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 |
|---|---|---|---|---|---|---|
destruct_call_inf H :=
let tac t := (destruct t) in
let T := type of H in
on_application f tac T. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_call_in | |
destruct_call_asf l :=
let tac t := (destruct t as l) in on_call f tac. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_call_as | |
destruct_call_as_inf l H :=
let tac t := (destruct t as l) in
let T := type of H in
on_application f tac T.
Tactic Notation "destruct_call" constr(f) := destruct_call f. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_call_as_in | |
fix_proto{A : Type} (a : A) := a.
Register fix_proto as program.tactic.fix_proto. | Definition | Corelib | [] | Corelib/Program/Tactics.v | fix_proto | |
destruct_rec_calls:=
match goal with
| [ H : fix_proto _ |- _ ] => destruct_calls H ; clear H
end. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_rec_calls | |
destruct_all_rec_calls:=
repeat destruct_rec_calls ; unfold fix_proto in *. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_all_rec_calls | |
autoinjectiontac :=
match goal with
| [ H : ?f ?a = ?f' ?a' |- _ ] => tac H
end. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | autoinjection | |
injectH := progress (inversion H ; subst*; clear_dups) ; clear H. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | inject | |
autoinjections:= repeat (clear_dups ; autoinjection ltac:(inject)). | Ltac | Corelib | [] | Corelib/Program/Tactics.v | autoinjections | |
destruct_nondepH := let H0 := fresh "H" in assert(H0 := H); destruct H0. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | destruct_nondep | |
bang:=
match goal with
| |- ?x =>
match x with
| context [False_rect _ ?p] => elim p
end
end. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | bang | |
add_hypothesisH' p :=
match type of p with
?X =>
match goal with
| [ H : X |- _ ] => fail 1
| _ => set (H':=p) ; try (change p with H') ; clearbody H'
end
end. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | add_hypothesis | |
replace_hypH c :=
let H' := fresh "H" in
assert(H' := c) ; clear H ; rename H' into H. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | replace_hyp | |
refine_hypc :=
let tac H := replace_hyp H c in
match c with
| ?H _ => tac H
| ?H _ _ => tac H
| ?H _ _ _ => tac H
| ?H _ _ _ _ => tac H
| ?H _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ _ => tac H
| ?H _ _ _ _ _ _ _ _ => tac H
end. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | refine_hyp | |
program_simplify:=
simpl; intros ; destruct_all_rec_calls ; repeat (destruct_conjs; simpl proj1_sig in * );
subst*; autoinjections ; try discriminates ;
try (solve [ red ; intros ; destruct_conjs ; autoinjections ; discriminates ]). | Ltac | Corelib | [] | Corelib/Program/Tactics.v | program_simplify | |
program_solve_wf:=
match goal with
| |- well_founded _ => auto with *
| |- ?T => match type of T with Prop => auto end
end.
Create HintDb program discriminated. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | program_solve_wf | |
program_simpl:= program_simplify ; try typeclasses eauto 10 with program ; try program_solve_wf.
#[global] Obligation Tactic := program_simpl.
#[export] Obligation Tactic := program_simpl. | Ltac | Corelib | [] | Corelib/Program/Tactics.v | program_simpl | |
Fix_F_sub(x : A) (r : Acc R x) : P x :=
F_sub x (fun y: { y : A | R y x} => Fix_F_sub (proj1_sig y)
(Acc_inv r (proj2_sig y))). | Fixpoint | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_F_sub | |
Fix_sub(x : A) := Fix_F_sub x (Rwf x).
Register Fix_sub as program.wf.fix_sub. | Definition | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_sub | |
Fix_F_eq:
forall (x:A) (r:Acc R x),
F_sub x (fun y:{y:A | R y x} => Fix_F_sub (`y) (Acc_inv r (proj2_sig y))) = Fix_F_sub x r.
Proof.
intros x r; destruct r using Acc_inv_dep; auto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_F_eq | |
Fix_F_inv: forall (x:A) (r s:Acc R x), Fix_F_sub x r = Fix_F_sub x s.
Proof.
intro x; induction (Rwf x); intros.
rewrite <- 2 Fix_F_eq; intros. apply F_ext; intros []; auto.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_F_inv | |
Fix_eq: forall x:A, Fix_sub x = F_sub x (fun y:{ y:A | R y x} => Fix_sub (proj1_sig y)).
Proof.
intro x; unfold Fix_sub.
rewrite <- (Fix_F_eq ).
apply F_ext; intros.
apply Fix_F_inv.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_eq | |
fix_sub_eq:
forall x : A,
Fix_sub x =
let f_sub := F_sub in
f_sub x (fun y: {y : A | R y x} => Fix_sub (`y)).
Proof.
exact Fix_eq.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | fix_sub_eq | |
MR(x y: T): Prop := R (m x) (m y).
Register MR as program.wf.mr. | Definition | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | MR | |
measure_wf: well_founded MR.
Proof.
unfold well_founded.
cut (forall (a: M) (a0: T), m a0 = a -> Acc MR a0).
+ intros H a.
apply (H (m a)); auto.
+ apply (@well_founded_ind M R wf (fun mm => forall a, m a = mm -> Acc MR a)).
intros ? H ? H0.
apply Acc_intro.
intros y H1.
... | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | measure_wf | |
F_unfoldx r:
Fix_F_sub A R P f x r =
f (fun y => Fix_F_sub A R P f (proj1_sig y) (Acc_inv r (proj2_sig y))).
Proof. intros. case r; auto. Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | F_unfold | |
Fix_F_sub_rect(Q: forall x, P x -> Type)
(inv: forall x: A,
(forall (y: A) (H: R y x) (a: Acc R y),
Q y (Fix_F_sub A R P f y a)) ->
forall (a: Acc R x),
Q x (f (fun y: {y: A | R y x} =>
Fix_F_sub A R P f (proj1_sig y) (Acc_inv a (proj2_sig y)))))
: forall x a, Q _ (Fix... | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_F_sub_rect | |
eq_Fix_F_subx (a a': Acc R x):
Fix_F_sub A R P f x a =
Fix_F_sub A R P f x a'.
Proof.
revert a'.
pattern x, (Fix_F_sub A R P f x a).
apply Fix_F_sub_rect.
intros ? H **.
rewrite F_unfold.
apply equiv_lowers.
intros.
apply H.
assumption.
Qed. | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | eq_Fix_F_sub | |
Fix_sub_rect(Q: forall x, P x -> Type)
(inv: forall
(x: A)
(H: forall (y: A), R y x -> Q y (Fix_sub A R Rwf P f y))
(a: Acc R x),
Q x (f (fun y: {y: A | R y x} => Fix_sub A R Rwf P f (proj1_sig y))))
: forall x, Q _ (Fix_sub A R Rwf P f x).
Proof.
unfold Fix_sub.
intros x.
... | Lemma | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | Fix_sub_rect | |
fold_subf :=
match goal with
| [ |- ?T ] =>
match T with
context C [ @Fix_sub _ _ _ _ _ ?arg ] =>
let app := context C [ f arg ] in
change app
end
end. | Ltac | Corelib | [
"Require Import Corelib.Init.Wf",
"Require Import Corelib.Program.Utils"
] | Corelib/Program/Wf.v | fold_sub | |
relation:= A -> A -> Prop.
Variable R : relation. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | relation | |
reflexive: Prop := forall x:A, R x x. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | reflexive | |
transitive: Prop := forall x y z:A, R x y -> R y z -> R x z. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | transitive | |
symmetric: Prop := forall x y:A, R x y -> R y x. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | symmetric | |
antisymmetric: Prop := forall x y:A, R x y -> R y x -> x = y. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | antisymmetric | |
equiv:= reflexive /\ transitive /\ symmetric. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | equiv | |
preorder: Prop :=
{ preord_refl : reflexive; preord_trans : transitive}. | Record | Corelib | [] | Corelib/Relations/Relation_Definitions.v | preorder | |
order: Prop :=
{ ord_refl : reflexive;
ord_trans : transitive;
ord_antisym : antisymmetric}. | Record | Corelib | [] | Corelib/Relations/Relation_Definitions.v | order | |
equivalence: Prop :=
{ equiv_refl : reflexive;
equiv_trans : transitive;
equiv_sym : symmetric}. | Record | Corelib | [] | Corelib/Relations/Relation_Definitions.v | equivalence | |
PER: Prop := {per_sym : symmetric; per_trans : transitive}. | Record | Corelib | [] | Corelib/Relations/Relation_Definitions.v | PER | |
inclusion(R1 R2:relation) : Prop :=
forall x y:A, R1 x y -> R2 x y. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | inclusion | |
same_relation(R1 R2:relation) : Prop :=
inclusion R1 R2 /\ inclusion R2 R1. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | same_relation | |
commut(R1 R2:relation) : Prop :=
forall x y:A,
R1 y x -> forall z:A, R2 z y -> exists2 y' : A, R2 y' x & R1 z y'. | Definition | Corelib | [] | Corelib/Relations/Relation_Definitions.v | commut | |
Setoid_Theory:= @Equivalence. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | Setoid_Theory | |
Build_Setoid_Theory:= @Build_Equivalence.
Register Build_Setoid_Theory as plugins.ring.Build_Setoid_Theory. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | Build_Setoid_Theory | |
Seq_reflA Aeq (s : Setoid_Theory A Aeq) : forall x:A, Aeq x x.
Proof.
unfold Setoid_Theory in s. intros ; reflexivity.
Defined. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | Seq_refl | |
Seq_symA Aeq (s : Setoid_Theory A Aeq) : forall x y:A, Aeq x y -> Aeq y x.
Proof.
unfold Setoid_Theory in s. intros ; symmetry ; assumption.
Defined. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | Seq_sym | |
Seq_transA Aeq (s : Setoid_Theory A Aeq) : forall x y z:A, Aeq x y -> Aeq y z -> Aeq x z.
Proof.
unfold Setoid_Theory in s. intros x y z H0 H1 ; transitivity y ; assumption.
Defined. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | Seq_trans | |
trans_stx :=
idtac "trans_st on Setoid_Theory is OBSOLETE";
idtac "use transitivity on Equivalence instead";
match goal with
| H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_trans _ _ H) with x; auto
end. | Ltac | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | trans_st | |
sym_st:=
idtac "sym_st on Setoid_Theory is OBSOLETE";
idtac "use symmetry on Equivalence instead";
match goal with
| H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_sym _ _ H); auto
end. | Ltac | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | sym_st | |
refl_st:=
idtac "refl_st on Setoid_Theory is OBSOLETE";
idtac "use reflexivity on Equivalence instead";
match goal with
| H : Setoid_Theory _ ?eqA |- ?eqA _ _ =>
apply (Seq_refl _ _ H); auto
end. | Ltac | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | refl_st | |
gen_st: forall A : Set, Setoid_Theory _ (@eq A).
Proof.
constructor; congruence.
Qed. | Definition | Corelib | [
"Require Export Corelib.Classes.SetoidTactics",
"Require Corelib.ssr.ssrsetoid"
] | Corelib/Setoids/Setoid.v | gen_st | |
rpredNR S (oppS : @opprPred R S) (kS : keyed_pred oppS) :
{mono -%%R: x / x \in kS}.
Because x \in kS will be displayed as x \in S (or x \is S, etc), the
canonical instance of opprPred will not normally be exposed (it will also
be erased by /= simplification). In addition each predicate structure
should have... | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | rpredN | |
addbb := if b then negb else id. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | addb | |
is_true: bool >-> Sortclass. (* Prop *) | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_true | |
prop_congr: forall b b' : bool, b = b' -> b = b' :> Prop.
Proof. by move=> b b' ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_congr | |
prop_congr:= apply: prop_congr. | Ltac | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | prop_congr | |
is_true_true: true. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_true_true | |
not_false_is_true: ~ false. Proof. by []. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | not_false_is_true | |
is_true_locked_true: locked true. Proof. by unlock. Qed.
#[global]
Hint Resolve is_true_true not_false_is_true is_true_locked_true : core. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_true_locked_true | |
isT:= is_true_true. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | isT | |
notF:= not_false_is_true. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | notF | |
negbTb : b = false -> ~~ b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbT | |
negbTEb : ~~ b -> b = false. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbTE | |
negbFb : (b : bool) -> ~~ b = false. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbF | |
negbFEb : ~~ b = false -> b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbFE | |
negbK: involutive negb. Proof. by case. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbK | |
negbNEb : ~~ ~~ b -> b. Proof. by case: b. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbNE | |
negb_inj: injective negb. Proof. exact: can_inj negbK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negb_inj | |
negbLRb c : b = ~~ c -> ~~ b = c. Proof. exact: canLR negbK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbLR | |
negbRLb c : ~~ b = c -> b = ~~ c. Proof. exact: canRL negbK. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | negbRL | |
contra(c b : bool) : (c -> b) -> ~~ b -> ~~ c.
Proof. by case: b => //; case: c. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contra | |
contraNN:= contra. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraNN | |
contraL(c b : bool) : (c -> ~~ b) -> b -> ~~ c.
Proof. by case: b => //; case: c. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraL | |
contraTN:= contraL. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraTN | |
contraR(c b : bool) : (~~ c -> b) -> ~~ b -> c.
Proof. by case: b => //; case: c. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraR | |
contraNT:= contraR. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraNT | |
contraLR(c b : bool) : (~~ c -> ~~ b) -> b -> c.
Proof. by case: b => //; case: c. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraLR | |
contraTT:= contraLR. | Definition | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraTT | |
contraTb : (~~ b -> false) -> b. Proof. by case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraT | |
wlog_negb : (~~ b -> b) -> b. Proof. by case: b => // ->. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | wlog_neg | |
contraFT(c b : bool) : (~~ c -> b) -> b = false -> c.
Proof. by move/contraR=> notb_c /negbT. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraFT | |
contraFN(c b : bool) : (c -> b) -> b = false -> ~~ c.
Proof. by move/contra=> notb_notc /negbT. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraFN | |
contraTF(c b : bool) : (c -> ~~ b) -> b -> c = false.
Proof. by move/contraL=> b_notc /b_notc/negbTE. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraTF | |
contraNF(c b : bool) : (c -> b) -> ~~ b -> c = false.
Proof. by move/contra=> notb_notc /notb_notc/negbTE. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraNF | |
contraFF(c b : bool) : (c -> b) -> b = false -> c = false.
Proof. by move/contraFN=> bF_notc /bF_notc/negbTE. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraFF | |
contra_not(P Q : Prop) : (Q -> P) -> (~ P -> ~ Q). Proof. by auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contra_not | |
contraPnot(P Q : Prop) : (Q -> ~ P) -> (P -> ~ Q). Proof. by auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraPnot | |
contraTnot(b : bool) (P : Prop) : (P -> ~~ b) -> (b -> ~ P).
Proof. by case: b; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraTnot | |
contraNnot(P : Prop) (b : bool) : (P -> b) -> (~~ b -> ~ P).
Proof. rewrite -{1}[b]negbK; exact: contraTnot. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraNnot | |
contraPT(P : Prop) (b : bool) : (~~ b -> ~ P) -> P -> b.
Proof. by case: b => //= /(_ isT) nP /nP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraPT | |
contra_notT(P : Prop) (b : bool) : (~~ b -> P) -> ~ P -> b.
Proof. by case: b => //= /(_ isT) HP /(_ HP). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contra_notT | |
contra_notN(P : Prop) (b : bool) : (b -> P) -> ~ P -> ~~ b.
Proof. rewrite -{1}[b]negbK; exact: contra_notT. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contra_notN | |
contraPN(P : Prop) (b : bool) : (b -> ~ P) -> (P -> ~~ b).
Proof. by case: b => //=; move/(_ isT) => HP /HP. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraPN | |
contraFnot(P : Prop) (b : bool) : (P -> b) -> b = false -> ~ P.
Proof. by case: b => //; auto. Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraFnot | |
contraPF(P : Prop) (b : bool) : (b -> ~ P) -> P -> b = false.
Proof. by case: b => // /(_ isT). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contraPF | |
contra_notF(P : Prop) (b : bool) : (b -> P) -> ~ P -> b = false.
Proof. by case: b => // /(_ isT). Qed. | Lemma | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | contra_notF | |
ofsum-style datatypes into bool, which makes it possible
to use ssr's boolean if rather than Rocq's "generic" if. **) | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | of | |
isSomeT (u : option T) := if u is Some _ then true else false. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | isSome | |
is_inlA B (u : A + B) := if u is inl _ then true else false. | Coercion | Corelib | [
"Require Import ssreflect ssrfun"
] | Corelib/ssr/ssrbool.v | is_inl |
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