aT}) [f : aT -> rT] [g : rT -> aT] :
{in A, cancel f g} -> {in A, pcancel f (fun y : rT => Some (g y))}.
Proof. by move=> fK x Ax; rewrite fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | can_in_pcan | |
pcan_in_inj[rT aT : Type] [A : {pred aT}]
[f : aT -> rT] [g : rT -> option aT] :
{in A, pcancel f g} -> {in A &, injective f}.
Proof. by move=> fK x y Ax Ay /(congr1 g); rewrite !fK// => -[]. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pcan_in_inj | |
in_inj_compA B C (f : B -> A) (h : C -> B) (P : pred B) (Q : pred C) :
{in P &, injective f} -> {in Q &, injective h} -> {homo h : x / Q x >-> P x} ->
{in Q &, injective (f \o h)}.
Proof.
by move=> Pf Qh QP x y xQ yQ xy; apply Qh => //; apply Pf => //; apply QP.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_inj_comp | |
can_in_comp[A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B -> A] [h : C -> B] [f' : A -> B] [h' : B -> C] :
{homo h : x / x \in D' >-> x \in D} ->
{in D, cancel f f'} -> {in D', cancel h h'} ->
{in D', cancel (f \o h) (h' \o f')}.
Proof. by move=> hD fK hK c cD /=; rewrite fK ?hK ?hD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | can_in_comp | |
pcan_in_comp[A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B -> A] [h : C -> B] [f' : A -> option B] [h' : B -> option C] :
{homo h : x / x \in D' >-> x \in D} ->
{in D, pcancel f f'} -> {in D', pcancel h h'} ->
{in D', pcancel (f \o h) (obind h' \o f')}.
Proof. by move=> hD fK hK c cD /=; rewrite fK/= ?hK ... | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pcan_in_comp | |
pred_oappT (D : {pred T}) : pred (option T) :=
[pred x | oapp (mem D) false x]. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | pred_oapp | |
ocan_in_comp[A B C : Type] (D : {pred B}) (D' : {pred C})
[f : B -> option A] [h : C -> option B] [f' : A -> B] [h' : B -> C] :
{homo h : x / x \in D' >-> x \in pred_oapp D} ->
{in D, ocancel f f'} -> {in D', ocancel h h'} ->
{in D', ocancel (obind f \o h) (h' \o f')}.
Proof.
move=> hD fK hK c cD /=; rewrite ... | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | ocan_in_comp | |
in1_sig: {in D1, {all1 P1}} -> forall x : sig D1, P1 (sval x).
Proof. by move=> DP [x Dx]; have := DP _ Dx. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in1_sig | |
in2_sig: {in D1 & D2, {all2 P2}} ->
forall (x : sig D1) (y : sig D2), P2 (sval x) (sval y).
Proof. by move=> DP [x Dx] [y Dy]; have := DP _ _ Dx Dy. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in2_sig | |
in3_sig: {in D1 & D2 & D3, {all3 P3}} ->
forall (x : sig D1) (y : sig D2) (z : sig D3), P3 (sval x) (sval y) (sval z).
Proof. by move=> DP [x Dx] [y Dy] [z Dz]; have := DP _ _ _ Dx Dy Dz. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in3_sig | |
sub_in2T d d' (P : T -> T -> Prop) :
sub_mem d d' -> forall Ph : ph {all2 P}, prop_in2 d' Ph -> prop_in2 d Ph.
Proof. by move=> /= sub_dd'; apply: sub_in11. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in2 | |
sub_in3T d d' (P : T -> T -> T -> Prop) :
sub_mem d d' -> forall Ph : ph {all3 P}, prop_in3 d' Ph -> prop_in3 d Ph.
Proof. by move=> /= sub_dd'; apply: sub_in111. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in3 | |
sub_in12T1 T d1 d1' d d' (P : T1 -> T -> T -> Prop) :
sub_mem d1 d1' -> sub_mem d d' ->
forall Ph : ph {all3 P}, prop_in12 d1' d' Ph -> prop_in12 d1 d Ph.
Proof. by move=> /= sub1 sub; apply: sub_in111. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in12 | |
sub_in21T T3 d d' d3 d3' (P : T -> T -> T3 -> Prop) :
sub_mem d d' -> sub_mem d3 d3' ->
forall Ph : ph {all3 P}, prop_in21 d' d3' Ph -> prop_in21 d d3 Ph.
Proof. by move=> /= sub sub3; apply: sub_in111. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | sub_in21 | |
equivalence_relP_inT (R : rel T) (A : pred T) :
{in A & &, equivalence_rel R}
<-> {in A, reflexive R} /\ {in A &, forall x y, R x y -> {in A, R x =1 R y}}.
Proof.
split=> [eqiR | [Rxx trR] x y z *]; last by split=> [|/trR-> //]; apply: Rxx.
by split=> [x Ax|x y Ax Ay Rxy z Az]; [rewrite (eqiR x x) | rewrite (eqiR ... | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | equivalence_relP_in | |
monoW: {mono f : x / aP x >-> rP x} -> {homo f : x / aP x >-> rP x}.
Proof. by move=> hf x ax; rewrite hf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoW | |
mono2W:
{mono f : x y / aR x y >-> rR x y} -> {homo f : x y / aR x y >-> rR x y}.
Proof. by move=> hf x y axy; rewrite hf. Qed.
Hypothesis fgK : cancel g f. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mono2W | |
homoRL:
{homo f : x y / aR x y >-> rR x y} -> forall x y, aR (g x) y -> rR x (f y).
Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homoRL | |
homoLR:
{homo f : x y / aR x y >-> rR x y} -> forall x y, aR x (g y) -> rR (f x) y.
Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homoLR | |
homo_mono:
{homo f : x y / aR x y >-> rR x y} -> {homo g : x y / rR x y >-> aR x y} ->
{mono g : x y / rR x y >-> aR x y}.
Proof.
move=> mf mg x y; case: (boolP (rR _ _))=> [/mg //|].
by apply: contraNF=> /mf; rewrite !fgK.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homo_mono | |
monoLR:
{mono f : x y / aR x y >-> rR x y} -> forall x y, rR (f x) y = aR x (g y).
Proof. by move=> mf x y; rewrite -{1}[y]fgK mf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoLR | |
monoRL:
{mono f : x y / aR x y >-> rR x y} -> forall x y, rR x (f y) = aR (g x) y.
Proof. by move=> mf x y; rewrite -{1}[x]fgK mf. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoRL | |
can_mono:
{mono f : x y / aR x y >-> rR x y} -> {mono g : x y / rR x y >-> aR x y}.
Proof. by move=> mf x y /=; rewrite -mf !fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | can_mono | |
mono1W_in:
{in aD, {mono f : x / aP x >-> rP x}} ->
{in aD, {homo f : x / aP x >-> rP x}}.
Proof. by move=> hf x hx ax; rewrite hf. Qed.
#[deprecated(since="Coq 8.16", note="Use mono1W_in instead.")]
Abbreviation mono2W_in := mono1W_in. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mono1W_in | |
monoW_in:
{in aD &, {mono f : x y / aR x y >-> rR x y}} ->
{in aD &, {homo f : x y / aR x y >-> rR x y}}.
Proof. by move=> hf x y hx hy axy; rewrite hf. Qed.
Hypothesis fgK : {in rD, {on aD, cancel g & f}}.
Hypothesis mem_g : {homo g : x / x \in rD >-> x \in aD}. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoW_in | |
homoRL_in:
{in aD &, {homo f : x y / aR x y >-> rR x y}} ->
{in rD & aD, forall x y, aR (g x) y -> rR x (f y)}.
Proof. by move=> Hf x y hx hy /Hf; rewrite fgK ?mem_g// ?inE; apply. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homoRL_in | |
homoLR_in:
{in aD &, {homo f : x y / aR x y >-> rR x y}} ->
{in aD & rD, forall x y, aR x (g y) -> rR (f x) y}.
Proof. by move=> Hf x y hx hy /Hf; rewrite fgK ?mem_g// ?inE; apply. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homoLR_in | |
homo_mono_in:
{in aD &, {homo f : x y / aR x y >-> rR x y}} ->
{in rD &, {homo g : x y / rR x y >-> aR x y}} ->
{in rD &, {mono g : x y / rR x y >-> aR x y}}.
Proof.
move=> mf mg x y hx hy; case: (boolP (rR _ _))=> [/mg //|]; first exact.
by apply: contraNF=> /mf; rewrite !fgK ?mem_g//; apply.
Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homo_mono_in | |
monoLR_in:
{in aD &, {mono f : x y / aR x y >-> rR x y}} ->
{in aD & rD, forall x y, rR (f x) y = aR x (g y)}.
Proof. by move=> mf x y hx hy; rewrite -{1}[y]fgK ?mem_g// mf ?mem_g. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoLR_in | |
monoRL_in:
{in aD &, {mono f : x y / aR x y >-> rR x y}} ->
{in rD & aD, forall x y, rR x (f y) = aR (g x) y}.
Proof. by move=> mf x y hx hy; rewrite -{1}[x]fgK ?mem_g// mf ?mem_g. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | monoRL_in | |
can_mono_in:
{in aD &, {mono f : x y / aR x y >-> rR x y}} ->
{in rD &, {mono g : x y / rR x y >-> aR x y}}.
Proof. by move=> mf x y hx hy; rewrite -mf ?mem_g// !fgK ?mem_g. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | can_mono_in | |
homo_sym: {homo f : x y / aR x y >-> rR x y} ->
{homo f : y x / aR x y >-> rR x y}.
Proof. by move=> fR y x; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homo_sym | |
mono_sym: {mono f : x y / aR x y >-> rR x y} ->
{mono f : y x / aR x y >-> rR x y}.
Proof. by move=> fR y x; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mono_sym | |
homo_sym_in: {in aD &, {homo f : x y / aR x y >-> rR x y}} ->
{in aD &, {homo f : y x / aR x y >-> rR x y}}.
Proof. by move=> fR y x yD xD; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homo_sym_in | |
mono_sym_in: {in aD &, {mono f : x y / aR x y >-> rR x y}} ->
{in aD &, {mono f : y x / aR x y >-> rR x y}}.
Proof. by move=> fR y x yD xD; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mono_sym_in | |
homo_sym_in11: {in aD & aD', {homo f : x y / aR x y >-> rR x y}} ->
{in aD' & aD, {homo f : y x / aR x y >-> rR x y}}.
Proof. by move=> fR y x yD xD; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | homo_sym_in11 | |
mono_sym_in11: {in aD & aD', {mono f : x y / aR x y >-> rR x y}} ->
{in aD' & aD, {mono f : y x / aR x y >-> rR x y}}.
Proof. by move=> fR y x yD xD; apply: fR. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | mono_sym_in11 | |
onW_can: cancel g f -> {on aD, cancel g & f}.
Proof. by move=> fgK x xaD; apply: fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onW_can | |
onW_can_in: {in rD, cancel g f} -> {in rD, {on aD, cancel g & f}}.
Proof. by move=> fgK x xrD xaD; apply: fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onW_can_in | |
in_onW_can: cancel g f -> {in rD, {on aD, cancel g & f}}.
Proof. by move=> fgK x xrD xaD; apply: fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_onW_can | |
onS_can: (forall x, g x \in aD) -> {on aD, cancel g & f} -> cancel g f.
Proof. by move=> mem_g fgK x; apply: fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onS_can | |
onS_can_in: {homo g : x / x \in rD >-> x \in aD} ->
{in rD, {on aD, cancel g & f}} -> {in rD, cancel g f}.
Proof. by move=> mem_g fgK x x_rD; apply/fgK/mem_g. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | onS_can_in | |
in_onS_can: (forall x, g x \in aD) ->
{in rT, {on aD, cancel g & f}} -> cancel g f.
Proof. by move=> mem_g fgK x; apply/fgK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | in_onS_can | |
inj_can_sym_in_on:
{homo f : x / x \in aD >-> x \in rD} -> {in aD, {on rD, cancel f & g}} ->
{in rD &, {on aD &, injective g}} -> {in rD, {on aD, cancel g & f}}.
Proof. by move=> fD fK gI x x_rD gx_aD; apply: gI; rewrite ?inE ?fK ?fD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inj_can_sym_in_on | |
inj_can_sym_on: {in aD, cancel f g} ->
{on aD &, injective g} -> {on aD, cancel g & f}.
Proof. by move=> fK gI x gx_aD; apply: gI; rewrite ?inE ?fK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inj_can_sym_on | |
inj_can_sym_in: {homo f \o g : x / x \in rD} -> {on rD, cancel f & g} ->
{in rD &, injective g} -> {in rD, cancel g f}.
Proof. by move=> fgD fK gI x x_rD; apply: gI; rewrite ?fK ?fgD. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | inj_can_sym_in | |
Reflexive(R : A -> A -> Prop) :=
reflexivity : forall x : A, R x x. | Class | Corelib | [] | Corelib/ssr/ssrclasses.v | Reflexive | |
eq_Reflexive{A : Type} : Reflexive (@eq A) := @eq_refl A.
#[global] | Instance | Corelib | [] | Corelib/ssr/ssrclasses.v | eq_Reflexive | |
iff_Reflexive: Reflexive iff := iff_refl. | Instance | Corelib | [] | Corelib/ssr/ssrclasses.v | iff_Reflexive | |
foo_defx1 .. xn := big_foo_expression.
Fact foo_key : unit. Proof. by #[# #]#. Qed. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | foo_def | |
foo:= locked_with foo_key foo_def.
Canonical foo_unlockable := #[#unlockable fun foo#]#.
This minimizes the comparison overhead for foo, while still allowing
rewrite unlock to expose big_foo_expression.
#[#elaborate x#]# == triggers Rocq elaboration to fill the holes of the term x
... | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | foo | |
abstract_lock:= unit. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | abstract_lock | |
abstract_key:= tt. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | abstract_key | |
abstract(statement : Type) (id : nat) (lock : abstract_lock) :=
let: tt := lock in statement.
Declare Scope ssr_scope. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | abstract | |
external_view: Type := tactic_view of Type. | Inductive | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | external_view | |
putvT sT (v1 v2 : vT) (s : sT) : Prop := Put. | Variant | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | put | |
getvT sT v s (p : @put vT sT v v s) := let: Put _ _ _ := p in s. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | get | |
get_byvT sT of sT -> vT := @get vT sT. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | get_by | |
my_type:= a_type.
my_type doesn't effectively inherit the struct structure from a_type. Our
solution is to redeclare the instance as follows
Canonical my_type_struct := Eval hnf in #[#struct of my_type#]#.
The special notation #[#str of _ #]# must be defined for each Structure "str"
with constructor "Str", typic... | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | my_type | |
clone_strs :=
let: Str _ x y ... z := s return {type of Str for s} -> str in
fun k => k _ x y ... z. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | clone_str | |
argumentTypeT P & forall x : T, P x := T. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | argumentType | |
dependentReturnTypeT P & forall x : T, P x := P. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | dependentReturnType | |
returnTypeaT rT & aT -> rT := rT. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | returnType | |
p_str: Type := p_Str {p_repr :> p_type; p_op : p_repr -> ...}
should be given as | Structure | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | p_str | |
indt_type(p : p_str) := Indt ... . | Inductive | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | indt_type | |
indt_of(p : p_str) & phantom p_type p := indt_type p. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | indt_of | |
indtp x y ... z : {indt p} := @Indt p x y ... z. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | indt | |
phantomT (p : T) : Prop := Phantom.
Arguments phantom : clear implicits.
Arguments Phantom : clear implicits. | Variant | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | phantom | |
phant(p : Type) : Prop := Phant. | Variant | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | phant | |
protect_term(A : Type) (x : A) : A := x.
Register protect_term as plugins.ssreflect.protect_term. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | protect_term | |
ssr_converseR (r : R) := (Logic.I, r). | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | ssr_converse |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.