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 |
|---|---|---|---|---|---|---|
foo:= nosimpl bar, foo (or foo t') will NOT be expanded by
the /= and //= switches unless it is in a forcing context (e.g., in
match foo t' with ... end, foo t' will be reduced if this allows the
match to be reduced). Note that nosimpl bar is simply notation for a
a term that beta-iota reduces to bar; h... | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | foo | |
master_key: unit. Proof. exact tt. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | master_key | |
lockedA := let: tt := master_key in fun x : A => x.
Register master_key as plugins.ssreflect.master_key.
Register locked as plugins.ssreflect.locked. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | locked | |
lockA x : x = locked x :> A. Proof. unlock; reflexivity. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | lock | |
done:=
trivial; hnf; intros; solve
[ do ![solve [trivial | simple refine (@sym_equal _ _ _ _); trivial]
| discriminate | contradiction | split]
| match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. | Ltac | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | done | |
ssrdone0:=
trivial; hnf; intros; solve
[ do ![solve [trivial | apply: sym_equal; trivial]
| discriminate | contradiction ]
| match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. | Ltac | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | ssrdone0 | |
unlockableT v := Unlockable {unlocked : T; _ : unlocked = v}. | Structure | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | unlockable | |
unlockT x C : @unlocked T x C = x. Proof. by case: C. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | unlock | |
locked_withk := let: tt := k in fun T x => x : T. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | locked_with | |
locked_withET k x : unkeyed (locked_with k x) = x :> T.
Proof. by case: k. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | locked_withE | |
unlock_withT k x : unlocked (locked_with_unlockable k x) = x :> T.
Proof. exact: unlock. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | unlock_with | |
ssr_have(Plemma : Prop) (Pgoal : Prop)
(step : Plemma) (rest : Plemma -> Pgoal) : Pgoal.
Proof. exact: rest step. Qed.
Register ssr_have as plugins.ssreflect.ssr_have.
Polymorphic Lemma ssr_have_upoly@{s1 s2;u1 u2}
(Plemma : Type@{s1;u1}) (Pgoal : Type@{s2;u2})
(step : Plemma) (rest : Plemma -> Pgoal) : Pgoal... | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | ssr_have | |
nary_congruence_statement(n : nat)
: (forall B, (B -> B -> Prop) -> Prop) -> Prop :=
match n with
| O => fun k => forall B, k B (fun x1 x2 : B => x1 = x2)
| S n' =>
let k' A B e (f1 f2 : A -> B) :=
forall x1 x2, x1 = x2 -> (e (f1 x1) (f2 x2) : Prop) in
fun k => forall A, nary_congruence_sta... | Fixpoint | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | nary_congruence_statement | |
nary_congruencen (k := fun B e => forall y : B, (e y y : Prop)) :
nary_congruence_statement n k.
Proof.
have: k _ _ := _; rewrite {1}/k.
elim: n k => [|n IHn] k k_P /= A; first exact: k_P.
by apply: IHn => B e He; apply: k_P => f x1 x2 <-.
Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | nary_congruence | |
ssr_congr_arrowPlemma Pgoal : Plemma = Pgoal -> Plemma -> Pgoal.
Proof. by move->. Qed.
Arguments ssr_congr_arrow : clear implicits.
Register nary_congruence as plugins.ssreflect.nary_congruence.
Register ssr_congr_arrow as plugins.ssreflect.ssr_congr_arrow. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | ssr_congr_arrow | |
iffLR: P -> Q. Proof. by case: eqPQ. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | iffLR | |
iffRL: Q -> P. Proof. by case: eqPQ. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | iffRL | |
iffLRn: ~P -> ~Q. Proof. by move=> nP tQ; case: nP; case: eqPQ tQ. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | iffLRn | |
iffRLn: ~Q -> ~P. Proof. by move=> nQ tP; case: nQ; case: eqPQ tP. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | iffRLn | |
abstract_contextT (P : T -> Type) x :
(forall Q, Q = P -> Q x) -> P x.
Proof. by move=> /(_ P); apply. Qed. | Lemma | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | abstract_context | |
over:= over_rel. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | over | |
over:=
by [ apply: Under_rel.under_rel_done
| rewrite over
]. | Ltac | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | over | |
UnderE:= Under_relE. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | UnderE | |
call_of(condition : unit) (result : bool) := Call {callee : Type}. | Structure | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | call_of | |
maybeProp(T : Type) := tt. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | maybeProp | |
callT := Call (maybeProp T) false T. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | call | |
test_of(result : bool) := Test {condition :> unit}. | Structure | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | test_of | |
test_Prop(P : Prop) := Test true (maybeProp P). | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | test_Prop | |
test_negative:= Test false tt. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | test_negative | |
type:=
Check {result : bool; test : test_of result; frame : call_of test result}. | Structure | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | type | |
checkresult test frame := @Check result test frame. | Definition | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | check | |
callee: call_of >-> Sortclass. | Coercion | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | callee | |
frame: type >-> call_of.
Abbreviation notProp T := (@check false test_negative (call T)). | Coercion | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | frame | |
vm_compute_eq{T : Type} (x y : T) := vm_compute : x = y.
#[global]
Hint Extern 0 (@vm_compute_eq _ _ _) =>
vm_compute; reflexivity : typeclass_instances. | Class | Corelib | [
"Require Import ssrmatching",
"Require Export ssrunder"
] | Corelib/ssr/ssreflect.v | vm_compute_eq | |
inference, as in the implementation of
the mxdirect predicate in matrix.v.
- The empty type:
void == a notation for the Empty_set type of the standard library.
of_void T == the canonical injection void -> T.
- Sigma types:
tag w == the i of w : {i : I & T i}.
... | Structure | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | inference | |
fooT sT & T -> sort sT := ...
We can avoid specifying sT directly by calling foo (@id T), or specify
the call completely while still ensuring the consistency of T and sT, by
calling @foo T sT idfun. The phant_id type allows us to extend this trick
to non-Type canonical projections. It also allows us to sidestep
de... | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | foo | |
r:= R {x; y : T(x)}.
if we need to build an r from a given y0 while inferring some x0, such
that y0 : T(x0), we pose | Record | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | r | |
mk_r.. y .. (x := ...) y' & phant_id y y' := R x y'.
Calling @mk_r .. y0 .. id will cause Rocq to use y' := y0, while checking
the dependent type constraint y0 : T(x0).
- Extensional equality for functions and relations (i.e. functions of two
arguments):
f1 =1 f2 == f1 x is equal to f2 x for all x.
... | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | mk_r | |
foox y .. := #[#rec e1, ... #]#.
+ proofs about foo | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | foo | |
pair_of_andP Q (PandQ : P /\ Q) := (proj1 PandQ, proj2 PandQ). | Coercion | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | pair_of_and | |
all_pairI T U (w : forall i : I, T i * U i) :=
(fun i => (w i).1, fun i => (w i).2). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_pair | |
applyaT rT (f : aT -> rT) x u := if u is Some y then f y else x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | apply | |
defaultT := apply (fun x : T => x). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | default | |
bindaT rT (f : aT -> option rT) := apply f None. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | bind | |
mapaT rT (f : aT -> rT) := bind (fun x => Some (f x)). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | map | |
liftaT rT (f : aT -> rT) := fun x => Some (f x). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | lift | |
esym:= sym_eq. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | esym | |
nesym:= sym_not_eq. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | nesym | |
etrans:= trans_eq. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | etrans | |
congr1:= f_equal. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | congr1 | |
congr2:= f_equal2. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | congr2 | |
all_equal_toT (x0 : T) := forall x, unkeyed x = x0. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_equal_to | |
unitE: all_equal_to tt. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | unitE | |
wrappedT := Wrap {unwrap : T}.
Canonical wrap T x := @Wrap T x.
Prenex Implicits unwrap wrap Wrap.
Delimit Scope function_scope with FUN.
Open Scope function_scope. | Structure | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | wrapped | |
simpl_fun(aT rT : Type) := SimplFun of aT -> rT. | Variant | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | simpl_fun | |
fun_of_simpl(f : simpl_fun aT rT) := fun x => let: SimplFun lam := f in lam x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | fun_of_simpl | |
fun_of_simpl: simpl_fun >-> Funclass. | Coercion | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | fun_of_simpl | |
SimplFunDeltaaT rT (f : aT -> aT -> rT) := [fun z => f z z]. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | SimplFunDelta | |
eqfun(f g : B -> A) : Prop := forall x, f x = g x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | eqfun | |
eqrel(r s : C -> B -> A) : Prop := forall x y, r x y = s x y. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | eqrel | |
freflf : eqfun f f. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | frefl | |
fsymf g : eqfun f g -> eqfun g f. Proof. by move=> eq_fg x. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | fsym | |
ftransf g h : eqfun f g -> eqfun g h -> eqfun f h.
Proof. by move=> eq_fg eq_gh x; rewrite eq_fg. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | ftrans | |
rreflr : eqrel r r. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | rrefl | |
comp(f : B -> A) (g : C -> B) x := f (g x). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | comp | |
catcompg f := comp f g. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | catcomp | |
pcomp(f : B -> option A) (g : C -> option B) x := obind f (g x). | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | pcomp | |
eq_compf f' g g' : f =1 f' -> g =1 g' -> comp f g =1 comp f' g'.
Proof. by move=> eq_ff' eq_gg' x; rewrite /comp eq_gg' eq_ff'. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | eq_comp | |
compA{A B C D : Type} (f : B -> A) (g : C -> B) (h : D -> C) :
f \o (g \o h) = (f \o g) \o h.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | compA | |
idfunT x : T := x.
Arguments idfun {T} x /. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | idfun | |
phant_idT1 T2 v1 v2 := phantom T1 v1 -> phantom T2 v2. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | phant_id | |
obindEapp(fo : aT -> option rT) : obind fo = oapp fo None.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | obindEapp | |
omapEbind: omap f = obind (olift f).
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | omapEbind | |
omap_id(x : option rT) : omap id x = x. Proof. by case: x. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | omap_id | |
eq_omap(h : aT -> rT) : f =1 h -> omap f =1 omap h.
Proof. by move=> Ef [?|] //=; rewrite Ef. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | eq_omap | |
omapEapp: omap f = oapp (olift f) None.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | omapEapp | |
oappEmap(y0 : rT) x : oapp f y0 x = odflt y0 (omap f x).
Proof. by case: x. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | oappEmap | |
omap_comp: omap (g \o f) =1 omap g \o omap f.
Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | omap_comp | |
oapp_compx : oapp (g \o f) x =1 (@oapp _ _)^~ x g \o omap f.
Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | oapp_comp | |
oapp_comp_f(x : rT) : oapp (g \o f) (g x) =1 g \o oapp f x.
Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | oapp_comp_f | |
olift_comp: olift (g \o f) = olift g \o f.
Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | olift_comp | |
of_voidT (x : void) : T := match x with end. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | of_void | |
tag:= projT1. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tag | |
tagged: forall w, T_(tag w) := @projT2 I [eta T_]. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tagged | |
Taggedx := @existT I [eta T_] i x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | Tagged | |
tag2(w : @sigT2 I T_ U_) := let: existT2 _ _ i _ _ := w in i. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tag2 | |
tagged2w : T_(tag2 w) := let: existT2 _ _ _ x _ := w in x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tagged2 | |
tagged2'w : U_(tag2 w) := let: existT2 _ _ _ _ y := w in y. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tagged2' | |
Tagged2x y := @existT2 I [eta T_] [eta U_] i x y. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | Tagged2 | |
tag_of_tag2I T_ U_ (w : @sigT2 I T_ U_) :=
Tagged (fun i => T_ i * U_ i)%type (tagged2 w, tagged2' w). | Coercion | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tag_of_tag2 | |
all_tagI T U :
(forall x : I, {y : T x & U x y}) ->
{f : forall x, T x & forall x, U x (f x)}.
Proof. by move=> fP; exists (fun x => tag (fP x)) => x; case: (fP x). Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_tag | |
all_tag2I T U V :
(forall i : I, {y : T i & U i y & V i y}) ->
{f : forall i, T i & forall i, U i (f i) & forall i, V i (f i)}.
Proof. by case/all_tag=> f /all_pair[]; exists f. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_tag2 | |
svalP(u : sig P) : P (sval u). Proof. by case: u. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | svalP | |
s2val(u : sig2 P Q) := let: exist2 _ _ x _ _ := u in x. | Definition | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | s2val | |
s2valPu : P (s2val u). Proof. by case: u. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | s2valP | |
s2valP'u : Q (s2val u). Proof. by case: u. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | s2valP' | |
tag_of_sigI P (u : @sig I P) := Tagged P (svalP u). | Coercion | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | tag_of_sig | |
sig_of_sig2I P Q (u : @sig2 I P Q) :=
exist (fun i => P i /\ Q i) (s2val u) (conj (s2valP u) (s2valP' u)). | Coercion | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | sig_of_sig2 | |
all_sigI T P :
(forall x : I, {y : T x | P x y}) ->
{f : forall x, T x | forall x, P x (f x)}.
Proof. by case/all_tag=> f; exists f. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_sig | |
all_sig2I T P Q :
(forall x : I, {y : T x | P x y & Q x y}) ->
{f : forall x, T x | forall x, P x (f x) & forall x, Q x (f x)}.
Proof. by case/all_sig=> f /all_pair[]; exists f. Qed. | Lemma | Corelib | [
"Require Import ssreflect"
] | Corelib/ssr/ssrfun.v | all_sig2 |
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